Degeneracy loci, virtual cycles and nested Hilbert schemes II
Amin Gholampour, Richard P. Thomas
TL;DR
This paper develops a method to compute invariants of nested Hilbert schemes and related sheaf-counting theories by expressing them as degeneracy loci and applying virtual cycle techniques, enabling explicit calculations.
Contribution
It introduces a novel approach to represent nested Hilbert schemes as virtual resolutions of degeneracy loci, facilitating the calculation of various enumerative invariants.
Findings
Provides explicit formulas for invariants like VW, SW, local PT, and local DT.
Shows how to modify obstruction theories to produce virtual cycles for sheaf-counting.
Offers an effective computational framework using Thom-Porteous-like Chern class formulas.
Abstract
We express nested Hilbert schemes of points and curves on a smooth projective surface as "virtual resolutions" of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa-Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom-Porteous-like Chern class formulae.
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Degeneracy loci, virtual cycles and nested Hilbert schemes II
Amin Gholampour and Richard P. Thomas
Abstract.
We express nested Hilbert schemes of points and curves on a smooth projective surface as “virtual resolutions” of degeneracy loci of maps of vector bundles on smooth ambient spaces.
We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa-Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom-Porteous-like Chern class formulae.
1. Introduction and summary of results
Overview. It is usually hard to calculate with the virtual fundamental class of a moduli space . In the rare situation that (and its perfect obstruction theory) is cut out of a smooth ambient space by a section of a vector bundle , the push forward of the virtual cycle is the Euler class of , making calculation possible.
In [GT1] we described a generalisation to the case where is instead the deepest degeneracy locus of a map of vector bundles on a smooth ambient space. This we applied to nested Hilbert schemes of points on smooth surfaces, reproducing the virtual cycles of [GSY1] — which agree with those arising in Vafa-Witten theory [TT1] or reduced local DT theory [GSY2].
That technique does not extend to other degeneracy loci, but it does apply to their resolutions (1.2). We apply this to nested Hilbert schemes of points and curves on smooth surfaces with their virtual cycles [GSY1] coming from Vafa-Witten theory [TT1] or reduced DT theory [GSY2]. The result is an effective new tool for computing sheaf-theoretic enumerative invariants of projective surfaces, such as those arising in Seiberg-Witten, Vafa-Witten, local DT and local PT theories.
Degeneracy loci. Let be a smooth complex quasi-projective variety and
[TABLE]
a map of vector bundles of ranks and over . For a positive integer , consider the th degeneracy locus of ,
[TABLE]
with scheme structure given by viewing it as the zero locus of
[TABLE]
That is, is the th Fitting scheme of the first cohomology sheaf of the complex (1.1). In particular it depends on only up to quasi-isomorphism, and we often denote it .
For the nested Hilbert scheme applications in this paper we will take only . On a first read-through of the abstract construction the reader may find it simpler to do the same, substituting projective bundles for the Grassmann bundles that follow.
Resolutions. Over points with there is a unique -dimensional subspace of on which vanishes. Taking the set of all such -dimensional subspaces for every point of defines a natural space dominating ,
[TABLE]
whose fibre over is . Its scheme structure111A more efficient description of is as the relative Grassmannian of -dimensional quotients of the sheaf . comes from viewing as the vanishing locus of the composition
[TABLE]
Here denotes the universal rank subbundle on (or its restriction to ) whose fibre at (1.2) is . If is appropriately transverse then each has codimension , singular locus , and is a resolution of singularities. For arbitrary we call the virtual resolution of since it is virtually smooth:
Proposition 1**.**
Expressing as the zero locus of the section (1.3) of the bundle endows it with a perfect obstruction theory
[TABLE]
of virtual dimension . It depends only on the quasi-isomorphism class of the complex (1.1). The resulting virtual cycle
[TABLE]
when pushed forward to , has class given by the Thom-Porteous formula222Here as in [Fu, Chapter 14].
[TABLE]
This is a rewriting of parts of [Fu, Chapter 14] in the language of [BF]. It is useful if we recognise a moduli space as having the same perfect obstruction theory as a . Then invariants defined by integration against can be calculated more easily on the smooth space if the integrand can be expressed as a pullback from .
The drawback is that the map contracts333Except when is the deepest degeneracy locus — i.e. when . Then and is an embedding. This is the case studied in [GT1]. the exceptional locus of , so most integrands (such as the one in (1.15) below) will not be pullbacks from . We could instead work in , to which pushes forward to give , but this description is not an invariant of the quasi-isomorphism class of .
However, in examples like (1.10, 1.12) below each kernel naturally embeds in (the fibre over of) a certain vector bundle ,
[TABLE]
The global condition is that surjects onto over (2.18). Then naturally embeds in , and we can push forward the virtual class.
Theorem 1**.**
Under the embedding the pushforward of the virtual cycle to A_{\operatorname{vd}}\big{(}\!\operatorname{Gr}(r,B)\big{)}is given by
[TABLE]
where is the universal quotient bundle over .
Nested Hilbert schemes. For simplicity consider first 2-step Hilbert schemes of nested subschemes (of dimensions 0 and 1) of a fixed smooth projective surface . Given and integers we set
[TABLE]
When we use to denote the Hilbert scheme of divisors in class . Conversely when we get the nested Hilbert scheme of points studied in [GT1]; we review this briefly first.
Nested Hilbert schemes of points. For simplicity we set for now. For points of we have
[TABLE]
Therefore is the degeneracy locus of the 2-term complex444This uses our simplifying assumption . (For general we modify in [GT1], removing terms to make it 2-term.) The notation will be defined in Section 4; the are the universal ideal sheaves on and denotes the projection down to . of vector bundles
[TABLE]
which, when restricted to the point , has 0 th cohomology (1.7). Moreover, (1.7) shows all higher degeneracy loci are empty, so we are in the situation of Footnote 3 with . Thus the Thom-Porteous formula gives
[TABLE]
In [GT1] we also showed the relevant perfect obstruction theory agrees (in K-theory at least) with that coming from Vafa-Witten theory. Therefore555Siebert’s formula [S, Theorem 4.6] for the virtual class shows it depends on the perfect obstruction theory only through the K-theory class of its virtual tangent bundle. Vafa-Witten invariants can be written as integrals over the smooth ambient space , as exploited to great effect in [La1, La2] for instance.
Nested Hilbert schemes of curves and points. When we consider
[TABLE]
In contrast to (1.7) this can become arbitrarily big, with different elements corresponding (up to scale) to different nested subschemes of . Therefore, the corresponding nested Hilbert scheme dominates the degeneracy locus of the complex of vector bundles
[TABLE]
which, when restricted to the point , has 0 th cohomology (1.8). Since a point of the nested Hilbert scheme is a one dimensional subspace of the kernel (1.8) of the complex (1.9), we see that
[TABLE]
is the virtual resolution of the degeneracy locus with . Since
[TABLE]
the [math] th cohomology of embeds in . So we get an example of (1.4) with and the trivial vector bundle with fibre . The embedding of Theorem 1 then becomes the following.
Theorem 2**.**
Suppose for now. Then has a virtual cycle of dimension which, under the embedding
[TABLE]
pushes forward to
[TABLE]
Reduced virtual cycle. For general with possibly nonzero we work instead with
[TABLE]
which, when restricted to the point , has 0 th cohomology (1.8). Here is a Poincaré line bundle over , normalised (by tensoring by the pullback of if necessary) so that is trivial on some fixed basepoint .
