Good Wild Harmonic Bundles and Good Filtered Higgs Bundles
Takuro Mochizuki

TL;DR
This paper establishes a correspondence between good wild harmonic bundles and filtered flat bundles, providing new insights into their structure and potential applications in constructing Frobenius manifolds.
Contribution
It proves the Kobayashi-Hitchin correspondence for good wild harmonic bundles and filtered $ abla$-flat bundles, extending the theory to cases with group action homogeneity.
Findings
Established the Kobayashi-Hitchin correspondence for good wild harmonic bundles.
Analyzed the correspondence in the presence of group action homogeneity.
Suggested a new approach to constructing Frobenius manifolds.
Abstract
We prove the Kobayashi-Hitchin correspondence between good wild harmonic bundles and polystable good filtered -flat bundles satisfying a vanishing condition. We also study the correspondence for good wild harmonic bundles with the homogeneity with respect to a group action, which is expected to provide another way to construct Frobenius manifolds.
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\FirstPageHeading
\ShortArticleName
Good Wild Harmonic Bundles and Good Filtered Higgs Bundles
\ArticleName
Good Wild Harmonic Bundles
and Good Filtered Higgs Bundles††This paper is a contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji Saito for his 77th birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Saito.html
\Author
Takuro MOCHIZUKI
\AuthorNameForHeading
T. Mochizuki
\Address
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan \Email[email protected]
\ArticleDates
Received July 16, 2020, in final form June 28, 2021; Published online July 17, 2021
\Abstract
We prove the Kobayashi–Hitchin correspondence between good wild harmonic bundles and polystable good filtered -flat bundles satisfying a vanishing condition. We also study the correspondence for good wild harmonic bundles with the homogeneity with respect to a group action, which is expected to provide another way to construct Frobenius manifolds.
\Keywords
wild harmonic bundles; Higgs bundles; -flat bundles; Kobayashi–Hitchin correspondence
\Classification
53C07; 58E15; 14D21; 81T13
*Dedicated to Professor Kyoji Saito
on the occasion of his 77th birthday*
1 Introduction
Let be a smooth projective variety with a simple normal crossing hypersurface . Let be an ample line bundle on . We shall prove the following theorem, that is the Kobayashi–Hitchin correspondence for good wild harmonic bundles and good filtered -flat bundles.
Theorem 1.1** **(Corollary
2.24).
The following objects are equivalent:
- •
Good wild harmonic bundles on .
- •
-Polystable filtered -flat bundles \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} on satisfying
[TABLE]
We shall recall the precise definitions of the objects in Section 2.
In [51], we have already proved that good wild harmonic bundles on induce -polystable good filtered -flat bundles satisfying the vanishing condition. Note that [math]-flat bundles are equivalent to Higgs bundles, and -flat bundles are flat bundles in the ordinary sense. Moreover, we studied an analogue of Theorem 1.1 in the case , i.e., the correspondence between good wild harmonic bundles and -polystable good filtered flat bundles satisfying a similar vanishing condition [51, Theorem 16.1.1]. It was applied to the study of the correspondence between semisimple algebraic holonomic -modules and pure twistor -modules.
In this paper, as a complement, we shall explain the proof for all . There is no new essential difficulty to prove Theorem 1.1 after our studies [46, 47, 48, 49, 51] on the basis of [62, 63]. Moreover, in some parts of the proof, the arguments can be simplified in the Higgs case. However, because the Higgs case is also particularly important, it would be useful to explain a rather detailed proof. We shall also explain the correspondences in homogeneous cases which would be useful in a generalized Hodge theory.
1.1 Kobayashi–Hitchin correspondences
1.1.1 Kobayashi–Hitchin correspondence for vector bundles
We briefly recall a part of the history of this type of correspondences. (See also [25, 35, 41].) For a holomorphic vector bundle on a compact Riemann surface , we set , which is called the slope of . A holomorphic bundle is called stable (resp. semistable) if (resp. ) holds for any holomorphic subbundle such that . It is called polystable if it is a direct sum of stable subbundles with the same slope. This stability, semistability and polystability conditions were introduced by Mumford [56] for the construction of the moduli spaces of vector bundles with reasonable properties. Narasimhan and Seshadri [58] established the equivalence between unitary flat bundles and polystable bundles of degree [math] on compact Riemann surfaces.
Let be a compact connected Kähler manifold. For any torsion-free -module , the slope of with respect to is defined as
[TABLE]
If the cohomology class of is the first Chern class of an ample line bundle , then is also denoted by . Then, a torsion-free -module is called -stable if holds for any saturated coherent subsheaf such that . This condition was first studied by Takemoto [71, 72]. It is also called -stability, or slope stability. Slope semistability and slope polystability are naturally defined.
Bogomolov [4] introduced the -stability condition for torsion-free sheaves on connected projective surfaces, and he proved the inequality of the Chern classes for any -semistable bundle of rank . We do not recall the precise definition of -stability condition here, but we note that if a holomorphic vector bundle on a complex projective manifold is slope semistable, then it is -semistable. (See [4, Section 7] for more details.) Gieseker [19] gave a different proof of the inequality for slope semistable bundles. The inequality is called Bogomolov–Gieseker inequality or Bogomolov inequality.
Inspired by these works, Kobayashi [32] introduced the concept of Hermitian–Einstein condition for metrics of holomorphic vector bundles. Let \big{(}E,\overline{\partial}_{E}\big{)} be a holomorphic vector bundle on a Kähler manifold . Let be a Hermitian metric of . Let denote the curvature of the Chern connection , associated with and . Then, is called Hermitian–Einstein if , where denotes the trace-free part of . In particular, he proved in [32] that if a holomorphic vector bundle on a compact Kähler manifold has a Hermitian–Einstein metric, then it is -semistable. Kobayashi [33, 34] and Lübke [40] proved that a holomorphic vector bundle on a compact connected Kähler manifold satisfies the slope polystability condition if it has a Hermitian–Einstein metric. Moreover, Lübke [39] established the so called Kobayashi–Lübke inequality for the first and the second Chern forms associated with Hermitian–Einstein metrics, which is reduced to the inequality \mathop{\rm Tr}\nolimits\bigl{(}\big{(}R(h)^{\bot}\big{)}^{2}\bigr{)}\omega^{\dim X-2}\geq 0 in the form level. In particular, it implies the Bogomolov–Gieseker inequality for holomorphic vector bundles \big{(}E,\overline{\partial}_{E}\big{)} with a Hermitian–Einstein metric on compact Kähler manifolds . Moreover, if and are satisfied for such \big{(}E,\overline{\partial}_{E},h\big{)}, and if we impose is flat, then the Kobayashi–Lübke inequality implies that , i.e., is flat.
Independently, in [36], Hitchin proposed a problem to ask an equivalence of the stability condition and the existence of a metric such that , under the vanishing of the first Chern class of the bundle. (See [25] for more precise explanation.) It clearly contains the most important essence. He also suggested possible applications of the vanishings. His problem stimulated Donaldson whose work on this topic brought several breakthroughs to whole geometry.
In [14], Donaldson introduced the method of global analysis to reprove the theorem of Narasimhan–Seshadri. In [15], by using the method of the heat flow associated with the Hermitian–Einstein condition, he established the equivalence of the slope polystability condition and the existence of a Hermitian–Einstein metric for holomorphic vector bundles on any complex projective surface. The important concept of Donaldson functional was also introduced in [15].
Eventually, Donaldson [16] and Uhlenbeck–Yau [73] established the equivalence on any dimensional complex projective manifolds. Note that Uhlenbeck–Yau proved it for any compact Kähler manifolds, more generally. The correspondence is called with various names; Kobayashi–Hitchin correspondence, Hitchin–Kobayashi correspondence, Donaldson–Hitchin–Uhlenbeck–Yau correspondence, etc. In this paper, we call it the Kobayashi–Hitchin correspondence.
As a consequence of the Kobayashi–Hitchin correspondence and the Kobayashi–Lübke inequality, we also obtain an equivalence between unitary flat bundles and slope polystable holomorphic vector bundles satisfying and . Note that Mehta and Ramanathan [43, 44] deduced the equivalence on complex projective manifolds directly from the equivalence in the surface case due to Donaldson [15].
1.1.2 Higgs bundles and -flat bundles
Such correspondences have been also studied for vector bundles equipped with some additional structure, which are also called Kobayashi–Hitchin correspondences in this paper. One of the most rich and influential is the case of Higgs bundles, pioneered by Hitchin and Simpson.
Let \big{(}E,\overline{\partial}_{E}\big{)} be a holomorphic vector bundle on a compact Riemann surface . A Higgs field of \big{(}E,\overline{\partial}_{E}\big{)} is a holomorphic section of . Let be a Hermitian metric of . We obtain the Chern connection and its curvature . Let denote the adjoint of . In [24], Hitchin introduced the following equation, called the Hitchin equation,
[TABLE]
Such \big{(}E,\overline{\partial}_{E},\theta,h\big{)} is called a harmonic bundle. In particular, he studied the case . Among many deep results in [24], he proved that a Higgs bundle \big{(}E,\overline{\partial}_{E},\theta\big{)} has a Hermitian metric satisfying (1.1) if and only if it is polystable of degree [math]. Here, a Higgs bundle \big{(}E,\overline{\partial}_{E},\theta\big{)} is called stable (resp. semistable) if (resp. ) holds for any holomorphic subbundle such that and that , and a Higgs bundle is called polystable if it is a direct sum of stable Higgs subbundles with the same slope. By this equivalence and another equivalence due to Donaldson [17] between irreducible flat bundles and twisted harmonic maps, Hitchin obtained that the moduli space of polystable Higgs bundles of degree [math] and the moduli space of semisimple flat bundles are isomorphic. His work revealed that the moduli spaces of Higgs bundles and flat bundles have extremely rich structures.
The higher dimensional case was studied by Simpson [62]. Note that Simpson started his study independently motivated by a new way to construct variations of Hodge structure, which we shall mention later in Section 1.2.1. For a holomorphic vector bundle \big{(}E,\overline{\partial}_{E}\big{)} on a complex manifold with arbitrary dimension, a Higgs field is defined to be a holomorphic section of satisfying the additional condition . Suppose that has a Kähler form. Let be a Hermitian metric of . Let denote the curvature of the connection . A Hermitian metric of a Higgs bundle \big{(}E,\overline{\partial}_{E},\theta\big{)} is called Hermitian–Einstein if . When is compact, the slope stability, semistability and polystability conditions for Higgs bundles are naturally defined in terms of the slopes of Higgs subsheaves. Simpson established that a Higgs bundle \big{(}E,\overline{\partial}_{E},\theta\big{)} on a compact Kähler manifold has a Hermitian–Einstein metric if and only if it is slope polystable. Moreover, he generalized the Kobayashi–Lübke inequality for the Chern forms to the context of Higgs bundles, which is reduced to the inequality \mathop{\rm Tr}\nolimits\bigl{(}\big{(}F(h)^{\bot}\big{)}^{2}\bigr{)}\omega^{\dim X-2}\geq 0 in the form level for any Hermitian–Einstein metric of \big{(}E,\overline{\partial}_{E},\theta\big{)}. Here, the condition is essential. In particular, it implies that if \big{(}E,\overline{\partial}_{E},\theta\big{)} on a compact Kähler manifold satisfies and , then a Hermitian–Einstein metric of \big{(}E,\overline{\partial}_{E},\theta\big{)} is a pluri-harmonic metric, i.e., the connection is flat. It is equivalent to the following:
[TABLE]
A Higgs bundle \big{(}E,\overline{\partial}_{E},\theta\big{)} with a pluri-harmonic metric is called a harmonic bundle. This equivalence and another important equivalence due to Corlette [11] induce an equivalence between semisimple flat bundles and polystable Higgs bundles \big{(}E,\overline{\partial}_{E},\theta\big{)} satisfying and on any connected compact Kähler manifold.
After the work of Corlette, Donaldson, Hitchin and Simpson, it turned out that the moduli space of flat bundles on a complex projective manifold has a hyper-Kähler metric. In particular, it induces the twistor space of the moduli space , which is a complex analytic space with a fibration , such that the fiber over is the moduli space of flat bundles, and that the fiber over [math] is the moduli space of Higgs bundles with vanishing rational Chern classes. The notion of -connections was introduced and developed by Deligne and Simpson [65, 66] for a more complex analytic construction of the twistor space . They obtain the family of the moduli spaces of -flat bundles on , and the family of the moduli spaces \mathcal{M}^{\mu}\big{(}X^{\dagger}\big{)} of -flat bundles on the conjugate of . They proved that the twistor space can be obtained as the gluing of the two families and \coprod_{\mu}\mathcal{M}^{\mu}\big{(}X^{\dagger}\big{)} by the natural identification of \mathcal{M}^{\lambda}(X)=\mathcal{M}^{\mu}\big{(}X^{\dagger}\big{)} for .
These correspondences are not only really interesting, but also provide a starting point of the further investigations. Simpson pursued the comparison of flat bundles, Higgs bundles and more generally -flat bundles in deeper levels [64, 66], and developed the non-abelian Hodge theory [65]. In particular, he explained that the Kobayashi–Hitchin correspondences for -flat bundles can be studied in a unified way [64]. For more recent study on the moduli spaces of -connections, see [10, 26, 27, 67], for example.
1.1.3 Filtered case
It is interesting to generalize such correspondences for objects on complex quasi-projective manif̃olds. We need to impose a kind of boundary condition, that is parabolic structure.
Mehta and Seshadri [45] introduced the concept of parabolic structure of vector bundles on compact Riemann surfaces. Let be a compact Riemann surface with a finite subset . Let be a holomorphic vector bundle on . A parabolic structure of at is a tuple of filtrations indexed by satisfying . Set , and
[TABLE]
We set . For any subbundle , filtrations on are induced as . Then, is called stable if for any subbundle with . Semistability and polystability conditions are also defined naturally. Then, Mehta and Seshadri proved an equivalence of irreducible unitary flat bundles on and stable parabolic vector bundles with on .
For some purposes, it is more convenient to replace parabolic bundles with filtered bundles introduced by Simpson [62, 63]. Let denote the sheaf of meromorphic functions on which may have poles along . Let be a locally free -module. A filtered bundle over is a tuple of lattices such that , the restriction of to a neighbourhood of depends only on , \mathcal{P}_{{\boldsymbol{a}}+{\boldsymbol{n}}}\mathcal{V}=\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}\big{(}\sum n_{P}P\big{)} for any and , for any , there exists such that . Let denote . Then, is equipped with the parabolic structure induced by the images of . It is easy to observe that filtered bundles are equivalent to parabolic bundles. We set for the filtered bundle .
Simpson [62, 63] generalized the theorem of Mehta-Seshadri to the correspondences of tame harmonic bundles and regular filtered -flat bundles on compact Riemann surfaces. A harmonic bundle \big{(}E,\overline{\partial}_{E},\theta,h\big{)} on is called tame on if the closure of the spectral curve of in is proper over . A regular filtered -flat bundle consists of a filtered bundle equipped with a flat -connection such that for any . Stability, semistable and polystable conditions are naturally defined in terms of the slope. Then, Simpson established the equivalence of tame harmonic bundles on and polystable regular filtered -flat bundles \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} satisfying . Note that filtered bundles express the growth order of the norms of holomorphic sections with respect to the metrics. We should mention that the study of the asymptotic behaviour of tame harmonic bundles is much harder than that of the asymptotic behaviour of unitary flat bundles. Hence, it is already hard to prove that tame harmonic bundles induce regular filtered -flat bundles.
There are several directions to generalize. One is a generalization in the context of tame harmonic bundles on higher dimensional varieties. Let be a smooth connected projective variety with a simple normal crossing hypersurface and an ample line bundle . Then, there should be equivalences of tame harmonic bundles on and -polystable regular filtered -flat bundles on satisfying and for each . In [2], Biquard studied the case where is smooth. In [37, 38, 70], Li, Narasimhan, Steer and Wren studied the correspondence for parabolic bundles without flat -connections. In [30], Jost and Zuo studied the correspondence between semisimple flat bundles and tame harmonic bundles. In [46, 47, 48, 49], the author obtained the satisfactory equivalences for tame harmonic bundles. Note that Donagi and Pantev recently proposed an attractive application of the Kobayashi–Hitchin correspondence for tame harmonic bundles to the study of geometric Langlands theory [13].
In another natural direction of generalization, we should consider more singular objects than regular filtered Higgs or flat bundles. A harmonic bundle \big{(}E,\overline{\partial}_{E},\theta,h\big{)} on is called wild if the closure of the spectral variety of in the projective completion of is complex analytic. For the analysis, we should impose that the spectral variety of the harmonic bundle satisfies some non-degeneracy condition along . (See Section 2.7.1.) This is not essential because the condition is always satisfied once we replace by its appropriate blow up. The notion of regular filtered -flat bundle is appropriately generalized to the notion of good filtered -flat bundle. The results of Simpson should be generalized to equivalences of good wild harmonic bundles and -polystable good filtered -flat bundles \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} satisfying and . Sabbah [59] studied the correspondence between semisimple meromorphic flat bundles and wild harmonic bundles in the one dimensional case. Biquard and Boalch [3] obtained generalization for wild harmonic bundles in the one dimensional case. Boalch informed the author that wild generalization in the context of the Higgs case was not expected in those days.
As mentioned, the author studied the wild harmonic bundles on any dimensional varieties in [51]. We obtained that good wild harmonic bundles induce -polystable good filtered -flat bundles satisfying the vanishing conditions. Moreover, we proved that the construction induces an equivalence of good wild harmonic bundles and slope polystable good filtered flat bundles satisfying the vanishing condition. Such an equivalence for meromorphic flat bundles is particularly interesting because we may apply it to prove a conjecture of Kashiwara [31] on semisimple algebraic holonomic -modules. See a survey paper [54] for more details on this application.
In [51], we did not give a proof of the equivalence for wild harmonic bundles in the case because it is rather obvious that a similar argument can work after [46, 47, 48, 49, 51] on the basis of [62, 63]. But, because the Higgs case is also important, it would be better to have a reference in which a rather detailed proof is explained. It is one reason why the author writes this manuscript. As another reason, in the next subsection, we shall explain an application to the correspondence for good wild harmonic bundles with homogeneity, which is expected to be useful in the generalized Hodge theory.
1.2 Homogeneity with respect to group actions
1.2.1 Variation of Hodge structure
As mentioned, Simpson [62] was motivated by the construction of polarized variation of Hodge structure. Let us recall the definition of polarized complex variation of Hodge structure given in [62], instead of the original definition of polarized variation of Hodge structure due to Griffiths. A complex variation of Hodge structure of weight is a graded -vector bundle equipped with a flat connection satisfying the Griffiths transversality condition, i.e., \nabla^{0,1}(V^{p,q})\subset\Omega^{0,1}\otimes\bigl{(}V^{p+1,q-1}\oplus V^{p,q}\bigr{)} and \nabla^{1,0}(V^{p,q})\subset\Omega^{1,0}\otimes\bigl{(}V^{p-1,q+1}\oplus V^{p,q}\bigr{)}, where denote the -part of . A polarization of a complex variation of Hodge structure is a flat Hermitian pairing satisfying the following conditions: the decomposition is orthogonal with respect to , \big{(}\sqrt{-1}\big{)}^{p-q}\langle\cdot,\cdot\rangle is positive definite on .
A polarization of pure Hodge structure typically appears when we consider the Gauss–Manin connection associated with a smooth projective morphism . Namely, the family of vector spaces H^{w}\big{(}f^{-1}(y)\big{)} naturally induces a flat bundle on . With the Hodge decomposition, it is a variation of Hodge structure of weight . A relatively ample line bundle induces a polarization on the variation of Hodge structure.
Simpson discovered a completely different way to construct a polarized variation of Hodge structure. Let be a complex variation of Hodge structure. Note that induces holomorphic structures of . We set . Then, \big{(}V=\bigoplus V^{p,q},\overline{\partial}_{V}\big{)} is a graded holomorphic vector bundle. We also note that induces linear maps , and hence . It is easy to check that is a Higgs field of \big{(}V,\overline{\partial}_{V}\big{)}. Such a graded holomorphic bundle with a Higgs field such that is called a Hodge bundle of weight . In general, we cannot construct a complex variation of Hodge structure from a Hodge bundle. However, Simpson discovered that if a Hodge bundle on a compact Kähler manifold satisfies the stability condition and the vanishing condition, then there exists a flat connection and a flat Hermitian pairing such that is a complex variation of Hodge structure which induces the Hodge bundle, is a polarization of . Indeed, according to the equivalence of Simpson between Higgs bundles and harmonic bundles, there exists a pluri-harmonic metric of . It turns out that the flat connection satisfies the Griffiths transversality. Moreover, the decomposition is orthogonal with respect to , and flat Hermitian paring is constructed by the relation \big{(}\sqrt{-1}\big{)}^{p-q}\langle\cdot,\cdot\rangle_{V^{p,q}}=h_{|V^{p,q}}.
Note that a Hodge bundle is regarded as a Higgs bundle \big{(}V,\overline{\partial}_{V},\theta\big{)} with an -homogeneity, i.e., \big{(}V,\overline{\partial}_{V}\big{)} is equipped with an -action such that for any . It roughly means that semistable Hodge bundles correspond to the fixed points in the moduli space of semistable Higgs bundles with respect to the natural -action induced by t\big{(}E,\overline{\partial}_{E},\theta\big{)}=\big{(}E,\overline{\partial}_{E},t\theta\big{)}.
By the deformation \big{(}E,\overline{\partial}_{E},\alpha\theta\big{)} , any semistable Higgs bundles is deformed to an -fixed point in the moduli space, i.e., a semistable Hodge bundle as . Note that the Higgs field of the limit is not necessarily [math]. Hence, by the equivalence between Higgs bundles and flat bundles, it turns out that any flat bundle is deformed to a flat bundle underlying a polarized variation of Hodge structure.
In particular, Simpson [62] applied these ideas to construct uniformizations of some types of project̄ive manifolds. He also applied it to prove that some type of discrete groups cannot be the fundamental group of any projective manifolds in [64].
1.2.2 TE-structure
We recall that a complex variation of Hodge structure on induces a TE-structure in the sense of Hertling [21], i.e., a holomorphic vector bundle on with a meromorphic flat connection
[TABLE]
where . Indeed, for a complex variation of Hodge structure , are holomorphic subbundles with respect to . Thus, we obtain a decreasing filtration of holomorphic subbundles satisfying the Griffiths transversality . Let denote the projection. We obtain the induced flat bundle . By the Rees construction, extends to a locally free -module , on which is a meromorphic flat connection satisfying the condition \widetilde{\nabla}\mathcal{V}\subset\mathcal{V}\otimes\mathcal{O}_{\mathcal{X}}\big{(}\mathcal{X}^{0}\big{)}\otimes\Omega^{1}_{\mathcal{X}}\big{(}\log\mathcal{X}^{0}\big{)}.
It is recognized that a TE-structure appears as a fundamental piece of interesting structures in various fields of mathematics. For instance, TE-structure is an ingredient of Frobenius manifold, which is important in the theory of primitive forms and flat structures due to K. Saito [61], the topological field theory of Dubrovin [18], the -geometry of Cecotti–Vafa [7, 8], the Gromov–Witten theory, the theory of Landau–Ginzburg models, etc. For the construction of Frobenius manifolds, it is an important step to obtain TE-structures. Abstractly, TE-structure is also an important ingredient of semi-infinite variation of Hodge structure [1, 9, 28], TERP structure [21, 22, 23], integrable variation of twistor structure [60], etc. (See also [50, 53].)
