Algebraic integrability of foliations with numerically trivial canonical bundle

introduction

the algebraic holonomy group of ${\mathcal{E}}$ (cf. Definition 2.13) is connected.

Suppose further that ${\mathcal{E}}$ is pseudoeffective (cf. Definition 2.1). Then

Nakayama [Nak99, Thm.B] and Druel [Dru17, Thm.6.1] obtained similar results for vector bundles of small rank. The recent progress on algebraic integrability of foliations by Campana-Pǎun [CP15] and Druel [Dru17] yields an immediate application:

By [BK08, Prop.5] the stability of all the $S^{[l]}{\mathcal{F}}$ is equivalent to assuming that ${\mathcal{F}}$ is stable and the algebraic holonomy group is $\mbox{SL}(F_{x})$ or $\mbox{Sp}(F_{x})$. It seems possible that the stability of ${\mathcal{F}}$ is enough to imply the algebraicity of leaves (cf. also [Tou08, LPT11, LPT13, PT13] for classification results of foliations with $c_{1}({\mathcal{F}})=0$).

The proof of Theorem 1.1 is surprisingly simple. Druel’s proof [Dru17, Thm.6.1] uses the stability of ${\mathcal{E}}$ to describe the components of the restricted base locus $B_{-}(\zeta)$ (see Section 3.A) that are divisors or generically finite over the base. Our key observation is that the systematic use of the symmetric powers $S^{[l]}{\mathcal{E}}$ allows to control irreducible components of $B_{-}(\zeta)$ of any codimension. An intersection computation essentially reduces Theorem 1.1 to the following:

B. Minimal models with trivial canonical class

The main motivation for our study of stable sheaves with numerically trivial determinant is to extend the Beauville-Bogomolov decomposition [Bea83] to singular spaces. Following [GKP16b], let us explain the notions of singular Calabi-Yau and singular irreducible symplectic varieties.

Definition. Let $X$ be a normal projective variety of dimension $n\geq 2$ with at most canonical singularities such that $\omega_{X}\simeq{\mathcal{O}}_{X}$.

$X$ is a Calabi-Yau variety if $H^{0}(Y,\Omega^{[q]}_{Y})=0$ for all integers $1\leq q\leq n-1$ and all covers $Y\to X$, étale in codimension $1$;

$X$ is irreducible symplectic if there exists a holomorphic $2$-form $\sigma\in H^{0}(X,\Omega^{}_{X})$ such that for all covers $\gamma:Y\to X$, étale in codimension $1$, the exterior algebra of holomorphic reflexive forms is generated by the reflexive pull-back $\gamma^{[*]}(\sigma).$

The Beauville-Bogomolov decomposition for a Ricci flat compact Kähler manifold $X$ states that after étale cover $X$ is a product of a torus, Calabi-Yau and irreducible symplectic manifolds. In the last years there has been an intensive effort [GKP16c, Dru17, GGK17, DG17] to generalise this statement to minimal models. Theorem 1.1 allows to complete this challenge:

Theorem. Let $X$ be a normal projective variety with at most klt singularities such that $c_{1}(K_{X})=0$.

into normal projective varieties with trivial canonical bundles, such that

the $Y_{j}$ are (singular) Calabi-Yau varieties;

the $Z_{k}$ are (singular) irreducible symplectic varieties.

Although this significantly improves results from earlier papers, one should note that Theorem 1.5 is based to equal parts on a tripod consisting of Druel’s algebraic integrability theorem [Dru17, Thm.1.4], the holonomy decomposition of Greb-Guenancia-Kebekus [GGK17, Thm.B and Prop.D] and our Theorem 1.1. For the proof we simply follow the arguments of [Dru17, Thm.1.6].

Theorem. Let $X$ be a normal projective variety with at most canonical singularities. Suppose that $X$ is smooth in codimension two and $c_{1}(K_{X})=0$. Assume that the tangent sheaf $T_{X}$ is strongly stable in the sense of [GKP16c, Defn.7.2].

Then both the reflexive cotangent sheaf $\Omega_{X}^{}$ and the tangent sheaf $T_{X}$ are not pseudoeffective. In particular if $X$ is Calabi-Yau or irreducible symplectic manifold (in the sense of Definition 1.4), then $\Omega_{X}^{}$ and $T_{X}$ are not pseudoeffective (cf. Definition 2.1).