Again a point of is a one dimensional subspace of the kernel (1.8) of the complex (1.11), so
[TABLE]
is the virtual resolution of the degeneracy locus with . This gives a description of as a projective cone over ; see Proposition 4.7, generalising [DKO, Lemma 2.15].
We can no longer take to have fibre over since this may jump in dimension. Instead we fix a sufficiently ample divisor and let be the bundle with fibre :
[TABLE]
Therefore the inclusions
[TABLE]
give the required embeddings of (1.4). Thus such that restricts to , and Theorem 1 gives the following.
Theorem 3**.**
Fix a surface and .
* If for all effective then inherits a reduced virtual cycle of dimension .*
* If for all then via it pushes forward to*
[TABLE]
where is the projection, and for any .
The condition ensures that is 2-term. (We handle in the next Section.) When , this theorem was proven in [KT1, Theorem A.7]. Here — as in [KT1, GSY1] — we call the virtual cycle the reduced cycle \big{[}S^{[n_{1},n_{2}]}_{\beta}\big{]}^{\mathrm{red}} since it differs from the one appearing in Vafa-Witten theory666Or indeed reduced DT theory, confusingly. Removing one copy of from the DT obstruction space of gives the Vafa-Witten/reduced DT obstruction theory. Removing two copies gives the degeneracy locus obstruction theory of Proposition 1. by an term in the obstruction theory.
The apparent difference between the formulae in Theorems 2 and 3 will be explained by the formula (2.25) in Theorem 2.24.
Non-reduced virtual cycle. In general, when need not vanish, (1.11) is a 3-term complex with nonzero cohomology sheaf. So in order to apply our theory we use the splitting trick from [GT1, Section 6.1] to remove from after pulling back to an affine bundle over . (Since an affine bundle has the same Chow groups as its base, no information is lost from the virtual cycle.)
We only manage this on a neighbourhood of , making a 2-term complex there. By Proposition 1 this is enough to prove
Theorem 4**.**
This construction defines a virtual fundamental class
[TABLE]
where . It agrees with the virtual cycle of [GSY1].
Computing virtual cycles by comparison. Since the above neighbourhood need not be all of , we get no useful push forward formula for (1.13) in general.777The exceptions are when — then (1.13) is the reduced virtual class of Theorem 3 — or when for all effective , so (1.13) is zero as the perfect obstruction theory which produced it can be reduced to give the class \big{[}S^{[n_{1},n_{2}]}_{\beta}\big{]}^{\mathrm{red}}. This problem is down to the curve class alone — if it does not arise, whereas if the problem already arises for the Hilbert scheme of pure curves in class . However,
[TABLE]
is a well-studied class that has been understood by other methods [DKO, K] due to its importance in Seiberg-Witten theory. Separating out the parts of the obstruction theory governing the curve and the points, and applying the degeneracy locus technique to the latter only, we are able to prove the following comparison result.
Let denote the universal curve. We call
[TABLE]
the Carlsson-Okounkov K-theory class [CO] on .
Theorem 5**.**
Pushing the virtual and reduced classes forwards along
[TABLE]
gives
[TABLE]
and, if for all effective ,
[TABLE]
When the first of these results was proved in [K] and the second follows from [KT1, Appendix A].
-step Hilbert schemes. The generalization of Theorems 3 and 5 to -step nested Hilbert schemes
[TABLE]
is straightforward. See Section 5 for details.
Vafa-Witten invariants. Vafa-Witten invariants [TT1] are made up of “instanton contributions” and “monopole contributions”. The former are virtual Euler characteristics of moduli spaces of rank sheaves on , as studied in [GK] for instance. The latter are integrals over moduli spaces of chains of sheaves on of total rank with nonzero maps between them.
When and is prime a vanishing result [Th, Theorem 5.23] implies the only nonzero monopole contributions come from moduli spaces of chains of rank 1 sheaves. After tensoring with a line bundle, these are nested Hilbert schemes. For instance in rank 2 the relevant integrals take the form
[TABLE]
where and h:=c_{1}\big{(}\mathcal{O}_{\mathbb{P}(B)}(1)\big{)} and
[TABLE]
Using Theorem 5 we can express this in terms of (a) integrals over the smooth space , and (b) integrals over . The latter give the Seiberg-Witten invariants [DKO, CK],
[TABLE]
Theorem 6**.**
Suppose . Then in H_{*}\big{(}S^{[n_{1}]}\times S^{[n_{2}]}\times\operatorname{Pic}_{\beta}(S)\big{)},
[TABLE]
where so is the generator of .
When then may not be of “Seiberg-Witten simple type”, which means we have to consider higher dimensional Seiberg-Witten moduli spaces. Letting denote the Abel-Jacobi map, the higher Seiberg-Witten invariants [DKO] are
[TABLE]
Theorem 7**.**
If then in ,
[TABLE]
* If then, setting ,*
[TABLE]
* If not,*
[TABLE]
and both sides vanish when . (Cf. Theorem 6.)
Now set , so that the condition that for some is the condition that is effective. Seiberg-Witten theory has a duality under . This gives interesting dualities between invariants of nested Hilbert schemes under . Define the map
[TABLE]
by replacing in and then composing with .
Theorem 8** (Duality).**
In ,
[TABLE]
where .
Theorem 9**.**
If the contributions of the nested Hilbert schemes to the Vafa-Witten invariants of are invariants of its oriented diffeomorphism type.
When this can fail because only the unordered pair is invariant under oriented diffeomorphisms, rather than the individual invariant .
Laarakker. In [La1] (stable case) and [La2] (general case) Laarakker uses these results to compute Vafa-Witten invariants [TT1, TT2] and refined Vafa-Witten invariants [Th] on surfaces with (and for now).
By the comparison result for virtual cycles of Theorem 5 the contributions from points and curves split, with the curves contributing Seiberg-Witten invariants — i.e. well-understood integrals over linear systems of curves. Laarakker evaluates the contributions of Hilbert schemes of points via the method of [EGL]. The result depends only on the curve class and the cobordism class of the surface — and thus only on and . Therefore these contributions can be calculated on K3 surfaces and toric surfaces (despite these not having !).
The results are very general, but the absolute simplest to state is for minimal of general type with and , such that is not divisible by 2.
Theorem 10**.**
[La1]** Let be as above. The monopole branch contributions888See [Th] for definitions. The monopole branch contributions come from -fixed Higgs pairs on with and . of rank 2 Higgs pairs with to the refined Vafa-Witten generating series can be written
[TABLE]
where
[TABLE]
are universal functions, independent of .
Furthermore, K3 calculations [GK, Th] determine completely in terms of modular forms. And can be predicted by modularity and the results of Göttsche-Kool [GK] on the instanton contributions; he checks this prediction in low degree by toric computations.
Acknowledgements. We thank Martijn Kool and Ties Laarakker for help and useful conversations, Bhargav Bhatt for pointing out the Jouanolou trick, and two thorough referees for useful suggestions. A.G. acknowledges partial support from NSF grant DMS-1406788. R.T. is partially supported by EPSRC grant EP/R013349/1.
Notation. Given a map , we often use the same letter to denote its basechange by any map , i.e. . In particular projections down the surface are all denoted by . We often suppress pullback maps on sheaves when it shouldn’t cause confusion.