1.2.3 Homogeneous harmonic bundles
As Simpson applied his Kobayashi–Hitchin correspondence to construct complex variations of Hodge structure, we may apply Theorem 1.1 to construct TE-structures with some additional structure. It is done through harmonic bundles with homogeneity as in the Hodge case.
Let be a complex manifold equipped with an -action. Let \big{(}E,\overline{\partial}_{E}\big{)} be an -equivariant holomorphic vector bundle. Let be a Higgs field of \big{(}E,\overline{\partial}_{E}\big{)}, which is homogeneous with respect to the -action, i.e., for some . Let be an -invariant pluri-harmonic metric of \big{(}E,\overline{\partial}_{E},\theta\big{)}. Then, as studied in [53, Section 3], we naturally obtain a TE-structure. More strongly, it is equipped with a grading in the sense of [9, 28], and it also underlies a polarized integrable variation of pure twistor structure of weight [math] [60]. Moreover, if there exists an -equivariant isomorphism between \big{(}E,\overline{\partial}_{E},\theta,h\big{)} and its dual, the TE-structure is enhanced to a semi-infinite variation of Hodge structure with a grading [1, 9, 28]. If the -action on is trivial, this is the same as the construction of a variation of Hodge structure from a Hodge bundle with a pluri-harmonic metric for which the Hodge decomposition is orthogonal.
Let be a simple normal crossing hypersurface of . From an -homogeneous good wild harmonic bundle \big{(}E,\overline{\partial}_{E},\theta,h\big{)} on , as mentioned above, we obtain a TE-structure with a grading on . Moreover, it extends to a meromorphic TE-structure on as studied in [53, Section 3]. We obtain the mixed Hodge structure as the limit objects at the boundary, which is useful for the study of more detailed properties of the TE-structure.
1.2.4 An equivalence
Let be a complex projective manifold with a simple normal crossing hypersurface and an ample line bundle , equipped with a -action. A good filtered Higgs bundle is called -homogeneous if is -equivariant and for some . Then, we obtain the following theorem by using Theorem 1.1. (See Section 8.1.2 for the precise definition of the stability condition in this context.)
Theorem 1.2** (Corollary 8.11).**
There exists an equivalence between the following objects:
- •
-polystable -homogeneous good filtered Higgs bundles on satisfying
[TABLE]
- •
-homogeneous good wild harmonic bundles on .
As mentioned in Section 1.2.3, Theorem 1.2 allows us to obtain a meromorphic TE-structure on with a grading from a -polystable -equivariant good filtered Higgs bundle satisfying the vanishing condition. We already applied it to a classification of solutions of the Toda equations on [52]. It seems natural to expect that this construction would be another way to obtain Frobenius manifolds.
Although we explained the homogeneity with respect to an -action, Theorem 1.2 is generalized for -homogeneous good wild harmonic bundles as explained in Section 8, where is any compact Lie group.
2 Good filtered -flat bundles and wild harmonic bundles
2.1 Filtered sheaves and filtered -flat sheaves
2.1.1 Filtered sheaves
Let denote a complex manifold with a simple normal crossing hypersurface . Let denote a decomposition such that each is smooth. Note that are not necessarily connected. For any , a holomorphic coordinate neighbourhood around is called admissible if . For such an admissible coordinate neighbourhood, there exists the map determined by . We obtain the map by \kappa_{P}({\boldsymbol{a}})=\big{(}a_{\rho(1)},\ldots,a_{\rho(\ell(P))}\big{)}.
Let denote the sheaf of meromorphic functions which may have poles along . Let be any coherent torsion free -module. A filtered sheaf over is defined to be a tuple of coherent -submodules satisfying the following conditions:
- •
if , i.e., for any .
- •
for any .
- •
\mathcal{P}_{{\boldsymbol{a}}+{\boldsymbol{n}}}\mathcal{E}=\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}\bigl{(}\sum_{i\in\Lambda}n_{i}H_{i}\bigr{)} for any and .
- •
For any there exists such that .
- •
For any , we take an admissible coordinate neighbourhood around . Then, for any , depends only on .
For any coherent -submodule , we obtain a filtered sheaf over by . If is saturated, i.e., is torsion-free, we obtain a filtered sheaf over by \mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}^{\prime\prime}:=\mathop{\rm Im}\nolimits\bigl{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}\longrightarrow\mathcal{E}^{\prime\prime}\bigr{)}.
A morphism of filtered sheaves is defined to be a morphism of -modules such that for any .
Remark 2.1**.**
The concept of filtered bundles on curves was introduced by Mehta and Seshadri [45] and Simpson [62, 63]. A higher dimensional version was first studied by Maruyama and Yokogawa [42] for the purpose of the construction of the moduli spaces.
2.1.2 Restriction and gluing
Let be any open subset. We set . Let be the irreducible decomposition. For any , we have such that is a connected component of . For any , we set .
Let be a filtered sheaf over . We shall define a filtered sheaf over the -module . Let . For any , we choose such that for any , and we obtain the following -submodule of the stalk :
[TABLE]
It is independent of the choice of as above. There uniquely exists a coherent -submodule of such that , and for any , the stalk of at is equal to . Thus, we obtain a filtered sheaf over , which is denoted as .
Let be an open covering. We set . For any filtered sheaf over , we obtain filtered sheaves over as the restriction. Conversely, let \mathcal{P}_{\ast}\big{(}\mathcal{E}_{|X^{(k)}}\big{)} be filtered sheaves over such that \mathcal{P}_{\ast}\big{(}\mathcal{E}_{|X^{(k)}}\big{)}_{|X^{(k)}\cap X^{(\ell)}}=\mathcal{P}_{\ast}\big{(}\mathcal{E}_{|X^{(\ell)}}\big{)}_{|X^{(k)}\cap X^{(\ell)}} for any .
Lemma 2.2**.**
There uniquely exists a filtered sheaf over such that \mathcal{P}_{\ast}\mathcal{E}_{|X^{(k)}}=\mathcal{P}_{\ast}\big{(}\mathcal{E}_{|X^{(k)}}\big{)} for any .
Proof.
Let . For any , there exists such that . Let be the irreducible decomposition. For any , we have such that is a connected component of . Thus, we obtain a map . For any , let be the image of by the induced map , and we obtain the following -submodule of :
[TABLE]
There uniquely exists a coherent -submodule of such that , and for any , the stalk of at is equal to . Thus, we obtain a filtered sheaf over with the desired property. The uniqueness is also clear. ∎
2.1.3 Reflexive filtered sheaves
A filtered sheaf on is called reflexive if each is a reflexive -module. Note that it is equivalent to the “reflexive and saturated” condition in [46, Definition 3.17] by the following lemma.
Lemma 2.3**.**
Suppose that is reflexive. Let . We take , and let be determined by and . Then, is a torsion-free -module.
Proof.
Let be a section of on an open set . There exists an open subset and a section of on such that and that induces . Note that there exists of codimension such that is a section of . Because is reflexive, there exists a section of on such that . Hence, we obtain that is a section of , i.e., . ∎
The following lemma is clear.
Lemma 2.4**.**
Let be a reflexive filtered sheaf on . Then a coherent -submodule is saturated if and only if the induced filtered sheaf is reflexive.
2.1.4 Filtered -flat sheaves
Let be any complex number. Let be a coherent torsion-free -module. A -connection is a -linear morphism of sheaves such that for any local sections and of and , respectively. Note that an -morphism is induced. If , it is called a flat -connection. When is equipped with a flat -connection, a -flat subsheaf of means a coherent -submodule such that . A pair of a filtered sheaf over and a flat -connection of is called a filtered -flat connection. It is called reflexive if is reflexive.
2.2 -stability condition for filtered -flat sheaves
Let be a connected projective manifold with a simple normal crossing hypersurface . Let be an ample line bundle.
2.2.1 Slope of filtered sheaves
Let be a filtered sheaf on . Recall the definition of the parabolic first Chern class . Let be the generic point of . Note that -modules depends only on , which is denoted by . We obtain -modules \mathop{\rm Gr}\nolimits^{\mathcal{P}}_{a}(\mathcal{E}_{\eta_{i}}):=\mathcal{P}_{a}(\mathcal{E}_{\eta_{i}})\big{/}\mathcal{P}_{<a}(\mathcal{E}_{\eta_{i}}). Then, we set
[TABLE]
Here, denote the cohomology class induced by . It is easy to see that is independent of the choice of . We set
[TABLE]
It is called the slope of with respect to . The following is proved in [46, Lemma 3.7].
Lemma 2.5**.**
Let be a morphism of filtered sheaves which is generically an isomorphism, i.e., the induced morphism at the generic point of is an isomorphism. Then, \mu_{L}\big{(}\mathcal{P}_{\ast}\mathcal{E}^{(1)}\big{)}\leq\mu_{L}\big{(}\mathcal{P}_{\ast}\mathcal{E}^{(2)}\big{)} holds. If the equality holds, is an isomorphism in codimension one, i.e., there exists an algebraic subset such that the codimension of is larger than , is an isomorphism.
2.2.2 -stability condition
A filtered -flat sheaf on is called -stable (resp. -semistable) if the following holds:
- •
Let be any -flat -submodule such that . Then, (resp. ) holds.
A filtered -flat sheaf \big{(}\mathcal{P}_{\ast}\mathcal{E},\mathbb{D}^{\lambda}\big{)} is called -polystable if the following holds:
- •
\big{(}\mathcal{P}_{\ast}\mathcal{E},\mathbb{D}^{\lambda}\big{)} is -semistable.
- •
\big{(}\mathcal{P}_{\ast}\mathcal{E},\mathbb{D}^{\lambda}\big{)}=\bigoplus\big{(}\mathcal{P}_{\ast}\mathcal{E}_{i},\mathbb{D}^{\lambda}_{i}\big{)}, where each \big{(}\mathcal{P}_{\ast}\mathcal{E}_{i},\mathbb{D}^{\lambda}_{i}\big{)} is -stable.
The following is standard. (See [46, Section 3.1.3] and [49, Section 2.1.4].)
Lemma 2.6**.**
Suppose that is a -polystable reflexive filtered -flat sheaf. Then, there exists a unique decomposition \big{(}\mathcal{P}_{\ast}\mathcal{E},\mathbb{D}^{\lambda}\big{)}=\bigoplus_{i=1}^{N}\big{(}\mathcal{P}_{\ast}\mathcal{E}_{i},\mathbb{D}^{\lambda}_{i}\big{)}\otimes{\mathbb{C}}^{m(i)} such that \big{(}\mathcal{P}_{\ast}\mathcal{E}_{i},\mathbb{D}^{\lambda}_{i}\big{)} are -stable, , \big{(}\mathcal{P}_{\ast}\mathcal{E}_{i},\mathbb{D}^{\lambda}_{i}\big{)}\not\simeq\big{(}\mathcal{P}_{\ast}\mathcal{E}_{j},\mathbb{D}^{\lambda}_{j}\big{)} .
Remark 2.7**.**
In [46, Section 3.1.3], “the inequality ” should be corrected to “the inequality ”.
2.3 Filtered bundles
2.3.1 Filtered bundles in the local case
We recall the notion of filtered bundle in the local case. We shall explain it in the global case in Section 2.3.3. Let be a neighbourhood of in . We set , and for some . Let be a locally free -module. A filtered bundle over is a tuple of locally free -submodules such that the following holds:
- •
if , i.e., for any .
- •
There exists a frame of and tuples such that
[TABLE]
where we set for any .
Clearly, a filtered bundle over is a filtered sheaf over .
Remark 2.8**.**
We set and \widetilde{\mathcal{R}}:=\mathcal{R}\big{[}z_{1}^{-1},\ldots,z_{\ell}^{-1}\big{]}. For a free -module , a filtered bundle over is defined to be a tuple \mathcal{P}_{\ast}\widehat{\mathcal{V}}:=\bigl{(}\mathcal{P}_{{\boldsymbol{a}}}\widehat{\mathcal{V}}\mid{\boldsymbol{a}}\in{\mathbb{R}}^{\ell(P)}\bigr{)} of free -submodules satisfying similar conditions as above.
2.3.2 Pull back, push-forward and descent with respect to ramified coverings
in the local case
Let be given by \varphi(\zeta_{1},\ldots,\zeta_{n})=\big{(}\zeta_{1}^{m_{1}},\ldots,\zeta_{\ell}^{m_{\ell}},\zeta_{\ell+1},\ldots,\zeta_{n}\big{)}. We set , and . The induced ramified covering is also denoted by .
For any , we set . For any filtered bundle on , we define a filtered bundle on as follows:
[TABLE]
We set . Thus, we obtain the pull back functor from the category of filtered bundles on to the category of filtered bundles on .
For any , we set . For any filtered bundle on , we obtain the following filtered bundle
[TABLE]
In this way, we obtain a functor from the category of filtered bundles on to the category of filtered bundles on .
We set G:=\prod_{i=1}^{\ell}\big{\{}\mu_{i}\in{\mathbb{C}}^{\ast}\mid\mu_{i}^{m_{i}}=1\big{\}}. We define the action of on by
[TABLE]
We identify as the Galois group of the ramified covering . Let be a -equivariant filtered bundles on . Then, is equipped with an induced -action. We obtain a filtered bundle on as the -invariant part of , which is called the descent of with respect to the -action. In this way, we obtain a functor from the category of -equivariant filtered bundles on to the category of filtered bundles on .
For a filtered bundle on , the pull back is a -equivariant filtered bundle on , and its descent is naturally isomorphic to .
2.3.3 Filtered bundles in the global case
We use the notation in Section 2.1.1. Let be a locally free -module. A filtered bundle \mathcal{P}_{\ast}\mathcal{V}=\bigl{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}\mid{\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}\bigr{)} over be a sequence of locally free -submodules of such that the following holds:
- •
For any , we take an admissible coordinate neighbourhood around . Then, for any , depends only on , denoted as .
- •
The sequence \big{(}\mathcal{P}^{(P)}_{{\boldsymbol{b}}}(\mathcal{V}_{|X_{P}})\mid{\boldsymbol{b}}\in{\mathbb{R}}^{\ell(P)}\big{)} is a filtered bundle over in the sense of Section 2.3.1.
In other words, a filtered bundle is a filtered sheaf (see Section 2.1.1) satisfying the condition in Section 2.3.1 locally around any point of .
Remark 2.9**.**
The higher dimensional version of filtered bundles was introduced in [47, 48] with a different formulation. See also [5, 6]. In this paper, we essentially follow Iyer and Simpson [29].
2.3.4 The induced bundles and filtrations
For any , let be the element whose -th component is [math] or . We also set and \partial H_{I}:=H_{I}\cap\bigl{(}\bigcup_{j\in\Lambda\setminus I}H_{j}\bigr{)}.
Let be a filtered bundle on . Take . Let . For any , we set . We set
[TABLE]
It is naturally regarded as a locally free -module. Moreover, it is a subbundle of . In this way, we obtain a filtration of indexed by . We shall also denote it as just if there is no risk of confusion.
We obtain the induced filtrations of if . Let denote the image of by the projection . Set . For any , we set
[TABLE]
By the condition of filtered bundles, the following compatibility condition holds.
- •
Let be any point of . There exist a neighbourhood of in and a non-canonical decomposition
[TABLE]
such that the following holds for any :
[TABLE]
Indeed, there exists a frame of around with tuples of real numbers satisfying (2.2), where is replaced with . There exists the bijection determined by , by which we identify with . Let be the subbundle of generated by satisfying . Then, we obtain the decomposition (2.3).
For any , we obtain the following locally free -modules:
[TABLE]
Here, means that for any and that . Clearly, if and , we obtain .
2.3.5 The induced filtered bundles
For , we choose such that , and we obtain the following -module:
[TABLE]
It is independent of the choice of as above. We obtain the irreducible decomposition . For any , there exists such that is a connected component of . Let . For , there exists such that , . We obtain an -submodule
[TABLE]
Note that \mathcal{P}_{{\boldsymbol{d}}}\bigl{(}{}^{I}\!\mathop{\rm Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{V})\bigr{)}_{P} is independent of the choice of . There uniquely exists an -submodule \mathcal{P}_{{\boldsymbol{d}}}\bigl{(}{}^{I}\!\mathop{\rm Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{V})\bigr{)}\subset{}^{I}\!\mathop{\rm Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{V}) whose stalk at are equal to \mathcal{P}_{{\boldsymbol{d}}}\bigl{(}{}^{I}\!\mathop{\rm Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{V})\bigr{)}_{P}. Thus, we obtain the following filtered bundle over on :
[TABLE]
2.3.6 First and second Chern characters for filtered bundles
Let be a filtered bundle over . Take any . As recalled in Section 2.2.1, we obtain the parabolic first Chern class:
[TABLE]
To explain the second parabolic Chern character in , let us introduce some notation. Let be the set of the irreducible components of . For , let denotes the induced cohomology class, and let denote the restriction of to . Moreover, denotes the Gysin map induced by . Then, the second parabolic Chern character is given as follows.
[TABLE]
Remark 2.10**.**
The higher Chern character for filtered sheaves was defined by Iyer and Simpson [29] in a systematic way. In this paper, we adopt the definition of in [46].
2.4 Good filtered -flat bundles
Let be a complex manifold with a simple normal crossing hypersurface .
2.4.1 Good set of irregular values at
Let be any point of . We take an admissible holomorphic coordinate neighbourhood around . Let . If , we set . If there exists such that , , then we set . Otherwise, is not defined.
For any , we take a lift . If is defined, we set . Otherwise, is not defined. Note that it is independent of the choice of a lift .
Let be a finite subset. We say that is a good set of irregular values if the following conditions are satisfied:
- •
is defined for any .
- •
is defined for any .
- •
is totally ordered with respect to the order . Here, we define if for any .
2.4.2 Good filtered -flat bundles
Let be a locally free -module with a flat -connection. Let be a filtered bundle over . For any , let denote the completion of the local ring with respect to the maximal ideal. Note that Remark 2.8 has a natural generalization to filtered -flat bundles. We say that \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is unramifiedly good at if the following holds:
- •
There exist a good set of irregular values and a decomposition of filtered -flat bundles
[TABLE]
such that are logarithmic with respect to the lattices for any and , i.e.,
[TABLE]
Here, denote lifts of to .
We say that \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is good at if the following holds:
- •
There exist a neighbourhood of in and a covering map ramified over such that \varphi_{P}^{\ast}\big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is unramifiedly good at . (See Section 2.3.2 for the pull back of filtered bundles.)
We say that \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is good (resp. unramifiedly good) if it is good (resp. unramifiedly good) at any point of .
2.5 Prolongation of holomorphic vector bundles
with a Hermitian metric
Let be any complex manifold with a simple normal crossing hypersurface . Let \big{(}E,\overline{\partial}_{E}\big{)} be a holomorphic vector bundle on with a Hermitian metric . Let us recall the construction of -module and -modules .
Let . For any open subset , let be the space of holomorphic sections of satisfying the following condition:
- •
For any point of , let be an admissible holomorphic coordinate neighbourhood around such that is relatively compact in . Set . (See Section 2.1.1.) Then,
[TABLE]
holds on for any .
We obtain an -module . We set which is an -module. Note that in general, are not necessarily coherent -modules.
Definition 2.11**.**
Let be a filtered bundle over . Let \big{(}E,\overline{\partial}_{E}\big{)} be the holomorphic vector bundle obtained as the restriction of to . A Hermitian metric is called adapted to if in \iota_{\ast}(E)=\iota_{\ast}\big{(}\mathcal{V}_{|X\setminus H}\big{)}, where denotes the inclusion.
2.5.1 A sufficient condition
We mention a useful sufficient condition for to be a filtered bundle, although we do not use it in this paper. Let be a Kähler metric satisfying the following condition [12]:
- •
For any , take an admissible holomorphic coordinate neighbourhood around such that is isomorphic to by the coordinate system. Set . Then, is mutually bounded with the restriction of the Poincaré metric
[TABLE]
A Hermitian metric of \big{(}E,\overline{\partial}_{E}\big{)} is called acceptable if the curvature of the Chern connection is bounded with respect to and . The following theorem is proved in [51, Theorem 21.3.1].
Theorem 2.12**.**
If is acceptable, then is a filtered bundle, and is a locally free -module.
2.6 Harmonic bundles
2.6.1 Pluri-harmonic metrics for -flat bundles
Let be any complex manifold. Let be a -vector bundle on . Let denote the space of -sections of . We set . In this context, a -connection of is a differential operator such that \mathbb{D}^{\lambda}(fs)=f\mathbb{D}^{\lambda}(s)+\big{(}\lambda\partial_{Y}+\overline{\partial}_{Y}\big{)}f\otimes s for any and . We obtain a section . A -connection is called flat if .
Let \big{(}E,\mathbb{D}^{\lambda}\big{)} be a -flat bundle on . We decompose into the -part and the -part. Then, \big{(}E,{\rm d}_{E}^{\prime\prime}\big{)} is a holomorphic vector bundle. Let be a Hermitian metric of . From and , we obtain the differential operator such that is a Chern connection. From and , we obtain the -operator determined by . As in [49, Section 2.2.1], we obtain the operators
[TABLE]
Note that \mathbb{D}^{\lambda}=\overline{\partial}_{E,h}+\theta_{E,h}+\lambda\big{(}\partial_{E,h}+\theta^{\dagger}_{E,h}\big{)}. We set \mathbb{D}^{\lambda\star}_{E,h}:=\delta^{\prime}_{E,h}-\delta^{\prime\prime}_{E,h}=\partial_{E,h}+\theta_{E,h}^{\dagger}\allowbreak-\overline{\lambda}\big{(}\overline{\partial}_{E,h}+\theta_{E,h}\big{)}, and G(h):=\bigl{[}\mathbb{D}^{\lambda},\mathbb{D}^{\lambda\star}_{E,h}\bigr{]}.
Definition 2.13**.**
is called a pluri-harmonic metric of \big{(}E,\mathbb{D}^{\lambda}\big{)} if . Such a tuple \big{(}E,\mathbb{D}^{\lambda},h\big{)} is called a harmonic bundle.
If , because \big{(}1+|\lambda|^{2}\big{)}\big{(}\overline{\partial}_{E,h}+\theta_{E,h}\big{)}=\mathbb{D}^{\lambda}-\lambda\mathbb{D}_{E,h}^{\lambda\star}, and \big{(}\mathbb{D}^{\lambda}\big{)}^{2}=\big{(}\mathbb{D}_{E,h}^{\lambda\star}\big{)}^{2}=0, we obtain
[TABLE]
Hence, implies that \big{(}E,\overline{\partial}_{E,h},\theta_{E,h}\big{)} is a Higgs bundle. The metric is a pluri-harmonic metric for \big{(}E,\overline{\partial}_{E,h},\theta_{E,h}\big{)}. Conversely, if is a pluri-harmonic metric for a Higgs bundle \big{(}E,\overline{\partial}_{E},\theta\big{)}, we obtain the Chern connection associated with and , and the adjoint of with respect to . We obtain a flat -connection . The metric is a pluri-harmonic metric for .
Remark 2.14**.**
If , a flat [math]-connection is equivalent to a Higgs bundle \big{(}E,\overline{\partial}_{E},\theta\big{)} by the relation . In this case, we obtain G(h)=\bigl{[}\overline{\partial}_{E}+\theta,\partial_{E,h}+\theta_{h}^{\dagger}\bigr{]}, and hence is equal to the curvature of the connection .
