A Bochner principle and its applications to Fujiki class $\mathcal C$ manifolds with vanishing first Chern class

TL;DR

This paper establishes a Bochner vanishing theorem for certain Fujiki class C manifolds with vanishing first Chern class, leading to results on the local homogeneity of affine geometric structures and properties of holomorphic Riemannian metrics.

Contribution

It introduces a Bochner type vanishing theorem for Fujiki class C manifolds with nef classes and applies it to analyze geometric structures and Riemannian metrics on these manifolds.

Findings

Holomorphic geometric structures are locally homogeneous on a Zariski open subset.

Rigid geometric structures imply an infinite fundamental group.

Manifolds with holomorphic Riemannian metrics have finite covers by complex tori.

Abstract

We prove a Bochner type vanishing theorem for compact complex manifolds $Y$ in Fujiki class $\mathcal C$, with vanishing first Chern class, that admit a cohomology class $[\alpha] \in H^{1,1}(Y,\mathbb R)$ which is numerically effective (nef) and has positive self-intersection (meaning $\int_Y \alpha^n \,>\, 0$, where $n\,=\,\dim_{\mathbb C} Y$). Using it, we prove that all holomorphic geometric structures of affine type on such a manifold $Y$ are locally homogeneous on a non-empty Zariski open subset. Consequently, if the geometric structure is rigid in the sense of Gromov, then the fundamental group of $Y$ must be infinite. In the particular case where the geometric structure is a holomorphic Riemannian metric, we show that the manifold $Y$ admits a finite unramified cover by a complex torus with the property that the pulled back holomorphic Riemannian metric on the torus is translation invariant.

Equations

$$H_{\partial \overline{\partial }}^{1,1}(X)=\{\alpha \in {\mathcal{C}}^{\infty }(X,{\Omega }_{X,\mathbb{C}}^{1,1})\mid d\alpha =0\}/\{\partial \overline{\partial }u\mid u\in {\mathcal{C}}^{\infty }(X,\mathbb{C})\}.$$

$$H_{\partial \overline{\partial }}^{1,1}(X,\mathbb{R}):=H_{\partial \overline{\partial }}^{1,1}(X)\cap H^2(X,\mathbb{R}).$$

$$f^{\ast }{\omega }_{\epsilon }⩾-\epsilon f^{\ast }{\omega }_Y⩾-\epsilon {\omega }_X.$$

$${\int }_X(f^{\ast }\alpha {)}^n={\int }_Y{\alpha }^n>0.$$

$$f:X⟶Y$$

$$\phi =a\log (|f_1{|}^2+\dots +|f_r{|}^2)+\psi .$$

$$E_{nK}([\alpha ]):=\bigcap _{T\in [\alpha ]}{E}_{+}(T),$$

$$\mathrm{Amp}([\alpha ]):=X\setminus E_{nK}([\alpha ]).$$

$$E_{nK}(f^{\ast }[\alpha ])=f^{-1}(E_{nK}([\alpha ]))\cup \mathrm{Exc}(f),$$

$$\tilde{V}\subset E_{nK}(f^{\ast }[\alpha ]).$$

$$f{|}_W:W⟶f(W),$$

$$f:X⟶Y.$$

$$K_X=f^{\ast }{K}_Y+E,$$

$$E=\sum _{i=1}^r{a}_i{E}_i$$

$$T\in f^{\ast }[\alpha ]$$

$$-\mathrm{Ric}(T)=[E].$$

$$⟨T^n⟩=|s_E{|}_h^{2}dV,$$

$$Z\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\varnothing \ne f(\mathrm{Amp}(f^{\ast }[\alpha ]))\subseteq \Omega \subseteq \mathrm{Amp}([\alpha ])$$

$$f(\mathrm{Amp}(f^{\ast }[\alpha ]))\subseteq \mathrm{Amp}([\alpha ]),$$

$$\mathrm{Ric}({\omega }_{t,\epsilon })=t{\omega }_X-{\theta }_{\epsilon },$$

$$\theta_{\varepsilon}\,=\,\sum_{i=1}^{r}a_{i}\Big{(}\Theta_{h_{i}}(E_{i})+dd^{c}\log(|s_{i}|^{2}_{h_{i}}+\varepsilon^{2})\Big{)}\,,$$

$$\begin{array}{cccc}{\mathrm{ev}}_y:& H^0(Y,T_Y^{\otimes p}\otimes {T_Y^{\ast }}^{\otimes q})& ⟶& (T_Y^{\otimes p}\otimes {T_Y^{\ast }}^{\otimes q}{)}_y\\ & \tau & ⟼& \tau (y)\end{array}$$

$$\hat{X}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\hat{f}}^{\ast }{\pi }^{\ast }\tau {|}_{{\Omega }^{\prime }}={\hat{\pi }}^{\ast }{f}^{\ast }\tau {|}_{{\Omega }^{\prime }}.$$

$$d{d}^c\log (|\sigma {|}^2+1)=\frac{1}{|\sigma {|}^2+1}\left(|D^{\prime }\sigma {|}^2-\frac{|⟨D^{\prime }\sigma ,\sigma ⟩{|}^2}{|\sigma {|}^2+1}-⟨{\Theta }_h(\mathcal{E})\sigma ,\sigma ⟩\right).$$

$${\int }_X\frac{⟨{\Theta }_h(\mathcal{E})\sigma ,\sigma ⟩}{|\sigma {|}^2+1}\wedge {\omega }_{t,\epsilon }^{n-1}={\int }_X\frac{1}{|\sigma {|}^2+1}\left(|D^{\prime }\sigma {|}^2-\frac{|⟨D^{\prime }\sigma ,\sigma ⟩{|}^2}{|\sigma {|}^2+1}\right)\wedge {\omega }_{t,\epsilon }^{n-1}.$$

$${\int }_X\frac{⟨{\Theta }_h(\mathcal{E})\sigma ,\sigma ⟩}{|\sigma {|}^2+1}\wedge {\omega }_{t,\epsilon }^{n-1}⩾{\int }_X\frac{|D^{\prime }\sigma {|}^2}{(|\sigma {|}^2+1{)}^2}\wedge {\omega }_{t,\epsilon }^{n-1}.$$

$$t^{\otimes p}(v_1\otimes \cdots \otimes v_p):=\sum _{i=1}^p{v}_1\otimes \cdots \otimes v_{i-1}\otimes t(v_i)\otimes v_{i+1}\otimes \cdots \otimes v_p.$$

$$n{\Theta }_h(\mathcal{E})\wedge {\omega }_{t,\epsilon }^{n-1}=-(♯\mathrm{Ric}\omega {)}^{\otimes p}{\omega }^n,$$

Full text

1. 1 Introduction

2. 2 Fujiki class $\mathcal{C}$ manifolds and Bochner principle

3. 2.1 Positivity property of $(1,1)$-classes.

4. 2.2 Non-Kähler locus of a cohomology class

5. 2.3 Singular Ricci-flat metrics

6. 2.4 Flatness of tensors

7. 3 Geometric structures on manifolds in class $\mathcal{C}$

8. 3.1 Holomorphic geometric structures

9. 3.2 A criterion for local homogeneity

10. 3.3 A non-affine type example

11. 3.4 Holomorphic Riemannian metric

1. Introduction

Yau’s celebrated theorem [Ya], proving Calabi’s conjecture, endows any compact Kähler manifold $X$ with vanishing real first Chern class (meaning $c_{1}(TX)\,=\,0$ in $H^{2}(X,\,\mathbb{R})$) with a Ricci flat Kähler metric. Such metrics constitute an extremely useful tool for studying the geometry of these manifolds, known as Calabi-Yau manifolds. For example, by the well-known Bochner principle, any holomorphic tensor on $X$ must be parallel with respect to any Ricci flat Kähler metric [Be]. The study of the holonomy of such a Ricci flat Kähler metric furnishes, in particular, an elegant proof of the Beauville–Bogomolov decomposition theorem that asserts that, up to a finite unramified cover, a Calabi-Yau manifold $X$ is biholomorphic to the product of a complex torus with a compact complex simply connected manifold with trivial first Chern class [Be, Bog].

In the special case where the second real Chern class of the Calabi-Yau manifold $X$ also vanishes, any Ricci flat Kähler metric on $X$ has vanishing sectional curvature. In that case, as a consequence of Bieberbach’s theorem, $X$ actually admits a finite unramified cover which is a complex torus [Be]. Notice that a compact Kähler manifold bearing a holomorphic affine connection on its holomorphic tangent bundle has vanishing real Chern classes [At, p. 192–193, Theorem 4], and hence it admits an unramified cover by a compact complex torus [IKO].