For a vector bundle over , we denote by the projective bundle of lines in . For a coherent sheaf on , we denote by the projective cone of quotient lines of . When is locally free then of course .
1.1. Splitting trick
Since we are ultimately interested in Chern class formulae, we often only care about the topological type of a given algebraic vector bundle. In particular, we will often wish to split nontrivial extensions of vector bundles, which does not change their topological type. One way to do this is by working with bundles; we prefer to stay within algebraic geometry and use the Jouanolou trick [J] instead.
Namely, if is quasi-projective, there exists an affine bundle such that is an affine variety. Therefore, on pulling locally free sheaves back by , they become projective -modules and any extensions become split. But is a homotopy equivalence, inducing an isomorphism of Chow groups [Kr, Corollary 2.5.7]
[TABLE]
so formulae for cycles upstairs on induce similar formulae on .
A very explicit such affine bundle was used in [GT1, Section 6.1], and all the necessary pullbacks were laboriously documented. In this paper, for the sake of clarity and economy of notation, we will often just say “by the Jouanolou trick” and give no further details. So we will pretend we are working on , suppressing mention of the fact that we have replaced by its homotopic affine model and then used the isomorphism (1.16) to recover results on at the end.
2. Virtual resolutions of degeneracy loci
Fix a smooth complex quasi-projective variety and
[TABLE]
a map of vector bundles of ranks and over . For each positive integer , we let
[TABLE]
denote the th degeneracy locus of , with scheme structure defined by the vanishing of
[TABLE]
This is the th Fitting scheme [Ei, Section 20.2] of the first cohomology sheaf of the complex (2.1). In particular it depends on only up to quasi-isomorphism, and we often denote it . Then define
[TABLE]
to be the relative Grassmannian of dimensional quotients of the cokernel sheaf of . This is the moduli space of dimensional subspaces of fibres which are annihilated by , in the following sense.
Consider the functor
[TABLE]
Let denote the relative Grassmannian, with universal subbundle .
Proposition 2.5**.**
The functor (2.4) is represented by . It embeds
[TABLE]
as the zero locus of defined by the composition
[TABLE]
Finally, the projection has scheme theoretic image inside .
Remarks. Clearly is a set-theoretic surjection — the fibre over is — but it need not be onto as a map of schemes. For instance on has , the thickened origin, whereas is a reduced over the origin.
By the definition of the functor (2.4), comes equipped with a universal subbundle . It is the restriction of the universal subbundle on , and the dual of the universal quotient bundle on . Its fibre over the point \big{(}x,U\subseteq\ker\hskip 0.6pt(\sigma_{x})\big{)} is .
Proof.
Since represents the functor
[TABLE]
it is sufficient to prove this is isomorphic to (2.4). A locally free quotient of coker gives a complex
[TABLE]
which is exact at . Since is locally free, the dual complex
[TABLE]
is exact at after any basechange — thus is a subbundle, giving an element of the set (2.4). The converse is easier: any complex (2.10) with a subbundle dualises to a complex (2.9) exact at .
The functor (2.4) is a subfunctor of the functor taking to the set rank subbundles represented by . This induces an embedding of -schemes under which the universal subbundle pulls back to the dual of the universal quotient bundle. Its image is contained in the zero locus of (2.7) by applying (2.4) to . So .
Conversely, setting to be the projection, the restriction of (2.7) to gives a rank subbundle which factors through by the definition of as the zero locus. This gives an element of the set (2.4), classified by a map . The two maps are mutual inverses by construction.
Finally factors through over by (2.7). Therefore p^{*}\mbox{\Large\wedge}^{e_{0}-r+1}\sigma (2.2) factors through \mbox{\Large\wedge}^{e_{0}-r+1}p^{*}E_{0}/\hskip 1.0pt\mathcal{U}. But this is zero because , so projects into the zero locus of (2.2). ∎
Perfect obstruction theory. Let denote the ideal of generated by (2.7). Then inherits the standard perfect obstruction theory
[TABLE]
That is, the complex on the bottom row is a representative of the truncated cotangent complex and the top row is the virtual cotangent bundle of the obstruction theory.
Proposition 2.12**.**
The virtual cotangent bundle is a cone
[TABLE]
of virtual dimension . It depends only on the quasi-isomorphism class of the complex (1.1). The pushforward of the resulting virtual cycle
[TABLE]
to is given by the Thom-Porteous formula
[TABLE]
Proof.
Modifying (2.11) by an acyclic complex, it is quasi-isomorphic to the total complex
[TABLE]
where is the composition . This diagram was shown to commute in [GT1, Claim 2, proof of Theorem 3.6].
Therefore \mathbb{L}^{\operatorname{vir}}_{\widetilde{D}_{r}}\cong\operatorname{Cone}\!\big{(}\hskip 1.0pt\mathcal{U}\otimes p^{*}E_{\bullet}^{\vee}|_{\widetilde{D}_{r}}\longrightarrow\operatorname{Cone}(\phi)\big{)}[-1]. To determine we fit into the following commutative diagram, all of whose rows and columns are exact triangles:
[TABLE]
We calculate the lower right hand arrow. By definition it is the composition from the top left to the bottom right of
[TABLE]
Here the upper row is the rightmost vertical exact triangle in (2.14), while the lower row is the exact triangle along the top row of (2.14). The first arrow in (2.15) is well known to be the relative (to ) Atiyah class of the bundle . So using its absolute Atiyah class we get the commutative diagram
[TABLE]
which shows the composition is zero. Therefore the bottom row of (2.14) gives , implying the claimed result.
Finally, since the perfect obstruction theory arises from ’s description as the zero locus of a section of the bundle , the resulting virtual cycle is the localised top Chern class [Fu, Section 14.1] of that bundle. Its pushforward to is
[TABLE]
further pushing down shows the pushforward of to is given by
[TABLE]
by [Fu, Theorem 14.4]. ∎
This proves Proposition 1 in the Introduction. But in passing from down to we have contracted back to and so lost information in general. (The exception is the case studied in [GT1] where is the largest integer for which is nonempty; then is an embedding.)
We could work in , in which the class of the virtual cycle is , but this is not a quasi-isomorphism invariant of the complex . In examples such as (1.10, 1.12) we can replace by a more canonical bundle , so now we assume we have chosen such that
[TABLE]
For basechange reasons the precise condition is most easily stated via duals.
Choice. We choose a vector bundle on and a surjection
[TABLE]
For some purposes we will require the surjection (2.18) extends to all of ,
[TABLE]
For instance if this is immediate from (2.18).
Note that by restricting both exact sequences and (2.18) to and dualising we recover (2.17).
The surjection of (2.18) induces an embedding since lies over . By Proposition 2.5 this is an embedding
[TABLE]
such that . Here and denote the universal sub- and quotient bundles on , and is the embedding (2.6). We will compare these via a third embedding,
[TABLE]
fitting into the Cartesian diagram
[TABLE]
Here the vertical maps are flat and the section . As usual we suppress various pullback or basechange maps.
Lemma 2.22**.**
The section in (2.21) is a regular embedding, with image cut out by the regular section of defined by the composition
[TABLE]
Proof.
Since on we see that applied to (2.23) gives the same as applied instead to the composition
[TABLE]
This is zero by the definition of , so lies in the zero locus of (2.23). We are left with showing that .