2.6.2 The case
Let denote the decomposition into -parts. If , we clearly obtain . If , we obtain the converse.
Proposition 2.15**.**
Suppose . If , we obtain , i.e., is a pluri-harmonic metric of \big{(}E,\mathbb{D}^{\lambda}\big{)}.
Proof.
As in [49, Lemma 2.28], the following holds:
[TABLE]
It is easy to check that , \big{(}\theta_{E,h}^{\dagger}\big{)}^{2}=-(\theta_{E,h}^{2})^{\dagger} and \big{(}\overline{\partial}_{E,h}\theta_{E,h}\big{)}^{\dagger}=\partial_{E,h}\theta^{\dagger}_{E,h}.
From the flatness , we obtain
[TABLE]
From (2.6) and (2.7), we obtain
[TABLE]
Because , we obtain
[TABLE]
Note that is equivalent to and . To obtain , it is enough to prove \mathop{\rm Tr}\nolimits\big{(}\theta_{E,h}^{2}\big{(}\theta_{E,h}^{\dagger}\big{)}^{2}\big{)}=0. Indeed, there exists depending on such that for any Kähler form of we obtain \mathop{\rm Tr}\nolimits\big{(}\theta_{E,h}^{2}\big{(}\theta_{E,h}^{\dagger}\big{)}^{2}\big{)}\omega^{\dim Y-2}=C\bigl{|}\theta_{E,h}^{2}\bigr{|}^{2}_{h,\omega}\omega^{\dim Y}. Hence, the vanishing \mathop{\rm Tr}\nolimits\big{(}\theta_{E,h}^{2}\big{(}\theta_{E,h}^{\dagger}\big{)}^{2}\big{)}=0 implies \theta_{E,h}^{2}=\big{(}\theta_{E,h}^{\dagger}\big{)}^{2}=0 and .
From (2.9) and , we obtain \partial_{E,h}\overline{\partial}_{E,h}\theta_{E,h}^{\dagger}=\overline{\lambda}^{-1}\big{(}1-|\lambda|^{2}\big{)}\partial_{E,h}\bigl{(}(\theta^{\dagger}_{E,h})^{2}\bigr{)}=0. We also have . Hence, we obtain the following equality:
[TABLE]
From (2.8), and , we obtain
[TABLE]
Hence, we obtain the following:
[TABLE]
We obtain the claim of the proposition from (2.10) and (2.11). ∎
By using Proposition 2.15 we can improve [49, Corollary 2.30] as follows.
Corollary 2.16**.**
If , the pluri-harmonicity of the metric is equivalent to the vanishing , i.e., .
Remark 2.17**.**
In [49, Lemma 2.29], the claim \big{[}\,\overline{\partial}_{V,h},\partial_{V,h}\big{]}+\big{[}\theta_{V,h},\theta_{V,h}^{\dagger}\big{]}=0 is incorrect, in general. The author thanks Pengfei Huang for pointing out it.
Remark 2.18**.**
If , the claim of Proposition 2.15 also follows from a Bochner type formula [48, Proposition 21.39], which originally goes back to the study of Simpson [64] in the context of harmonic bundles, the study of Corlette [11] in the context of harmonic metrics for flat bundles on Riemannian manifolds, and the study of Siu [69] in the context of harmonic maps.
2.7 Wild harmonic bundles
2.7.1 Higgs case
Let be a complex manifold with a simple normal crossing hypersurface . Let \big{(}E,\overline{\partial}_{E},\theta,h\big{)} be a harmonic bundle on . It is called wild on if the following holds:
- •
Let denote the spectral cover of , i.e., denotes the support of the coherent -module induced by \big{(}E,\overline{\partial}_{E},\theta\big{)}. Then, the closure of in the relatively projective completion of with respect to is complex analytic.
A wild harmonic bundle \big{(}E,\overline{\partial}_{E},\theta,h\big{)} is called unramifiedly good at if the following holds:
- •
There exists a good set of irregular values , a neighbourhood , and a decomposition
[TABLE]
such that the closure of the spectral cover of in is proper over , where denote lifts of to .
A wild harmonic bundle \big{(}E,\overline{\partial}_{E},\theta,h\big{)} is called good at if the following holds:
- •
There exist a neighbourhood and a covering ramified along such that the pull back \varphi_{P}^{-1}\big{(}E,\overline{\partial}_{E},\theta,h\big{)}_{|X_{P}} is unramifiedly good wild at any point of .
We say that \big{(}E,\overline{\partial}_{E},\theta,h\big{)} is good wild (resp. unramifiedly good wild) on if it is good wild (resp. unramifiedly good wild) at any point of .
Note that not every wild harmonic bundle on is necessarily good on . But, the following is known [55, Corollary 15.2.8].
Theorem 2.19**.**
Let \big{(}E,\overline{\partial}_{E},\theta,h\big{)} be a wild harmonic bundle on . Then, there exists a proper birational morphism of complex manifolds such that is simple normal crossing, , \varphi^{-1}\big{(}E,\overline{\partial}_{E},\theta,h\big{)} is good wild on .
2.7.2 The case of -flat bundles
A -flat bundle \big{(}E,\mathbb{D}^{\lambda}\big{)} with a pluri-harmonic metric on is called (good, unramifiedly good) wild if the associated Higgs bundle with a pluri-harmonic metric \big{(}E,\overline{\partial}_{E,h},\theta_{E,h},h\big{)} is a (good, unramifiedly good) wild harmonic bundle.
2.7.3 Prolongation of good wild harmonic bundles to good filtered -flat bundles
The following is one of the fundamental theorems in the study of wild harmonic bundles [51, Theorem 7.4.3].
Theorem 2.20**.**
If \big{(}E,\mathbb{D}^{\lambda},h\big{)} is a good wild harmonic bundle on , then \big{(}\mathcal{P}^{h}_{\ast}E,\mathbb{D}^{\lambda}\big{)} is a good filtered -flat bundle on .
The following is a consequence of the norm estimate for good wild harmonic bundles [51, Theorem 11.7.2].
Theorem 2.21**.**
Let \big{(}E,\mathbb{D}^{\lambda},h_{i}\big{)} be good wild harmonic bundles on such that . Then, are mutually bounded around any point of .
2.7.4 Prolongation of good wild harmonic bundles in the projective case
Suppose that is projective and connected. Let be any ample line bundle on . The following is proved in [51, Propositions 13.6.1 and 13.6.4].
Proposition 2.22**.**
Let \big{(}E,\mathbb{D}^{\lambda},h\big{)} be a good wild harmonic bundle on .
- •
\big{(}\mathcal{P}^{h}_{\ast}E,\mathbb{D}^{\lambda}\big{)}* is -polystable with \mu_{L}\big{(}\mathcal{P}^{h}_{\ast}E\big{)}=0.*
- •
We obtain and .
- •
Let be another pluri-harmonic metric of \big{(}E,\mathbb{D}^{\lambda},h\big{)} such that . Then, there exists a decomposition of the -flat bundle \big{(}E,\mathbb{D}^{\lambda}\big{)}=\bigoplus\big{(}E_{j},\mathbb{D}^{\lambda}_{j}\big{)} such that the decomposition is orthogonal with respect to both and , for some .
- •
Let \big{(}\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1}\big{)} be any direct summand of \big{(}\mathcal{P}^{h}_{\ast}E,\mathbb{D}^{\lambda}\big{)}. Let \big{(}E_{1},\mathbb{D}^{\lambda}_{1}\big{)} be the -flat bundle on obtained as the restriction of \big{(}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1}\big{)}, and let be the metric of induced by . Then, \big{(}E_{1},\mathbb{D}^{\lambda}_{1},h_{1}\big{)} is a harmonic bundle. In particular, we obtain and .
2.8 Main existence theorem in this paper
Let be a smooth connected projective complex manifold with a simple normal crossing hypersurface . Let be any ample line bundle on . Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a good filtered -flat bundle on . Let \big{(}E,\overline{\partial}_{E},\mathbb{D}^{\lambda}\big{)} be the -flat bundle obtained as the restriction of \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} to .
Theorem 2.23**.**
Suppose that \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is -polystable, and that the following vanishing holds:
[TABLE]
Then, there exists a pluri-harmonic metric of \big{(}E,\overline{\partial}_{E},\mathbb{D}^{\lambda}\big{)} such that \big{(}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X\setminus H}\simeq\big{(}E,\mathbb{D}^{\lambda}\big{)} extends to \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}\simeq\big{(}\mathcal{P}^{h}_{\ast}E,\mathbb{D}^{\lambda}\big{)}.
We proved the claim of the theorem in the case in [51, Theorem 16.1.1]. We shall explain the proof in Sections 3–7. Note that the one dimensional case is due to Biquard–Boalch [3].
Corollary 2.24**.**
There exists the equivalence of the following objects for each :
- •
Good wild harmonic bundles on .
- •
-polystable good filtered -flat bundles \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} satisfying the condition .
Remark 2.25**.**
One of the referees raised the following interesting question. Let be a big and nef line bundle on such that there exists a positive current representing whose restriction to is a smooth Kähler form with at most Poincaré growth near . We do not assume that is ample. We can define the slope for a filtered sheaves, by using which we can introduce a stability condition for good filtered -flat bundles. Then, we may ask whether a statement similar to Theorem 2.23 holds. This question might also be related with a generalization of Kobayashi–Hitchin correspondence to the context of -modules.
2.8.1 Outline of the proof
Theorem 2.23
Let us explain a rough outline of the proof. We shall omit some technical details. Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a -stable good filtered -flat bundle on such that . Let \big{(}V,\mathbb{D}^{\lambda}\big{)} be the -flat bundle on obtained as the restriction of \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} to .
In the case , we shall apply the argument in [63] as follows. For each , we take a holomorphic coordinate neighbourhood \big{(}X_{P},z_{P}\big{)} around . We take a Kähler metric such that are mutually bounded with for some . If is sufficiently small, there exists a Hermitian metric of such that , and is bounded with respect to and , is flat. (See Corollary 3.28. Though we state it as a corollary of Proposition 3.27, which also deals with a perturbation, it is easy to deduce it directly from the estimate in the tame case [63].) Moreover for any filtered -flat subsheaf , is equal to the analytic degree of \big{(}\mathcal{V}^{\prime},\mathbb{D}^{\lambda}\big{)}_{|X\setminus H} with respect to and . Then, by [62, Theorem 1], if \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is stable of degree [math], there exists a harmonic metric of \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} such that and are mutually bounded (Theorem 4.1). Let us note that the proof allows us to obtain the inequality for the Donaldson functional (Proposition 4.4). This inequality is useful for the study of the continuity of the family of harmonic metrics of some family of good filtered -flat bundles (Proposition 4.5).
For the higher dimensional case, we use the same strategy in [46, 49] and [51]. It is a key step to study the case . There are two naive ideas which are not available as they are.
One is to apply [62, Theorem 1] by constructing a Hermitian metric of such that , is dominated in an appropriate way, is flat. For the construction of such a Hermitian metric , a compatibility condition seems necessary between the nilpotent parts of the induced endomorphisms and on . (See Section 3.5.3 for the endomorphisms \mathop{\rm Res}\nolimits_{i}\big{(}\mathbb{D}^{\lambda}\big{)}.) Once we prove the existence of a pluri-harmonic, it turns out that such a compatibility condition is satisfied. However, before proving the existence, it is not clear whether such a compatibility condition is satisfied. As a result, it is difficult to construct a Hermitian metric with the desired property, in general.
The other is to use Mehta–Ramanathan type theorem (Proposition 3.8), according to which there exists such that for the [math]-set of a generic section of H^{0}\big{(}X,L^{\otimes m}\big{)}, the restriction \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y} is also stable. Hence, if we fix a flat metric of adapted to , there exists a harmonic metric of \big{(}V,\mathbb{D}^{\lambda}\big{)}_{|Y\setminus H} adapted to such that . If we can prove that there exists a Hermitian metric of such that for such generic hypersurfaces , then should be the desired pluri-harmonic metric for \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}. But, the existence of such is not clear.
Roughly speaking, we combine these two ideas as follows. For any small , there exists a filtered bundle over such that \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is a -stable good filtered -flat bundle, \mathop{\rm Res}\nolimits_{i}\big{(}\mathbb{D}^{\lambda}\big{)} are semisimple for , \det\big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}\big{)}=\det(\mathcal{P}_{\ast}\mathcal{V}), the difference of and are dominated by . (See Section 3.7.2 for more precise conditions.) The last condition implies that \lim\limits_{\epsilon\to 0}\int\mathop{\rm ch}\nolimits_{2}\big{(}\mathcal{P}^{(\epsilon)}_{\ast}\big{)}=0. For \big{(}\mathcal{P}_{\ast}^{(\epsilon)}\mathcal{V},\mathbb{D}^{\lambda}\big{)}, we can construct such that , G\big{(}h^{(\epsilon)}_{\mathop{\rm in}\nolimits}\big{)} is dominated in an appropriate way, \det\big{(}h^{(\epsilon)}_{\mathop{\rm in}\nolimits}\big{)}=h_{\det(\mathcal{V})}. By [62, Theorem 1], there exists a Hermitian–Einstein metric of \big{(}V,\mathbb{D}^{\lambda}\big{)} such that and are mutually bounded, and that \det\big{(}h^{(\epsilon)}_{\mathop{\rm HE}\nolimits}\big{)}=h_{\det(\mathcal{V})}. (See Section 3.1 for Hermitian–Einstein metrics of Higgs bundles.) Moreover, G\big{(}h^{(\epsilon)}_{\mathop{\rm HE}\nolimits}\big{)}\to 0 in as . Hence, we would like to construct the desired pluri-harmonic metric as . If is sufficiently small, \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y} is also stable for the [math]-set of a generic section of H^{0}\big{(}X,L^{\otimes m}\big{)}, and hence \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y} has a harmonic metric such that \det\big{(}h^{(\epsilon)}_{Y}\big{)}=h_{\det(\mathcal{V})|Y\setminus H}. By the continuity of a family of harmonic metrics mentioned above, the sequence is convergent to as (Proposition 4.5). Because is not necessarily a harmonic metric of \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y}, it is not necessarily equal to . But, because the -norm of G\big{(}h^{(\epsilon)}_{\mathop{\rm HE}\nolimits|Y}\big{)} is dominated by , we can deduce the convergence of the sequence to as (Proposition 4.8). Hence, we obtain the convergence of almost everywhere, and the limit satisfies for the [math]-set of generic section of H^{0}\big{(}X,L^{\otimes m}\big{)}. Thus, we can prove the theorem in the case . (See Section 7.2 for a more precise argument.)
In the case , we use an induction on . By the Mehta–Ramanathan type theorem, there exists such that for the [math]-sets of generic sections of H^{0}\big{(}X,L^{\otimes m}\big{)}, \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y_{i}} and \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y_{1}\cap Y_{2}} are -stable. By fixing a flat metric for \det\big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}, there exist pluri-harmonic metric of \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y_{i}} such that . Because \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y_{1}\cap Y_{2}} is also -stable, we obtain that . Hence, there exists a Hermitian metric of for a finite subset , such that for the [math]-set of a generic section of H^{0}\big{(}X,L^{\otimes m}\big{)}. It is easy to see that is the desired pluri-harmonic metric. (See Section 7.3 for a more precise argument.)
3 Preliminaries
3.1 Hermitian–Einstein metrics of -flat bundles
Let be a Kähler manifold with a Kähler form . Let \big{(}E,\mathbb{D}^{\lambda}\big{)} be a -flat bundle on with a Hermitian metric. Recall that is called a Hermitian–Einstein metric of the -flat bundle if , where denote the trace-free part of , and denote the adjoint of the multiplication by (see [35, Section 3.2]). The following is a generalization of Kobayashi–Lübke inequality to the context of -flat bundles due to Simpson [62, Proposition 3.4].
Proposition 3.1** (Simpson).**
If is a Hermitian–Einstein metric, there exists depending only on such that the following holds:
[TABLE]
As a result, if \mathop{\rm Tr}\nolimits\bigl{(}\big{(}G(h)^{\bot}\big{)}^{2}\bigr{)}\omega^{n-2}=0, then we obtain .
3.2 Rank one case
Let be an dimensional smooth connected projective variety with a simple normal crossing hypersurface . Let be a Kähler form. Let be the irreducible decomposition. Let be a -Hermitian metric of the line bundle . Let denote the section of induced by the inclusion .
Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a good filtered -flat bundle on of rank one. For each , there uniquely exists such that . Let be the constant determined by
[TABLE]
The following proposition is standard.
Proposition 3.2**.**
There exists a Hermitian metric of the line bundle such that , is a Hermitian metric of of -class. Such a metric is unique up to the multiplication by a positive constant. Moreover, if , then holds, and hence is a pluri-harmonic metric of \big{(}E,\mathbb{D}^{\lambda}\big{)}.
Proof.
Note that G(h)=\big{(}1+|\lambda|^{2}\big{)}R(h) holds in the rank one case. (See [49, Lemma 2.31].) Let be a -metric of . We obtain the metric of on . It is well known that naturally extends to a closed -form on of -class which represents . By the condition of , we obtain \int_{X}\bigl{(}\sqrt{-1}\Lambda_{\omega}R(h_{0})-\big{(}1+|\lambda|^{2}\big{)}^{-1}A\bigr{)}\omega^{n}=0. Note that . Hence, there exists an -valued -function such that \sqrt{-1}\Lambda_{\omega}R(h_{0}{\rm e}^{\varphi_{0}})-\big{(}1+|\lambda|^{2}\big{)}^{-1}A=0. The metric has the desired property. The uniqueness is clear.
Suppose that . In the rank one case, a Hermitian metric of is a pluri-harmonic metric of \big{(}E,\mathbb{D}^{\lambda}\big{)}, if and only if . Because the cohomology class of is [math], there exists an -valued -function such that by the standard -lemma. By the uniqueness, we obtain the second claim of the lemma. ∎
For the metric in Proposition 3.2, induces a closed -form on of -class which represents .
3.3 -subobject and socle for reflexive filtered -flat sheaves
Let and be as in Section 3.2. Let be an ample line bundle on . For any coherent -module , we set .
3.3.1 -subobjects
Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a reflexive filtered -flat sheaf on . For any , let denote the family of saturated coherent subsheaves of such that and that is a -flat subsheaf of . Any induces a reflexive filtered sheaf by for any . We set . Thus, we obtain a function on .
Lemma 3.3**.**
The image f_{A}\bigl{(}\mathcal{S}(\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V},A)\bigr{)} is a finite subset of . In particular, has the maximum.
Proof.
According to [20, Lemma 2.5], the family is bounded. Hence, by using the flattening stratifications [57, Section 8], it is easy to see that there exists a finite decomposition such that is constant on each . ∎
It is standard that any reflexive filtered -flat sheaf has a -subobject, i.e., the following holds.
Proposition 3.4**.**
For any reflexive filtered -flat sheaf \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}, there uniquely exists a non-zero -flat subsheaf such that the following holds for any non-zero reflexive -flat subsheaf :
- •
* holds.*
- •
If holds, then we obtain .
Proof.
By the formula (2.1), there exists such that the following holds for any saturated subsheaf :
[TABLE]
We set . Let denote the maximum of . Then, it is easy to see that for any saturated -flat subsheaf . Moreover, if , then \big{(}\mathcal{P}_{\ast}\mathcal{V}^{\prime},\mathbb{D}^{\lambda}_{\mathcal{V}^{\prime}}\big{)} is -semistable, where denote the flat -connection induced by .
Suppose that the -flat subsheaves satisfy . We obtain the subsheaf . Because is a quotient of , we obtain a filtered sheaf over . induced by . Then, by the -semistability of \big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)}, we obtain that . Let denote the saturated subsheaf of generated by . We obtain a filtered sheaf by . Because the natural morphism is generically an isomorphism, we obtain by Lemma 2.5. Hence, we obtain . Then, the claim of the lemma is clear. ∎
3.3.2 Socle
Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a -semistable reflexive filtered -flat sheaf on . Let denote the family of saturated -flat subsheaves such that the induced filtered -flat sheaf \big{(}\mathcal{P}_{\ast}\mathcal{V}^{\prime},\mathbb{D}^{\lambda}_{\mathcal{V}^{\prime}}\big{)} is -stable with . Let be the saturated -submodule of generated by . It is a -flat subsheaf of .
Proposition 3.5**.**
\big{(}\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{\mathcal{V}_{1}}\big{)}* is equal to the direct sum \bigoplus_{k=1}^{\ell}\big{(}\mathcal{P}_{\ast}\mathcal{V}^{(k)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(k)}}\big{)} of a tuple of -stable filtered -flat subsheaves of \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}. In particular, \big{(}\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1}\big{)} is -polystable. The filtered -flat subsheaf \big{(}\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{\mathcal{V}_{1}}\big{)} is called the socle of \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}.*
Proof.
Let be saturated -flat subsheaves of such that \mu_{L}\big{(}\mathcal{P}_{\ast}\mathcal{V}^{(i)}\big{)}=\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}), \big{(}\mathcal{P}_{\ast}\mathcal{V}^{(1)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(1)}}\big{)} is -semistable, \big{(}\mathcal{P}_{\ast}\mathcal{V}^{(2)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(2)}}\big{)} is -stable.
Lemma 3.6**.**
Either or holds.
Proof.
Let us consider the morphism , where denote the inclusions. Let denote the kernel. We obtain a filtered sheaf over by . The projection induces . It induces a morphism of filtered -flat sheaves g\colon\big{(}\mathcal{P}_{\ast}\mathcal{K},\mathbb{D}^{\lambda}_{\mathcal{K}}\big{)}\longrightarrow\big{(}\mathcal{P}_{\ast}\mathcal{V}^{(2)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(2)}}\big{)}. We set . Because \bigoplus_{i=1,2}\big{(}\mathcal{P}_{\ast}\mathcal{V}^{(i)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(i)}}\big{)} and \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} are -semistable with the same slope , we obtain that \big{(}\mathcal{P}_{\ast}\mathcal{K},\mathbb{D}^{\lambda}_{\mathcal{K}}\big{)} is also -semistable with .
Suppose that , i.e., . Because is a subsheaf of , we also obtain a filtered sheaf induced by . Because , we obtain a filtered sheaf over induced by . Then, we obtain
[TABLE]
Because \big{(}\mathcal{P}_{\ast}\mathcal{V}^{(2)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(2)}}\big{)} is -stable and because , we obtain that , i.e., and are generically isomorphic. Because \mu_{L}(\mathcal{P}_{\ast}\mathcal{I})=\mu_{L}\big{(}\mathcal{P}_{\ast}\mathcal{V}^{(2)}\big{)}, Lemma 2.5 implies that is an isomorphism in codimension . Hence, there exists a closed algebraic subset such that the codimension of is larger than , . Because is reflexive we obtain that . ∎
Let us study the case where . Let denote the saturated -flat subsheaf of generated by . Let denote the filtered sheaf over induced by .
Lemma 3.7**.**
\big{(}\mathcal{P}_{\ast}\mathcal{V}^{(3)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(3)}}\big{)}* is -semistable, and the induced morphism is an isomorphism in codimension one.*
Proof.