This result was proven for surfaces in [Nak04, Thm.IV.4.15] [BDPP13, Thm.7.8] and for threefolds in [Dru17, Cor.6.5].

C. Almost nef sheaves

While Theorem 1.1 is sufficiently strong for the proof of the decomposition theorem, it is in general not easy to control the stability of all the symmetric powers. We therefore also consider a weaker positivity notion:

Definition. Let $X$ be a normal projective variety, and let ${\mathcal{E}}$ be a reflexive sheaf on $X$. We say that ${\mathcal{E}}$ is almost nef, if there exist at most countably many proper subvarieties $S_{j}\subsetneq X$ such that the following holds: let $C\subset X$ be a curve such that ${\mathcal{E}}|_{C}={\mathcal{E}}\otimes{\mathcal{O}}_{C}$ is not nef, then $C$ is contained in $\cup_{j\in J}S_{j}$.

Using completely different techniques we prove the following

Based on analytic techniques a slightly weaker statement was shown in [CH17, Prop.2.11]. In particular we obtain a positive answer to a question asked in [DPS01], without any assumption on the stability:

Then $T_{X}$ is not almost nef, i.e. there exists a dominating family of irreducible curves $C_{t}\subset X_{\operatorname*{nons}}$ such that $T_{X}|_{C_{t}}$ is not nef.

This statement was shown for smooth threefolds in [BDPP13, Thm.7.7]. The assumption of Theorem 1.9 is too weak to use the techniques from Theorem 1.1. Nevertheless we expect that the stronger conclusion of Theorem 1.6 also holds for minimal models with trivial canonical class such that the decomposition does not contain an abelian factor.

Acknowledgements. We thank S. Cantat and P. Graf for some very useful references. This work was partially supported by the Agence Nationale de la Recherche grant project FoliageANR-16-CE40-0008 and by the DFG project "Zur Positivität in der komplexen Geometrie".

Notation, basic facts and proof of Proposition 1.3

We work over the complex numbers, for general definitions we refer to [Har77]. We use the terminology of [Deb01] and [KM98] for birational geometry and notions from the minimal model program and [Laz04a] for notions of positivity. Manifolds and varieties will always be supposed to be irreducible. Given a normal variety $X$ we denote by $T_{X}:=\Omega_{X}^{*}$ its tangent sheaf. The sheaf of reflexive differentials of degree $q\in\{1,\ldots,\dim X\}$ is given by

A finite map $\gamma:X^{\prime}\rightarrow X$ between normal varieties is quasi-étale if its ramification divisor is empty (or equivalently, by purity of branch, $\gamma$ is étale over the smooth locus of $X$).

for all $q\geq 0$ and all locally free sheaves ${\mathcal{G}}$ on $X$.

This vanishing will be crucial in our argumentation.

Definition. Let $X$ be a normal variety, and let ${\mathcal{E}}$ be a reflexive sheaf on $X$.

Let $X_{0}\subset X$ be the locus where $X$ is smooth and ${\mathcal{E}}$ is locally free, and let

By [Nak04, III.5.10.3] there exists an effective divisor $\Lambda$ supported on $D$ such that

Using the defining property of a tautological class, the arguments of [Dru17, Lemma 2.7] apply literally to show the following:

Definition. [DPS94] Let $X$ be a normal projective variety, and let ${\mathcal{E}}$ be a locally free sheaf on $X$. We say that ${\mathcal{E}}$ is numerically flat if both ${\mathcal{E}}$ and ${\mathcal{E}}^{*}$ are nef. This is equivalent to assuming that both ${\mathcal{E}}$ and $\det{\mathcal{E}}^{*}$ are nef.

- By [DPS94] we know that if ${\mathcal{E}}$ is numerically flat, then all the Chern classes vanish and ${\mathcal{E}}$ is semi-stable for any ample polarization.

- Let ${\mathcal{E}}$ be a locally free sheaf on $X$ that is flat, i.e. ${\mathcal{E}}$ is given by a linear representation of $\pi_{1}(X)$. If $c_{1}(\det{\mathcal{E}})=0$, then ${\mathcal{E}}$ is nef (e.g. [JR13, Thm.1.1]), hence numerically flat.