Using the Bochner principle, it was proved in [Du1] that holomorphic geometric structures of affine type (their definition is recalled in the paragraph following Definition 3.1) on any compact Kähler manifold $X$ with vanishing first Chern class are in fact locally homogeneous. Consequently, if the geometric structure satisfies the condition that it is rigid in the sense of Gromov, [DG, Gr], then $X$ admits a finite unramified cover which is a complex torus.

The aim in this paper is to generalize the above mentioned results to the broader context of Fujiki class $\mathcal{C}$ manifolds [Fu2, Fu1]. Recall that a compact complex manifold $Y$ is in Fujiki class $\mathcal{C}$ if it is the image of a holomorphic map from a compact Kähler manifold. By an important result of Varouchas [Va], this is equivalent to the assertion that $Y$ admits a surjective holomorphic map from a compact Kähler manifold such that the map is a bimeromorphism.

Let $Y$ be a compact connected complex manifold, of complex dimension $n$, with trivial first (real) Chern class. We assume that $Y$

• •

lies in Fujiki class $\mathcal{C}$, and

• •

admits a numerically effective (nef) cohomology class $[\alpha]\,\in\,H^{1,1}(Y,\,\mathbb{R})$ that has positive self-intersection, meaning $\int_{Y}\alpha^{n}\,>\,0$ with $\alpha$ being a real closed smooth $(1,1)$–form representing the cohomology class $[\alpha]$.

By a result of Demailly and Păun in [DP], the combination of the two conditions above is equivalent to the condition that $Y$ admits a cohomology class $[\alpha]\,\in\,H^{1,1}_{\partial\overline{\partial}}(Y,\,\mathbb{R})$ which is both nef and big (see Section 2.1, e.g. Remark 2.5 and Corollary 2.6).

Under the above assumptions we prove the following Bochner type theorem for holomorphic tensors on $Y$ (see Corollary 2.17).

Theorem A**.**

Let $Y$ be a complex compact manifold with trivial first Chern class. Assume that $Y$ admits a nef and big cohomology class $[\alpha]\in H^{1,1}_{\partial\overline{\partial}}(Y,\,\mathbb{R})$.

Then, there exists a closed, positive $(1,1)$-current $\omega\in[\alpha]$ which induces a genuine Ricci-flat Kähler metric on a non-empty Zariski open subset $\Omega\subset Y$.

Furthermore, given any global holomorphic tensor $\tau\in H^{0}(Y,\,{T_{Y}}^{\otimes p}\otimes{T_{Y}^{*}}^{\otimes q})$ with $p,q\,\geq\,0$, the restriction $\tau|_{\Omega}$ is parallel with respect to $\omega|_{\Omega}$.

The existence of the singular Ricci-flat metric $\omega$ is essentially due to [BEGZ]; see Section 2.3. The statement about parallelism is new, although the techniques involved in its proof are mostly borrowed from [Gue] (see also [CP1]). Using this Bochner principle, we deduce the following result (see Theorem 3.4).

Theorem B**.**

Under the assumptions of Theorem A, the following holds:

1. (1)

There exists Zariski open subset $\emptyset\neq\Omega\subset Y$ such that any holomorphic geometric structure of affine type on $Y$ is locally homogeneous on $\Omega$. 2. (2)

If $Y$ admits a rigid holomorphic geometric structure of affine type, then the fundamental group of $Y$ is infinite.

For the particular case of a holomorphic Riemannian metric, we deduce using the above mentioned Bochner principle the following result (see Theorem 3.5).

Theorem C**.**

Under the assumptions of Theorem A, assume additionally that $Y$ admits a holomorphic Riemannian metric $g$.

Then, there is a finite unramified cover $\gamma\,:\,{\mathbb{T}}\,\longrightarrow\,Y$, where $\mathbb{T}$ is a complex torus, such that the pulled back holomorphic Riemannian metric $\gamma^{*}g$ on the torus ${\mathbb{T}}$ is translation invariant.

The Zariski open subset $\Omega$ involved in Theorem B can be chosen to contain the smooth locus of the singular Ricci-flat metric $\omega$ from Theorem A. We would actually conjecture that we could choose $\Omega=Y$. This was proved to be true in [BD] for Moishezon manifolds (meaning manifolds that are bimeromorphic to some complex projective manifold [Mo]) using deep positivity properties proved recently in [CP2] for tensor powers of the cotangent bundle of projective manifolds with pseudoeffective canonical class (see also the expository article [Cl]). However it is still not known whether this holds in the set-up of Kähler manifolds. It should also be mentioned that it was proved in [BD] that a compact simply connected manifold in Fujiki class $\mathcal{C}$ does not admit any holomorphic affine connection on its holomorphic tangent bundle. A compact complex manifold in Fujiki class $\mathcal{C}$ bearing a holomorphic affine connection has in fact trivial real Chern classes (its proof is identical to the proof in the Kähler case [At]). We would conjecture that a compact complex manifold in Fujiki class $\mathcal{C}$ with trivial Chern classes (in the real cohomology) admits a finite unramified cover which is a complex torus.

2. Fujiki class $\mathcal{C}$ manifolds and Bochner principle

2.1. Positivity property of $(1,1)$-classes.

Let $X$ be a compact complex manifold of dimension $n$. We set $d^{c}\,:=\,\frac{\sqrt{-1}}{2\pi}(\overline{\partial}-\partial)$ the real operator so that $dd^{c}\,=\,\frac{\sqrt{-1}}{\pi}\partial\overline{\partial}$. We shall denote by $H^{1,1}_{\partial\overline{\partial}}(X)$ the $\partial\overline{\partial}$–cohomology, in other words,

[TABLE]

Moreover, set

[TABLE]

Definition 2.1**.**

Let $X$ be a compact complex manifold of dimension $n$, and let $[\alpha]\,\in\,H^{1,1}_{\partial\overline{\partial}}(X,\,{\mathbb{R}})$ be a cohomology class represented by a smooth, closed $(1,1)$-form $\alpha$. We recall the standard terminology:

1. (1)

$[\alpha]$ is nef if for any $\varepsilon>0$, there exists a smooth representative $\omega_{\varepsilon}\,\in\,[\alpha]$ such that $\omega_{\varepsilon}\,\geqslant\,-\varepsilon\omega_{X}$, where $\omega_{X}$ is some fixed (independent of $\varepsilon$) hermitian metric on $X$. 2. (2)

$[\alpha]$ has positive self-intersection if $\int_{X}\alpha^{n}\,>\,0$. 3. (3)

$[\alpha]$ is big if there exists a Kähler current $T\,\in\,[\alpha]$, i.e., if there exists a closed $(1,1)$-current $T\,=\,\alpha+dd^{c}u\,\in\,[\alpha]$ such that $T\,\geqslant\,\omega_{X}$ in the sense of currents, where $u\in L^{1}(X)$ and $\omega_{X}$ is some hermitian metric on $X$.

If $X$ is a compact Kähler manifold or more generally a Fujiki manifold (see Definition 2.4), then the $\partial\overline{\partial}$–cohomology coincides with the usual $\overline{\partial}$–cohomology (also called Dolbeault cohomology). Moreover, Demailly and Păun proved the following fundamental theorem which is the key result towards their numerical characterization of the Kähler cone:

Theorem 2.2** ([DP, p. 1259, Theorem 2.12]).**

Let $X$ be a compact Kähler manifold, and let $[\alpha]\,\in\,H^{1,1}(X,\,\mathbb{R})$ be a nef class with positive self-intersection. Then, $[\alpha]$ is a big class.

Its converse is true as well, meaning all nef and big classes on a compact Kähler manifold have positive self-intersection [Bou1, p. 1054, Lemma 4.2]. Let us now recall a few basic facts about the behavior of these notions under bimeromorphisms.

Lemma 2.3**.**

Let $f\,:\,X\,\longrightarrow\,Y$ be a surjective, bimeromorphic morphism between two compact complex manifolds. Let $[\alpha]\,\in\,H^{1,1}_{\partial\overline{\partial}}(Y,\,\mathbb{R})$ and $[\beta]\,\in\,H^{1,1}_{\partial\overline{\partial}}(X,\,\mathbb{R})$ be cohomology classes.

1. (1)

If $[\beta]$ is big, then so is $f_{*}[\beta]$. 2. (2)

If $[\alpha]$ is nef and has positive self-intersection, then $f^{*}[\alpha]$ is also nef and has positive self-intersection. 3. (3)

If $[\alpha]$ is big and $X$ is Kähler, then $f^{*}[\alpha]$ is big.

Proof.

Let $\omega_{Y}$ be a hermitian metric on $Y$, and let $\omega_{X}$ be a hermitian metric on $X$ such that $\omega_{X}\,\geqslant\,f^{*}\omega_{Y}$.