We obtain \mu_{0}=\mu_{L}\big{(}\mathcal{P}_{\ast}\big{(}\mathcal{V}^{(1)}\oplus\mathcal{V}^{(2)}\big{)}\big{)}\leq\mu_{L}\big{(}\mathcal{P}_{\ast}\mathcal{V}^{(3)}\big{)}\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{V})=\mu_{0}. Hence, we obtain that \mu_{L}\big{(}\mathcal{P}_{\ast}\mathcal{V}^{(3)}\big{)}=\mu_{0} and that \big{(}\mathcal{P}_{\ast}\mathcal{V}^{(3)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(3)}}\big{)} is -semistable. Because is generically an isomorphism, and because they have the same slope, is an isomorphism in codimension one by Lemma 2.5. ∎
By Lemma 3.7, it is easy to observe that there exists a finite sequence of reflexive -flat subsheaves such that the induced filtered -flat sheaves \big{(}\mathcal{P}_{\ast}\mathcal{V}^{\prime}_{j},\mathbb{D}^{\lambda}_{\mathcal{V}_{j}^{\prime}}\big{)} are -stable, the image of the induced morphism is generically an isomorphism. Because \mu_{0}=\mu_{L}\bigl{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}\bigr{)}\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{1})\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{V})=\mu_{0}, we obtain that \mu_{L}\bigl{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}\bigr{)}=\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{1})=\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}). Hence, is an isomorphism in codimension one by Lemma 2.5. Because both and are reflexive, we obtain that . Thus, we obtain Proposition 3.5. ∎
3.4 Mehta–Ramanathan type theorems
Let be a smooth connected -dimensional projective variety with a simple normal crossing hypersurface . Let be the irreducible decomposition. Let be an ample line bundle on .
3.4.1 Restriction to general curves
Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a reflexive filtered -flat sheaf on . There exists a Zariski closed subset with such that the singular locus of is contained in , is a filtered bundle on .
Let be a smooth curve in such that , intersects with the smooth part of transversally. Set . We obtain a locally free -module . It is equipped with the induced flat -connection . Let . For any , there exists such that . We choose such that , and we obtain an -submodule \mathcal{P}_{{\boldsymbol{b}}}\big{(}\mathcal{V}_{|Y}\big{)}_{P}:=\mathcal{P}_{{\boldsymbol{a}}(P,{\boldsymbol{b}})}(\mathcal{V})_{P} of \big{(}\mathcal{V}_{|Y}\big{)}_{P}, which is independent of the choice of as above. There exists a locally free -module \mathcal{P}_{{\boldsymbol{b}}}\big{(}\mathcal{V}_{|Y}\big{)}\subset\mathcal{V}_{|Y} whose stalk at is \mathcal{P}_{{\boldsymbol{b}}}\big{(}\mathcal{V}_{|Y}\big{)}_{P}. Thus, we obtain a filtered -flat bundle \big{(}\mathcal{P}_{\ast}\big{(}\mathcal{V}_{|Y}\big{)},\mathbb{D}^{\lambda}_{|Y}\big{)} which is denoted by \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y}.
3.4.2 The stability condition
Proposition 3.8**.**
A reflexive filtered -flat sheaf \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} on is -stable (resp. -semistable) if and only if the following holds:
- •
For any , there exists such that \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y} is -stable resp. -semistable, where denotes a generic -dimensional complete intersection of hypersurfaces of .
Proof.
The case is already studied in [51, Section 13.2]. The case is reduced to the case . As for the case , we can prove the claim of the proposition by the argument in [46, Section 3.4], which closely follows the arguments of Mehta–Ramanathan [43, 44] and Simpson [64]. We use for a large instead of in [46, Section 3.4]. (See also [51, Section 13.2].) ∎
3.4.3 Restrictions of morphisms and the polystability condition
Let us give a complement on the restriction of morphisms of reflexive filtered -flat sheaves to generic complete intersection curves, which is a variant of [64, Lemma 3.9]. Let \big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}\big{)} be reflexive filtered -flat sheaves on . Let \mathop{\rm Hom}\nolimits\bigl{(}\big{(}\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1}\big{)},\big{(}\mathcal{P}_{\ast}\mathcal{V}_{2},\mathbb{D}^{\lambda}_{2}\big{)}\bigr{)} denote the vector space of morphisms of filtered -flat sheaves \big{(}\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1}\big{)}\longrightarrow\big{(}\mathcal{P}_{\ast}\mathcal{V}_{2},\mathbb{D}^{\lambda}_{2}\big{)}. We shall prove a refined claim (Proposition 3.16) of the following proposition in Sections 3.4.4–3.4.6.
Proposition 3.9**.**
There exists a positive integer such that the restriction
[TABLE]
is an isomorphism for a generic -dimensional complete intersection of hypersurfaces of .
Before going to the proof of Proposition 3.9, we state a variant of Proposition 3.8 on the -polystability condition.
Corollary 3.10**.**
A reflexive filtered -flat sheaf \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} on is -polystable if and only if the following holds:
- •
For any , there exists such that \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y} is -polystable, where denotes the -dimensional complete intersection of generic hypersurfaces of .
Proof.
If \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is -polystable, we obtain a decomposition \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}=\bigoplus\big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)} into -stable filtered -flat sheaves. Applying Proposition 3.8 to each stable component, we obtain the “only if” claim.
Let be an integer larger than in Proposition 3.9 for \mathop{\rm Hom}\nolimits\bigl{(}\big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)},\big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}\bigr{)}. Suppose that there exists such that \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y} is -polystable for a generic -dimensional complete intersection of hypersurfaces of . We obtain the decomposition
[TABLE]
into stable filtered Higgs bundles. Let denote the endomorphisms of \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y} obtained by composing the projection \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y}\longrightarrow\big{(}\mathcal{P}_{\ast}\mathcal{V}_{Y,i},\mathbb{D}^{\lambda}_{Y,i}\big{)} with respect to the decomposition (3.1), with the inclusion \big{(}\mathcal{P}_{\ast}\mathcal{V}_{Y,i},\mathbb{D}^{\lambda}_{Y,i}\big{)}\longrightarrow\big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|Y}. Note that they satisfy , and . By Proposition 3.9, there uniquely exist the endomorphisms of \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} such that . By Proposition 3.9 again, they satisfy , and . Let denote the image of . We define for any . Because are compatible with and the filtration , we obtain the decomposition \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}=\bigoplus\big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)}. By the construction, \big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)}_{|Y}=\big{(}\mathcal{P}_{\ast}\mathcal{V}_{Y,i},\mathbb{D}^{\lambda}_{Y,i}\big{)} are stable. Hence, \big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)} are -stable with \mu_{L}\big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)}=\mu_{L}\big{(}\mathcal{P}_{\ast}\mathcal{V}_{j},\mathbb{D}^{\lambda}_{j}\big{)} , i.e., \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is -polystable. ∎
3.4.4 General Enriques–Severi lemma due to Mehta–Ramanathan
To prove Proposition 3.9, we recall the general Enriques–Severi lemma in [43]. Recall . For a positive integer , let denote the projective space of lines in H^{0}\big{(}X,L^{\otimes m}\big{)}. For sequences with , we set . There exists the correspondence variety , i.e., Z_{{\boldsymbol{m}}}=\bigl{\{}(x,s_{1},\ldots,s_{t})\in X\times S_{{\boldsymbol{m}}}\mid s_{i}(x)=0,\,1\leq i\leq t\bigr{\}}. For any , we set .
Let be a coherent reflexive -module on . For any with , and for any , we set . For any integer , let .
According to [43, Proposition 1.5], there exists a non-empty Zariski open subset such that the following holds.
- •
is smooth.
- •
For any , is a reflexive -module.
In the proof of [43, Proposition 3.2], the following proposition is proved.
Proposition 3.11**.**
Let . There exists a positive integer depending only on such that the following holds:
- •
For any with , there exists a non-empty Zariski open subset such that for any and any .
Corollary 3.12**.**
Let . There exists a positive integer depending only on such that the following holds:
- •
For any with , there exists a non-empty Zariski open subset such that for any and any .
Proof.
Let be a positive integer as in Proposition 3.11. We also assume for any . We use an induction on . For with , we set . By the assumption of the induction and Proposition 3.11, there exists a non-empty Zariski open subset such that for any and any .
For any , let denote the image in by the projection . There exists the exact sequence
[TABLE]
By [43, Proposition 1.5], there exists a Zariski open subset such that if then we obtain the following exact sequence from (3.2) by taking the tensor product with :
[TABLE]
We shrink so that contains the image of by the projection . Let . For any , we obtain and because . Hence, we obtain . ∎
Corollary 3.13**.**
Let . There exists depending only on such that the following holds:
- •
For any with , there exists a non-empty Zariski open subset such that the natural morphism is an isomorphism for any .
Proof.
It is enough to apply the argument in the first paragraph of the proof of [43, Proposition 3.2] with Proposition 3.11 and Corollary 3.12. (This is essentially pointed out in [64, Lemma 3.9].) ∎
Let denote the conormal bundle of in for any .
Corollary 3.14**.**
Let . There exists depending only on such that the following holds:
- •
For any with , there exists a non-empty Zariski open subset such that H^{0}\big{(}X_{s},T^{\ast}_{X_{s}}X\otimes F_{s}\big{)}=0 for any .
Proof.
Because , the claim follows from Corollary 3.12. ∎
3.4.5 Flat sections of reflexive filtered -flat sheaves
Let be a simple normal crossing hypersurface of with the irreducible decomposition . Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a reflexive filtered -flat sheaf on . Note that there exists a Zariski closed subset with such that contains the singular locus of , is a filtered bundle on . For any and for any , we set .
According to [43, Proposition 1.5], there exists a non-empty Zariski open subset such that the following holds:
- •
is smooth.
- •
For any , holds, and intersects with in transversally. Moreover, are locally free -modules.
There exists a non-negative integer such that induces a morphism of sheaves : . For , we set
[TABLE]
The flat -connection induces such that . Thus, we obtain a complex of sheaves \mathcal{C}^{\bullet}_{N}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V},\mathbb{D}^{\lambda}\big{)} on . Clearly, the following holds:
[TABLE]
For any , we obtain the filtered -flat bundle \big{(}\mathcal{P}_{\ast}\mathcal{V}_{s},\mathbb{D}^{\lambda}_{s}\big{)}:=\big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X_{s}}. Let denote the image of induced by the natural map . Let denote the inclusion. We obtain the natural morphism of complexes of sheaves \mathcal{C}_{N}^{\bullet}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V},\mathbb{D}^{\lambda}\big{)}\longrightarrow\iota_{s\ast}\mathcal{C}_{N}^{\bullet}\big{(}\mathcal{P}_{{\boldsymbol{a}}(s)}\mathcal{V}_{s},\mathbb{D}^{\lambda}_{s}\big{)}, which induces
[TABLE]
The following proposition is essentially [64, Lemma 3.9].
Proposition 3.15**.**
There exists a positive integer such that the following claim holds for any with and a non-empty Zariski open subset .
- •
For any , the natural morphism (3.3) is an isomorphism.
Proof.
According to Corollary 3.13, if is sufficiently large, there exists a non-empty Zariski open subset such that the following natural morphisms are isomorphisms for any :
[TABLE]
There exists the following exact sequence:
[TABLE]
According to Corollary 3.14, if is sufficiently large, there exists a non-empty Zariski open subset such that the following holds for any :
[TABLE]
Hence, the natural morphism
[TABLE]
is injective for any . We obtain the injectivity of the following natural morphism for any :
[TABLE]
Then, we obtain the claim of the proposition. ∎
3.4.6 Morphisms of reflexive filtered -flat sheaves
Let be reflexive filtered sheaves with meromorphic flat -connection on . Let \mathop{\rm Hom}\nolimits\bigl{(}\big{(}\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1}\big{)},\big{(}\mathcal{P}_{\ast}\mathcal{V}_{2},\mathbb{D}^{\lambda}_{2}\big{)}\bigr{)} denote the vector space of morphisms of filtered -flat sheaves \big{(}\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1}\big{)}\longrightarrow\big{(}\mathcal{P}_{\ast}\mathcal{V}_{2},\mathbb{D}^{\lambda}_{2}\big{)}.
Proposition 3.16**.**
There exists a positive integer such that the following claim holds for any with and for a non-empty Zariski open subset .
- •
For any , let \big{(}\mathcal{P}_{\ast}\mathcal{V}_{i,s},\mathbb{D}^{\lambda}_{i,s}\big{)} denote the induced filtered -flat bundles on . Then, the natural morphism
[TABLE]
is an isomorphism.
Proof.
For any , let denote the subsheaf of the -module determined as follows for any open subset :
[TABLE]
It is easy to see that are reflexive -modules. Thus, we obtain a reflexive filtered sheaf with the induced flat -connection . We can easily observe that
[TABLE]
for any large , where . Then, the claim follows from Proposition 3.15. ∎
3.5 Good filtered -flat bundles and ramified coverings
3.5.1 Pull back
Let be any complex manifold with a simple normal crossing hypersurface . Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a good filtered -flat bundle. We set .
For any point , let denote an admissible holomorphic coordinate neighbourhood around . We set and . We set \big{(}\mathcal{P}_{\ast}\mathcal{V}_{P},\mathbb{D}^{\lambda}_{P}\big{)}:=\big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X_{P}}.
By using the coordinate system, we may regard as an open subset of . Let : be given by \varphi_{P}(\zeta_{1},\ldots,\zeta_{n})=\big{(}\zeta_{1}^{e},\ldots,\zeta_{\ell(P)}^{e},\zeta_{\ell(P)+1},\ldots,\zeta_{n}\big{)}. We set , and . We set . It is identified with the Galois group of the ramified covering by the action as in Section 2.3.2.
We obtain the -equivariant good filtered -flat bundle \big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P},\widetilde{\mathbb{D}}^{\lambda}_{P}\big{)}:=\varphi_{P}^{\ast}\big{(}\mathcal{P}_{\ast}\mathcal{V}_{P},\mathbb{D}^{\lambda}_{P}\big{)} on \big{(}\widetilde{X}_{P},\widetilde{H}_{P}\big{)}.
Lemma 3.17**.**
\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P},\widetilde{\mathbb{D}}^{\lambda}_{P}\big{)}* is unramifiedly good.*
Proof.
See [51, Lemma 2.2.7]. ∎
3.5.2 The associated graded bundles
We obtain the -equivariant filtered bundles {}^{1}\!\mathop{\rm Gr}\nolimits^{F}_{c}\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P}\big{)} on \big{(}\widetilde{H}_{P,1},\partial\widetilde{H}_{P,1}\big{)}. There exists the -equivariant decomposition
[TABLE]
where acts on \mathbb{G}_{\ell}{}^{1}\!\mathop{\rm Gr}\nolimits^{F}_{c}\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P}\big{)} as the multiplication by .
Lemma 3.18**.**
The pull back naturally induces the isomorphism (\varphi_{|\widetilde{H}_{P,1}})^{\ast}\bigl{(}{}^{1}\!\mathop{\rm Gr}\nolimits^{F}_{c/e}(\mathcal{P}_{\ast}\mathcal{V}_{P})\bigr{)}\simeq\mathbb{G}_{0}{}^{1}\!\mathop{\rm Gr}\nolimits^{F}_{c}\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P}\big{)}. As a result, is the descent of \mathbb{G}_{0}{}^{1}\!\mathop{\rm Gr}\nolimits^{F}_{c}\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P}\big{)}. More generally, the pull back and the multiplication by induces an isomorphism (\varphi_{|\widetilde{H}_{P,1}})^{\ast}\bigl{(}{}^{1}\!\mathop{\rm Gr}\nolimits^{F}_{(c+\ell)/e}(\mathcal{P}_{\ast}\mathcal{V}_{P})\bigr{)}\simeq\mathbb{G}_{\ell}{}^{1}\!\mathop{\rm Gr}\nolimits^{F}_{c}\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P}\big{)}.
Clearly, there exist a similar decomposition {}^{i}\!\mathop{\rm Gr}\nolimits^{F}_{c}\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P}\big{)}=\bigoplus_{\ell=0}^{e-1}\mathbb{G}_{\ell}{}^{i}\!\mathop{\rm Gr}\nolimits^{F}_{c}\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P}\big{)} and isomorphisms for any .
3.5.3 Residues
Let us recall that we obtain the endomorphisms \mathop{\rm Res}\nolimits_{j}\big{(}\mathbb{D}^{\lambda}\big{)} on by using Lemma 3.18. (See [51, Section 2.5.2] for more detailed explanations.)
Let be any point of . First, let us construct the residues \mathop{\rm Res}\nolimits_{1}\big{(}\mathbb{D}^{\lambda}_{P}\big{)} on . At any , we obtain the formal decomposition \big{(}\mathcal{P}_{{\boldsymbol{a}}}\widetilde{\mathcal{V}}_{P},\mathbb{D}^{\lambda}_{P}\big{)}\otimes\mathcal{O}_{\widetilde{X}_{P},\widehat{Q}}=\bigoplus\big{(}\mathcal{P}_{{\boldsymbol{a}}}\widetilde{\mathcal{V}}_{\mathfrak{a}},\widetilde{\mathbb{D}}^{\lambda}_{\mathfrak{a}}\big{)} as in (2.4). For , we obtain the endomorphisms \mathop{\rm Res}\nolimits_{1}\big{(}\widetilde{\mathbb{D}}_{P}^{\lambda}\big{)}_{Q} of {}^{1}\!\mathop{\rm Gr}\nolimits^{F}_{c}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\widetilde{\mathcal{V}}\big{)}_{|Q} as the residue of \bigoplus\bigl{(}\widetilde{\mathbb{D}}^{\lambda}_{\mathfrak{a}}-{\rm d}\widetilde{\mathfrak{a}}\mathop{\rm id}\nolimits_{\widetilde{\mathcal{V}}_{\mathfrak{a}}}\bigr{)} at . According to [51, Lemma 2.5.2], by varying , we obtain the endomorphism \mathop{\rm Res}\nolimits_{1}\big{(}\widetilde{\mathbb{D}}_{P}^{\lambda}\big{)} of the filtered bundle {}^{1}\!\mathop{\rm Gr}\nolimits^{F}_{c}\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P}\big{)}. It is -equivariant. Hence, we obtain \mathop{\rm Res}\nolimits_{1}\big{(}\mathbb{D}^{\lambda}_{P}\big{)} on as the descent of on \mathbb{G}_{0}{}^{1}\!\mathop{\rm Gr}\nolimits^{F}_{ec}\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P}\big{)}. The factor comes from the relation . Similarly, we obtain \mathop{\rm Res}\nolimits_{i}\big{(}\mathbb{D}^{\lambda}_{P}\big{)} for .
It is easy to see that there exists a globally defined endomorphism \mathop{\rm Res}\nolimits_{j}\big{(}\mathbb{D}^{\lambda}\big{)} on which is equal to the endomorphisms constructed locally around as above.
3.5.4 Parabolic weights
We introduce some notation. We set {\mathcal{P}{\rm ar}}(\mathcal{P}_{\ast}\mathcal{V},j):=\bigl{\{}b\in{\mathbb{R}}\mid{}^{j}\!\mathop{\rm Gr}\nolimits^{F}_{b}(\mathcal{P}_{\ast}\mathcal{V})\neq 0\bigr{\}} for . Because we shall often use the pull back by a ramified covering as in Section 3.5.1, for a fixed , it is convenient to consider
[TABLE]
Note that {\mathcal{P}{\rm ar}}\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P},j\big{)}=\bigl{\{}eb\mid b\in\widetilde{{\mathcal{P}{\rm ar}}}(\mathcal{P}_{\ast}\mathcal{V}_{P},j)\bigr{\}} for . (See Lemma 3.18.) We define
[TABLE]
If is finite, we also set .
For each , we set
[TABLE]
We remark the following obvious lemma.
Lemma 3.19**.**
For each , there exists such that for any .
3.6 Approximation by model filtered -flat bundles
3.6.1 Model filtered -flat bundles
Let be a complex manifold. Let be a neighbourhood of in . We set . Let be a positive integer. Let be the standard complex coordinate of . Consider induced by . We set and . The induced morphism is also denoted by . Let denote the group of the -th roots of , which is naturally identified with the Galois group of the ramified covering .
Let be a finite subset of H^{0}\big{(}\widetilde{Y},\mathcal{O}_{\widetilde{Y}}\big{(}{\ast}\widetilde{H}\big{)}\big{)} which is preserved by the -action. Let and be finite subsets of and , respectively. Let be finite dimensional -vector spaces equipped with a nilpotent endomorphism . Note that may be [math]. We suppose that is a -representation such that it is -equivariant as a vector bundle over , commutes with the -action.
We set \widetilde{\mathcal{V}}_{\mathfrak{a},a,\alpha}:=\mathcal{O}_{\widetilde{Y}}\big{(}{\ast}\widetilde{H}\big{)}\otimes V_{\mathfrak{a},a,\alpha}. We define the filtered bundle over by setting
[TABLE]
for any , where . We define the flat -connection on by setting
[TABLE]
for any , which we regard as a section of in a natural way. Thus, we obtain a -equivariant filtered -flat bundle \bigoplus_{\mathfrak{a},a,\alpha}\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{\mathfrak{a},a,\alpha},\widetilde{\mathbb{D}}^{\lambda}_{\mathfrak{a},a,\alpha}\big{)}, called a model filtered -flat bundle. If induces a good set of irregular values in \mathcal{O}_{\widetilde{Y}}\big{(}{\ast}\widetilde{H}\big{)}_{Q}/\mathcal{O}_{\widetilde{Y},Q} at each , then \bigoplus_{\mathfrak{a},a,\alpha}\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{\mathfrak{a},a,\alpha},\widetilde{\mathbb{D}}^{\lambda}_{\mathfrak{a},a,\alpha}\big{)} is an unramifiedly good filtered -flat bundle. It induces a filtered -flat bundle on as the descent, which is also called a model filtered -flat bundle.
3.6.2 Approximation of good filtered -flat bundles
Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be any good filtered -flat bundle on . Assume the following condition:
Condition 3.20**.**
For each , the conjugacy classes of \mathop{\rm Res}\nolimits\big{(}\mathbb{D}^{\lambda}\big{)} on are independent of . Note that this condition is trivially satisfied if .
We set . Let and \big{(}\widetilde{Y},\widetilde{H}\big{)} be as in Section 3.6.1. We set \big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}},\widetilde{\mathbb{D}}^{\lambda}\big{)}:=\varphi^{\ast}\big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}. For each , there exists a decomposition
[TABLE]
as in (2.5). We obtain the vector spaces \mathop{\rm Gr}\nolimits^{F}_{a}\big{(}\mathcal{P}_{0}\widetilde{\mathcal{V}}_{\mathfrak{a}}\big{)}_{|Q} equipped with the endomorphisms \mathop{\rm Res}\nolimits\big{(}\widetilde{\mathbb{D}}^{\lambda}\big{)}. Condition 3.20 is equivalent to the following.
- •
The conjugacy classes of \mathop{\rm Res}\nolimits\big{(}\widetilde{\mathbb{D}}^{\lambda}\big{)} on \mathop{\rm Gr}\nolimits^{F}_{a}\big{(}\mathcal{P}_{0}\widetilde{\mathcal{V}}_{\mathfrak{a}}\big{)}_{|Q} are independent of for any .
In particular, the condition implies that there exists a decomposition
[TABLE]
on , where \mathop{\rm Res}\nolimits\big{(}\widetilde{\mathbb{D}}^{\lambda}\big{)}-\alpha\mathop{\rm id}\nolimits are nilpotent on \mathbb{E}_{\alpha}\mathop{\rm Gr}\nolimits^{F}_{a}\big{(}\mathcal{P}_{0}\widetilde{\mathcal{V}}_{\mathfrak{a}}\big{)}.