- Strictly speaking, both statements are established so far only when $X$ is smooth. However, it is not difficult to derive the assertions in the normal case by passing to a desingularisation; see [GKP16b] for the techniques.

Definition. Let $X$ be a normal projective variety of dimension $n$ that is smooth in codimension two, and let $H$ be an ample divisor on $X.$ Let ${\mathcal{E}}$ be a reflexive free coherent sheaf on $X$.

For $m\gg 0$ let $D_{j}$ be general divisors in $|mH|$ and set

observing that ${\mathcal{E}}_{S}$ is locally free.

In this paper we will use the standard notion of slope-(semi-)stability of torsion-free sheaves ${\mathcal{F}}$ with respect to an ample line bundle $H$ as defined in [MP97, Part I, Lect.III], and denote by $\mu_{H}({\mathcal{F}})$ the slope of ${\mathcal{F}}$ with respect to $H$. Miyaoka has shown the following useful basic fact:

Proposition. [Miy87], [Laz04b, Prop.6.4.11] Any semistable vector bundle ${\mathcal{E}}$ over a smooth curve with $c_{1}({\mathcal{E}})=0$ is nef.

The behaviour of stability under restrictions will play an important role.

Remark. The abbreviation MR stands of course for Mehta-Ramanathan, alluding to the well-known fact [MR82, Fle84] that for $m\gg 0$ and a general $D_{j}\in|mH|$, the restriction ${\mathcal{F}}_{C}$ is indeed semistable.

For lack of reference we include the following singular version of the restriction theorem of Mehta-Ramanathan [MR84, Thm.4.3].

Remark. As in Definition 2.9 we will call $C$ a MR-general curve.

so the divisors $D_{i}^{\prime}$ are strict transforms of general divisors $D_{i}\in|mH|$. Since $X$ is normal, the intersection $C:=D_{1}\cap\ldots\cap D_{n-1}$ is in the smooth locus of $X$, thus the curves $C^{\prime}$ and $C$ can be identified. Since ${\mathcal{E}}$ is torsion-free, hence locally free in codimension one, the sheaves $(\mu^{*}{\mathcal{E}})^{**}$ and ${\mathcal{E}}$ identify in a neighbourhood of $C=C^{\prime}$, so ${\mathcal{E}}_{C}=(\mu^{*}{\mathcal{E}})^{**}|_{C^{\prime}}$ is stable. ∎

(2) More generally, the restriction to $D_{1}\cap\ldots\cap D_{k}$ with $1\leq k\leq n-1$ is $H$-stable. Indeed if the intermediate restriction is not stable, then the restriction to a MR-general curve is not stable.

We recall the notion of the algebraic holonomy, introduced by Balaji and Kollár [BK08]. For convenience, given a reflexive sheaf ${\mathcal{E}}$ and $x\in X,$ we set

where $m_{x}$ is the maximal ideal at $x.$

Definition. Let $X$ be a normal projective variety of dimension $n$, and let $H$ be an ample line bundle on $X.$ Let ${\mathcal{E}}$ be a reflexive sheaf on $X$ that is $H$-stable with slope $\mu_{H}({\mathcal{E}})=0.$ Fix a smooth point $x\in X$ such that ${\mathcal{E}}$ is free near $x$. The algebraic holonomy group of ${\mathcal{E}}$ at $x$ is the (unique) smallest subgroup

such that the following holds: for any smooth curve $C$ with fixed point $c\in C$ and any morphism $g:C\to X$ with $g(c)=x$ and ${\mathcal{E}}$ locally free near $g(C),$ and such that $g^{*}({\mathcal{E}})$ is poly-stable, the Narasimhan-Seshadri representation $\rho:\pi_{1}(D,c)\to GL(g^{*}(E_{x}))$ has image in $H_{x}({\mathcal{E}}).$

For details and explanations we refer to [BK08]. The following useful lemma is well-known to specialists:

By [NS65, Sect.12, Cor.1] the vector bundle ${\mathcal{E}}$ is defined by an irreducible unitary representation $\rho:\pi_{1}(C,x)\rightarrow\mbox{U}(E_{x})$, so a symmetric power $S^{m}{\mathcal{E}}$ is defined by the unitary representation $S^{m}\rho$ induced on the symmetric power $S^{m}E_{x}$. By [NS65, Sect.12, Cor.2] the bundle $S^{m}{\mathcal{E}}$ is stable if and only if the representation $S^{m}\rho$ is irreducible. Suppose that this is not the case. Since the image of $\rho$ is dense in the algebraic holonomy group $H_{x}({\mathcal{E}})$ [BK08, Thm.1(2)], the induced representation of $H_{x}({\mathcal{E}})$ on $S^{m}E_{x}$ is reducible. Yet it is a classical result of the representation theory of Lie groups [Wey49] that the symmetric representations of $\mbox{SL}(E_{x})$ or $\mbox{Sp}(E_{x})$ are irreducible. ∎

B. Subvarieties of projectivised bundles

We will now prove the key lemma of this paper.