By the definition of , pulling back the composition (2.23) by shows the composition along the top row of the following diagram is zero. This uniquely fills in the dotted arrow to the lower short exact sequence.
[TABLE]
Since is a vector bundle injection it is an isomorphism. And since is pulled back from , this gives
[TABLE]
Thus this subbundle is represented by both of the maps and , but any subbundle is classified by a unique map . So , which gives .
Since codim it follows that is a regular embedding. ∎
Theorem 2.24**.**
Let . The pushforward of to is given by
[TABLE]
If (2.19) holds this also equals
[TABLE]
Proof.
By Proposition 2.5 and flat basechange around the diagram (2.21),
[TABLE]
Therefore Lemma 2.22 gives
[TABLE]
Pushing down gives
[TABLE]
by [Fu, Proposition 14.2.2].
To prove (2.25) we need to lift the surjection of (2.19) to a map ; see (2.27) below for how to do this after using the Jouanolou trick of Section 1.1. This then gives for all , which means that
[TABLE]
is a subbundle. Therefore its cokernel is a rank vector bundle over , equivalent in K-theory to . Thus we can apply [Fu, Example 14.4.12] to rewrite (2.26) as
[TABLE]
This formula should be compared with (2.16). ∎
Digression: description as a deepest degeneracy locus
As usual fix a smooth quasi-projective variety carrying a 2-term complex of bundles
[TABLE]
Almost by definition the virtual resolution of the th degeneracy locus can be seen as the deepest degeneracy locus of the complex (1.3) up on . Such deepest degeneracy loci (those for which is maximal — i.e. is empty) were studied in [GT1].
Since is not a quasi-isomorphism invariant we would again prefer to replace with (an affine bundle over) . For this we require (2.19) to hold, giving
[TABLE]
Using the Jouanolou trick described in Section 1.1 we may, on replacing by an affine bundle over it (and pulling everything back to it), assume that is a projective -module. Then we may pick a lift .
Over , with its universal quotient bundle (suppressing some pullback maps as usual) we consider the composition
[TABLE]
At a closed point (i.e. a point and an -dimensional subspace ) we have
[TABLE]
where in the last expression we have used to identify as a subspace of by (2.17). In particular is at most -dimensional — so is a deepest degeneracy locus — and the locus of points where it is precisely -dimensional is
[TABLE]
This bijection of sets generalises to -points of : given we get a bijection between the sets
[TABLE]
[TABLE]
The functors taking to either of these two sets are therefore isomorphic. The first is (2.4) and is represented by by Proposition 2.5. The second is represented by . Therefore (2.29) is an isomorphism of schemes.
Furthermore we can compare the perfect obstruction theory of Proposition 2.12 for to the deepest degeneracy locus perfect obstruction theory of [GT1, Theorem 3.6] for . In both cases the K-theory class of the virtual cotangent bundle is the restriction of
[TABLE]
so their virtual cycles agree. Finally, the Thom-Porteous formula for of Theorem 2.24 and the Thom-Porteous formula for of [GT1, Theorem 3.6] both give
[TABLE]
as the pushforward of the virtual cycle to .
Theorem 2.30**.**
Suppose (2.19) holds. Then after pulling back to an affine bundle, the embedding of (2.20) is the deepest degeneracy locus of the map of vector bundles (2.28),
[TABLE]
The two resulting virtual cycles and Thom-Porteous formulae agree. ∎
Generalised Carlsson-Okounkov vanishing
Proposition 2.31**.**
Under the assumptions of Theorem 2.30,
[TABLE]
Proof.
On an affine bundle over recall the map (2.28)
[TABLE]
of the last Section. By (2.19) it is an injective map of bundles and so is quasi-isomorphic to a rank vector bundle (its cokernel). Therefore its higher Chern classes vanish on the affine bundle, and so on by (1.16). ∎
Taking and (in the notation of Section 4) over the product of Hilbert schemes of points this recovers the original Carlsson-Okounkov vanishing of [CO] and [GT1, Corollary 8.1], at least when to ensure that is 2-term. The general surface can be handled by another splitting trick to remove from ; see [GT1, Section 8].
3. Comparison
Given a map of 2-term complexes we can try to compare the virtual cycles of Section 2. We begin with maps of a specific shape.
Proposition 3.1**.**
Suppose given a commutative diagram of vector bundles
[TABLE]
over a smooth quasi-projective variety , with a vector bundle injection and a surjection. Then there is an embedding
[TABLE]
such that the tautological bundle on pulls back to that on and
[TABLE]
Proof.
Recall that (and its perfect obstruction theory) is cut out of by the section of (2.7). Therefore its virtual cycle is given by the localised top Chern class of [Fu, Section 14.1],
[TABLE]
where is the zero section of and .
Now basechange by the embedding induced by the vector bundle injection . This is a regular embedding cut out by the canonical section of on .999This is a special case of Proposition 2.5 applied to the 2-term complex of bundles in place of . Therefore [Fu, Proposition 14.1(d)(ii)] gives
[TABLE]
Set . Suppressing pull back maps, on we have the exact sequence
[TABLE]
in which the section of the second bundle projects to in the third. Therefore (3.3) and Lemma 3.4 below give the identity
[TABLE]
where j\colon\widetilde{D}_{r}(F_{\bullet})\hookrightarrow u^{-1}\big{(}\widetilde{D}_{r}(E_{\bullet})\big{)} and . Pushing forward by i\colon u^{-1}\big{(}\widetilde{D}_{r}(E_{\bullet})\big{)}\hookrightarrow\widetilde{D}_{r}(E_{\bullet}) and using gives
[TABLE]
We then apply [Fu, Example 6.3.4] to the fibre square
[TABLE]
in which the bottom row is the regular embedding cut out by the canonical section of . The result is
[TABLE]
Substituting in K-theory gives the result. ∎
Lemma 3.4**.**
Let be an exact sequence of vector bundles over a quasi-projective variety . Given a section of we let be the induced section of . Then the resulting localised top Chern classes of and are related by the identity
[TABLE]
where is the embedding of zero schemes and .
Proof.
The total spaces of the bundles fit in the fibre square
[TABLE]
with both maps being flat. Letting denote the zero section of , the top Chern class of localised by is
[TABLE]
where denotes the refined intersection product of [Fu, Chapter 8], lying in the Chow group of the scheme-theoretic intersection .
So by the projection formula [Fu, Example 8.1.7],
[TABLE]
Now cap with the top Chern class of to get its zero section:
[TABLE]
But is the inclusion . ∎
Proposition 3.1 now implies the following comparison results.
Theorem 3.5**.**
Let be a smooth quasi-projective variety. Suppose
[TABLE]
is an exact triangle in with being 2-term complexes of vector bundles supported in degrees . If is quasi-isomorphic to a rank vector bundle then there is an embedding such that
[TABLE]
Similarly if is a rank vector bundle then and
[TABLE]
Proof.
In the first case fix a representative vector bundle and a locally free resolution of , sufficiently negative that the morphism in is represented by a genuine map of complexes. Then trim by removing and replacing by . Similarly remove and replace by . The upshot is a quasi-isomorphic 2-term complex of bundles with a map .
Then since we can realise the exact triangle (3.6) by the following short exact sequence of vertical complexes
[TABLE]
Therefore the map of 2-term complexes takes the form of (3.2) with and , and Proposition 3.1 now gives the result.