Proof of 1: If $[\beta]$ is big, then there exists a closed positive current $T\,\in\,[\beta]$ and also a real number $\varepsilon_{0}\,>\,0$, such that $T\,\geqslant\,\varepsilon_{0}\,\omega_{X}\,\geqslant\,\varepsilon_{0}f^{*}\omega_{Y}$. Then, $f_{*}T\in f_{*}[\beta]$ satisfies the condition $f_{*}T\,\geqslant\,\varepsilon_{0}\,\omega_{Y}$, and consequently $f_{*}[\beta]$ is big.

Proof of 2: If $[\alpha]$ is nef, then for every $\varepsilon\,>\,0$, there exists a smooth form $\omega_{\varepsilon}\,\in\,[\alpha]$ such that $\omega_{\varepsilon}\,\geqslant\,-\varepsilon\,\omega_{Y}$. Therefore, the pullback $f^{*}\omega_{\varepsilon}\,\in\,f^{*}[\alpha]$ satisfies the condition

[TABLE]

From this it follows that $f^{*}[\alpha]$ is nef. Moreover, we have

[TABLE]

Proof of 3: Take $[\alpha]$ to be big. Let $T\,\in\,[\alpha]$ be a Kähler current. In view of Demailly’s regularization theorem, we may assume that $T$ has analytic singularities (see Section 2.2). Since $T$ can locally be written as $\alpha+dd^{c}\varphi$ with $\varphi$ quasi-plurisubharmonic, we may set the pullback of $T$ to be $f^{*}T\,=\,f^{*}\alpha+dd^{c}(\varphi\circ f)$. Then, $f^{*}T$ is a positive current lying in the class $f^{*}[\alpha]$, and $f^{*}T$ is a Kähler metric on a Zariski open set of $X$. Now from [Bou1, p. 1057, Theorem 4.7] it follows that $f^{*}[\alpha]$ is big. ∎

Definition 2.4**.**

A compact complex manifold $Y$ of dimension $n$ is said to be in Fujiki class $\mathcal{C}$ if there exist a compact Kähler manifold $X$ and a surjective meromorphic map

[TABLE]

([Fu2, Fu1], [Va, p. 50, Definition 3.1]). By [Va, p. 51, Theorem 5], for any $Y$ in Fujiki class $\mathcal{C}$, the above pair $(X,\,f)$ can be so chosen that the map $f$ from the compact Kähler manifold $X$ is a bimeromorphism. In other words, a Fujiki class $\mathcal{C}$ manifold admits a Kähler modification.

Remark 2.5**.**

A manifold $Y$ in Fujiki class $\mathcal{C}$ admits a Kähler current (big class). Indeed, if $f\,:\,X\,\longrightarrow\,Y$ is a Kähler modification, and $\omega$ is a Kähler form on $X$, then $f_{*}\omega$ is a Kähler current on $Y$ (see proof of Lemma 2.3 (1)). In the opposite direction, it was proved in [DP, p. 1250, Theorem 0.7] that any complex compact manifold $Y$ admitting a Kähler current is necessarily in Fujiki class $\mathcal{C}$.

Corollary 2.6**.**

Let $Y$ be a compact complex manifold in Fujiki class $\mathcal{C}$. Let $[\alpha]\,\in\,H^{1,1}_{\partial\overline{\partial}}(Y,\,\mathbb{R})$ be a nef class. Then, $[\alpha]$ is big if and only if $[\alpha]$ has positive self-intersection.

Proof.

Let $f\,:\,X\,\longrightarrow\,Y$ be a Kähler modification.

Assume that $[\alpha]$ has positive self-intersection. Then the pullback $f^{*}[\alpha]$ is nef and has positive self-intersection by Lemma 2.3 (2). Now Theorem 2.2 says that $f^{*}[\alpha]$ is big. Therefore, Lemma 2.3 (1) implies that $[\alpha]$ is big.

Conversely, assume that $[\alpha]$ is big. By Lemma 2.3 (3), the pullback $f^{*}[\alpha]$ is big as well and if follows from [Bou1, p. 1054, Lemma 4.2] that $f^{*}[\alpha]$ has positive self-intersection. As $\int_{Y}\alpha^{n}\,=\,\int_{X}(f^{*}\alpha)^{n}\,>\,0$, the result is proved. ∎

Remark 2.7**.**

Demailly and Păun conjectured the following ([DP, p. 1250, Conjecture 0.8]): If a complex compact manifold $Z$ possesses a nef cohomology class $[\alpha]$ which has positive self-intersection, then $Z$ lies in the Fujiki class $\mathcal{C}$. This conjecture would imply that a nef class $[\alpha]\,\in\,H^{1,1}_{\partial\overline{\partial}}(Z,\,\mathbb{R})$ on a compact complex manifold $Z$ is big if and only if $[\alpha]$ has positive self-intersection (see Corollary 2.6).

2.2. Non-Kähler locus of a cohomology class

Definition 2.8**.**

Let $X$ be a complex manifold and $U\,\subset\,X$ an open subset.

1. (1)

A plurisubharmonic function (psh for short) $\varphi$ on $U$ is said to have analytic singularities if there exist holomorphic functions $f_{1},\,\ldots,\,f_{r}\,\in\,\mathcal{O}_{X}(U)$, a smooth function $\psi\,\in\,\mathcal{C}^{\infty}(U)$, and $a\,\in\,\mathbb{R}_{+}$, such that

[TABLE] 2. (2)

Let $T$ be a closed, positive $(1,1)$-current on $X$. Then $T$ is said to have analytic singularities if it can be expressed, locally, as $T\,=\,dd^{c}\varphi$, where $\varphi$ is a psh function with analytic singularities. The singular set for $T$ is denoted by $E_{+}(T)$; it is a proper Zariski closed subset of $X$.

It follows from the fundamental regularization theorems of Demailly, [De], that any big class $[\alpha]$ in a compact complex manifold $X$ contains a Kähler current $T$ with analytic singularities. Note that such a current $T$ is smooth on a non-empty Zariski open subset of $X$, and $T$ induce a Kähler metric on this Zariski open subset. Following [Bou2], we define:

Definition 2.9**.**

Let $X$ be a compact complex manifold, and let $[\alpha]\,\in\,H^{1,1}_{\partial\overline{\partial}}(X,\,{\mathbb{R}})$ be a big cohomology class. The non-Kähler locus of $[\alpha]$ is

[TABLE]

where the intersection is taken over all positive currents with analytic singularities. The ample locus $\mathrm{Amp}([\alpha])$ of $[\alpha]$ is the complement of the non-Kähler locus, meaning

[TABLE]

Boucksom proved the following: Given any big class $[\alpha]\,\in\,H^{1,1}_{\partial\overline{\partial}}(X,\,{\mathbb{R}})$, there exists a positive current $T\,\in\,[\alpha]$ with analytic singularities such that $E_{+}(T)\,=\,E_{nK}([\alpha])$; in particular, $E_{nK}(\alpha)$ is a proper Zariski closed subset of $X$ (see [Bou2, p. 59, Theorem 3.17]).

The following Proposition builds upon a result of Collins and Tosatti [CT, p. 1168, Theorem 1.1] showing that the non-Kähler locus of a nef and big class $[\alpha]$ on a Fujiki manifold $X$ coincides with its null locus $\mathrm{Null}([\alpha])$ defined as the reunion of all irreducible subvarieties $V\,\subseteq\,X$ such that $\int_{V}\alpha^{\dim V}\,=\,0$.

Proposition 2.10**.**

Let $f\,:\,X\,\longrightarrow\,Y$ be a bimeromorphic morphism between two compact complex manifolds belonging to the Fujiki class $\mathcal{C}$. Let $[\alpha]\,\in\,H^{1,1}_{\partial\overline{\partial}}(X,\,\mathbb{R})$ be a nef and big cohomology class. Then

[TABLE]

where $\mathrm{Exc}(f)$ is the exceptional locus of $f$, i.e., the singular locus of the Jacobian of $f$.

Proof.

By [CT, p. 1168, Theorem 1.1], it is enough to prove the analogous result for null loci. Let $n\,:=\,\dim_{\mathbb{C}}X$.

Let $E\,\subset\,\mathrm{Exc}(f)$ be an irreducible component; it has dimension $n-1$, while $\dim f(E)\,\leqslant\,n-2$. Therefore, $\int_{E}(f^{*}\alpha)^{n-1}\,=\,\int_{f(E)}\alpha^{n-1}\,=\,0$ and $E\,\subset\,E_{nK}(f^{*}[\alpha])$. Next, if $V\,\subset\,E_{nK}([\alpha])$ is a $k$-dimensional subvariety not included in $f(\mathrm{Exc}(f))$, let $\widetilde{V}$ be its strict transform. We have $f^{-1}(V)\,=\,\widetilde{V}\cup F$ with $F\,\subset\,\mathrm{Exc}(f)$ and $f$ inducing a bimeromorphic morphism $f|_{\widetilde{V}}\,:\,\widetilde{V}\,\longrightarrow\,V$. From the identity $\int_{\widetilde{V}}(f^{*}\alpha)^{k}\,=\,\int_{V}\alpha^{k}$ it follows that

[TABLE]

In summary, $f^{-1}(E_{nK}([\alpha]))\cup\mathrm{Exc}(f)\subset E_{nK}(f^{*}[\alpha])$.