Suppose that the locally free sheaf $S^{l}{\mathcal{E}}$ is stable. Then exactly one of the following holds:

$Z$ is covered by curves $C^{\prime}$ such that $\zeta\cdot C^{\prime}=0$ and such that the map $C^{\prime}\rightarrow C$ is étale.

and ${\rm rk}S^{l}{\mathcal{E}}>{\rm rk}\varphi_{*}({\mathcal{O}}_{Z}(l))$. Since $S^{l}{\mathcal{E}}$ is nef by Proposition 2.8, its quotient $\varphi_{*}({\mathcal{O}}_{Z}(l))$ is also nef. If $\varphi_{*}({\mathcal{O}}_{Z}(l))$ were not ample, by Hartshorne’s theorem [Laz04b, Thm.6.4.15] there would exist a quotient

such that $c_{1}(Q)=0$. Yet this quotient would destabilise $S^{l}{\mathcal{E}}$, so $\varphi_{*}({\mathcal{O}}_{Z}(l))$ has to be ample. Since ${\mathcal{O}}_{Z}(l)$ is $\varphi$-globally generated, we have a surjective morphism

Since $\varphi^{*}\varphi_{*}({\mathcal{O}}_{Z}(l))$ is semiample, its quotient ${\mathcal{O}}_{Z}(l)$ is a semiample line bundle. Denote by $\tau:Z\rightarrow B$ the morphism with connected fibers defined by some positive multiple of ${\mathcal{O}}_{Z}(l)$. Then $\tau$ is not birational if and only if

Suppose now that $\zeta^{d}\cdot Z=0$, so $\tau$ is not birational. The restriction of $\zeta$ to any $\tau$-fibre is numerically trivial, so the curves $C^{\prime}$ contained in the $\tau$-fibres define a covering family such that $\zeta\cdot C^{\prime}=0$.

Let us now show that for any curve $C^{\prime}\subset Z$ such that $\zeta\cdot C^{\prime}=0$ the map $f:=\pi|_{C^{\prime}}$ is étale. Note that since ${\mathcal{O}}_{Z}(l)$ is semiample, the restriction ${\mathcal{O}}_{C^{\prime}}(l)$ to $C^{\prime}$ is a torsion line bundle. Thus for a sufficiently divisible $m\gg 0$ the exact sequence

Since $S^{m}{\mathcal{E}}$ is nef, its quotient $f_{*}({\mathcal{O}}_{C^{\prime}})$ is a nef vector bundle. Now we conclude with Lemma 2.16. ∎

Lemma. Let $C$ be a smooth projective curve, and let $f:C^{\prime}\rightarrow C$ be a surjective morphism from a projective, integral curve. If $f_{*}({\mathcal{O}}_{C^{\prime}})$ is nef, then $C^{\prime}$ is smooth and $f$ is étale.

where $Q$ is a torsion sheaf that is non-zero if and only if $\nu$ is an isomorphism. Pushing down to $C$ we obtain an exact sequence

and a Grothendieck-Riemann-Roch computation shows that $c_{1}(f_{*}({\mathcal{O}}_{C^{\prime}}(m)))=0$. Thus $S^{m}{\mathcal{E}}$ is not stable, a contradiction. ∎

Reflexive sheaves with pseudoeffective tautological class

where $A$ is an arbitrary ample divisor. Note that if $A$ is very ample then

we define the asymptotic multiplicity or vanishing order of $D$ along $\Gamma$, as defined in [Nak04, III, Lemma 1.6]; see also [FL17, Defn.2.15]. By definition it is a numerical invariant of $D$.