When is a vector bundle a similar argument represents the exact triangle as
[TABLE]
So now and are both cut out of by sections and of and respectively, as in (2.7). Since the former is a subbundle of the latter we find and the claimed formula is the excess intersection formula101010Equivalently, the obstruction sheaf of is an extension of by the obstruction sheaf of , from which the formula also follows. of [Fu, Theorem 6.3] applied to the fibre diagram
[TABLE]
with excess normal bundle \hskip 1.0pt\mathcal{U}^{*}\otimes E_{1}\big{/}\hskip 1.0pt\mathcal{U}^{*}\otimes F_{1}\cong\hskip 1.0pt\mathcal{U}^{*}\otimes H. ∎
4. Nested Hilbert schemes
A nested Hilbert scheme represents the functor which takes a base scheme to the set of families of ideals
[TABLE]
with each flat over and of fixed topological class on each fibre. By [Ko, Proof of Lemma 6.13] the double duals are locally free of the form for some Cartier divisor flat over . Therefore can be written as for some subscheme , flat over of relative dimension 0. Writing then (4.1) takes the form
[TABLE]
As we will see in Section 6 the Hilbert schemes of pure divisors have been heavily studied for their relationship to Seiberg-Witten theory; see [DKO] for instance. Taking the product with a (smooth) Hilbert scheme of points gives the Hilbert scheme of ideals , where length . So to understand nested Hilbert schemes what remains is to find a way to impose, one by one, the inclusions
[TABLE]
on these products of Hilbert schemes. Taking double duals implies is effective, so (4.3) is equivalent to the two inclusions
[TABLE]
It is sufficient to study the first, since the second is a special case of it. So we are reduced to studying the 2-step nested Hilbert scheme
[TABLE]
which represents the functor mapping to the set of families of ideals
[TABLE]
with flat over and such that on closed fibres we have and colength . The map to double duals recovers and , giving a classifying map to a smooth space
[TABLE]
Since (4.4) is the same data as a 1-dimensional subspace of we see that, at the level of points, (4.5) is the virtual resolution of the degeneracy locus of the complex over which restricts to at . To make a scheme theoretic statement we use
[TABLE]
with its universal ideal sheaves and a fixed choice of Poincaré line bundle111111We always normalise (by tensoring it by the pullback of if necessary) so that is trivial on some fixed basepoint . . Using basechange, the Nakayama lemma and the fact that for on any fibre of , we can trim
[TABLE]
to be a 3-term complex of locally free sheaves. Using its stupid truncation gives the following result, which will be modified later to give versions invariant under quasi-isomorphisms.
Proposition 4.7**.**
The nested Hilbert scheme (4.5) is the virtual resolution of the degeneracy locus of (4.6), giving isomorphisms
[TABLE]
under which the tautological bundle on corresponds to the dual of the tautological quotient bundle on .
Proof.
By the case of Proposition 2.5, represents the functor which maps an -scheme to the set of line subbundles over factoring through
[TABLE]
where is the basechange of . By adjunction this is equivalent to the set of line bundles and maps
[TABLE]
which are nonzero on restriction to any fibre . We need to show this is the same as the functor (4.4) represented by , i.e. that (4.8) is equivalent to the set of families
[TABLE]
with and flat over of the correct topological type on each fibre.
Firstly, taking double duals in (4.8) gives a map for some divisor which does not contain any fibres and so is flat over . This gives (4.9) as required. Conversely, given a family (4.9), note that is isomorphic to on each fibre, so may be written globally as , where is the line bundle . This gives (4.8), with the map nonzero on every fibre (since is flat over and so does not contain any fibre).
The second claimed isomorphism follows from the definition (2.3) of : it is the space of 1-dimensional quotients of the cokernel of ,
[TABLE]
by relative Serre duality down . ∎
4.1. Embedding and reduced virtual cycle.
Throughout this Section we require to be of Hodge type (1,1). Until now everything has been true (but vacuous) when this assumption fails, and the results deformation invariant. The constructions of this Section are only invariant under deformations of inside the Noether-Lefschetz locus in which is of type (1,1).
As discussed in the Introduction, each fibre of admits a natural embedding
[TABLE]
Here we have fixed, once and for all, cutting out a divisor sufficiently positive that
[TABLE]
(When and is effective we may take and the proof of Theorem 3 below reduces to a proof of Theorem 2 instead.) So we let be the vector bundle
[TABLE]
Then (4.10) gives the embedding of (1.4) at each point . The global version (2.19) is the composition of surjections
[TABLE]
By (2.20) we get an embedding
[TABLE]
Suppose now that, as in the statements of Theorems 2 and 3,
[TABLE]
The resulting surjection
[TABLE]
and basechange show that has over the image of in , and therefore also over a neighbourhood thereof. Therefore in (4.6) we can take to be a 2-term complex of vector bundles
[TABLE]
Proposition 4.7 then gives
[TABLE]
and Proposition 2.12 endows it with a perfect obstruction theory which we call the reduced obstruction theory.121212This is because (4.22) shows that at the level of virtual tangent bundles it is obtained from the Vafa-Witten perfect obstruction theory by removing a copy of .
Theorem 4.21**.**
Suppose for every effective divisor in class . The above construction gives a reduced obstruction theory with virtual tangent bundle
[TABLE]
in K-theory, and reduced virtual cycle
[TABLE]
where and .
If for all then the pushforward of (4.23) to is
[TABLE]
where for any .
Proof.
The virtual tangent bundle is described by the case of Proposition 2.12 with , giving the restriction of
[TABLE]
to . Since we derive (4.22) from
[TABLE]
When for all then is 2-term over the whole of , so we may apply Theorem 2.24 with to give the formula for the pushforward. ∎
Remark. This proves Theorems 2 and 3 from the Introduction. Since the virtual class depends only on the K-theory class of the virtual tangent bundle, Theorem 4.21 shows that \big{[}S^{[n_{1},n_{2}]}_{\beta}\big{]}^{\operatorname{red}} coincides with the reduced virtual cycles of [KT1] (when ) and [GSY1, Theorem 3] (when the latter is defined).
4.2. Comparison of reduced cycles.
Next we will compare the reduced cycles on and . Note that and are the virtual resolutions of the degeneracy loci of
[TABLE]
respectively. These are related by the obvious diagram
[TABLE]
with the vertical column an exact triangle. Here , so
[TABLE]
by relative Serre duality down . Thus is a vector bundle.
So we now use the Jouanolou trick of Section 1.1, pulling back to an affine bundle over , so that becomes a projective -module. In particular its connecting homomorphism to in the vertical exact triangle (4.24) is zero over the open neighbourhood (4.20) over which it is supported in degrees and . (Recall our assumption (4.19) that for every effective of class .) So over we may choose the splittings marked with dotted arrows in (4.24).
Lemma 4.25**.**
* is quasi-isomorphic to a rank vector bundle.*
Proof.
Taking of (4.24) at a closed point of gives
[TABLE]
Since this diagram commutes the diagonal arrow is an injection. Thus, by the Nakayama lemma, , and any basechange of it, has .