Now, let $W\subset E_{nK}(f^{*}[\alpha])$ be an irreducible $k$-dimensional subvariety not included in $\mathrm{Exc}(f)$. The morphism $f$ induces a bimeromorphic morphism

[TABLE]

and $\int_{f(W)}\alpha^{k}\,=\,\int_{W}(f^{*}\alpha)^{k}\,=\,0$. Therefore, $f(W)\,\subset\,E_{nK}([\alpha)])$, which completes the proof of the Proposition. ∎

2.3. Singular Ricci-flat metrics

Let $Y$ be a compact complex manifold of dimension $n$ such that the first Chern class $c_{1}(T_{Y})$ vanishes in $H^{2}(Y,\,\mathbb{R})$. We assume that $Y$ admits a big cohomology class $[\alpha]\,\in\,H^{1,1}_{\partial\overline{\partial}}(Y,\,\mathbb{R})$. In particular, $Y$ lies in Fujiki class $\mathcal{C}$ (see Remark 2.5). We fix once and for all a Kähler modification

[TABLE]

It follows from Lemma 2.3 (3) that the class $f^{*}[\alpha]$ is big. The Jacobian of $f$ induces the following identity

[TABLE]

where

[TABLE]

is an effective $\mathbb{Z}$-divisor on $X$ contracted by $f$, meaning $\mathrm{codim}_{Y}f(E)\,\geqslant\,2$, and each $E_{i}$ is irreducible. More precisely, $\mathrm{Supp}(E)$ coincides with the exceptional locus of $f$, that is the complement of the locus of points on $X$ in a neighborhood of which $f$ induces a local biholomorphism.

Since $c_{1}(K_{Y})\,=\,0\,\in\,H^{2}(Y,\,\mathbb{R})$, it follows from (2.1) that $K_{X}$ is numerically equivalent to the effective divisor $E$. Moreover, the class $f^{*}[\alpha]\,\in\,H^{1,1}(X,\,\mathbb{R})$ is big by Lemma 2.3  (3). From [BEGZ, p. 200, Theorem A] we know that there exists a unique closed positive current

[TABLE]

with finite energy such that

[TABLE]

In terms of Monge–Ampère equations, this means that the non-pluripolar Monge–Ampère measure of $T$, denoted by $\langle T^{n}\rangle$, can be expressed as

[TABLE]

where

• •

$h$ is a smooth hermitian metric on $\mathcal{O}_{X}(E)$ with Chern curvature tensor denoted by $\Theta_{h}(E)$,

• •

$s_{E}\,\in\,H^{0}(X,\mathcal{O}_{X}(E))$ satisfies the condition $\mathrm{div}(s_{E})\,=\,E$, and

• •

$dV$ is the smooth volume form on $X$ satisfying $\mathrm{Ric}\,(dV)\,=\,-\Theta_{h}(E)$ and normalized such that $\int_{X}|s_{E}|_{h}^{2}dV\,=\,\mathrm{vol}(f^{*}\alpha)\,=\,\mathrm{vol}(\alpha)$.

Proposition-Definition 2.11**.**

Let $Y$ be a compact complex manifold with trivial first Chern class endowed with a big cohomology class $[\alpha]\in H^{1,1}_{\partial\overline{\partial}}(Y,\,\mathbb{R})$. Let $f\,:\,X\,\longrightarrow\,Y$ be a Kähler modification, and let $T\in f^{*}[\alpha]$ be the associated singular Ricci-flat metric. Then

1. (1)

There exists a closed, positive $(1,1)$-current $\omega\in[\alpha]$ on $Y$ such that $T=f^{*}\omega$. 2. (2)

The current $\omega$ is independent of the choice of the Kähler modification of $Y$.

We call $\omega$ the singular Ricci-flat metric in $[\alpha]$.

Proof.

For the first item, let us write $T=f^{*}\alpha+dd^{c}u$ for some $(f^{*}\alpha)$-psh function $u$ on $X$. The restriction of $u$ to any fiber of $f$ is psh, but the fibers of $f$ are connected. By the maximum principle, $u$ is constant on the fibers of $f$, hence can be written as $u\,=\,\pi^{*}v$ for some $\alpha$-psh function $v$ on $Y$. Then, $T\,=\,f^{*}\omega$ with $\omega\,=\,\alpha+dd^{c}v$.

Now, let $f^{\prime}\,:\,X^{\prime}\,\longrightarrow\,Y$ be another Kähler modification, and let $T^{\prime}\,:=\,(f^{\prime})^{*}\omega^{\prime}$ be the associated singular Ricci-flat current. Let $Z$ be a desingularization of $X^{\prime}\times_{Y}X$, so that we have the following Cartesian square where all maps are bimeromorphic

[TABLE]

Let $h\,:=\,f\circ g^{\prime}\,=\,f^{\prime}\circ g$. By [BEGZ, p. 201, Theorem B] and [BEGZ, Proposition 1.12], both currents $g^{*}T^{\prime}$ and $(g^{\prime})^{*}T$ have minimal singularities in $h^{*}[\alpha]$ and are solutions of the non-pluripolar Monge-Ampère equation $-\mathrm{Ric}\,(\cdotp)\,=\,[K_{Z/Y}]$. By uniqueness of the solution of that equation [BEGZ, Theorem A], if follows that $h^{*}\omega\,=\,h^{*}\omega^{\prime}$. Taking the direct image of the previous equality, it is deduced that $\omega\,=\,\omega^{\prime}$. ∎

The regularity properties of the current $T$ (or $\omega$) are quite mysterious in general. For instance, it is not known whether $T$ is smooth on a Zariski open set. However, things become simpler once we assume additionally that the cohomology class $[\alpha]$ is nef.

Proposition-Definition 2.12**.**

In the set-up of Proposition-Definition 2.11, assume additionally that the cohomology class $[\alpha]$ is nef. Then, the singular Ricci-flat current $\omega\in[\alpha]$ is smooth on a non-empty Zariski open subset of $Y$.

We define $\Omega$ to be the largest Zariski open subset in restriction of which $\omega$ is a genuine Kähler form. Then

[TABLE]

for any Kähler modification $f\,:\,X\,\longrightarrow\,Y$ (see Definition 2.9).

Proof.

Let $f\,:\,X\,\longrightarrow\,Y$ be a Kähler modification. As $[\alpha]$ is nef, $f^{*}[\alpha]$ is nef as well (see Lemma 2.3 (2)) and the proof of [BEGZ, p. 201, Theorem C] applies verbatim to give that $T$ is smooth outside $E_{nK}(f^{*}[\alpha])\smallsetminus\mathrm{Supp}(E)$. Note that Proposition 2.10 shows that this locus is just $E_{nK}(f^{*}[\alpha])$. In particular, this shows that the current $\omega\in[\alpha]$ on $Y$ is a genuine Ricci-flat Kähler metric on the non-empty Zariski open set

[TABLE]

which concludes the proof. ∎

We will need a refinement of the result above which will be explained next.

Let $t,\,\varepsilon\,>\,0$; by Yau’s theorem (Calabi’s conjecture) [Ya], there exists a unique Kähler metric $\omega_{t,\varepsilon}\,\in\,f^{*}{[\alpha]}+t[\omega_{X}]$ solving the equation

[TABLE]

where $\theta_{\varepsilon}\,\in\,c_{1}(E)$ is a regularization of the current defined by integration along $E$. For instance, such a smooth closed $(1,1)$–form $\theta_{\varepsilon}$ can be constructed as follows: pick smooth hermitian metrics $h_{i}$ on $\mathcal{O}_{X}(E_{i})$ (see (2.2)), and choose a holomorphic section $s_{i}\,\in\,H^{0}(X,\mathcal{O}_{X}(E_{i}))$ for each $i$ cutting out $E_{i}$ in the sense that $\mathrm{div}(s_{i})\,=\,E_{i}$; then set

[TABLE]

where $\Theta_{h_{i}}(E_{i})$ is the Chern curvature of the hermitian line bundle $(\mathcal{O}_{X}(E_{i}),h_{i})$.

Now, the proof of [BEGZ, p. 201, Theorem C] can be adapted without any significant changes to obtain the following result.