We extend this definition to higher codimension, following [Nak04, III, Defn. 2.2].

be the composition of an embedded resolution of $Z$, and the blow-up of the strict transform of $Z$. Let $E_{Z}\subset Y$ be the unique prime divisor mapping onto $Z$. Then we define

It is easy to check that this definition does not depend on the choice of $f,$ using [Nak04, III, Lemma 5.15],

Note that by [Nak04, III, Lemma 1.7(2)] we have

where $A$ is an arbitrary ample divisor on $P$. Thus $\sigma_{Z}(D)$ is the asymptotic vanishing order of $D$ in the generic point of the subvariety $Z$.

Suppose now that $Z$ is an irreducible component of $B_{-}(D)$. By [Dru17, Lemma 6.13] we have

By [Nak04, III, Lemma 1.7(2)] (applied to $\sigma_{E_{Z}}(\bullet)$, cf. also [Nak04, p.93, Remark (1)]) the function

is lower semicontinuous and continuous when restricted to the big cone.

The following technical lemma, an analogue of [Dru17, Lemma 6.13], will be very important.

Lemma. Let $X$ be a normal projective variety, and let $H$ be an ample line bundle on $X$. Let ${\mathcal{E}}$ be a reflexive sheaf of rank $r$ on $X$ that is $H$-semistable with $\mu_{H}({\mathcal{E}})=0$. Let

for every irreducible component $W\subset B_{-}(\zeta)$ of codimension at most $k-1$ the image $\pi(W)$ has codimension at least $2$.

Let $Z\subset B_{-}(\zeta)$ be an irreducible component of codimension $k$, and let $C\subset X$ be a very general MR-general smooth curve (with respect to $H$).

for any collection of nef divisors $H_{1},\ldots,H_{r-k}$ on $\pi^{-1}(C)$.

for every irreducible component $W\subset B_{-}(\zeta)$ of codimension at most $k-1$,

If $\pi(Z)$ has codimension at least two in $X$, then the intersection $Z\cap\pi^{-1}(C)$ is empty, and consequently the assertion of Lemma 3.4 is trivially true. Thus we can assume from now on that

We set $a_{1}:=\sigma_{Z}(\zeta)$ and observe that $a_{1}>0$ by (3). Since $\sigma_{Z}$ is a lower semicontinuous function we may suppose, possibly enlarging $n_{0}$, that

has pure codimension $k$ in $\pi^{-1}(C)$. Since $C$ is general, the intersection $Z\cap\pi^{-1}(C)$ is reduced, thus by (5)

where $\mbox{mult}_{Z\cap\pi^{-1}(C)}(\bullet)$ is the order of vanishing in any general point of $Z\cap\pi^{-1}(C)$. Consequently

is an effective cycle of pure codimension $k$ in $\pi^{-1}(C)$. Since

for any collection of nef divisor $H_{i}$ on $\pi^{-1}(C)$. Since $\frac{a_{1}}{2}$ does not depend on $n\geq n_{0}$ the statement now follows by setting $a:=(\frac{a_{1}}{2})^{k}$ and passing to the limit $n\to\infty$. ∎

B. Proof of Theorem 1.1

Let us start by showing that it is enough to prove that

In this case the existence of the quasi-étale cover such that $\nu^{[*]}({\mathcal{E}})$ is numerically flat follows from [GKP16a, Thm.1.20] (cf. also Remark 2.6).

For the proof of (6), observe first that it is sufficient to show that

Claim. Let $W\subset B_{-}(\zeta)$ be an irreducible component. Then ${\rm codim}_{X}(\pi(W))\geq 2.$

Since the class of $S$ is a positive multiple of $H^{n-2}$ this proves Theorem 1.1.

Note that for $k=1$ the unique subvariety of $P$ having codimension $k-1=0$ is $P$ itself. Since $\zeta$ is pseudoeffective by assumption, the total space $P$ is not in $B_{-}(\zeta)$. Hence the induction hypothesis holds for $k=1$.

Arguing by contradiction we suppose that there exists an irreducible component $Z\subset B_{-}(\zeta)$ of codimension $k$ such that

where $W$ is any of the varieties appearing in the induction hypothesis.