Taking instead gives part of the diagram
[TABLE]
Again this commutes and the rows are exact. Since surjects onto , in which lies, the diagonal arrow is surjective. Thus has and is a vector bundle as claimed. ∎
So it follows immediately from Theorem 3.5 and the Jouanolou trick that
[TABLE]
On the tautological composition gives the universal section
[TABLE]
In particular on . Thus represents the Carlsson-Okounkov K-theory class of (1.14):
[TABLE]
Hence we have proved the second formula in Theorem 5:
Corollary 4.27**.**
If for all effective divisors in class then
[TABLE]
**
4.3. Comparison of virtual cycles.
To recover the full perfect obstruction theory of [GSY1] or Vafa-Witten theory on from (the virtual resolution of) a degeneracy locus we would have to modify the 3-term complexes and by terms to make them 2-term. This cannot be done over .131313A referee suggested the following illustrative example. Let be the fibre class in an elliptic surface and take . Then and its Vafa-Witten virtual tangent bundle is easily worked out to be (using (4.40), for example). To see as of a complex over point we must take , but this has virtual tangent bundle .
However, up on (the product of with) we can use the universal section (4.26) of — notice the twist! — together with a choice of splitting , to induce maps
[TABLE]
The lower row is an exact triangle. We apply the Jouanolou trick of Section 1.1, pulling back from to an affine bundle over it. Here becomes a projective -module, so the right hand diagonal arrow becomes zero. Thus we can fill in the dotted arrow, giving a commutative diagram
[TABLE]
Here the right hand column is an exact triangle, with a vector bundle Serre dual to
[TABLE]
On second cohomology sheaves we claim the diagram gives surjections
[TABLE]
The claim for the maps follows from zeros in the obvious exact sequences in which each sits. It then follows from the diagram that and are also onto. So if we use the notation
[TABLE]
for any of the maps or in (4.28), then each has and can be represented by a 2-term complex of vector bundles. The bottom row and right hand column of (4.28) now give the diagram
[TABLE]
which should be compared to (4.24). Again the column is an exact triangle, and again is a projective -module on the affine bundle so has vanishing connecting homomorphism to the complex \tau\big{(}R\pi_{*}\hskip 0.6pt\mathcal{L}_{\beta}(1)\big{)}[2] supported in degrees and . Thus we may split the vertical exact triangle and fill in the dotted arrows on .
Extension to ambient space. Let be the bundle (4.11). Applying (4.18) when gives an embedding and so
[TABLE]
where as in (4.5).
We apply the Jouanolou trick to , giving an affine bundle over it whose total space is an affine variety. The restriction of this affine bundle to is therefore also an affine variety, to which the constructions of the last Section 4.3 apply. As usual we suppress pullbacks to these affine bundles.
All of the terms in (4.28) extend to (the affine bundle over) , as do the horizontal and vertical arrows (canonically). As for the diagonal arrows, their targets are quasi-isomorphic to complexes of vector bundles , and is a projective -module on an affine variety, so the arrows can be represented by maps . Again since is a projective -module these maps can be extended to (the affine bundle over) . So we get an extension of the diagram (4.30) to (the affine bundle over) .
Since the surjectivity (4.29) is an open condition, all the terms of this extended diagram can be represented by 2-term complexes of vector bundles over a neighbourhood of (the affine bundle over) inside (the affine bundle over) . By shrinking if necessary, we may assume that it is affine.141414Let be an open neighbourhood of a closed subvariety of an affine variety . Then is disjoint from so we get a surjection . Let map to . Then is an affine neighbourhood of .
So finally the splitting map H\to\tau\big{(}R\curly H\!om_{\pi}(\mathcal{I}_{1},\mathcal{L}_{\beta}(1))\big{)} of the vertical exact sequence in (4.30) extends from (the affine bundle over) to its affine neighbourhood since is a projective -module (and hence has vanishing connecting homomorphism to the complex \tau\big{(}R\pi_{*}\hskip 0.6pt\mathcal{L}_{\beta}(1)\big{)}[2] supported in degrees and ). So we can fill in the dotted arrows, giving
[TABLE]
[TABLE]
Lemma 4.25 now applies verbatim to the map of (4.30) to show that
[TABLE]
Next we describe a nice form of these complexes on the open set .
Lemma 4.33**.**
Over each of the terms in (4.31) has a 2-term resolution by vector bundles
[TABLE]
which, on dualising, induces the surjection B^{*}\to\to h^{0}\big{(}(\tau(\mathcal{F})(-1))^{\vee}\big{)} of (4.17). Moreover the arrows in (4.31) may be taken to be genuine maps of complexes between these resolutions.
Proof.
The final claim follows from the fact that on any affine variety like , maps in the derived category between complexes of locally free sheaves are quasi-isomorphic to genuine maps of complexes.
We start by working downstairs on . The composition
[TABLE]
induces the inclusion on restricting to any point and taking . Therefore its cone has cohomology in degrees after basechange to any point, so it can be represented by a 2-term complex of vector bundles , meaning we can write
[TABLE]
By construction this resolution dualises to induce the surjection of (4.17).
Now pullback to (the affine bundle over) , twist by and use the map of (4.28) (which we already noted extends to the affine bundle over ). It can be represented by a genuine map of complexes, so we can represent its cone as
[TABLE]
over the affine open on which the second arrow is a surjection. This gives the result for .
For \mathcal{F}=\tau\big{(}R\curly H\!om_{\pi}(\mathcal{I}_{1},\mathcal{I}_{2}\otimes\mathcal{L}_{\beta}(1))\big{)} we can now write the exact triangle defined by (4.31) as
[TABLE]
with the last arrow a genuine map of complexes of vector bundles on . Thus the first term can be written as the 2-term complex of vector bundles
[TABLE]
A similar argument applies to \tau\big{(}R\curly H\!om_{\pi}(\mathcal{I}_{1},\mathcal{L}_{\beta}(1))\big{)}. ∎
So we can now see (the affine bundle over) as the zeros in of the section of defined by the composition
[TABLE]
where is the representative (4.34) of \tau\big{(}R\curly H\!om_{\pi}(\mathcal{I}_{1},\mathcal{I}_{2}\otimes\mathcal{L}_{\beta})\big{)}. For the confused reader we summarise the construction.
Summary. By Proposition 4.7, is the virtual resolution of the degeneracy locus of the complex over . Since this is 3-term in general this does not give it a natural virtual cycle, but it does express it as the zero locus in of the composition161616Changing , for instance to when this is locally free, gives the same zero locus but a different perfect obstruction theory.
[TABLE]
We cannot modify by an term on to get a 2-term degeneracy locus construction directly. But we can after pulling back to and twisting by , replacing (4.37) by
[TABLE]
on the open set where the last arrow is a surjection. This is (4.36).
Theorem 4.39**.**
* is the zero locus of (4.36). The virtual cycle of the resulting perfect obstruction theory agrees with the non-reduced virtual cycles of [GSY1, TT1] whenever they are defined.171717The Vafa-Witten moduli space of [TT1] depends on a stability condition. But even in the unstable case, where we get a stack with possibly nontrivial stabilisers, we can pass to the -fixed locus and cut down the automorphisms to those which commute with the action. The nested Hilbert scheme then appears as a connected component, with the -fixed part of the Vafa-Witten obstruction theory [TT1] defining a perfect obstruction theory on it which recovers those of [GSY1] and Theorem 4.39. It satisfies*
[TABLE]
under push forward by .
Proof.