Fix . Let be determined by \varphi\big{(}\widetilde{P}\big{)}=P. We set V_{\mathfrak{a},a,\alpha}:=\mathbb{E}_{\alpha}\mathop{\rm Gr}\nolimits^{F}_{a}\big{(}\mathcal{P}_{0}\widetilde{\mathcal{V}}_{\mathfrak{a}}\big{)}_{|\widetilde{P}}. Let be the nilpotent part of \mathop{\rm Res}\nolimits\big{(}\widetilde{\mathbb{D}}^{\lambda}\big{)} on . For a neighbourhood of in , we set , and . We may assume that any \mathfrak{a}\in\mathcal{I}\big{(}\widetilde{P}\big{)} has a lift in H^{0}\big{(}\widetilde{Y}_{P},\mathcal{O}_{\widetilde{Y}_{P}}\big{(}{\ast}\widetilde{H}_{P}\big{)}\big{)}. From the set \{(\mathfrak{a},a,\alpha)\}\subset\mathcal{I}\big{(}\widetilde{P}\big{)}\times]{-}1,0]\times{\mathbb{C}} and the tuples , we obtain a model filtered -flat bundle
[TABLE]
on \big{(}\widetilde{Y}_{P},\widetilde{H}_{P}\big{)}. It is unramifiedly good and naturally -equivariant. As the descent, we obtain a good filtered -flat bundle \big{(}\mathcal{P}_{\ast}\mathcal{V}_{0},\mathbb{D}_{0}^{\lambda}\big{)} on .
Lemma 3.21** (assume Condition 3.20).**
For any positive integer , there exist a neighbourhood of in and an isomorphism of filtered bundles such that the following holds:
- •
We set and A:=\big{(}\widetilde{\Phi}_{m}\big{)}^{\ast}\big{(}\widetilde{\mathbb{D}}^{\lambda}\big{)}-\widetilde{\mathbb{D}}_{0}^{\lambda} on . Let be the decomposition such that
[TABLE]
Then, we obtain the following for any :
[TABLE]
Here, .
Proof.
By [51, Proposition 2.4.4], for any large inter , there exists a -equivariant decomposition of filtered bundles
[TABLE]
on \big{(}\widetilde{Y}_{P},\widetilde{H}_{P}\big{)} such that the following holds.
- •
Let denote the projection of onto , and let denote the inclusion of into . Then, for any and for any , we obtain
[TABLE]
- •
For , we set . Then, for any , we obtain
[TABLE]
Then, the claim of the lemma is clear. ∎
3.7 Perturbation of good filtered -flat bundles
3.7.1 Curve case
Let be a Riemann surface with a finite subset . Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a good filtered -flat bundle on . We set . We choose such that .
For any , and for any , let be a map such that , if then . We define by
[TABLE]
We take as in Lemma 3.19. For each and , we obtain the endomorphism \mathop{\rm Res}\nolimits_{P}\big{(}\mathbb{D}^{\lambda}\big{)} of \mathop{\rm Gr}\nolimits^{F}_{b}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|P}\big{)}. Let W_{\bullet}\mathop{\rm Gr}\nolimits^{F}_{b}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|P}\big{)} denote the weight filtration associated with the nilpotent part of . For any , we obtain the subspace W_{k}\bigl{(}F_{b}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|P}\big{)}\bigr{)} as the pull back of W_{k}\mathop{\rm Gr}\nolimits^{F}_{b}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|P}\big{)} by the projection F_{b}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|P}\big{)}\longrightarrow\mathop{\rm Gr}\nolimits^{F}_{b}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|P}\big{)}. We define the filtration on indexed by as follows:
[TABLE]
We obtain the corresponding filtered bundle . Note the following lemma.
Lemma 3.22**.**
\big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}* is a good filtered -flat bundle.*
Proof.
It is enough to consider the case where is a neighbourhood of in . We obtain , , , and \big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}},\widetilde{\mathbb{D}}^{\lambda}\big{)} as in Section 3.5.1. We set and . For , we construct W_{k}F_{b}\big{(}\mathcal{P}_{\widetilde{a}}\widetilde{\mathcal{V}}_{|\widetilde{P}}\big{)} as above.
For b\in{\mathcal{P}{\rm ar}}\big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}},\widetilde{a}\big{)}, note that , and we set
[TABLE]
We set
[TABLE]
We obtain the corresponding -equivariant filtered bundle . We can easily observe that by using .
There exists the decomposition
[TABLE]
as in (2.4). We apply the same procedure to each by using , and we obtain filtered bundles for which are logarithmic. Because , we obtain that \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\widetilde{\mathcal{V}},\widetilde{\mathbb{D}}^{\lambda}\big{)} is unramifiedly good. Hence, we obtain that \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is good. ∎
Suppose that is compact and connected. We clearly obtain \lim\limits_{\epsilon\to 0}c_{1}\big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}\big{)}=c_{1}(\mathcal{P}_{\ast}\mathcal{V}). The following is also standard.
Lemma 3.23**.**
Suppose that \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is stable. Then, if is sufficiently small, \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is also stable.
Proof.
For any positive integer , and for any , let be the set of real numbers expressed as
[TABLE]
where are non-negative integers such that . Let denote the set of real numbers expressed as \frac{1}{s}\bigl{(}m+\sum_{P\in D}c_{P}\bigr{)}, where and . Then, is discrete in . Hence, there exists such that if satisfies , we obtain . Then, the claim of the lemma is clear. ∎
3.7.2 Surface case
Let be a complex projective surface with a simple normal crossing hypersurface . Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a good filtered -flat bundle on . We shall explain a similar perturbation of good filtered -flat bundles. Set . We choose such that .
For any , let be a map such that , if then . We define by .
We take for as in Lemma 3.19. The eigenvalues of the endomorphism \mathop{\rm Res}\nolimits_{i}\big{(}\mathbb{D}^{\lambda}\big{)} on \mathop{\rm Gr}\nolimits^{F}_{b}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{i}}\big{)} are constant on because are compact. We obtain the well defined nilpotent part of \mathop{\rm Res}\nolimits_{i}\big{(}\mathbb{D}^{\lambda}\big{)}. There exists a finite subset such that the conjugacy classes of the nilpotent part of are constant. We obtain the weight filtration of \mathop{\rm Gr}\nolimits^{F}_{b}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{i}\setminus Z_{i}}\big{)} by algebraic vector subbundles whose restriction to are the weight filtration of . It uniquely extends to a filtration of \mathop{\rm Gr}\nolimits^{F}_{b}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{i}}\big{)} by algebraic subbundles, which is also denoted by .
For any , let W_{k}F_{b}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{i}}\big{)} denote the subbundle of obtained as the pull back of W_{k}\mathop{\rm Gr}\nolimits^{F}_{b}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{i}}\big{)} by the projection F_{b}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{i}}\big{)}\longrightarrow\mathop{\rm Gr}\nolimits^{F}_{b}\big{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{i}}\big{)}. We define the filtration on indexed by as follows:
[TABLE]
We obtain the corresponding filtered bundle over . As in the curve case (see Lemma 3.22), we obtain the following.
Lemma 3.24**.**
\big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}* is a good filtered -flat bundle.*
We clearly have \lim\limits_{\epsilon\to 0}c_{1}\big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}\big{)}=c_{1}(\mathcal{P}_{\ast}\mathcal{V}) and \lim\limits_{\epsilon\to 0}\mathop{\rm ch}\nolimits_{2}\big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}\big{)}=\mathop{\rm ch}\nolimits_{2}(\mathcal{P}_{\ast}\mathcal{V}). The following is standard, and similar to Lemma 3.23. (See also [46, Proposition 3.28].)
Lemma 3.25**.**
Let be an ample line bundle on . Suppose that \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is -stable. Then, if is sufficiently small, \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is also -stable.
3.8 Some families of auxiliary metrics
on a punctured disc
3.8.1 Regular model case
Let be a finite dimensional vector space over with a nilpotent endomorphism . Let . Let be a neighbourhood of [math] in . We set . We set . From and , we obtain a model filtered -flat bundle \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} by applying the construction in Section 3.6.1 in the case and .
Fix . For any , we take such that , and we obtain the regular filtered -flat bundle \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} as in Section 3.7.1. We set . We consider the Kähler metric g_{\epsilon}:=\big{(}\eta^{2}|z|^{2\eta-2}+\epsilon^{2}|z|^{2\epsilon-2}\big{)}\,{\rm d}z\,{\rm d}\overline{z} of .
Proposition 3.26**.**
There exist Hermitian metrics of for such that the following holds:
- •
.
- •
* is a Hermitian metric of \big{(}E,\mathbb{D}^{\lambda}\big{)}, and in the -sense locally on .*
- •
There exist such that the following conditions are satisfied for any :
[TABLE]
- •
Let be the -endomorphisms of determined by the condition for . Then, there exists such that \big{|}B^{(\epsilon)}_{i}\big{|}_{h^{(\epsilon)}}\leq C_{3} holds for any .
Moreover, for any , depends only on , where are naturally regarded as holomorphic sections of .
In the case , such a family of Hermitian metrics is constructed in [49, Sections 4.3 and 4.4.1]. We shall explain the case in Section 3.8.4.
3.8.2 General case
Let be a finite set. Let be a finite subset of . Let be a finite subset of . Let () be finite dimensional -vector spaces equipped with a nilpotent endomorphism . Set and . Take such that
[TABLE]
As in Section 3.8.1, we obtain the regular filtered -flat bundles \big{(}\mathcal{P}_{\ast}\mathcal{V}_{i,a,\alpha},\mathbb{D}^{\lambda}_{i,a,\alpha}\big{)} from . For , we take such that , and we obtain \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}_{i,a,\alpha},\mathbb{D}^{\lambda}_{i,a,\alpha}\big{)}. We set . We obtain the metrics of as in Proposition 3.26. We set and .
Fix a positive integer and a positive number . We consider the following data:
- •
For each , let denote a polynomial \sum_{j=1}^{m}\mathfrak{a}(i)_{j}z^{-j}\in z^{-1}{\mathbb{C}}\big{[}z^{-1}\big{]} such that .
- •
Let be a holomorphic section of with the decomposition , where
[TABLE]
If , we obtain , and if , we obtain .
We define the flat -connection on as follows:
[TABLE]
Let be the Kähler metrics of as in Section 3.8.1.
Proposition 3.27**.**
There exists a constant depending only on and such that
[TABLE]
3.8.3 A consequence
Let be a neighbourhood of [math] in . We set . Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be any good filtered -flat bundle on . Let \big{(}E,\mathbb{D}^{\lambda}\big{)} be the -flat bundle obtained as the restriction of \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} to . We set . Let be as in Section 3.6, and we set \big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}},\widetilde{\mathbb{D}}^{\lambda}\big{)}:=\varphi^{\ast}\big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}. We take such that . Let be the Kähler metric of . By using a special case of Proposition 3.27, we obtain the following corollary.
Corollary 3.28**.**
There exists a Hermitian metric of \big{(}E,\mathbb{D}^{\lambda}\big{)} such that , is bounded on .
Proof.
Let \big{(}\mathcal{P}_{\ast}\mathcal{V}_{0},\mathbb{D}_{0}^{\lambda}\big{)} be a model filtered -flat bundle with an isomorphism as in Lemma 3.21, where is a sufficiently large integer. We recall that \big{(}\mathcal{P}_{\ast}\mathcal{V}_{0},\mathbb{D}_{0}^{\lambda}\big{)} is obtained as the descent of the -equivariant model filtered -flat bundle \big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{0},\widetilde{\mathbb{D}}^{\lambda}_{0}\big{)}=\bigoplus_{\mathfrak{a},a,\alpha}\big{(}\mathcal{P}_{\ast}\mathcal{V}_{\mathfrak{a},a,\alpha},\mathbb{D}^{\lambda}_{\mathfrak{a},a,\alpha}\big{)}, and is induced by a -equivariant isomorphism . Let be the Hermitian metric of as in Proposition 3.26 with . By the isomorphism , it induces a -equivariant Hermitian metric of . Applying Proposition 3.27 to \big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}},\widetilde{\mathbb{D}}^{\lambda}\big{)} with , we obtain the boundedness of G\big{(}\widetilde{h}\big{)} with respect to and . Because is -equivariant, we obtain the Hermitian metric of which has the desired property. ∎
3.8.4 Proof of Proposition 3.26
Let with the nilpotent map determined by and . We obtain and the Higgs field determined by . Let \big{(}E_{2},\overline{\partial}_{E_{2}},\theta_{2}\big{)} be the Higgs bundle on obtained as the restriction of . For any , we set . We also set .
Lemma 3.29**.**
We obtain . There exists such that
[TABLE]
on for any .
Proof.
As proved in [49, Section 4.2], holds. We set g_{1}(\epsilon):=-\epsilon\log|z|^{2}-\big{(}1-|z|^{2\epsilon}\big{)} for any and for . It is easy to check that and . Hence, we obtain . For and , we set . Then, . Then, we can check that and . Then, we obtain the second claim of the lemma. ∎
Let be the -metric of given by
[TABLE]
Lemma 3.30**.**
\big{(}E_{2},\overline{\partial}_{E_{2}},\theta_{2},h_{2}^{(\epsilon)}\big{)}* are harmonic bundles. Moreover, the family of metrics satisfies the condition in Proposition 3.26 for .*
Proof.
Let be the matrix valued function on determined by . Then, the following holds:
[TABLE]
Let be the matrix valued function representing with respect to the frame , i.e., . Let denote the adjoint of with respect to . Let denote the matrix valued function representing . The following holds:
[TABLE]
Hence, we obtain
[TABLE]
It implies that
[TABLE]
It is exactly the Hitchin equation for \big{(}E_{2},\overline{\partial}_{E_{2}},\theta_{2},h_{2,\epsilon}\big{)}. The other claim is easy to see. ∎
For each , we set . We set and . We obtain the regular filtered Higgs bundles \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}_{\ell},\theta_{\ell}\big{)}. Note that \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}_{\ell},\theta_{\ell}\big{)} is naturally isomorphic to the -th symmetric product of \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}_{2},\theta_{2}\big{)}. Hence, induce harmonic metrics of \big{(}E_{\ell},\overline{\partial}_{E_{\ell}},\theta_{\ell}\big{)}:=(\mathcal{V}_{\ell},\theta_{\ell})_{|X\setminus H} satisfying the conditions in Proposition 3.26 for \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}_{\ell},\theta_{\ell}\big{)}.
Let , and be as in Section 3.8.1. There exist integers such that
[TABLE]
We obtain . We obtain the harmonic metrics for \big{(}E,\overline{\partial}_{E},\theta\big{)}. We can easily check that they satisfy the conditions in Proposition 3.26 for . ∎
3.8.5 Proof of Proposition 3.27
We set and . Let and denote the adjoint of and with respect to , respectively. We obtain the decompositions and . Note that \bigl{[}\mathbb{D}^{\lambda\mathop{\rm reg}\nolimits},\Phi_{h^{(\epsilon)}}^{\dagger}\bigr{]}=\bigl{[}\mathbb{D}^{\lambda\mathop{\rm reg}\nolimits\star}_{h^{(\epsilon)}},\Phi\bigr{]}=\bigl{[}\Phi,\Phi^{\dagger}_{h^{(\epsilon)}}\bigr{]}=0. By the assumption, we obtain
[TABLE]
We also obtain \bigl{|}\big{[}A,A^{\dagger}_{h^{(\epsilon)}}\big{]}\bigr{|}_{h^{(\epsilon)}}\leq 2C^{2}C_{1}^{2}|z|^{4\eta-4\mathop{\rm rank}\nolimits(\mathcal{V})\epsilon}. Because
[TABLE]
we obtain
[TABLE]
Hence, we obtain the desired estimate for G\big{(}h^{(\epsilon)}\big{)}=\bigl{[}\mathbb{D}^{\lambda},\mathbb{D}^{\lambda\star}_{h^{(\epsilon)}}\bigr{]}. ∎
3.9 Estimate of the curvature for Hermitian–Einstein metrics
of a Higgs bundle
Let be a complex surface. Let \big{(}E,\overline{\partial}_{E},\theta\big{)} be a Higgs bundle on . Let be a sequence of Kähler metric on which is convergent to a Kähler metric in the -sense locally on . Let be Hermitian–Einstein metrics of the Higgs bundle. We assume the following:
- •
.
Let be the Chern connection of \big{(}E,\overline{\partial}_{E},h_{i}\big{)}. Let denote the curvature of . The following proposition is a refinement contained in the argument in [46, Section 9.1.1].
Proposition 3.31**.**
For any relatively compact open subset , and for any , the -norms of with respect to and are bounded.
Proof.
Let be any point of . Let be a holomorphic coordinate neighbourhood of around . Let us describe as . Let be a relatively compact neighbourhood of in . According to [46, Lemma 2.13], there exist , which are independent of , such that the following inequalities hold on :
[TABLE]
(Note that is denoted as in [46].) Let denote a positive number. After rescaling the coordinate system, we may assume the following on :
[TABLE]
There exists such that for we obtain . Because the -norms are scale invariant, we obtain
[TABLE]
Let be a relatively compact neighbourhood of in . If is sufficiently small, by the theorem of Uhlenbeck [74, Corollary 2.2], there exists an orthonormal frame of for each such that the connection form of with respect to satisfies is , on for a positive constant independently from , satisfies , where denotes the adjoint of with respect to the metric . Let and represent and with respect to the frame . Then, satisfies
[TABLE]
Let be a relatively compact neighbourhood of in . By the argument of Donaldson in the proof of [15, Corollary 23], we obtain that are for any on , and that there exists such that on , where are independent of . In particular, there exists independently from such that \big{\|}R(h_{i})_{|X_{P,2}}\big{\|}_{L^{p}}\leq C_{5,p}.
Let be the decomposition into the -part and the -part. Because , we obtain \overline{\partial}\Theta_{i}+\big{[}A_{i}^{0,1},\Theta_{i}\big{]}=0. Hence, there exist independently from such that \big{\|}\Theta_{i|X_{P,2}}\big{\|}_{L_{2}^{p}}\leq C_{6,p}.
Note that and . Let denote the formal adjoint of with respect to and . Because \Lambda_{g_{i,P}}R(h_{i})+\Lambda_{g_{i,P}}\big{[}\theta_{i},\theta_{i}^{\dagger}\big{]}=0, there exists such that \bigl{\|}\overline{\partial}^{\ast}_{E,h_{i},g_{i,P}}R(h_{i})_{|X_{P,2}}\bigr{\|}_{L_{1}^{p}}<C_{7,p}. Let be a relatively compact neighbourhood of in . There exists independently from such that \bigl{\|}R(h_{i})_{|X_{P,3}}\bigr{\|}_{L_{2}^{p}}<C_{8,p}. It implies the claim of the proposition. ∎
4 Existence and continuity of harmonic metrics
in the curve case
4.1 Existence of Hermitian–Einstein metric
Let be a compact Riemann surface. Let be a finite subset. Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a stable good filtered -flat bundle on . Let \big{(}E,\mathbb{D}^{\lambda}\big{)} be the -flat bundle on obtained as the restriction of \big{(}\mathcal{V},\mathbb{D}^{\lambda}\big{)}. Let be any Kähler form of . Let be a Hermitian metric of such that \Lambda_{\omega}R(h_{\det(E)})=2\pi\deg(\mathcal{P}_{\ast}\mathcal{V})\big{(}\int_{X}\omega\big{)}^{-1}, is adapted to , i.e., . (See Proposition 3.2.)
Theorem 4.1** (Biquard–Boalch).**
There exists a unique Hermitian–Einstein metric of \big{(}E,\mathbb{D}^{\lambda}\big{)} adapted to such that . If , is a harmonic metric.
Proof.
It is enough to prove the case . We explain an outline of the proof based on the fundamental theorem of Simpson [62, Theorem 1] (and its variant [49, Proposition 2.49]) because we obtain a consequence on the Donaldson functional from the proof, which will be useful in the proof of Proposition 4.5 below. Set . Take such that . (See Section 3.5.4 for .)
Let \big{(}X_{P},z_{P}\big{)} be an admissible coordinate neighbourhood around . Set . We take a Kähler metric of satisfying the following condition:
- •
is mutually bounded with on for each .
Recall that the Kähler manifold satisfies the assumptions given in [62, Section 2], according to [62, Proposition 2.4].
Lemma 4.2**.**
There exists a Hermitian metric of such that the following holds:
\big{(}E,{\rm d}^{\prime\prime}_{E},h_{0}\big{)}* is acceptable, and .* 2.
* is bounded with respect to and .* 3.
.
Proof.
By Corollary 3.28, we obtain a Hermitian metric of satisfying and . We define the function by . Then, induces a -function on . We set . Then, the metric has the desired property. ∎
For any -flat subbundle , let denote the Hermitian metric of induced by . Let denote the Higgs field of obtained as the restriction of . We obtain the Chern connection from the -part of and . Let denote the curvature of . We set
[TABLE]
Let denote the adjoint of the multiplication by the Kähler form associated with . Because is bounded with respect to and , is well defined in by the Chern–Weil formula [62, Lemma 3.2] (see also [49, Lemma 2.34]):
[TABLE]
Here, denotes the orthogonal projection with respect to .
Lemma 4.3**.**
* holds. Namely, is analytically stable in the sense of [63, Section 6] see also [49, Section 2.3].*
Proof.
By [63, Lemma 6.1], we have \deg(E,h_{0})=\deg\big{(}\mathcal{P}^{h_{0}}_{\ast}E\big{)}=0. Let be a -flat subbundle on . By [63, Lemma 6.2], if , extends to a filtered subbundle , and \deg(E^{\prime},h_{0})=\deg\big{(}\mathcal{P}^{h_{0}^{\prime}}_{\ast}E^{\prime}\big{)} holds. Because is assumed to be stable, we obtain \deg(E^{\prime},h_{0})/\mathop{\rm rank}\nolimits E^{\prime}<\deg\big{(}\mathcal{P}^{h_{0}}_{\ast}E\big{)}/\mathop{\rm rank}\nolimits E=0. Hence, \big{(}E,\overline{\partial}_{E},\theta,h_{0}\big{)} is analytically stable. ∎
According to the existence theorem of Simpson [62, Theorem 1] (see also [49, Proposition 2.49]), there exists a Hermitian–Einstein metric of \big{(}E,\mathbb{D}^{\lambda}\big{)} such that and that and are mutually bounded. We already know the uniqueness as in Proposition 2.22. Thus, we obtain Theorem 4.1. ∎
4.1.1 Complement on the Donaldson functional
Let and be as in the proof of Theorem 4.1. Let be the space of -Hermitian metrics of satisfying the following condition:
- •
Let be the endomorphism of such that , is self-adjoint with respect to both and . Then, \sup_{Q\in X\setminus D}|u_{1}|_{h_{0}}(Q)+\|\mathbb{D}^{\lambda}u_{1}\|_{L^{2}}+\bigl{\|}\mathbb{D}^{\lambda}\mathbb{D}^{\lambda\star}_{h_{0}}u_{1}\bigr{\|}_{L^{1}}<\infty. Here, we use the -norms induced by and .
The Donaldson functional is defined as in [62, Section 5] and [49, Section 2.4].
Proposition 4.4**.**
Let be the Hermitian–Einstein metric in Theorem 4.1. Then, is contained in , and holds.
Proof.
Let be the automorphism of which is self-adjoint with respect to both and , and determined by . The theorem of Simpson [62, Theorem 1] (see also [49, Proposition 2.49]) implies that and are bounded, and that is with respect to and . By [62, Lemma 3.1] (see also [49, Section 2.2.5]), we also obtain is . Hence, is contained in . In the proof of [62, Theorem 1] and [49, Proposition 2.39], the metric is constructed as the limit of a subsequence of the heat flow for which holds. Because by the construction, we obtain , and hence . ∎
4.2 Continuities of some families of Hermitian metrics
4.2.1 Setting
Family of curves.