Theorem 2.13** ([BEGZ, Theorem C]).**

In the set-up of Proposition-Definition 2.11, assume furthermore that $[\alpha]$ is nef, and let $\omega_{t,\varepsilon}\,=\,f^{*}{[\alpha]}+t[\omega_{X}]$ be the Kähler form solving Eq. (2.4). When $t\,\to\,0$ and $\varepsilon\,\to\,0$, the form $\omega_{t,\varepsilon}$ converges to the current $T$ in (2.3) in the weak topology of currents, and also in the $\mathcal{C}^{\infty}_{\rm loc}(\mathrm{Amp}(f^{*}[\alpha]))$ topology.

2.4. Flatness of tensors

It will be convenient in this section to refer to the following set-up.

Setting 2.14**.**

Let $Y$ be a compact complex manifold with trivial first Chern class endowed with a nef and big cohomology class $[\alpha]\,\in\,H^{1,1}_{\partial\overline{\partial}}(Y,\,\mathbb{R})$. We denote by $\omega$ the singular Ricci-flat metric from Proposition-Definition 2.11, and we let $\Omega$ be its smooth locus; see Proposition-Definition 2.12.

Theorem 2.15**.**

In Setting 2.14, let $\tau\,\in\,H^{0}(Y,\,{T_{Y}^{*}}^{\otimes p})$ with $p\,\geq\,0$.

Then, the restriction $\tau|_{\Omega}$ to $\Omega$ in (2.12) is parallel with respect to the Ricci-flat Kähler metric $\omega|_{\Omega}$.

Remark 2.16**.**

Since $c_{1}(T_{Y})\,=\,0$ in $H^{2}(Y,\,\mathbb{R})$, and $Y$ is in Fujiki class $\mathcal{C}$, from [To, Theorem 1.5] we know that the holomorphic line bundle $K_{Y}$ is of finite order. Therefore, there exists a finite unramified cover $\pi\,:\,\widehat{Y}\,\longrightarrow\,Y$ such that $K_{\widehat{Y}}$ is holomorphically trivial.

The previous remark leads to the following application of Theorem 2.15, valid for any holomorphic tensors.

Corollary 2.17**.**

In Setting 2.14, let $\tau\,\in\,H^{0}(Y,\,{T_{Y}}^{\otimes p}\otimes{T_{Y}^{*}}^{\otimes q})$ with $p,q\,\geq\,0$. Then, the restriction $\tau|_{\Omega}$ to $\Omega$ in (2.12) is parallel with respect to the Ricci-flat Kähler metric $\omega|_{\Omega}$.

In particular, the evaluation map

[TABLE]

is injective for all $y\,\in\,\Omega$ and all integers $p,\,q\,\geq\,0$.

Proof of Corollary 2.17.

Let $\tau$ be any holomorphic tensor on $Y$, not necessarily contravariant. Let $\pi\,:\,\widehat{Y}\,\longrightarrow\,Y$ be the finite unramified cover from Remark 2.16. As $\pi$ is a local biholomorphism, the pull-back $\pi^{*}\tau$ on $\widehat{Y}$ is well-defined. We can interpret $\pi^{*}\tau$ as a holomorphic contravariant tensor on $\widehat{Y}$. Indeed, a holomorphic trivialization of $K_{\widehat{Y}}$ produces a holomorphic isomorphism $T_{\widehat{Y}}\,\simeq\,\bigwedge^{n-1}T^{*}_{\widehat{Y}}$, where $n\,=\,\dim_{\mathbb{C}}Y$.

Now, let $f\,:\,X\,\longrightarrow\,Y$ be a Kähler modification, and let us set $T\,:=\,f^{*}\omega$ and $\Omega^{\prime}\,:=\,f(\mathrm{Amp}(f^{*}[\alpha]))\,\subset\,\Omega$. As $\Omega^{\prime}$ is dense in $\Omega$ for the usual topology, and both $\tau$ and $\omega$ are smooth on $\Omega$, it suffices to prove that $\tau|_{\Omega^{\prime}}$ is parallel with respect to $\omega|_{\Omega^{\prime}}$. Since $f$ is an isomorphism over $\Omega^{\prime}$, we may pull back $\tau|_{\Omega^{\prime}}$ by $f$ over this locus. This way we reduce the question to proving that $f^{*}\tau|_{\Omega^{\prime}}$ is parallel with respect to $T|_{f^{-1}(\Omega^{\prime})}$.

Let $\widehat{X}\,:=\,X\times_{Y}\widehat{Y}$, so that we have a Cartesian square

[TABLE]

The morphism $\widehat{\pi}$ is finite unramified (in particular, $\widehat{X}$ is compact Kähler) and $\widehat{f}$ is birational. By the observation at the beginning of the proof, we may apply Theorem 2.15 to show that $\widehat{f}^{*}\pi^{*}\tau$ is parallel with respect to the singular Ricci-flat Kähler metric $T_{\widehat{X}}\,\in\,\widehat{f}^{*}\pi^{*}[\alpha]$ on the locus $\mathrm{Amp}(\widehat{f}^{*}\pi^{*}[\alpha])$. As $T_{\widehat{X}}\,\in\,\widehat{\pi}^{*}f^{*}[\alpha]$, the functoriality property of Kähler-Einstein metrics with respect to finite morphisms (see e.g. [GGK, Proposition 3.5]) shows that $T_{\widehat{X}}\,=\,\widehat{\pi}^{*}T$. Over $\Omega^{\prime}$, the following identity holds

[TABLE]

Therefore, $\widehat{\pi}^{*}f^{*}\tau|_{\Omega^{\prime}}$ is parallel with respect to $\widehat{\pi}^{*}T|_{f^{-1}(\Omega^{\prime})}$, hence $f^{*}\tau|_{\Omega^{\prime}}$ is parallel with respect to $T|_{f^{-1}(\Omega^{\prime})}$. The Corollary now follows easily. ∎

Let us now prove Theorem 2.15. The arguments and computations in the proof are extensively borrowed from [Gue] (see also [CP1]).

Proof of Theorem 2.15.

Let us fix a Kähler modification $f\,:\,X\,\longrightarrow\,Y$, and let us set $\Omega^{\prime}:=f(\mathrm{Amp}(f^{*}[\alpha]))\subset\Omega$. By the same arguments as in the proof of Corollary 2.17, it is sufficient to prove that $\tau|_{\Omega^{\prime}}$ is parallel with respect to $\omega|_{\Omega^{\prime}}$ or, equivalently, that the restriction of $\sigma\,:=\,f^{*}\tau\,\in\,H^{0}(X,\,{T_{X}^{*}}^{\otimes p})$ to $f^{-1}(\Omega^{\prime})$ is parallel with respect to $(f^{*}\omega)|_{f^{-1}(\Omega^{\prime})}$.

Let $\mathcal{E}\,:=\,{T_{X}^{*}}^{\otimes p}$, and let $h\,=\,|\cdotp\!|$ be the hermitian metric on $\mathcal{E}$ induced by the Kähler metric $\omega_{t,\varepsilon}$ introduced in Eq. (2.4). Let $D\,=\,D^{\prime}+\overline{\partial}$ be the corresponding Chern connection for $(\mathcal{E},\,h)$.The curvature of this Chern connection for $(\mathcal{E},\,h)$ will be denoted by $\Theta_{h}(\mathcal{E})$. The following holds:

[TABLE]

Wedging (2.5) with $\omega_{t,\varepsilon}^{n-1}$ and then integrating it on $X$ yields:

[TABLE]

Since $|\langle D^{\prime}\sigma,\sigma\rangle|\,\leqslant\,|D^{\prime}\sigma|\cdotp|\sigma|$, we obtain the inequality

[TABLE]

First let us introduce a notation: if $V$ is a complex vector space of dimension $n$, $1\leqslant p\leqslant n$ is an integer, and $t\,\in\,\mathrm{End}(V)$, then we denote by $t^{\otimes p}$ the endomorphism of $V^{\otimes p}$ defined by

[TABLE]

it may be noted that if $V$ has an hermitian structure, and if $t$ is hermitian semipositive, then $t^{\otimes p}$ is also hermitian semipositive for the hermitian structure on $V^{\otimes p}$ induced by the hermitian structure on $V$; also, the inequality $\mathrm{tr}(t^{\otimes p})\,\leqslant\,n^{p}\,\mathrm{tr}(t)$ holds.

Now we can easily check the following identity:

[TABLE]

where $\sharp\mathrm{Ric}\,\omega$ is the endomorphism of $T_{X}^{*}$ induced by $\mathrm{Ric}\,\omega_{t,\varepsilon}$ via $\omega_{t,\varepsilon}$. As $\mathrm{Ric}\,\omega_{t,\varepsilon}\,=\,-\theta_{\varepsilon}$, we deduce that

[TABLE]

(the operator $\sharp$ is defined above).