Since \tau\big{(}R\curly H\!om_{\pi}(\mathcal{I}_{1},\mathcal{I}_{2}\otimes\mathcal{L}_{\beta})\big{)} differs from only in (even after any basechange), is unaffected. In other words, the zero loci of (4.36) and (4.37) are the same — and the latter is by Proposition 4.7.
The virtual tangent bundle differs from the one in Theorem 4.21 by only , so its K-theory class is
[TABLE]
agreeing with the ones from [GSY1, TT1]. Since virtual classes depend only on the K-theory class of the virtual tangent bundle, this makes the virtual classes of [GSY1, TT1] equal to the localised top Chern class
[TABLE]
of the bundle and its section defined by (4.36).
Setting we get
[TABLE]
where is the resolution of \tau\big{(}R\pi_{*}\hskip 0.6pt\mathcal{L}_{\beta}\big{)} on the open subset of (the affine bundle over) which is the image of (in the affine bundle over ), and is the induced section of . Pulling back to gives
[TABLE]
where as usual we have supressed the pull back map on and its section . Now the map of (4.31) induces, via Lemma 4.33, a map
[TABLE]
such that . By (4.32) is a surjection with kernel the vector bundle , so applying Lemma 3.4 and using (4.42) gives
[TABLE]
Remark. We have now proved Theorem 5 from the Introduction. When and for all effective in class — so the reduced class is defined — the virtual cycle is automatically zero by the identity
[TABLE]
of Theorem 3.5.
5. -step nested Hilbert schemes.
Fix curve classes and integers . As a set the -step nested Hilbert scheme is
[TABLE]
As a scheme it represents the functor described in (4.2) (with ). That description immediately implies that it can be written as the fibre product
[TABLE]
It follows that is the intersection of the virtual resolutions of the degeneracy loci of the complexes
[TABLE]
Here while is the projection and are (the pullbacks of) Poincaré line bundles and the universal ideal sheaves respectively.
Taking to be a divisor which is sufficiently positive that
[TABLE]
we get vector bundles
[TABLE]
and an embedding
[TABLE]
Suppose for every effective of class . Then each of the complexes (5.1) can be represented by a 2-term complex of vector bundles . So we can mimic the constructions of Section 2 and 4. We replace the projective bundles and by the fibre products and , and cut out the -step nested Hilbert scheme by the obvious section of , where . The upshot is a (multiply reduced) perfect obstruction theory on with virtual tangent bundle
[TABLE]
in K-theory, and reduced virtual class
[TABLE]
Theorem 5.4**.**
If for all and all then the push forward of the reduced cycle (5.3) by the embedding (5.2) is
[TABLE]
*where and denotes the pullback of . *
For general classes we follow the procedure of Section 4.3 to modify each of the complexes (5.1) by after pulling back to (the fibre product of the) and tensoring by . The result is a perfect obstruction theory with virtual tangent bundle
[TABLE]
in K-theory, and virtual cycle
[TABLE]
Just as in (4.43) this vanishes if and for all for some .
Iterating the construction of Section 4.3 we get the following comparison result, a direct generalisation of Theorem 5.
Theorem 5.6**.**
The pushforward of \big{[}S^{[n_{1},\dots,n_{\ell}]}_{\beta_{1},\dots,\beta_{\ell-1}}\big{]}^{\operatorname{vir}} by the embedding
[TABLE]
is given by the formula
[TABLE]
Moreover, if for all effective then the same formula holds with all virtual classes replaced by reduced classes .
Remark. Since (5.5) matches the virtual tangent bundle in [GSY1], the virtual class \big{[}S^{[n_{1},\dots,n_{\ell}]}_{\beta_{1},\dots,\beta_{\ell-1}}\big{]}^{\operatorname{vir}} is the same class that arises in higher rank Vafa-Witten theory [TT1] or the reduced local DT theory of [GSY2].
6. Computations
Vafa-Witten invariants. On a polarised surface , Vafa-Witten theory is an enumerative sheaf-counting theory [TT1]. The “instanton contributions” compute virtual Euler numbers of moduli spaces of Gieseker stable sheaves of fixed rank and determinant on , as studied by Göttsche-Kool [GK].
This leaves the “monopole contributions”, which are integrals over moduli spaces of chains of sheaves (of total rank ) with nonzero maps between them. When the individual sheaves have rank 1, we get nested ideal sheaves (all tensored by a line bundle). And when is prime and , a vanishing theorem [Th, Section 5.2] implies that all nonzero contributions come from nested Hilbert schemes. A similar result applies in the semistable case [TT2] and the refined version of both theories [Th].
The simplest case (for simplicity) is the monopole contribution to the Vafa-Witten invariants in rank 2 and fixed determinant when slope stability is the same as slope semistability. Here we get integrals over the virtual cycles of nested Hilbert schemes of pairs for any satisfying the slope stability181818Gieseker stability is slightly more subtle, with a further inequality on the length in the strictly slope semistable case. Laarakker [La1, Proposition 3.5] has proved Gieseker unstable Higgs pairs have so we can sum over all , independent of stability. condition . We then sum over all giving the same total Chern classes, and then multiply by a factor191919Thanks to Ties Laarakker for pointing out the factor in total rank . This counts the -torsion line bundles we can twist by to get another Higgs pair with the same fixed determinant. of .
The integrand is , where is the moving part of the Vafa-Witten obstruction theory and is the -equivariant Euler class. So by [TT1, Section 8] we want to compute the integral of
[TABLE]
over \big{[}S_{\beta}^{[n_{1},n_{2}]}\big{]}^{\operatorname{vir}}. Write as as in (4.26). Let on and its subschemes . Then each term takes the form
[TABLE]
where the have only nonzero weight spaces (so the quotient makes sense in localised equivariant cohomology) and ranks . Therefore, expanding in powers of with coefficients pulled back from , we get an expression
[TABLE]
where is part of the commutative diagram
[TABLE]
Here is the Abel-Jacobi map. So we should calculate the pushdown \rho_{*}\big{(}h^{i}\cap\big{[}S_{\beta}^{[n_{1},n_{2}]}\big{]}^{\operatorname{vir}}\big{)}. This will prove Theorems 6, 7 and 8 from the Introduction, and compute Vafa-Witten invariants in terms of integrals over the smooth spaces .
Let
[TABLE]
denote the Seiberg-Witten invariant [DKO, CK] of in class . Things are easiest when , i.e. .
Theorem 6.2**.**
Suppose . Then in H_{*}\big{(}S^{[n_{1}]}\times S^{[n_{2}]}\times\operatorname{Pic}_{\beta}(S)\big{)},
[TABLE]
*where so is the generator of . *
Proof.
If then , so by our comparison result Theorem 4.39 (or Theorem 5 in the Introduction) both sides of the claimed identity are zero. So we assume that .
By [DKO, Definition 3.19] this means that is a basic class. And by [DKO, Proposition 3.20] is of simple type, which implies that .
Thus is a 0-dimensional class. Firstly, this implies for , and so the second claimed result. And secondly, on restriction to
[TABLE]
the line bundle is topologically equivalent to (the pull back of) for any . In particular in the formula of Theorem 4.39,
[TABLE]
we may replace (1.14) by . Applying then gives
[TABLE]
where point is the generator of . ∎
Now we suppose , i.e. . Then may not be simple type, so we have to consider higher dimensional Seiberg-Witten moduli spaces and the higher Seiberg-Witten invariants [DKO]
[TABLE]
Theorem 6.4**.**
If then in ,
[TABLE]
Proof.