Let be a compact connected oriented real -dimensional -manifold with a finite subset . Let be a sequence of complex structures on such that the sequence is convergent to in the -sense. Assume that there exists a neighbourhood of in such that are independent of . Let denote the compact Riemann surfaces . Similarly, let denote the compact Riemann surface . Let be isomorphisms of complex vector bundles on such that , . We regard as isomorphisms of complex vector bundles .
For , let denote an admissible coordinate neighbourhood of in such that . We may regard \big{(}X_{P},z_{P}\big{)} as a holomorphic coordinate neighbourhood of in . Let be a positive integer, and set . As in Section 3.5.1, let be the ramified covering given by . Let denote the Galois group of the ramified covering .
Family of good filtered -flat bundles.
Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a stable good filtered -flat bundle of rank on with . Let \big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)} be stable good filtered -flat bundles of rank on with . For each , we set \big{(}\mathcal{P}_{\ast}\mathcal{V}_{P},\mathbb{D}^{\lambda}_{P}\big{)}:=\big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X_{P}} and \big{(}\mathcal{P}_{\ast}\mathcal{V}_{i,P},\mathbb{D}^{\lambda}_{i,P}\big{)}:=\big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}\big{)}_{|X_{P}}. Set \big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P},\widetilde{\mathbb{D}}^{\lambda}_{P}\big{)}:=\varphi_{P}^{\ast}\big{(}\mathcal{P}_{\ast}\mathcal{V}_{P},\mathbb{D}^{\lambda}_{P}\big{)} and \big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{i,P},\widetilde{\mathbb{D}}^{\lambda}_{i,P}\big{)}:=\varphi_{P}^{\ast}\big{(}\mathcal{P}_{\ast}\mathcal{V}_{i,P},\mathbb{D}^{\lambda}_{i,P}\big{)}. There exist -invariant subsets and the formal decompositions
[TABLE]
for each . Suppose moreover that there exist -invariant bijections such that the following holds:
- •
.
- •
and \mathop{\rm ord}\nolimits(\mathfrak{a}-\mathfrak{b})=\mathop{\rm ord}\nolimits\bigl{(}\rho_{i,P}(\mathfrak{a})-\rho_{i,P}(\mathfrak{b})\bigr{)}.
- •
in \zeta^{-1}{\mathbb{C}}\big{[}\zeta^{-1}\big{]}.
We fix such bijections . Let denote the projection . Similarly, let denote the projection .
-isomorphisms.
We set \big{(}E,\mathbb{D}^{\lambda}\big{)}:=\big{(}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X\setminus D} and \big{(}E_{i},\mathbb{D}^{\lambda}_{i}\big{)}:=\big{(}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)}_{|X_{i}\setminus D}. Let denote -metrics of adapted to such that . Let denote -metrics of adapted to such that . Let and denote the -parts of and . Suppose that there exist -isomorphisms satisfying the following conditions:
- •
in the -sense locally on .
- •
On , are holomorphic with respect to and , and extend to isomorphisms of filtered bundles .
- •
For each , we obtain \mathop{\rm Gr}\nolimits^{F}_{c}\big{(}f_{i|X_{P}}\big{)}\circ\mathop{\rm Res}\nolimits_{P}(\mathbb{D}^{\lambda})=\mathop{\rm Res}\nolimits_{P}\big{(}\mathbb{D}^{\lambda}_{i}\big{)}\circ\mathop{\rm Gr}\nolimits^{F}_{c}\big{(}f_{i|X_{P}}\big{)} on for any . Moreover, there exists for any such that for the induced isomorphisms \varphi_{P}^{\ast}\big{(}f_{i|X_{P}}\big{)}\colon\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{P}\otimes{\mathbb{C}}[\![\zeta]\!]\simeq\mathcal{P}_{\ast}\widetilde{\mathcal{V}}_{i,P}\otimes{\mathbb{C}}[\![\zeta]\!], we obtain
[TABLE]
and the sequences (4.1) are convergent to [math] as .
- •
in the -sense with respect to locally on .
Perturbation.
We take satisfying and . We take for as in Lemma 3.19. For any , by taking as in Section 3.7.1, we obtain families of good filtered -flat bundles \big{(}\mathcal{P}_{\ast}^{(\epsilon)}\mathcal{V},\mathbb{D}^{\lambda}\big{)} and \big{(}\mathcal{P}_{\ast}^{(\epsilon)}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)}. We assume the following for each :
[TABLE]
In particular, \deg(\mathcal{P}_{\ast}\mathcal{V})=\deg\big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}\big{)} and \deg(\mathcal{P}_{\ast}\mathcal{V}_{i})=\deg\big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}_{i}\big{)} hold. By making smaller, we may assume that \big{(}\mathcal{P}_{\ast}^{(\epsilon)}\mathcal{V},\mathbb{D}^{\lambda}\big{)} are stable for any .
4.2.2 Continuity of the family of harmonic metrics
According to Theorem 4.1, there exists a harmonic metric of \big{(}E_{i},\mathbb{D}^{\lambda}\big{)} adapted to such that . Similarly, there exists a harmonic metric of \big{(}E,\mathbb{D}^{\lambda}\big{)} adapted to such that . The following proposition is a variant of [49, Propositions 4.1 and 4.2].
Proposition 4.5**.**
For any sequence , the sequence is convergent to locally on in the -sense.
Proof.
For , let be the Kähler metric on such that the following holds on for any :
[TABLE]
Let denote the adjoint of the multiplication by the Kähler form associated with .
By the isomorphisms and the metrics , we obtain the Kähler metrics of . Let denote the adjoint of the multiplication by the Kähler form associated with .
There exists an approximation of \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X_{P}} by a model filtered -flat bundle as in Section 3.6. By using a family of Hermitian metrics for the model -flat bundle as in Proposition 3.26, and by using Proposition 3.27, we construct a family of metrics of such that the following holds:
- •
is adapted to .
- •
.
- •
locally on in the -sense as .
- •
There exists such that \big{|}G\big{(}h^{(\epsilon)}_{\mathop{\rm in}\nolimits}\big{)}\big{|}_{g_{X\setminus D,\epsilon},h^{(\epsilon)}_{\mathop{\rm in}\nolimits}}<C_{1} for any .
Let be the function on determined by \big{(}f_{i}^{-1}\big{)}^{\ast}(\det h_{0})={\rm e}^{\nu_{i}}\det h_{i,0}. We obtain the Hermitian metrics h_{i,\mathop{\rm in}\nolimits}^{(\epsilon)}:={\rm e}^{\nu_{i}/\mathop{\rm rank}\nolimits\mathcal{V}}\big{(}f_{i}^{-1}\big{)}^{\ast}\big{(}h^{(\epsilon)}_{\mathop{\rm in}\nolimits}\big{)} of . Then, by Proposition 3.27, we obtain the following:
- •
is adapted to .
- •
.
- •
locally on in the -sense as .
- •
By replacing with a larger constant, we may assume \big{|}G\big{(}h^{(\epsilon)}_{i,\mathop{\rm in}\nolimits}\big{)}\big{|}_{g_{X_{i}\setminus D,\epsilon},h^{(\epsilon)}_{i,\mathop{\rm in}\nolimits}}<C_{1} for any and any .
Lemma 4.6**.**
Let be automorphisms of which are self-adjoint with respect to such that the following holds:
- •
\mathop{\rm Tr}\nolimits\big{(}u^{(i)}\big{)}=0.
- •
h_{i,\mathop{\rm in}\nolimits}^{(\epsilon_{i})}{\rm e}^{u^{(i)}}\in\mathcal{H}\big{(}h_{in}^{(\epsilon_{i})}\big{)}, i.e., \sup\big{\|}u^{(i)}\big{\|}_{h^{(\epsilon_{i})}_{i,\mathop{\rm in}\nolimits}}+\big{\|}\mathbb{D}^{\lambda}_{i}u^{(i)}\big{\|}_{L^{2}}+\big{\|}\mathbb{D}^{\lambda}_{i}\mathbb{D}^{\lambda\star}_{i,h^{(\epsilon_{i})}_{i,\mathop{\rm in}\nolimits}}u^{(i)}\big{\|}_{L^{1}}<\infty, where the -norms are taken with respect to and . We do not assume that the estimate is uniform in .
- •
There exists such that \bigl{|}\Lambda_{i,\epsilon_{i}}G\big{(}h^{(\epsilon_{i})}_{i,\mathop{\rm in}\nolimits}{\rm e}^{u^{(i)}}\big{)}\bigr{|}_{h^{(\epsilon_{i})}_{i,\mathop{\rm in}\nolimits}}<C_{2} for any .
Then, there exists such that the following holds for any
[TABLE]
Proof.
By identifying the vector bundles and by , we apply the same argument as in the proof of [49, Lemma 2.45]. ∎
Let be the automorphism of which is self-adjoint with respect to and and determined by . Note that . Take any sequence . By Proposition 4.4 and Lemma 4.6, there exists a constant such that the following holds for any :
[TABLE]
Lemma 4.7**.**
\int\Lambda_{i,\epsilon_{i}}\bigl{(}\overline{\partial}_{X_{i}}\partial_{X_{i}}\mathop{\rm Tr}\nolimits\big{(}b_{i,1}^{(\epsilon_{i})}\big{)}\bigr{)}\omega_{X_{i}\setminus D,\epsilon_{i}}=0* holds.*
Proof.
We use Proposition 4.4. Because is with respect to and , we obtain that \Lambda_{i,\epsilon_{i}}\overline{\partial}_{X_{i}}\partial_{X_{i}}\mathop{\rm Tr}\nolimits\big{(}b_{i,1}^{(\epsilon_{i})}\big{)} is with respect to . Because is with respect to and , we obtain that \partial_{X_{i}}\mathop{\rm Tr}\nolimits\big{(}b_{i,1}^{(\epsilon_{i})}\big{)} is with respect to . Therefore, we obtain the claim of the lemma by using [62, Lemma 5.2]. ∎
By [62, Lemma 3.1], the following holds:
[TABLE]
Therefore, there exists such that the following holds for any :
[TABLE]
We also obtain
[TABLE]
Let \big{(}E_{i},\overline{\partial}^{(\epsilon_{i})}_{E_{i}},\theta_{i}^{(\epsilon_{i})}\big{)} be the Higgs bundles underlying \big{(}E_{i},\mathbb{D}^{\lambda},h_{i}^{(\epsilon_{i})}\big{)}. Then, there exists such that the following holds for any :
[TABLE]
Then, by applying the argument in [49, Section 4.5.3], we obtain the desired convergence of the sequence . ∎
4.2.3 Continuity of some families of Hermitian metrics
For , we set . We may naturally regard as a subset of . Fix . Let be a sequence of Kähler metrics of , such that and that
[TABLE]
Let be a sequence such that . The following proposition is a variant and a refinement of [49, Proposition 5.1].
Proposition 4.8**.**
Let be Hermitian metrics of satisfying the following conditions:
- •
.
- •
\big{\|}G\big{(}h_{i,1}^{(\epsilon_{i})}\big{)}\big{\|}_{L^{2},g_{i,\epsilon},h_{i,1}^{(\epsilon_{i})}}\to 0* as .*
- •
Let be the automorphism of which is self-adjoint with respect to and determined by . Then, and \big{(}s^{(i)}\big{)}^{-1} are bounded with respect to on , and are with respect to and . The estimates may depend on .
Then, the sequence \big{\{}f_{i}^{\ast}\big{(}s^{(i)}\big{)}\big{\}} is weakly convergent to in locally on . Moreover, there exists such that \big{|}s^{(i)}\big{|}_{h_{i}^{(\epsilon_{i})}}<A and \big{|}\big{(}s^{(i)}\big{)}^{-1}\big{|}_{h_{i}^{(\epsilon_{i})}}<A for any .
Proof.
This is essentially the same as [49, Proposition 5.1]. We explain an outline of the proof. We identify with by . We set
[TABLE]
We set . We set . The following holds.
[TABLE]
From the boundedness of and the -property of , we obtain \int\Delta_{g_{i,0}}\mathop{\rm Tr}\nolimits\big{(}\widetilde{s}^{(i)}\big{)}\mathop{\rm dvol}\nolimits_{g_{i,0}}=0 as in Lemma 4.7. We obtain the following for some and :
[TABLE]
Hence, the sequence is -bounded on any compact subset of . By taking an appropriate subsequence, it is weakly convergent in locally on . Let denote the weak limit of the sequence. We obtain . Because are self-adjoint and uniformly bounded with respect to , is self-adjoint and bounded with respect to . We can prove that by the same argument as in the proof of [49, Lemma 5.2]. Hence, is a non-zero endomorphism of \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}. It implies that is a multiplication by a positive constant .
Note that the sequence is convergent in for any locally on , and hence \det\big{(}\widetilde{s}^{(i)}\big{)} is convergent to \det\big{(}\widetilde{s}^{(\infty)}\big{)} in for any locally on . Because \det\big{(}s^{(i)}\big{)}=1, we obtain that the sequence is convergent to . In particular, it implies that the sequence is bounded. Then, we obtain the claim of the proposition. ∎
4.3 Tensor product of stable filtered -flat sheaves
Let us state a consequence of Theorem 4.1 on the tensor product of reflexive filtered -flat sheaves on arbitrary dimensional projective varieties.
Let be an -dimensional non-singular projective variety equipped with a very ample line bundle . Let be a simple normal crossing hypersurface of with the irreducible decomposition . Let \big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)} be reflexive filtered -flat sheaves on . We assume the following condition:
Condition 4.9**.**
There exists a Zariski closed subset with such that \big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)}_{|X\setminus Z} are good filtered -flat bundles on .
For example, if is logarithmic and if , Condition 4.9 is satisfied.
We set which is equipped with the induced flat -connection . Note that and that is a locally free -module. There exists the natural morphism
[TABLE]
The -modules and are coherent, and their supports are contained in . Hence, we obtain that . It implies that , i.e., is a reflexive -module. For , we set
[TABLE]
Let denote the coherent reflexive subsheaf of generated by . Thus, we obtain a reflexive filtered -flat sheaf \big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}},\widetilde{\mathbb{D}}^{\lambda}\big{)} on .
Proposition 4.10**.**
If \big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)} are -stable, then \big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}},\widetilde{\mathbb{D}}^{\lambda}\big{)} is -polystable.
Proof.
According to Propositions 3.8, 3.9 and Condition 4.9, there exists a positive integer such that the following holds for any general complete intersection curve of .
- •
\big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)}_{|Y} are stable good filtered -flat bundles.
- •
The natural morphism
[TABLE]
is an isomorphism.
Because \big{(}\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}\big{)}_{|Y} are stable good filtered -flat bundles, each -flat bundle are equipped with a Hermitian–Einstein metric adapted to the filtered bundle by Theorem 4.1. Because is a Hermitian–Einstein metric of the -flat bundle \big{(}\widetilde{\mathcal{V}},\widetilde{\mathbb{D}}^{\lambda}\big{)}_{|Y\setminus H} adapted to the filtered bundle , we obtain that \big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}},\widetilde{\mathbb{D}}^{\lambda}\big{)}_{|Y} is polystable. By Corollary 3.10, we obtain that \big{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}},\widetilde{\mathbb{D}}^{\lambda}\big{)} is -polystable. ∎
5 Preliminary existence theorem for Hermitian–Einstein
metrics
5.1 Statements
5.1.1 Kähler metrics
Let be a smooth projective surface with a simple normal crossing hypersurface . Let be an ample line bundle on . Let be the Kähler metric of such that the associated Kähler form represents .
We take Hermitian metrics of . Let denote the canonical section. Take . There exists such that the following form defines a Kähler form on for any :
[TABLE]
It is easy to observe that and that for any closed --form on .
5.1.2 Condition for good filtered -flat bundles and initial metrics
Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a good filtered -flat bundle on satisfying the following condition: We set .
Condition 5.1**.**
**
- •
There exists and such that for each .
- •
The nilpotent part of \mathop{\rm Res}\nolimits_{i}\big{(}\mathbb{D}^{\lambda}\big{)} on are [math] for any , and .
Let \big{(}E,\mathbb{D}^{\lambda}\big{)} denote the -flat bundle on obtained as the restriction of \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}.
Let be any point of . Let be an admissible coordinate neighbourhood around . There exists an open subset in such that the map given by is a ramified covering. We set . We obtain the induced good filtered -flat bundle \big{(}\mathcal{P}_{\ast}\varphi_{P}^{\ast}\mathcal{V},\varphi_{P}^{\ast}\mathbb{D}^{\lambda}\big{)} on such that {\mathcal{P}{\rm ar}}\big{(}\mathcal{P}_{\ast}\varphi_{P}^{\ast}\mathcal{V}\big{)}=\{m\cdot c_{i}\}+{\mathbb{Z}}.
Definition 5.2**.**
A Hermitian metric of is called strongly adapted to if there exists a Hermitian metric of \mathcal{P}_{mc_{i}}\big{(}\varphi_{P}^{\ast}\mathcal{V}\big{)} on such that .
Let be any point of . Let be a admissible coordinate neighbourhood around such that and . There exists an open subset in such that the map given by is a ramified covering. We set . We obtain the induced good filtered -flat bundle \big{(}\mathcal{P}_{\ast}\varphi_{P}^{\ast}\mathcal{V},\varphi^{\ast}\mathbb{D}^{\lambda}\big{)} on such that {\mathcal{P}{\rm ar}}\big{(}\mathcal{P}_{\ast}\varphi_{P}^{\ast}\mathcal{V},1\big{)}=\{m\cdot c_{i}\}+{\mathbb{Z}} and {\mathcal{P}{\rm ar}}\big{(}\mathcal{P}_{\ast}\varphi_{P}^{\ast}\mathcal{V},2\big{)}=\{m\cdot c_{j}\}+{\mathbb{Z}}.
Definition 5.3**.**
A Hermitian metric of is called strongly adapted to if there exists a -Hermitian metric of such that .
Definition 5.4**.**
A Hermitian metric of is called strongly adapted to if the following holds:
- •
For any , there exists a neighbourhood of such that is strongly adapted to in the sense of Definitions 5.2 and 5.3.
Lemma 5.5**.**
Let be a Hermitian metric of strongly adapted to . Then, the following holds:
[TABLE]
Proof.
It is the equality (36) in the proof of [46, Proposition 4.18]. ∎
For each , we choose . Set . We take a Hermitian metric of such that induces a Hermitian metric of of -class.
Proposition 5.6**.**
There exists a Hermitian metric of such that the following holds:
- •
* is strongly adapted to .*
- •
* is bounded with respect to and , where .*
- •
The following holds:
[TABLE]
- •
.
Such a Hermitian metric is called an initial metric of \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}.
The case was studied in [51, Sections 14.1, 14.2 and Lemma 14.4.2]. The case can be argued in the essentially same way. We shall explain the construction in the case in Section 5.4 after preliminaries in Sections 5.2–5.3.
5.1.3 Preliminary existence theorem for Hermitian–Einstein metrics
Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a good filtered -flat bundle satisfying Condition 5.1. Let be an initial metric for \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} as in Proposition 5.6. We shall prove the following theorem in Section 5.5.
Theorem 5.7**.**
Suppose that \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is -stable. Then, there exists a Hermitian–Einstein metric of \big{(}E,\mathbb{D}^{\lambda}\big{)} with respect to the Kähler form \big{(}\epsilon:=m^{-1}\big{)} satisfying the following conditions:
* and are mutually bounded.* 2.
\mathbb{D}^{\lambda}\big{(}h_{\mathop{\rm HE}\nolimits}\cdot h_{\mathop{\rm in}\nolimits}^{-1}\big{)}* is with respect to and .* 3.
* holds. In particular, the following holds:*
[TABLE] 4.
The following equality holds:
[TABLE]
5.2 Around cross points
Let X_{0}:=\bigl{\{}(z_{1},z_{2})\in{\mathbb{C}}^{2}\mid|z_{i}|<1\bigr{\}}. We set and . Let be a good filtered Higgs bundle on . We choose , and set . We also choose any Hermitian metric of such that is a Hermitian metric of of -class.
5.2.1 Unramified case
Suppose that satisfies the following condition:
Condition 5.8**.**
**
- •
There exists such that , .
- •
There exists a decomposition of good filtered Higgs bundles
[TABLE]
such that \theta_{\mathfrak{a},\alpha}-\big{(}{\rm d}\mathfrak{a}+\sum\alpha_{i}{\rm d}z_{i}/z_{i}\big{)}\mathop{\rm id}\nolimits_{\mathcal{V}_{\mathfrak{a},{\boldsymbol{\alpha}}}} induce holomorphic Higgs fields of .
We take any holomorphic frame of compatible with the decomposition (5.3). For , we obtain determined by . Let be the metric of determined by and . Note that \partial_{h_{0}}{\boldsymbol{v}}={\boldsymbol{v}}\bigl{(}-\sum_{k=1,2}c_{k}{\rm d}z_{k}/z_{k}\bigr{)}\,I, where denotes the identity matrix. Hence, \big{[}\partial_{h_{0}},\overline{\partial}\,\big{]}=0 holds. We obtain the description \theta{\boldsymbol{v}}={\boldsymbol{v}}\bigl{(}\Lambda_{0}+\Lambda_{1}\bigr{)} such that the following holds:
- •
and .
- •
are holomorphic -forms for any and . Moreover, holds unless .
We obtain \theta^{\dagger}_{h_{0}}{\boldsymbol{v}}={\boldsymbol{v}}\big{(}\overline{\Lambda}_{0}+{}^{t}\!\overline{\Lambda}_{1}\big{)} and \big{[}\theta,\theta_{h_{0}}^{\dagger}\big{]}{\boldsymbol{v}}={\boldsymbol{v}}\big{[}\Lambda_{1},{}^{t}\!\overline{\Lambda}_{1}\big{]}, where the entries of \big{[}\Lambda_{1},{}^{t}\!\overline{\Lambda}_{1}\big{]} are on . We have , where any entries of are holomorphic -forms, and unless .
Note that there exists a -function on such that . We set .
Lemma 5.9**.**
\big{[}\theta,\theta_{h_{\mathop{\rm in}\nolimits}}^{\dagger}\big{]}, and are bounded with respect to and .
5.2.2 Ramified case
Let be given by . We set , and . We set , which acts on by .
Suppose that satisfies Condition 5.8 on . We construct a -metric of as in the previous subsection. We may assume that is -invariant. Note that there exists a -invariant -function on such that . We set . Because it is -invariant, we obtain the induced metric of .
Let denote the Kähler metric on . Because is a covering map, it induces a Kähler metric of .
Lemma 5.10**.**
\bigl{[}\theta,\theta_{h_{\mathop{\rm in}\nolimits}}^{\dagger}\bigr{]}, and are bounded with respect to \big{(}h_{\mathop{\rm in}\nolimits},\varphi_{\ast}g_{X_{0}^{\prime}}\big{)}.
5.2.3 An estimate
We set Y(\epsilon):=\bigl{\{}(z_{1},z_{2})\in X_{0}\mid\min(|z_{i}|)=\epsilon\bigr{\}}.
Lemma 5.11**.**
We obtain \lim\limits_{\epsilon\to 0}\int_{Y(\epsilon)}\mathop{\rm Tr}\nolimits\big{(}\theta\overline{\partial}\theta^{\dagger}\big{)}=0 and \lim\limits_{\epsilon\to 0}\int_{Y(\epsilon)}\mathop{\rm Tr}\nolimits\big{(}\theta^{\dagger}\partial\theta\big{)}=0.