We can write $\theta\,=\,\sum a_{i}\theta_{i,\varepsilon}$, where $\theta_{i,\varepsilon}\,:=\,a_{i}\left(\frac{\varepsilon^{2}|D^{\prime}s_{i}|^{2}}{(|s_{i}|^{2}+\varepsilon^{2})^{2}}+\frac{\varepsilon^{2}\Theta_{h_{i}}(E_{i})}{|s_{i}|^{2}+\varepsilon^{2}}\right)$. In order to simplify the notation, we drop the index $i$ and set

[TABLE]

so that $\theta_{i,\varepsilon}\,=\,a(\beta+\gamma)$; remember that these forms are smooth as long as $\varepsilon\,>\,0$.

Let us start with $\gamma$: there exists a constant $C\,>\,0$ such that

[TABLE]

As both of the two operations $\sharp$ and $p$-th tensor power preserve positivity, we get that

[TABLE]

But $\sharp\omega_{X}$ is a positive endomorphism whose trace is $\mathrm{tr}_{\omega_{t,\varepsilon}}\omega_{X}$, and therefore we have $(\sharp\omega_{X})^{\otimes p}\,\leqslant\,n^{p}\mathrm{tr}_{\omega_{t,\varepsilon}}(\omega_{X})\cdot\mathrm{Id}$. Consequently,

[TABLE]

for some $C\,>\,0$ which is independent of $t$ and $\varepsilon$. From Lemma 2.18 below and using the dominated convergence theorem, we deduce that the integral

[TABLE]

converges to [math] when $\varepsilon$ goes to zero.

We now have to estimate the term involving $\beta$. We know that $\beta$ is non-negative, so $(\sharp\beta)^{\otimes p}\omega_{t,\varepsilon}^{n}\,\leqslant\,n^{p+1}\,\beta\wedge\omega_{t,\varepsilon}^{n-1}\,\mathrm{Id}$, and hence

[TABLE]

We already observed that the second integral converges to [math] when $\varepsilon\,\to\,0$. As for the first term, it is cohomological (independent of $\varepsilon$), equal to $t^{n-1}(E_{i}\cdotp[\omega_{X}]^{n-1})$ since $f^{*}[\alpha]$ is orthogonal to $E_{i}$, and thus it converges to [math] as $t$ goes to [math].

Now recall (2.6):

[TABLE]

When $\varepsilon$ and $t$ both go to [math], $\omega_{t,\varepsilon}$ converges weakly to $f^{*}\omega$. Moreover, $f^{*}\omega$ is smooth on $f^{-1}(\Omega^{\prime})$ (defined at the beginning of the proof), and the convergence is smooth on the compact subsets of $\Omega^{\prime}$ by Theorem 2.13. Therefore, by Fatou lemma, we deduce that $D^{\prime}\sigma\,=\,0$ on this locus, which was required to be shown (here $D^{\prime}$ denotes the Chern connection on $\mathcal{E}$ associated to $f^{*}\omega$ on this open subset $\Omega^{\prime}$). ∎

The proof of Theorem 2.15 involved the following result, which is proved in much greater generality in [Gue, Lemma 3.7]. We provide a simpler proof more suited to the present set-up.

Lemma 2.18**.**

For every fixed $t\,>\,0$, and any section $s\,\in\,{\mathcal{O}}_{X}(E_{i})$ for some component $E_{i}$ of $E$, the integral

[TABLE]

converges to [math] when $\varepsilon$ goes to [math].

Proof.

By [Ya, p. 360, Proposition 2.1], there is a constant $C_{t}>0$ independent of $\varepsilon>0$ such that $\omega_{t,\varepsilon}\,\leqslant\,C_{t}\,\omega_{X}$. The lemma thus follows from Lebesgue’s dominated convergence theorem. ∎

3. Geometric structures on manifolds in class $\mathcal{C}$

In this section we give two applications of Theorem 2.15 for manifolds in Fujiki class $\mathcal{C}$ bearing a holomorphic geometric structure.

3.1. Holomorphic geometric structures

Let us first recall the definition of a holomorphic (rigid) geometric structure as given in [DG, Gr].

Let $Y$ be a complex manifold of complex dimension $n$ and $k\,\geq\,1$ an integer. Denote by $R^{k}(Y)\,\longrightarrow\,Y$ the holomorphic principal bundle of $k$-frames of $Y$: it is the bundle of $k$-jets of local holomorphic coordinates on $Y$. Recall that the structure group of $R^{k}(Y)$ is the group $D^{k}$ of $k$-jets of local biholomorphisms of $\mathbb{C}^{n}$ fixing the origin. This $D^{k}$ is a complex algebraic group.

Definition 3.1**.**

A holomorphic geometric structure $\phi$ (of order $k$) on $Y$ is a holomorphic $D^{k}$–equivariant map from $R^{k}(Y)$ to a complex algebraic manifold $Z$ endowed with an algebraic action of the algebraic group $D^{k}$.

A holomorphic geometric structure $\phi$ as in Definition 3.1 is said to be of affine type if $Z$ in Definition 3.1 is a complex affine manifold.

Notice that holomorphic tensors are holomorphic geometric structures of affine type of order one. Holomorphic affine connections are holomorphic geometric structures of affine type of order two [DG]. In contrast, while holomorphic foliations and holomorphic projective connections are holomorphic geometric structure in the sense of Definition 3.1, they are not of affine type.

A holomorphic tensor which is the complex analog of a Riemannian metric is defined in the following way:

Definition 3.2**.**

A holomorphic Riemannian metric on a complex manifold $Y$ of complex dimension $n$ is a holomorphic section

[TABLE]

where $\text{S}^{i}$ stands for the $i$-th symmetric product, such that for every point $y\,\in\,Y$ the complex quadratic form $g(y)$ on $T_{y}Y$ is of (maximal) rank $n$.

Take a holomorphic Riemannian manifold $(Y,\,g)$ as above. The real part of $g$ is a pseudo-Riemannian metric $h$ of signature $(n,n)$ on the real manifold of dimension $2n$ underlying the complex manifold $Y$.

As in the set-up of (pseudo-)Riemannian manifolds, there exists a unique torsion-free holomorphic connection $\nabla$ on the holomorphic tangent $T_{Y}$ such that $g$ is parallel with respect to $\nabla$. It is called the Levi-Civita connection for $g$.

Given $(Y,\,g)$ as above, consider the curvature of the holomorphic Levi-Civita connection $\nabla$ for $g$. This curvature tensor vanishes identically if and only if $g$ is locally isomorphic to the standard flat (complex Euclidean) model $(\mathbb{C}^{n},\,dz_{1}^{2}+\ldots+dz_{n}^{2})$. In this flat case the real part $h$ of $g$ is also flat and it is locally isomorphic to $(\mathbb{R}^{2n},\,dx_{1}^{2}+\ldots+dx_{n}^{2}-dy_{1}^{2}-\ldots-dy_{n}^{2})$. For more details about the geometry of holomorphic Riemannian metrics the reader is referred to [Du2, Gh2].

A natural notion of (local) infinitesimal symmetry is the following (the terminology comes from the standard Riemannian setting).

Definition 3.3**.**

A (local) holomorphic vector field $\theta$ on $Y$ is a (local) Killing field for a holomorphic geometric structure of order $k$

[TABLE]

if the flow for the canonical lift of $\theta$ to $R^{k}(Y)$ preserves each of the fibers of the map $\phi$.

Consequently, the (local) flow of a Killing vector field for $\phi$ preserves $\phi$. It is evident that the Killing vector fields for $\phi$ form a Lie algebra with respect to the operation of Lie bracket.

The holomorphic geometric structure $\phi$ is called locally homogeneous on an open subset $\Omega$ of $Y$ if the holomorphic tangent bundle $T_{Y}$ is spanned by local Killing vector fields of $\phi$ in the neighborhood of every point in $\Omega$. This implies that for any pair of points $o\,,o^{\prime}\,\in\,\Omega$, there exists a (local) biholomorphism, from a neighborhood of $o$ to a neighborhood of $o^{\prime}$, that preserves $\phi$ and also sends $o$ to $o^{\prime}$.

A holomorphic geometric structure $\phi$ is called rigid of order $l$ in the sense of Gromov, [Gr], if any local biholomorphism $f$ preserving $\phi$ is determined uniquely by the $l$–jet of $f$ at any given point.

Holomorphic affine connections are rigid of order one in the sense of Gromov [DG, Gr]. Their rigidity comes from the fact that local biholomorphisms fixing a point and preserving a connection actually linearize in exponential coordinates around the fixed point, so they are completely determined by their differential at the fixed point.