We again use our comparison formula (6.3). By [Ma, Proposition 1]202020Manivel extends the formula from rank bundles to rank perfect complexes. In fact we showed in (4.32) that has the K-theory class of a vector bundle on (an affine bundle over) anyway.
[TABLE]
So (\operatorname{id}\times\operatorname{id}\times\,\mathsf{AJ})_{*}\big{(}h^{i}\,\cap (6.3)\big{)}=\rho_{*}\big{(}h^{i}\cap\big{[}S^{[n_{1},n_{2}]}_{\beta}\big{]}^{\operatorname{vir}}\big{)} is
[TABLE]
We can get simpler formulae by splitting into the cases that for all or not. By Serre duality this is the condition that is not effective or effective, respectively.
Theorem 6.5**.**
Suppose . Fix and . Then the following results hold in .
* If is effective,*
[TABLE]
and both sides vanish when . (Cf. Theorem 6.2.)
* If is not effective,*
[TABLE]
with .
Proof.
By the comparison result (6.3) both sides of the first identity vanish if or is not effective. So we may assume both are effective and . By [DKO, Corollary 3.15] this implies and .
So, just as in (6.3), is trivial on the 0-dimensional and we may replace in (6.3) by for one . So applying to (6.3) gives
[TABLE]
This deals with . When the left hand side of the required identity vanishes by and (6.3), while the right hand side vanishes by the generalised Carlsson-Okounkov vanishing of [GT1, Theorem 3].
For the second result we may assume that is type (1,1), otherwise is empty and both sides of the identity vanish. Since we also have for all the reduced cycle of Section 4.1 is defined, and — since — it equals the virtual cycle. Therefore by Theorem 4.21 (or Theorem 3 of the Introduction) the pushforward of the virtual cycle to is given by
[TABLE]
again by [Ma, Proposition 1]. Here .
We now cup with and push down (which restricts to the map of (6.1)). Using the fact that is the Segre class for , and zero otherwise, we find that \rho_{*}\big{(}h^{i}\cap\big{[}S^{[n_{1},n_{2}]}_{\beta}\big{]}^{\operatorname{vir}}\big{)} is
[TABLE]
[TABLE]
Finally we use the standard duality in Seiberg-Witten theory under to give interesting dualities between invariants of nested Hilbert schemes under and . Define the map
[TABLE]
by replacing in and then composing with .
Theorem 6.6** (Duality).**
In ,
[TABLE]
where .
Proof.
By Theorem 6.2 with and ,
[TABLE]
for a fixed . For Seiberg-Witten invariants have the standard duality
[TABLE]
(For instance, in [DKO] this is part of Conjecture 0.1, and it is then shown this would follow from proving on all minimal general type surfaces. This latter identity was proved in [CK].) And
[TABLE]
by Serre duality down the fibres of . This gives the first identity for . For it is trivial by Theorem 6.2.
To prove the second identity we distinguish 3 cases. If is not effective, i.e. for all , then the result is the last part of Theorem 6.5. Similarly if is not effective then we apply the last part of Theorem 6.5 with . By and Serre duality down the fibres of ,
[TABLE]
which gives the required result.
Finally we consider and both effective. If then the left hand side of the identity is zero by the comparison result Theorem 5. On the right hand side, we have
[TABLE]
The first term of the sum vanishes for , while the second term vanishes for by the generalised Carlsson-Okounkov vanishing of [GT1, Theorem 3]. When this makes the whole sum vanish.
So we may assume that and are both effective. This implies and by [DKO, Corollary 3.15].
So, just as in (6.3), is trivial on the 0-dimensional and we may replace in (6.3) by the pull back of . Thus pushing (6.3) down by gives \rho_{*}\big{[}S^{[n_{1},n_{2}]}_{\beta}\big{]}^{\operatorname{vir}} as
[TABLE]
Swapping and shows \rho^{\vee}_{*}\big{[}S^{[n_{2},n_{1}]}_{\beta^{\vee}}\big{]}^{\operatorname{vir}} is
[TABLE]
By Serre duality down and the identity , this is
[TABLE]
We have noted that , so . Thus, identifying via , [DKO, Theorem 3.16] gives
[TABLE]
Since is 1-dimensional, . And by the generalised Carlsson-Okounkov vanishing of [GT1, Theorem 3]. Therefore
[TABLE]
Substituting this into (6.8) and comparing with (6.7) yields
[TABLE]
Multiplying by and noting then gives the required result for .
For the restriction of to both and \big{[}S^{[n_{1},n_{2}]}_{\beta}\big{]}^{\operatorname{vir}} are trivial (the former by , the latter by the comparison result Theorem 5). ∎
Theorem 6.9**.**
If the nested Hilbert scheme contributions to the Vafa-Witten invariants are invariants of ’s oriented diffeomorphism type.
Proof.
We sketch the argument for 2-step nested Hilbert schemes; the general case is no more complicated. Handling stability as in Footnote 18, we express the invariants as a sum over with of terms
[TABLE]
Let be any line bundle with . Applying Theorem 6.2 gives
[TABLE]
where the integrand is
[TABLE]
The coefficient is proved in [DKO, CK] to be a Seiberg-Witten invariant, which is an oriented diffeomorphism invariant when .212121When this need not be quite true; only the unordered pair is invariant under oriented diffeomorphisms.
We would now like to apply the inductive method of [EGL], which deals with integrals over single Hilbert schemes . This is adapted in [GNY, Section 5]222222We thank a referee for suggesting this method and pointing out the reference [GNY]. to deal with products of Hilbert schemes , essentially by applying [EGL] to the disjoint union and its Hilbert schemes
[TABLE]
Let and denote the ideal sheaves on whose restrictions to are and respectively. Then (6.10) can be rewritten
[TABLE]
The induction writes this as an integral over , then and so on, finally giving an integral over . The first step pulls back along the generically finite map , divides by , then pushes down , in the diagram
[TABLE]
We split into two (unions of) connected components
[TABLE]
and restrict the diagram to each in turn; we describe the first. On each of its components is projection down the cone — where is the universal subscheme — multiplied by the identity on . As such it carries a line bundle ; it is the kernel of in the obvious notation.
By [GNY, Equation 5.4] and [EGL, Equation 10] the integrand pulls back to a rational function in and the Chern classes of
- (1)
pulled back from , where is a tensor product of powers of and , 2. (2)
pulled back from , 3. (3)
and pulled back from , and 4. (4)
the line bundle .
Taking the coefficient of gives a polynomial in these terms. Since (1)-(3) are pulled back from we can push down by using the projection formula and [EGL, Lemma 1.1] to replace powers of by polynomials in the Chern classes of and . This completes the first step.
The second (and later) steps are similar, with (6.11) replaced by
[TABLE]
The pull back and push down handles classes pulled back from either factor of just as before. The new issue is to deal with Chern classes of and on . Here we use [GNY, Equation 5.2] to express them as pull backs from . So the induction continues, just as in [EGL, Proposition 3.1].
The final result is an integral over which can be written as a polynomial in the numbers and . See also [GNY, Lemma 5.5] and [La1, Proposition 7.2], or [KT2, Section 3] when .
Since and the intersection form are oriented diffeomorphism invariants of , so are . This leaves , but — as an observant referee pointed out — we may assume this equals ; otherwise so and the integral (6.10) vanishes. ∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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