Proof.
It is enough to consider the case where Condition 5.8 is satisfied for . Let be any anti-holomorphic function on . Let us consider . We set and . We have
[TABLE]
It is of the form
[TABLE]
Here, is a holomorphic function. We consider the Taylor expansion of and . Then, the contributions of the terms
[TABLE]
to (5.4) is [math] unless and . If the equalities hold, we have and . Hence, we obtain . Similarly, we obtain . Similarly and more easily, we obtain . Then, the claim of the lemma follows. ∎
5.3 Around smooth points
We set X_{0}:=\bigl{\{}(z_{1},z_{2})\in{\mathbb{C}}^{2}\mid|z_{i}|<1\bigr{\}} and . Let be a -function such that induces a nowhere vanishing -function on . Let be a good filtered Higgs bundle on . Let \big{(}E,\overline{\partial}_{E},\theta\big{)} be the Higgs bundle obtained as the restriction of to . We choose and a Hermitian metric of such that induces a metric of .
5.3.1 Unramified case
Suppose that satisfies Condition 5.12.
Condition 5.12**.**
**
- •
There exists such that .
- •
There exists a decomposition of good filtered Higgs bundles
[TABLE]
- •
* are holomorphic Higgs fields of .*
We take -metrics of , and we set . We may assume that .
Let be any holomorphic frame of compatible with the decomposition. For each , and are determined by the condition that is a section of . There exist matrix valued --forms such that
[TABLE]
where denotes the identity matrix, and unless . Let denote the matrix valued holomorphic -form determined by . There exists the decomposition such that the following holds:
- •
if , and if .
- •
are holomorphic -forms, and unless .
There exists a matrix valued -form such that \theta_{h_{0}}^{\dagger}{\boldsymbol{v}}={\boldsymbol{v}}\big{(}\overline{\Lambda}_{0}+\Lambda_{2}\big{)}. Moreover, holds unless .
We have R(h_{0})=\big{(}{-}c\overline{\partial}\partial\log\sigma^{2}\big{)}I+\bigoplus R(h_{\mathfrak{a},\alpha}), where are . Note that and [\Lambda_{0},\Lambda_{i}]=\big{[}\Lambda_{0},\overline{\Lambda}_{i}\big{]}=0. Hence, , and are . We also have
[TABLE]
We set and . Then, it is easy to check that is a complex coordinate system. Clearly, . There exists a -function and a -form such that . We set .
Lemma 5.13**.**
\lim\limits_{\epsilon\to 0}\int_{Y(\epsilon)}\mathop{\rm Tr}\nolimits\big{(}\theta\overline{\partial}\theta^{\dagger}\big{)}=0* and \lim\limits_{\epsilon\to 0}\int_{Y(\epsilon)}\mathop{\rm Tr}\nolimits\big{(}\theta^{\dagger}\partial\theta\big{)}=0 hold.*
Proof.
It is enough to prove \lim\limits_{\epsilon\to 0}\int_{Y(\epsilon)}\mathop{\rm Tr}\nolimits\big{(}\theta\overline{\partial}\theta^{\dagger}\big{)}=0. It is easy to see that
[TABLE]
Let us study \int_{Y(\epsilon)}\mathop{\rm Tr}\nolimits\big{(}\Lambda_{0}\overline{\partial}\Lambda_{2}\big{)}. For any -function , we consider the following integral:
[TABLE]
We can rewrite the first term in the right hand side of (5.5) as follows, for some non-negative integer and for a -function :
[TABLE]
Take . We consider the expansion
[TABLE]
Here, are -functions of . The contributions
[TABLE]
are [math] unless . If , then holds. Hence, we obtain
[TABLE]
We rewrite the second term in the right hand side of (5.5) as follows, for some -functions and a non-negative integer :
[TABLE]
Take . Consider the expansions f_{i}=\sum(f_{i})_{k,m}(w_{2})\,w_{1}^{k}\overline{w}_{1}^{m}+O\big{(}|w_{1}|^{N}\big{)}. The contributions
[TABLE]
are [math] unless . If holds, then we have . The contributions
[TABLE]
are [math] unless . If holds, then we have . Hence, we obtain
[TABLE]
Similarly and more easily, we obtain for any and for any -function . Thus, we obtain the claim of the lemma. ∎
5.3.2 Ramified case
Let be given by \varphi(\zeta_{1},\zeta_{2})=\big{(}\zeta_{1}^{m},\zeta_{2}\big{)}. We set and . Let , which acts on by .
Suppose that satisfies Condition 5.12. We construct a Hermitian metric for as in the previous subsection. We may assume that is -invariant. There exists a -function on determined by . We set . Because is -invariant, we obtain a Hermitian metric of induced by . Let denote the Kähler metric of induced by .
Lemma 5.14**.**
, \big{[}\theta,\theta_{h_{\mathop{\rm in}\nolimits}}^{\dagger}\big{]}, and are bounded with respect to and . We also have
[TABLE]
5.4 Proof of Proposition 5.6
Let , and be as in Section 5.1.1. Let be a good filtered Higgs bundle on satisfying Condition 5.1. Note that is as in Section 5.2.2 around any cross point of , and is as in Section 5.3.2 around any smooth points of . There exists a Hermitian metric of such that , the restriction of around any points of are as in Section 5.2.2 or Section 5.3.2. By the construction, is strongly adapted to . By Lemmas 5.10 and 5.14, we obtain that , \big{[}\theta,\theta_{h_{\mathop{\rm in}\nolimits}}^{\dagger}\big{]}, and are bounded with respect to and . As in the proof of [46, Proposition 4.18], we have
[TABLE]
Then, we obtain (5.1) from Lemmas 5.11 and 5.14. Thus, we obtain Proposition 5.6. ∎
5.5 Proof of Theorem 5.7
Let be any coherent -flat -subsheaf. We assume that is saturated, i.e., is torsion-free. Let \big{(}E^{\prime},\mathbb{D}^{\lambda}_{E^{\prime}}\big{)} be the induced -flat sheaf on . There exists a discrete subset such that is a subbundle of . Let denote the metric of induced by . We obtain the Chern connection of \big{(}E^{\prime},d^{\prime\prime}_{E^{\prime}},h^{\prime}\big{)} and the operator from and . Let denote the curvature of . We obtain G(E^{\prime},h^{\prime}):=\bigl{[}\mathbb{D}^{\lambda}_{E^{\prime}},\mathbb{D}^{\lambda\star}_{E^{\prime},h^{\prime}}\bigr{]}. Following [62], we define
[TABLE]
It is well defined in by the Chern–Weil formula [62, Lemma 3.2]:
[TABLE]
Here, denotes the orthogonal projection of onto .
Lemma 5.15**.**
If , then extends to a filtered subsheaf of and
[TABLE]
holds. As a result, \big{(}E,\overline{\partial}_{E},\theta,h_{\mathop{\rm in}\nolimits}\big{)} is analytically stable in the sense of [62] see also [49, Section 2.3].
Proof.
If , we obtain . As studied in [37, 38] on the basis of [68], we obtain a coherent -submodule as an extension of . Moreover, as proved in [46, Lemma 4.20], we obtain the equality \deg_{\omega_{\epsilon}}(E^{\prime},h_{\mathop{\rm in}\nolimits})=\int_{X}c_{1}\big{(}\mathcal{P}^{h^{\prime}}_{\ast}E^{\prime}\big{)}\omega_{X}. ∎
According to the fundamental theorem of Simpson [62, Theorem 1] and its variant [49, Proposition 2.49], there exists a Hermitian–Einstein metric of \big{(}E,\mathbb{D}^{\lambda}\big{)} satisfying the conditions , and . By [62, Proposition 3.5] and [62, Lemma 7.4] (see also [49, Proposition 2.49]), we obtain
[TABLE]
It is equal to by Lemma 5.5 and Proposition 5.6. Thus, Theorem 5.7 is proved. ∎
6 Bogomolov–Gieseker inequality
Let be any dimensional smooth connected projective variety with a simple normal crossing hypersurface . Let be any ample line bundle on .
Theorem 6.1**.**
Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a -polystable good filtered -flat bundle on . Then, the Bogomolov–Gieseker inequality holds:
[TABLE]
Proof.
By the Mehta–Ramanathan type theorem (Proposition 3.8), it is enough to study the case , which we shall assume in the rest of the proof. We use the notation in Section 3.7.2. Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}=\bigoplus\big{(}\mathcal{P}_{\ast}\mathcal{V}_{j},\mathbb{D}^{\lambda}_{j}\big{)} be the decomposition into the stable components.
We set . We choose such that . We take for as in Lemma 3.19.
Let such that . For any , we set b(\epsilon):=\max\bigl{\{}d\in\epsilon{\mathbb{Z}}\mid d<b\bigr{\}}. We set
[TABLE]
We have . For any , we set . Then, we obtain and the following equalities:
[TABLE]
Moreover, we have .
Applying the construction in Section 3.7.2, we obtain good filtered -flat bundles \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}_{j},\mathbb{D}^{\lambda}_{j}\big{)} on . By the construction, they satisfy Condition 5.1. By Lemma 3.25, there exists such that \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}_{j},\mathbb{D}^{\lambda}_{j}\big{)} are -stable if . Let \big{(}E_{j},\mathbb{D}^{\lambda}_{j}\big{)} be the -flat bundle obtained as the restriction of \big{(}\mathcal{P}_{\ast}\mathcal{V}_{j},\mathbb{D}^{\lambda}_{j}\big{)} to . We use the Kähler metric of as in Section 5.1.1. There exist Hermitian–Einstein metrics of the -flat bundles \big{(}E_{j},\mathbb{D}^{\lambda}_{j}\big{)} as in Theorem 5.7 for the good filtered -flat bundles \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V}_{j},\mathbb{D}^{\lambda}_{j}\big{)}. Note that is a Hermitian–Einstein metric of \bigoplus\big{(}E_{j},\mathbb{D}^{\lambda}_{j}\big{)}.
By Proposition 3.1, the equality (5.2), and the equality \frac{\sqrt{-1}}{2\pi}\frac{1}{1+|\lambda|^{2}}\mathop{\rm Tr}\nolimits G\big{(}h^{(\epsilon)}_{\mathop{\rm HE}\nolimits}\big{)}=\frac{\sqrt{-1}}{2\pi}R(h_{\det E}), we obtain
[TABLE]
By taking the limit as , i.e., , we obtain the desired inequality. ∎
Corollary 6.2**.**
Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be a -polystable good filtered -flat bundle on . Suppose that
[TABLE]
Then, holds.
Moreover, for any decomposition \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}=\bigoplus\big{(}\mathcal{P}_{\ast}\mathcal{V}_{j},\mathbb{D}^{\lambda}_{j}\big{)} into -stable good filtered -flat bundles, we obtain and .
Proof.
On one hand, because of the Hodge index theorem and , we obtain
[TABLE]
and the equality holds if and only if . On the other hand, by the Bogomolov–Gieseker inequality and , we obtain
[TABLE]
Hence, we obtain .
Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}=\bigoplus\big{(}\mathcal{P}_{\ast}\mathcal{V}_{j},\mathbb{D}^{\lambda}_{j}\big{)} be a decomposition into -stable good filtered -flat bundles. We have . Hence, by the Hodge index theorem, we obtain
[TABLE]
By the Bogomolov–Gieseker type inequality, we obtain
[TABLE]
Because , we obtain
[TABLE]
Thus, we obtain the claim of the corollary. ∎
Remark 6.3**.**
Although was assumed to be ample in [51, Section 14.4, Corollary 14.5.1], it is not essential. Indeed, for any simple normal crossing hypersurface , there exists an ample simple normal crossing hypersurface such that . Let \big{(}\mathcal{P}^{\prime}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be the filtered -flat bundle on naturally induced by \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}. The Chern characters of and are equal, and hence the Bogomolov–Gieseker inequalities for and are equivalent.
7 Existence theorem of pluri-harmonic metrics
7.1 Statement
Let us prove Theorem 2.23. According to Corollary 6.2, it is enough to study the case where \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} is a -stable good filtered -flat bundle on such that
[TABLE]
Let \big{(}E,\mathbb{D}^{\lambda}\big{)} be the -flat bundle obtained as the restriction \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X\setminus H}. Let denote the pluri-harmonic metric of \big{(}\det(E),\mathbb{D}^{\lambda}_{\det(E)}\big{)} strongly adapted to . For the proof of Theorem 2.23, it is enough to prove the following theorem.
Theorem 7.1**.**
There exists a unique pluri-harmonic metric of the -flat bundle \big{(}E,\mathbb{D}^{\lambda}\big{)} such that and .
The proof is given in the rest of this section.
7.2 Surface case
Let us study the case . The following argument is essentially the same as the proof of [49, Theorem 5.5]. Let \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} be as in Section 7.1. We use the notation in the proof of Theorem 6.1. For large , we set . We have the perturbations \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}. We use the Kähler metrics of as in Section 5.1.1. There exist the Hermitian–Einstein metrics of \big{(}E,\mathbb{D}^{\lambda}\big{)} adapted to \big{(}\mathcal{P}^{(\epsilon)}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} such that \det\big{(}h^{(\epsilon)}_{\mathop{\rm HE}\nolimits}\big{)}=h_{\det(E)}.
Proposition 7.2**.**
For any sequence , we set . Then, after going to a subsequence, is convergent almost everywhere on , and the limit is a pluri-harmonic metric of the -flat bundle \big{(}E,\mathbb{D}^{\lambda}\big{)} adapted to such that .
7.2.1 Family of ample hypersurfaces
There exists a [math]-dimensional closed subset such that contains the singular points of , any has a neighbourhood in on which the conjugacy classes of are constant.
Take a sufficiently large integer . We set \mathfrak{Z}_{M}:=H^{0}\big{(}X,L^{\otimes\,M}\big{)}\setminus\{0\}. It is equipped with a natural -action. Let denote the projection of onto the -th component. There exists the universal section of p_{1}^{\ast}\big{(}L^{\otimes M}\big{)}. Let denote the scheme obtained as . Let and denote the morphism induced by . For each , let denote the fiber product of and the inclusion .
There exists the -invariant maximal Zariski open subset such that the induced morphism is smooth, is normal crossing for any , . Let denote the restriction of to . For any , we obtain the subspace of codimension . It determines a point in \mathbb{P}\big{(}T^{\ast}_{\texttt{P}_{1}(Q)}X\big{)}. Hence, we obtain the natural morphism . If is sufficiently large, and are surjective.
By the Mehta–Ramanathan type theorem (Proposition 3.8), there exists a non-empty -invariant Zariski open subset of such that the following holds:
- •
For each , \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X_{s}} is stable.
We set . Note that W_{M}:=X\setminus\texttt{P}^{\circ}_{1}\big{(}\mathfrak{X}_{M}^{\triangle}\big{)} is a finite set. For each , the intersection \widetilde{\texttt{P}}^{\circ}_{1}\big{(}\mathfrak{X}_{M}^{\triangle}\big{)}\cap\mathbb{P}(T^{\ast}_{P}X) in is Zariski dense in .
We set . Let denote the -flat bundle on obtained as the restriction of \big{(}E,\mathbb{D}^{\lambda}\big{)}. For each , there exists a pluri-harmonic metric of \big{(}E_{s},\mathbb{D}^{\lambda}_{s}\big{)} such that is adapted to , .
Let be the induced map. Let \mathfrak{H}_{M}^{\triangle}:=\big{(}\texttt{P}_{1}^{\triangle}\big{)}^{-1}(H). We set \big{(}E^{\triangle},\mathbb{D}^{\lambda}_{E^{\triangle}}\big{)}:=\big{(}\texttt{P}_{1}^{\triangle}\big{)}^{-1}\big{(}E,\mathbb{D}^{\lambda}\big{)} on . By Lemma 3.21 and Proposition 4.5, the family of pluri harmonic metrics induces a continuous Hermitian metric of . We also obtain Hermitian metrics h^{\triangle(\epsilon_{i})}:=\big{(}\texttt{P}_{1}^{\triangle}\big{)}^{-1}\big{(}h^{(\epsilon_{i})}_{\mathop{\rm HE}\nolimits}\big{)}.
7.2.2 Local holomorphic coordinate systems
Let . We take such that . The following is clear because \widetilde{\texttt{P}}_{1}\big{(}\mathfrak{X}^{\triangle}_{M}\big{)}\cap\mathbb{P}(T^{\ast}_{P}X) is dense in .
Lemma 7.3**.**
There exist and such that the following holds:
- •
* .*
- •
* and are transversal at .*
- •
, , \bigl{\{}s_{1}+s_{2}+as_{\infty}\mid|a|<\delta\bigr{\}} and \bigl{\{}s_{1}+\sqrt{-1}s_{2}+as_{\infty}\mid|a|<\delta\bigr{\}} are contained in .
We set . There exists a neighbourhood of in such that is a holomorphic coordinate system on . Note that \bigl{\{}\sum b_{i}x_{i}=c\bigr{\}}\cap U_{P} is equal to .
7.2.3 Proof of Proposition 7.2
Take a sequence in . We set . By Proposition 3.1, we obtain the following convergence:
[TABLE]
Let be a Hermitian metric for \big{(}\mathcal{P}^{(\epsilon_{i})}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)} as in Proposition 5.6. Let be the automorphism of determined by . Then, for each , is with respect to and .
Let denote the Kähler form of induced by . Let denote the restriction of to .
By Fubini’s theorem, after going to a subsequence, there exists a -invariant subset with the following property:
\lim\limits_{i\to\infty}\int_{X_{s}\setminus H_{s}}\bigl{|}G\big{(}h_{s}^{(\epsilon_{i})}\big{)}\bigr{|}^{2}_{h_{s}^{(\epsilon_{i})},\omega_{\epsilon_{i},s}}=0 holds for each . 2.
For each , \mathbb{D}^{\lambda}_{s}\big{(}b_{i|X_{s}\setminus H_{s}}\big{)} is with respect to and . 3.
The Lebesgue measure of is [math].
Note that the condition implies the following.
Lemma 7.4**.**
Let . Let be a harmonic metric of adapted to \big{(}\mathcal{P}_{\ast}^{(\epsilon_{i})}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X_{s}} such that \det\big{(}\widetilde{h}^{(\epsilon_{i})}_{s}\big{)}=h_{\det(E)|X_{s}\setminus H_{s}}. Let be the automorphism of determined by . Then, and are bounded with respect to , and \mathbb{D}^{\lambda}\big{(}\widetilde{b}_{i,s}\big{)} is with respect to and
Proof.
Let be the automorphism of determined by . Then, and are bounded with respect to , and is with respect to and , according to Proposition 4.4. Then, we obtain the claim of the lemma. ∎
Lemma 7.5**.**
There exists a -invariant subset such that the following holds:
- •
The measure of is [math].
- •
A subsequence of is convergent to at any point of .
Proof.
By Proposition 4.8, for any , the sequence is weakly convergent to in locally on . We set . We obtain \det\big{(}b_{s}^{(\epsilon_{i})}\big{)}=1, and converges to the identity locally on in for any . We set g^{(\epsilon_{i})}_{s}:=\bigl{|}b^{(\epsilon_{i})}_{s}\bigr{|}_{h_{s}} on . We obtain the function on from . By Lemma 3.21 and Proposition 4.8, for any compact subset , the restriction of to K\cap\bigl{(}\mathfrak{X}_{M}^{\triangle}\times_{\mathfrak{Z}_{M}^{\triangle}}\mathfrak{Z}_{M}^{\sharp}\bigr{)} are uniformly bounded. Note that the sequence \bigl{(}h^{\triangle(\epsilon_{i})}\big{(}h^{\triangle}\big{)}^{-1}\bigr{)}_{|K\cap X_{s}}=b_{s|K\cap X_{s}}^{(\epsilon_{i})} is convergent in for any . By Fubini’s theorem and Lebesgue theorem, we obtain the -convergence of h^{\triangle(\epsilon_{i})}\big{(}h^{\triangle}\big{)}^{-1} to the identity for any on K\cap\bigl{(}\mathfrak{X}_{M}^{\triangle}\times_{\mathfrak{Z}_{M}^{\triangle}}\mathfrak{Z}_{M}^{\sharp}\bigr{)}. Then, after going to a subsequence, we obtain the desired convergence. ∎
Remark 7.6**.**
If , the argument can be simplified. Indeed, by Proposition 3.31, the curvature R\big{(}h_{s}^{(\epsilon_{i})}\big{)} of are bounded locally on . Hence, we obtain that is weakly convergent to in . In particular, is convergent to in the -sense locally on .
There exists a subset X^{\sharp}\subset\texttt{P}^{\triangle}_{1}\big{(}\mathfrak{X}_{M}^{\sharp}\big{)} such that for any , the measure of \big{(}\texttt{P}^{\triangle}_{1}\big{)}^{-1}(P)\setminus\bigl{(}\mathfrak{X}_{M}^{\sharp}\bigr{)} is [math] in \big{(}\texttt{P}^{\triangle}_{1}\big{)}^{-1}(P), and that the measure of is [math] in . We obtain that the sequence is convergent to a Hermitian metric of .
Lemma 7.7**.**
For any and s\in\texttt{P}_{2}\bigl{(}\big{(}\texttt{P}_{1}^{\triangle}\big{)}^{-1}(P)\bigr{)}, we obtain .
Proof.
For any s\in\texttt{P}_{2}\bigl{(}\big{(}\texttt{P}^{\triangle}_{1}\big{)}^{-1}(P)\bigr{)}\cap\mathfrak{Z}_{M}^{\sharp}, we obtain . Then, by using the continuity of on , we obtain for any s\in\texttt{P}_{2}\bigl{(}\big{(}\texttt{P}^{\triangle}_{1}\big{)}^{-1}(P)\bigr{)}. ∎
Lemma 7.8**.**
Let . Then, for any s_{1},s_{2}\in\texttt{P}_{2}\bigl{(}\big{(}\texttt{P}^{\triangle}_{1}\big{)}^{-1}(P)\bigr{)}, we obtain .
Proof.
For any and for any s_{1},s_{2}\in\texttt{P}_{2}\bigl{(}\big{(}\texttt{P}^{\triangle}_{1}\big{)}^{-1}(P)\bigr{)}, we obtain . By the continuity of on , we obtain the claim of the lemma. ∎
Then, extends to a Hermitian metric of by setting for s\in\texttt{P}_{2}\big{(}\big{(}\texttt{P}^{\triangle}_{1}\big{)}^{-1}(P)\big{)}.
Lemma 7.9**.**
* induces a Hermitian metric of of -class. The -Hermitian metric is also denoted by .*
Proof.
Let be any point of . Let be a holomorphic coordinate neighbourhood as in Section 7.2.2. By using Proposition 4.5, we define the continuous Hermitian metric of by the condition that is equal to the restriction of . By the construction of , we obtain . Hence, we obtain that induces a continuous Hermitian metric of , and hold. Moreover, by Proposition 4.5, any derivative of with respect to and are continuous. We obtain that is . Thus, we obtain the claim of the lemma. ∎
We obtain the operator from and . We define G(h_{\infty}):=\bigl{[}\mathbb{D}^{\lambda},\mathbb{D}^{\lambda\star}_{h_{\infty}}\bigr{]} as a current.
Lemma 7.10**.**
* on .*
Proof.