A holomorphic Riemannian metric $g$ is also a rigid holomorphic geometric structure, because local biholomorphisms preserving $g$ also preserve the associated Levi-Civita connection. In contrast, holomorphic symplectic structures and holomorphic foliations are non-rigid geometric structures [DG, Gr].

3.2. A criterion for local homogeneity

Theorem 3.4**.**

Let $Y$ be a compact complex manifold in Fujiki class $\mathcal{C}$ with trivial first Chern class ($c_{1}(T_{Y})\,=\,0$ in $H^{2}(Y,\,\mathbb{R})$). Assume that there exists a nef cohomology class $[\alpha]\,\in\,H^{1,1}(Y,\,\mathbb{R})$ of positive self-intersection. Then the following two hold:

1. (1)

There exists a non-empty Zariski open subset $\Omega\,\subset\,Y$ such that any holomorphic geometric structure of affine type on $Y$ is locally homogeneous on $\Omega$. 2. (2)

If $Y$ admits a rigid holomorphic geometric structure of affine type, then the fundamental group of $Y$ is infinite.

Proof.

(1): In view of Corollary 2.6, the assumptions of Theorem 2.15 are satisfied by $Y$. Corollary 2.17 implies that there exists a Zariski open subset $\Omega\,\subset\,Y$ such that any holomorphic tensor on $Y$ is parallel with respect to some Kähler metric on $\Omega$. In particular, any holomorphic tensor on $Y$ vanishing at some point of $\Omega$ must be identically zero. Then Lemme 3.2 in [Du1, p. 565] gives that any holomorphic geometric structure of affine type on $Y$ is locally homogeneous on $\Omega$.

(2): To prove by contradiction, assume that the fundamental group of $Y$ is finite. Substituting the universal cover of $Y$ in place of $Y$, and considering the pull-back, to the universal cover, of the geometric structure on $Y$, we may assume $Y$ in the theorem to be simply connected.

Now $K_{Y}$ is trivial because $Y$ is simply connected; see Remark 2.16. A result, first proved by Nomizu in the Riemannian setting [No], and subsequently generalized by Amores [Am] and Gromov [Gr], gives the following: the condition that $Y$ is simply connected implies that any local (holomorphic) Killing field of a rigid holomorphic geometric structure on $Y$ extends to a global (holomorphic) Killing field (see also a nice exposition of it in [DG]).

In particular, using statement (1) in the theorem, we obtain that at any point of $z\,\in\,\Omega$, the fiber $T_{z}\Omega$ of the holomorphic tangent bundle $T_{Y}$ is spanned by globally defined holomorphic vector fields on $Y$. It was noted in the proof of (1) that any holomorphic tensor on $Y$ that vanishes at some point of $\Omega$ must be identically zero. Combining these we conclude that there are $n$ global holomorphic vector fields on $Y$, where $n\,=\,\dim_{\mathbb{C}}Y$, that span $T_{\Omega}$.

Fix $n$ global holomorphic vector fields $X_{1},\,\cdots,\,X_{n}$ on $Y$ that span $T_{\Omega}$. Also, fix a nontrivial holomorphic section $vol$ of the trivial canonical bundle $K_{Y}$. Then $vol(X_{1},\cdots,X_{n})$ is a global holomorphic function on $Y$. This function must be constant (by the maximum principle) and nonzero at points in $\Omega$. Since $vol(X_{1},\cdots,X_{n})$ is nowhere vanishing on $Y$, it follows that $X_{1},\,\cdots,\,X_{n}$ span the holomorphic tangent bundle $T_{Y}$ at all points of $Y$.

In other words, $Y$ is a parallelizable manifold. Hence by a theorem of Wang, [Wa, p. 774, Theorem 1], the complex manifold $Y$ must be biholomorphic to a quotient of a connected complex Lie group by a co-compact lattice in it. In particular, $Y$ is not simply connected. This gives the contradiction that we are seeking. ∎

3.3. A non-affine type example

It should be mentioned that statement (1) in Theorem 3.4 is not valid in general for holomorphic geometric structures of non-affine type. To see such an example, first recall that Ghys constructed in [Gh1] codimension one holomorphic foliations on complex tori which are not translation invariant. Such a foliation can be obtained in the following way. Consider a complex torus $T\,=\,\mathbb{C}^{n}/\Lambda$, with $\Lambda$ a lattice in $\mathbb{C}^{n}$ and assume that there exists a linear form $\widetilde{\pi}\,:\,\mathbb{C}^{n}\,\longrightarrow\,\mathbb{C}$ sending $\Lambda$ to a lattice $\Lambda^{\prime}$ in $\mathbb{C}$. Then $\widetilde{\pi}$ descends to a holomorphic fibration $\pi\,:\,T\,\longrightarrow\,\mathbb{C}/\Lambda^{\prime}$ over the elliptic curve $\mathbb{C}/\Lambda^{\prime}$. Choose a non-constant meromorphic function $u$ on the elliptic curve $\mathbb{C}/\Lambda^{\prime}$ and consider the meromorphic closed one-form $\Omega\,=\,\pi^{*}(udz)+\omega$ on $T$, where $\omega$ is any (translation invariant) holomorphic one-form on $T$ and $dz$ is a nontrivial holomorphic section of the canonical bundle of $\mathbb{C}/\Lambda^{\prime}$. It is easy to see that the foliation given by the kernel of $\Omega$ extend on all of $T$ as a nonsingular codimension one holomorphic foliation $\mathcal{F}$. This foliation is not invariant by all translations in $T$, more precisely, it is invariant only by those translations that are spanned by vectors lying in the kernel of $\widetilde{\pi}$. The subgroup of translations preserving $\mathcal{F}$ is a subtorus $T^{\prime}$ of complex codimension one in $T$ [Gh1].

On the other hand, since the holomorphic tangent bundle of $T$ is trivial, we have a family of global (commuting) holomorphic vector fields $X_{1},\,X_{2},\,\ldots,\,X_{n}$ on $T$ which span the holomorphic tangent bundle $T_{T}$ at any point in $T$.

The holomorphic geometric structure $\phi\,=\,(\mathcal{F},\,X_{1},\,\ldots,\,X_{n})$, obtained by juxtaposing Ghys’ foliation $\mathcal{F}$ and the vector fields $X_{i}$ is a holomorphic rigid geometric structure of non-affine type [DG, Gr]. Local Killing fields of $\phi$ commute with all $X_{i}$, so they are linear combinations of $X_{i}$; so they extend as globally defined holomorphic vector fields on $T$. Since the local Killing fields of $\phi$ are translations which must also preserve $\mathcal{F}$, they span the subtorus $T^{\prime}$ of $T$. In particular, the Killing algebra of $\phi$ has orbits of complex codimension one in $T$ and therefore $\phi$ is not locally homogeneous on any nontrivial open subset of $T$.

On the contrary, for holomorphic geometric structure of affine type, we think that the non-empty Zariski open set $\Omega$ in Theorem 3.4 is all of the manifold. This was proved to be true in [BD] for Moishezon manifolds (these are manifolds bimeromorphic to some complex projective manifold [Mo]).

3.4. Holomorphic Riemannian metric

Theorem 3.5**.**

Let $Y$ be a compact complex manifold in Fujiki class $\mathcal{C}$ admitting a holomorphic Riemannian metric $g$. Assume that there exists a cohomology class $[\alpha]\,\in\,H^{1,1}_{\partial\overline{\partial}}(Y,\,\mathbb{R})$ that is nef and has positive self-intersection. Then there is a finite unramified cover $\gamma\,:\,{\mathcal{T}}\,\longrightarrow\,Y$, where $\mathcal{T}$ is a complex torus, such that the pulled back holomorphic Riemannian metric $\gamma^{*}g$ on the torus ${\mathcal{T}}$ is translation invariant.

Proof.

Assume that $Y$ admits a holomorphic Riemannian metric $g$. We have $T_{Y}\,=\,T^{*}_{Y}$, because $g$ gives a holomorphic isomorphism of $T^{*}_{Y}$ with $T_{Y}$. This implies that the first Chern class of $T_{Y}$ vanishes. So Theorem 2.15 holds for $Y$ because of Corollary 2.6. By Theorem 2.15, there exists a non-empty Zariski open subset $\Omega\,\subset\,Y$ endowed with a Kähler metric $\omega$, such that the restriction of the holomorphic tensor $g$ to $\Omega$ is parallel with respect to the Levi-Civita connection on $T_{\Omega}$ for the Kähler metric $\omega$.

The following lemma proves that $g$ and $\omega$ are flat.