Let . Let be a holomorphic coordinate neighbourhood as in Section 7.2.2. We have the expression
[TABLE]
Because is equal to , we obtain for .
By considering the holomorphic coordinate system and the coefficient of in , we obtain . By considering the holomorphic coordinate system (z_{1},z_{2})=\big{(}x_{1}+\sqrt{-1}x_{2},x_{1}-\sqrt{-1}x_{2}\big{)} and the coefficient of in , we obtain . Therefore, we obtain that . ∎
Lemma 7.11**.**
We obtain on . As a result, is on . If moreover , then is a pluri-harmonic metric of \big{(}E,\mathbb{D}^{\lambda}\big{)}_{|X\setminus H\cup W_{M}}.
Proof.
The first claim immediately follows from Lemma 7.10. We obtain the second claim by the elliptic regularity and a standard bootstrapping argument. The last claim follows from Corollary 2.16. ∎
Lemma 7.12**.**
In the case , we obtain , i.e., is a pluri-harmonic metric of the Higgs bundle \big{(}E,\overline{\partial}_{E},\theta\big{)}_{|X\setminus(H\cup W_{M})}.
Proof.
Let us observe that the sequence is convergent to [math] almost everywhere on . It is enough to prove that is convergent to [math] for . Let be the automorphism of which is self-adjoint with respect to and determined by . By Proposition 4.8, the sequence \big{(}b_{s}^{(\epsilon_{i})}\big{)}^{-1}\partial_{h_{s}}\big{(}b_{s}^{(\epsilon_{i})}\big{)} is convergent to [math] weakly in locally on . By Proposition 3.31, the sequence \big{(}b_{s}^{(\epsilon_{i})}\big{)}^{-1}\partial_{h_{s}}\big{(}b_{s}^{(\epsilon_{i})}\big{)} is bounded in locally on for any .
Lemma 7.13**.**
\big{(}b_{s}^{(\epsilon_{i})}\big{)}^{-1}\partial_{h_{s}}\big{(}b_{s}^{(\epsilon_{i})}\big{)}* is convergent to [math] in locally on .*
Proof.
Let \big{(}b_{s}^{\prime(\epsilon_{i})}\big{)}^{-1}\partial_{h_{s}}\big{(}b_{s}^{\prime(\epsilon_{i})}\big{)} be any subsequence of \big{(}b_{s}^{(\epsilon_{i})}\big{)}^{-1}\partial_{h_{s}}\big{(}b_{s}^{(\epsilon_{i})}\big{)}. Because it is bounded in locally on , it contains a subsequence \big{(}b_{s}^{\prime\prime(\epsilon_{i})}\big{)}^{-1}\partial_{h_{s}}\big{(}b_{s}^{\prime\prime(\epsilon_{i})}\big{)} which is weakly convergent in locally on for any . By the Sobolev embedding theorem, the sequence \big{(}b_{s}^{\prime\prime(\epsilon_{i})}\big{)}^{-1}\partial_{h_{s}}\big{(}b_{s}^{\prime\prime(\epsilon_{i})}\big{)} is convergent in locally on . Because \big{(}b_{s}^{(\epsilon_{i})}\big{)}^{-1}\partial_{h_{s}}\big{(}b_{s}^{(\epsilon_{i})}\big{)} is convergent to [math] weakly in locally on , the limit should be [math]. Therefore, we obtain the claim of Lemma 7.13. ∎
As a result, is convergent to almost everywhere. Note that
[TABLE]
We also have \lim\limits_{i\to\infty}\int_{X}\mathop{\rm ch}\nolimits_{2}\big{(}\mathcal{P}^{(\epsilon_{i})}_{\ast}\mathcal{V}\big{)}=0. We have the following convergence almost everywhere on :
[TABLE]
Therefore, we obtain \int\bigl{|}\partial_{h_{\infty}}\theta\bigr{|}^{2}_{h_{\infty},\omega_{X}}=0 by Fatou’s lemma. ∎
Lemma 7.14**.**
* induces a -metric of on , and hence it is a pluri-harmonic metric of \big{(}E,\mathbb{D}^{\lambda}\big{)}.*
Proof.
It is enough to prove that is a -metric around any point of . We have only to apply the argument in [49, Lemma 5.15]. ∎
If , we obtain that \big{(}E,\overline{\partial}_{E},\theta,h_{\infty}\big{)} is a good wild harmonic bundle on , because is a good filtered Higgs bundle. If , the associated Higgs bundle \big{(}E,\overline{\partial}_{E},\theta\big{)} with the pluri-harmonic metric is a good wild harmonic bundle by [51, Proposition 13.5.1]. We obtain a good filtered -flat bundle \big{(}\mathcal{P}_{\ast}^{h_{\infty}}E,\mathbb{D}^{\lambda}\big{)} on . We put . For any P\in H\setminus\big{(}W_{M}\cup H^{[2]}\big{)}, there exists such that . By the construction, . Hence, we obtain . Let , which is a finite subset of . We obtain that . By Hartogs theorem, we obtain that . Thus, the proof of Proposition 7.2 is completed. ∎
7.3 Higher dimensional case
Let us prove Theorem 7.1 in the case by an induction on . Take a sufficiently large integer . We set \mathfrak{Z}_{M}:=H^{0}\big{(}X,L^{\otimes\,M}\big{)}\setminus\{0\}, and let be defined as as in Section 7.2.1. For any , set . Let denote the projectivization of the cotangent bundle of . If is sufficiently large, there exists a Zariski dense open subset such that the following holds:
- •
is smooth.
- •
\mathfrak{X}_{M}^{\circ}\cup\bigl{(}H\times\mathfrak{Z}_{M}^{\circ}\bigr{)} is simply normal crossing. Moreover the intersections of any tuple of irreducible components are smooth over .
- •
The induced map is surjective. Moreover, the induced morphism : is surjective.
Let denote the projection of onto the product of the -th component and the -th component. For , let \big{(}\mathfrak{X}_{M}^{\circ}\big{)}^{(j)} denote the pull back of by . There exists a Zariski dense open subset such that the following holds:
- •
Let \big{(}\mathfrak{X}_{M}^{\circ}\big{)}^{(j)}_{\mathfrak{U}_{M}} denote the fiber product of \big{(}\mathfrak{X}_{M}^{\circ}\big{)}^{(j)} and over . Then, \big{(}\mathfrak{X}_{M}^{\circ}\big{)}^{(2)}_{\mathfrak{U}_{M}}\cup\big{(}\mathfrak{X}_{M}^{\circ}\big{)}^{(3)}_{\mathfrak{U}_{M}}\cup(H\times\mathfrak{U}_{M}) is simply normal crossing. Moreover, the intersection of any tuple of irreducible components are smooth over .
By the Mehta–Ramanathan type theorem (Proposition 3.8), there exists a Zariski dense open subset such that the following holds:
- •
For , we set . Then, the restriction \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X_{{\boldsymbol{s}}}} is a -stable good filtered -flat bundle on .
Hence, there exists a Zariski dense open subset such that the following holds:
- •
For any , \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X_{s}} is a -stable good filtered -flat bundle on .
- •
For any , there exists a Zariski open subset such that the restrictions \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X_{(s_{i},s_{3})}} are -stable for any .
We set . Let denote the naturally induced morphism. Then, W_{M}:=X\setminus\texttt{P}_{2}^{\triangle}\big{(}\mathfrak{X}_{M}^{\triangle}\big{)} is a finite subset.
For any , there exists such that . Then, \big{(}\mathcal{P}_{\ast}\mathcal{V}_{s},\mathbb{D}^{\lambda}_{s}\big{)}:=\big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X_{s}} is -stable, and the following holds:
[TABLE]
There exists a pluri-harmonic metric of \big{(}E_{s},\overline{\partial}_{E_{s}},\mathbb{D}^{\lambda}_{s}\big{)}:=\big{(}E,\overline{\partial}_{E},\mathbb{D}^{\lambda}\big{)}_{|X_{s}\setminus H} adapted to such that . Take another such that . There exists a pluri-harmonic metric of \big{(}E_{s^{\prime}},\overline{\partial}_{E_{s^{\prime}}},\mathbb{D}^{\lambda}_{s^{\prime}}\big{)} adapted to such that .
Lemma 7.15**.**
.
Proof.
Suppose that is simply normal crossing. We set . It is smooth and connected. We obtain a good filtered -flat bundle \big{(}\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}\big{)}_{|X_{s,s^{\prime}}}, and and are adapted to . Let be the automorphism of which is self-adjoint with respect to both and , and determined by . There exists a decomposition
[TABLE]
which is orthogonal with respect to both and , and for some positive constants .
There exists . Then, and are -stable. Therefore, we have and . We obtain that . It implies that are , and hence .
In general, there exists such that , and are simply normal crossing. By the above consideration, we obtain . ∎
Therefore, we obtain Hermitian metrics of . By using the argument in Lemma 7.9, we can prove that they induce a Hermitian metric of of -class. We obtain from and as a current. Because \big{(}s\in\mathfrak{U}_{M}^{\triangle}\big{)} are pluri-harmonic metrics of \big{(}E,\mathbb{D}^{\lambda}\big{)}_{|X_{s}\setminus H}, we obtain that . It also implies that is on . By using the argument in [49, Lemma 5.15], we obtain that induces a pluri-harmonic metric of \big{(}E,\mathbb{D}^{\lambda}\big{)} on . Then, as in the proof of Proposition 7.2, we can conclude that \big{(}E,\mathbb{D}^{\lambda},h\big{)} is a good wild harmonic bundle, and that . Thus, we obtain Theorem 7.1. ∎
8 Homogeneity with respect to group actions
8.1 Preliminary
8.1.1 Homogeneous harmonic bundles
Let be a complex manifold. Let be a compact Lie group. Let be a -action on such that is holomorphic for any . Let be a homomorphism of Lie groups.
Let \big{(}E,\overline{\partial}_{E},\theta,h\big{)} be a harmonic bundle on . It is called -homogeneous if \big{(}E,\overline{\partial}_{E},h\big{)} is -equivariant and .
Remark 8.1**.**
According to s3, harmonic bundles are equivalent to polarized variation of pure twistor structure of weight , for any given integer . If is non-trivial, as studied in [53, Section 3], by choosing a vector in the Lie algebra of such that , we obtain the integrability of the variation of pure twistor structure from the homogeneity of harmonic bundles.
8.1.2 Homogeneous filtered Higgs sheaves
and the stability condition
with respect to the action
Let be a connected complex projective manifold with a simple normal crossing hypersurface . Let be a complex reductive algebraic group. Let be an algebraic -action on which preserves . Let be a homomorphism of complex algebraic groups.
Let be a filtered Higgs sheaf on . It is called -homogeneous if is -equivariant and for any .
Let be a -equivariant ample line bundle on . A -homogeneous filtered Higgs sheaf on is called -stable (resp. -semistable) with respect to the -action if the following holds:
- •
Let be a -invariant saturated Higgs subsheaf of such that . Then, (resp. ) holds.
A -homogeneous filtered Higgs sheaf on is called -polystable with respect to the -action if it is -semistable with respect to the -action and isomorphic to a direct sum of -homogeneous filtered sheaves , where each is -stable with respect to the -action.
Lemma 8.2**.**
* is -semistable if and only if is -semistable with respect to the -action.*
Proof.
The “only if” part is clear. Let us prove that the “if” part. Let be the -subobject as in Proposition 3.4. Because also has the same property, we obtain that is -invariant. Then, the claim of the proposition is clear. ∎
The following lemma is clear.
Lemma 8.3**.**
If is -stable, then is -stable with respect to the -action.
Lemma 8.4**.**
If is -stable with respect to the -action, then is -polystable.
Proof.
According to Lemma 8.2, is -semistable. Let be the socle of as in Proposition 3.5. Because also has the same property, is -invariant. Moreover, holds. Hence, we obtain . According to Proposition 3.5, is -polystable. ∎
Remark 8.5**.**
In general, even if is -stable with respect to the -action, is not necessarily -stable.
8.1.3 Actions of a complex reductive group
and its compact real form
Let be a complex projective manifold equipped with an algebraic action of a complex reductive group . Let be a -equivariant ample line bundle on . Let be a compact real form of .
Let \big{(}E,\overline{\partial}_{E}\big{)} be a -equivariant holomorphic vector bundle on . Then, as the restriction, we may naturally regard \big{(}E,\overline{\partial}_{E}\big{)} as a -equivariant holomorphic vector bundle on .
Lemma 8.6**.**
The above procedure induces an equivalence between -equivariant holomorphic vector bundles and -equivariant holomorphic vector bundles on .
Proof.
Let \big{(}E,\overline{\partial}_{E}\big{)} be a -equivariant holomorphic vector bundle on . There exists such that is globally generating. We set \mathcal{G}_{0}:=H^{0}\big{(}X,E\otimes L^{\otimes\,m_{0}}\big{)}\otimes\big{(}L^{\otimes\,m_{0}}\big{)}^{-1}. There exists a naturally induced epimorphism of -modules . Let denote the kernel. There exists such that is globally generating. We set \mathcal{G}_{1}:=H^{0}\big{(}X,\mathcal{K}\otimes L^{\otimes\,m_{1}}\big{)}\otimes\big{(}L^{\otimes\,m_{1}}\big{)}^{-1}. There exists a naturally induced epimorphism . Thus, we obtain a resolution of . Because is -equivariant, H^{0}\big{(}X,E\otimes L^{\otimes m_{0}}\big{)} is naturally a -representation, is a -equivariant holomorphic vector bundle on , and is -equivariant. Hence, is a -equivariant holomorphic vector bundle. Similarly H^{0}\big{(}X,\mathcal{K}\otimes L^{\otimes m_{2}}\big{)} is a -representation, and is -equivariant holomorphic vector bundle, and is -equivariant.
The -representations on H^{0}\big{(}X,E\otimes L^{\otimes\,m_{1}}\big{)} and H^{0}\big{(}X,\mathcal{K}\otimes L^{\otimes m_{2}}\big{)} naturally induce -representations on H^{0}\big{(}X,E\otimes L^{\otimes\,m_{1}}\big{)} and H^{0}\big{(}X,\mathcal{K}\otimes L^{\otimes m_{2}}\big{)}. Hence, are naturally algebraic -equivariant vector bundles on . Moreover, the morphism is -equivariant and algebraic. Hence, is a -equivariant algebraic vector bundle on . ∎
8.2 An equivalence
8.2.1 Good filtered Higgs bundles associated with
homogeneous good wild Higgs bundles
Let be a connected complex projective manifold with a simple normal crossing hypersurface . Let be a complex reductive group acting on . Let be a compact real form of . The actions of and on are denoted by . Let be a character. The induced homomorphism is also denoted by .
Let \big{(}E,\overline{\partial}_{E},\theta,h\big{)} be a -homogeneous harmonic bundle on which is good wild on . We obtain a good filtered Higgs bundle \big{(}\mathcal{P}^{h}_{\ast}E,\theta\big{)} on . Because each is naturally a -equivariant holomorphic vector bundle on , is naturally -equivariant by Lemma 8.6. Because for any , we obtain for any . Therefore, \big{(}\mathcal{P}_{\ast}^{h}E,\theta\big{)} is a -homogeneous good filtered Higgs bundle on .
Let be a -equivariant ample line bundle on .
Proposition 8.7**.**
\big{(}\mathcal{P}_{\ast}^{h}E,\theta\big{)}* is -polystable with respect to the -action, i.e., there exists a decomposition \big{(}E,\overline{\partial}_{E},\theta,h\big{)}=\bigoplus(E_{i},\overline{\partial}_{E_{i}},\theta_{i},h_{i}) of -homogeneous harmonic bundles such that each \big{(}\mathcal{P}_{\ast}^{h_{i}}E_{i},\theta_{i}\big{)} is -stable with respect to the -action.*
Proof.
Because \big{(}\mathcal{P}^{h}_{\ast}E,\theta\big{)} is -polystable, we obtain that \big{(}\mathcal{P}^{h}_{\ast}E,\theta\big{)} is -semistable with respect to the -action. Let be a -invariant saturated Higgs -submodule such that \mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{1})=\mu_{L}\big{(}\mathcal{P}_{\ast}^{h}E\big{)}=0. Let be the Higgs subsheaf of obtained as the restriction of to . Then, by the argument in the proof of [51, Proposition 13.6.1], we obtain that is a subbundle, and the orthogonal complement is also a holomorphic subbundle. Moreover, , and is -equivariant. Hence, we obtain a decomposition \big{(}E,\overline{\partial}_{E},\theta,h\big{)}=\big{(}E_{1},\overline{\partial}_{E_{1}},\theta_{1},h_{1}\big{)}\oplus\big{(}E_{2}\overline{\partial}_{E_{2}},\theta_{2},h_{2}\big{)} of -homogeneous harmonic bundles. Then, the claim of the proposition is clear. ∎
8.2.2 Uniqueness
Let \big{(}E,\overline{\partial}_{E},\theta,h\big{)} be a -homogeneous harmonic bundle on which is good wild on . Let be another pluri-harmonic metric of \big{(}E,\overline{\partial}_{E},\theta\big{)} such that is -invariant, . The following is clear from Proposition 2.22.
Proposition 8.8**.**
There exists a decomposition \big{(}E,\overline{\partial}_{E},\theta\big{)}=\bigoplus_{i=1}^{m}\big{(}E_{i},\overline{\partial}_{E_{i}},\theta_{i}\big{)} such that the decomposition is orthogonal with respect to both and , there exist such that , the decomposition is preserved by the -action.
8.2.3 Existence theorem
Let be a -homogeneous good filtered Higgs bundle on such that
[TABLE]
Let \big{(}E,\overline{\partial}_{E},\theta\big{)} be the Higgs bundle on obtained as the restriction of .
Theorem 8.9**.**
Suppose that is -stable with respect to the -action. Then, there exists a -invariant pluri-harmonic metric of \big{(}E,\overline{\partial}_{E},\theta\big{)} such that . If is another -invariant pluri-harmonic metric of \big{(}E,\overline{\partial}_{E},\theta\big{)}, there exists a positive constant such that .
Proof.
By Lemma 8.4, is -polystable. There exists the canonical decomposition
[TABLE]
where are -stable good filtered Higgs bundles such that , and are finite dimensional complex vector spaces. Let \big{(}E_{i},\overline{\partial}_{E_{i}},\theta_{i}\big{)} denote the Higgs bundle obtained as the restriction of to . There exist pluri-harmonic metrics of \big{(}E_{i},\overline{\partial}_{E_{i}},\theta_{i}\big{)} adapted to the filtered bundles . Let be Hermitian metrics of . We obtain a pluri-harmonic metric h^{(0)}=\bigoplus\big{(}h_{i}\otimes h^{(0)}_{U_{i}}\big{)} of \big{(}E,\overline{\partial}_{E},\theta\big{)} adapted to . By Proposition 2.22 and the uniqueness of the canonical decomposition, we obtain the following lemma.
Lemma 8.10**.**
For any pluri-harmonic metric of \big{(}E,\overline{\partial}_{E},\theta\big{)} adapted to , there uniquely exist Hermitian metrics of such that h^{(1)}=\bigoplus\big{(}h_{i}\otimes h^{(1)}_{U_{i}}\big{)}.
For any , we obtain a pluri-harmonic metric of \big{(}E,\overline{\partial}_{E},\kappa(k)\theta\big{)} adapted to . Because , k^{\ast}\big{(}h^{(0)}\big{)} is also a pluri-harmonic metric of \big{(}E,\overline{\partial}_{E},\theta\big{)} adapted to . Hence, there uniquely exist Hermitian metrics of such that . By using the Haar measure on with , we define the Hermitian metric of as follows:
[TABLE]
Then, is also a pluri-harmonic metric. By the construction, is -invariant.
Let be another -invariant pluri-harmonic metric of \big{(}E,\overline{\partial}_{E},\theta\big{)} adapted to . We obtain the decomposition \big{(}E,\overline{\partial}_{E},\theta\big{)}=\bigoplus\big{(}E_{i},\overline{\partial}_{E_{i}},\theta_{i}\big{)} as in Proposition 2.22, which induces a decomposition of good filtered Higgs bundles . Because both and are -invariant, the decompositions are also -invariant. Hence, the decomposition is -invariant. By the -stability of , we obtain , i.e., for . ∎
Corollary 8.11**.**
We obtain the equivalence between the isomorphism classes of the following objects:
- •
-homogeneous good wild harmonic bundles on .
- •
-homogeneous good filtered Higgs bundles such that it is -polystable with respect to the -action, , .
Acknowledgement
It is my great pleasure to dedicate this paper to Professor Kyoji Saito on the occasion of his 77th birthday. Not to mention his pioneering and profound achievements, I have always been impressed with his kindness and generosity towards the younger generation of mathematicians, as well as his insatiable quest for mathematics. I appreciate the editors of this special issue for the invitation.
I thank Carlos Simpson for his fundamental works on harmonic bundles which are most important in this study. I am grateful to the referees for their careful readings and helpful comments to improve this manuscript. I thank Philip Boalch and Andy Neitzke for their kind comments to a preliminary note on the proof in the one dimensional case. I thank Claude Sabbah for discussions on many occasions and for his kindness. I thank François Labourie for his comment on the definition of wild harmonic bundles. I thank Pengfei Huang for discussions. I am grateful to Ya Deng for asking questions related to Mehta–Ramanathan type theorems in Section 3.4 and tensor products in Section 4.3. I thank Akira Ishii and Yoshifumi Tsuchimoto for their constant encouragement. A part of this manuscript was prepared for lectures in the Oka symposium and ICTS program “Quantum Fields, Geometry and Representation Theory”. I thank the organizers for the opportunities.
I am partially supported by the Grant-in-Aid for Scientific Research (S) (No. 17H06127), the Grant-in-Aid for Scientific Research (S) (No. 16H06335), the Grant-in-Aid for Scientific Research (A) (No. 21H04429), the Grant-in-Aid for Scientific Research (C) (No. 15K04843), and the Grant-in-Aid for Scientific Research (C) (No. 20K03609), Japan Society for the Promotion of Science.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Barannikov S., Quantum periods. I. Semi-infinite variations of Hodge structures, Int. Math. Res. Not. 2001 (2001), 1243–1264, ar Xiv:math.AG/0006193 . · doi ↗
- 2[2] Biquard O., Fibrés de Higgs et connexions intégrables: le cas logarithmique (diviseur lisse), Ann. Sci. École Norm. Sup. (4) 30 (1997), 41–96. · doi ↗
- 3[3] Biquard O., Boalch P., Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004), 179–204, ar Xiv:math.DG/0111098 . · doi ↗
- 4[4] Bogomolov F.A., Holomorphic tensors and vector bundles on projective manifolds, Math. USSR Izv. 13 (1979), 499–555. · doi ↗
- 5[5] Borne N., Fibrés paraboliques et champ des racines, Int. Math. Res. Not. 2007 (2007), rnm 049, 38 pages, ar Xiv:math.AG/0604458 . · doi ↗
- 6[6] Borne N., Sur les représentations du groupe fondamental d’une variété privée d’un diviseur à croisements normaux simples, Indiana Univ. Math. J. 58 (2009), 137–180, ar Xiv:0704.1236 . · doi ↗
- 7[7] Cecotti S., Vafa C., Topological–anti-topological fusion, Nuclear Phys. B 367 (1991), 359–461. · doi ↗
- 8[8] Cecotti S., Vafa C., On classification of N = 2 𝑁 2 N=2 supersymmetric theories, Comm. Math. Phys. 158 (1993), 569–644, ar Xiv:hep-th/9211097 . · doi ↗