Lemma 3.6**.**

Let $U$ be an open subset of ${\mathbb{C}}^{n}$ in Euclidean topology, and let $\omega^{\prime}$ be a Kähler metric on $U$. Suppose that there exists a holomorphic Riemannian metric $g^{\prime}$ on $U$ such that the tensor $g^{\prime}$ is parallel with respect to the Levi-Civita connection for the Kähler metric $\omega^{\prime}$. Then the following three hold:

1. (1)

The Kähler metric $\omega^{\prime}$ is flat. 2. (2)

The holomorphic Levi-Civita connection for $g^{\prime}$ is flat. 3. (3)

The tensor $\omega^{\prime}$ is flat with respect to the holomorphic Levi-Civita connection for $g^{\prime}$.

Proof.

Take any $u\,\in\,U$. Using de Rham’s local splitting theorem, there exists a local decomposition of an open neighborhood $U^{u}\,\subset\,U$ of $u$ in ${\mathbb{C}}^{n}$ such that $(U^{u},\,\omega^{\prime})$ is a Riemannian product

[TABLE]

where $(U_{0},\,\omega_{0})$ is a flat Kähler manifold and $(U_{i},\,\omega_{i})$ is an irreducible Kähler manifold for every $1\,\leq\,i\,\leq\,p$ (the reader is referred to [GGK, Proposition 2.9] for more details on this local Kähler decomposition).

For any $v\,\in\,U^{u}$, let $Q_{v}$ be the complex bilinear form associated to the quadratic form $g^{\prime}(v)$ on $T_{v}U^{u}$. Write $v\,=\,(v_{0},\,v_{1},\ldots,v_{p})$ using (3.1). Since $g^{\prime}$ is parallel with respect to $\omega^{\prime}$, it follows that for all $0\,\leq\,i,\,j\,\leq\,p$,

[TABLE]

for all $w_{i}\,\in\,T_{v_{i}}U_{i}$, $w_{j}\,\in\,T_{v_{j}}U_{j}$, for any $h_{i}\,\in\,\text{GL}(T_{v_{i}}U_{i})$ in the holonomy group for $(U_{i},\,\omega_{i})$, and for any $h_{j}\,\in\,\text{GL}(T_{v_{j}}U_{j})$ in the holonomy group for $(U_{j},\,\omega_{j})$. Assume that $i\,\not=\,j$. So at least one of $i$ and $j$ is different from zero. Assume that $j\,\not=\,0$. From (3.2) if follows that

[TABLE]

for all $h_{j},\,h^{\prime}_{j}$ in the holonomy group for $(U_{j},\,g_{j})$ (set $h_{i}\,=\,\text{Id}$ in (3.2)). Since $j\,>\,0$, the holonomy group for $(U_{j},\,g_{j})$ is irreducible, which implies that the vector subspace of $T_{v_{j}}U_{j}$ generated by all elements of the form $h_{j}\cdot w_{j}-h^{\prime}_{j}\cdot w_{j}$, where $w_{j}\,\in\,T_{v_{j}}U_{j}$ and $h_{j},\,h^{\prime}_{j}$ are elements of the holonomy group for $(U_{j},\,\omega_{j})$, is entire $T_{v_{j}}U_{j}$. Using this, from (3.3) it follows that $TU_{i}$ and $TU_{j}$ are $g^{\prime}$–orthogonal for any $i\,\not=\,j$. Consequently, $g^{\prime}$ is non-degenerate when restricted to all $(U_{i},\,\omega_{i})$, $0\,\leq\,i\,\leq\,p$.

To prove the first statement of the lemma it suffices to show that $(U^{u},\,\omega^{\prime})\,=\,(U_{0},\,\omega_{0})$, meaning $p\,=\,0$ in (3.1).

To prove $p\,=\,0$ by contradiction, assume that $\omega^{\prime}$ admits an (irreducible) factor $(U_{1},\,\omega_{1})$. The parallel transport for the Levi-Civita connection for $\omega_{1}$ must preserve the restriction $g_{1}$ of $g^{\prime}$ to $TU_{1}$ and also preserve the real part $h_{1}$ of $g_{1}$. The real part $h_{1}$ of $g_{1}$ is a pseudo-Riemannian metric of signature $(n_{1},\,n_{1})$, where $n_{1}$ is the complex dimension of $U_{1}$. Consider the positive and the negative eigenspaces of $h_{1}$ with respect to $\omega_{1}$. Since the parallel transport for the Levi-Civita connection for $\omega_{1}$ preserves $g_{1}$, the holonomy of $\omega_{1}$ preserves the positive and the negative eigenspaces of $h_{1}$ with respect to $\omega_{1}$. This is a contradiction, because the factor $(U_{1},\,\omega_{1})$ is irreducible. This proves the first statement of the lemma.

To prove the second statement, since the Kähler metric $\omega^{\prime}$ is flat, there exists local holomorphic coordinates with respect to which $\omega^{\prime}$ is the standard hermitian metric on ${\mathbb{C}}^{n}$. Take such a holomorphic coordinate function on an open subset $U^{\prime}\,\subset\,U$. Therefore, on $U^{\prime}$ parallel transports for the Levi-Civita connection for $\omega^{\prime}$ are just translations in ${\mathbb{C}}^{n}$ in terms of this holomorphic coordinate function on $U^{\prime}$. Consequently, on $U^{\prime}$ the holomorphic Riemannian metric $g^{\prime}$ must be translation invariant (for the holomorphic coordinate function), because $g^{\prime}$ is invariant under the parallel transports for the Levi-Civita connection for $\omega^{\prime}$. Hence the holomorphic Levi-Civita connection for $g^{\prime}$ coincides with the standard affine connection on ${\mathbb{C}}^{n}$ in terms of the holomorphic coordinate function on $U^{\prime}$. In particular, the holomorphic Levi-Civita connection for $g^{\prime}$ is flat; this proves (2).

Since the holomorphic Levi-Civita connection for $g^{\prime}$ coincides with the standard affine connection on ${\mathbb{C}}^{n}$ in terms of a holomorphic coordinate function on $U^{\prime}$ with respect to which $\omega^{\prime}$ is the standard Kähler form on ${\mathbb{C}}^{n}$, it follows immediately that $\omega^{\prime}$ is flat with respect to the holomorphic Levi-Civita connection for $g^{\prime}$. This completes the proof of the lemma. ∎

Continuing with the proof of Theorem 3.5, let $\nabla^{g}$ be the holomorphic Levi-Civita connection on $Y$ for the holomorphic Riemannian metric $g$. The $C^{\infty}$ connection on $\Omega^{1,1}_{Y}\,=\,\Omega^{1,0}_{Y}\otimes\Omega^{0,1}_{Y}$ induced by $\nabla^{g}$ is flat because $\nabla^{g}$ is flat by Lemma 3.6(2). Note that from Lemma 3.6(3) it follows immediately that the section $\omega$ of this flat bundle $\Omega^{1,1}_{\Omega}$ is flat (covariant constant).

Since $\Omega$ is a non-empty Zariski open subset of $Y$, the natural homomorphism

[TABLE]

of fundamental groups is surjective, where $y_{0}\,\in\,\Omega$. Using this it can be deduced that the above flat section $\omega$ of $\Omega^{1,1}_{\Omega}$ for the connection $\nabla^{g}$ extends to a flat section of $\Omega^{1,1}_{Y}$. Indeed, this follows immediately from the fact that the flat sections of a flat vector bundle are precisely the invariants of the monodromy representation. Note that for a $\pi_{1}(Y,y_{0})$-module $V$, we have $V^{\pi_{1}(Y,y_{0})}\,=\,V^{\pi_{1}(\Omega,y_{0})}$, because the homomorphism in (3.4) is surjective. The flat section of $\Omega^{1,1}_{Y}$ (for the connection on it induced by $\nabla^{g}$) obtained by extending $\omega$ will be denoted by $\widehat{\omega}$.

Now consider the $\mathbb{C}$–linear homomorphism

[TABLE]

given by $\widehat{\omega}$. It is connection preserving (for the connections induced by $\nabla^{g}$), because the section $\omega$ is flat. Since $\widehat{\omega}^{\prime}$ is an isomorphism over $\Omega$ (as $\omega$ is a Kähler form), and $\widehat{\omega}^{\prime}$ is connection preserving, it follows that $\widehat{\omega}^{\prime}$ is an isomorphism over the entire $Y$.

Hence $\widehat{\omega}^{\prime}$ defines a (nonsingular) hermitian structure on $Y$. This hermitian structure is Kähler because its restriction to $\Omega$ is Kähler. This Kähler structure on $Y$ is flat because its restriction to $\Omega$ is flat by Lemma 3.6(1). Therefore, $Y$ admits a finite unramified cover by a complex torus $\mathcal{T}$ such that the pull-back of $g$ to $\mathcal{T}$ is translation invariant [Be], [Bog]. ∎