Since ${\mathcal{E}}_{C}$ is stable and $c_{1}({\mathcal{E}}_{C})=0$, the restricted tautological class $\zeta_{C}$ is nef by Proposition 2.8. Thus we may apply Lemma 3.4 with $H_{1}=\ldots=H_{r-k}=\zeta_{C}$: there is a real number $a>0$ such that

Since $c_{1}({\mathcal{E}}_{C})=0$, we have $\zeta_{C}^{r}=0$. Moreover, $\zeta_{C}$ being nef, we have

Thus we see that $\zeta|_{Z\cap\pi^{-1}(C)}$ is not big. Note that this already excludes the possibility that $\pi(Z)$ has codimension one in $X$: in this case $Z\cap\pi^{-1}(C)$ would be non-empty and contained in the $\pi$-fibres, so $\zeta_{C}$ would be ample on $Z\cap\pi^{-1}(C).$

Thus we will assume from now on that $\pi(Z)=X.$ Since $S^{l}{\mathcal{E}}_{C}$ is stable and since $\zeta|_{Z\cap\pi^{-1}(C)}$ is not big, Lemma 2.15 exhibits a curve $C^{\prime}\subset(Z\cap\pi^{-1}(C))$ such that

and $f:=\pi_{C^{\prime}}:C^{\prime}\rightarrow C$ is étale. Clearly the vector bundle $f^{*}{\mathcal{E}}_{C}$ is not stable since it has the numerically trivial quotient $f^{*}{\mathcal{E}}_{C}\twoheadrightarrow\zeta|_{C^{\prime}}$. On the other hand, the vector bundle ${\mathcal{E}}_{C}$ is stable and the holonomy group $H_{x}({\mathcal{E}})=H_{x}({\mathcal{E}}_{C})$ is connected by assumption, so the pull-back $f^{*}{\mathcal{E}}_{C}$ is stable by [Dru17, Lemma 6.22]. Thus we have reached a contradiction.

C. Proof of Corollary 1.2

By [Dru17, Prop.8.4], it suffices to show that ${\mathcal{F}}^{*}$ not pseudoeffective (cf. Definition 2.1). Suppose to the contrary that ${\mathcal{F}}^{*}$ is pseudoeffective. Since $X$ is simply connected, the algebraic holonomy group $H_{x}({\mathcal{F}}^{*})$ is connected [BK08, Prop.4]. Hence we may apply Theorem 1.1 and conclude that ${\mathcal{F}}^{*}$ is numerically flat. Thus we have $c_{2}({\mathcal{F}})=c_{2}({\mathcal{F}}^{*})=0$ [DPS94], contrary to our assumption.

Proof of Theorems 1.5 and 1.6

We start with the proof of the decomposition theorem.

We will follow the approach of Druel [Dru17].

By [Dru17, Thm.1.4] and [GGK17, Thm. B] there exists a quasi-étale finite cover

where $A$ is an abelian variety and $Z$ a normal projective variety with the following properties.

into reflexive integrable subsheaves ${\mathcal{E}}_{j}$ of rank $m_{j}$ which are strongly stable in the sense of [GKP16c, Defn.7.2] for any polarization.

By [GGK17, Thm. B and Prop.D] there exists a singular Ricci-flat Kähler metric, inducing a Riemannian metric $g$, such that for a smooth point $x\in X$, the decomposition

corresponds to the decomposition of $T_{X,x}$ into irreducible representations according to the action of the differential-geometric holonomy group $G$ of $g$ at $x$. Recall that $E_{j,x}~:=({\mathcal{E}}_{j})_{x}/m_{x}({\mathcal{E}}_{j})_{x}$ with $m_{x}$ the maximal ideal in $x$. Moreover the differential-geometric holonomy groups $G_{j}$ of the direct factors ${\mathcal{E}}_{j}$ are either $SU(m_{j})$ or $Sp(\frac{m_{j}}{2})$, the representation $\rho_{j}:G_{j}\rightarrow\mbox{GL}(E_{j,x})$ being the standard one.

By [Dru17, Prop.4.10], it suffices to show that the leaves of the foliations ${\mathcal{E}}_{j}$ are algebraic. By [Dru17, Prop. 8.4] it is furthermore sufficient to show that ${\mathcal{E}}_{j}^{*}$ is not pseudoeffective (cf. Definition 2.1).

Proof of the claim. Since ${\mathcal{F}}$ is strongly stable in the sense of [GKP16c, Defn.7.2] we may suppose by [BK08, Lemma 40], [Dru17, Lemma 6.20], possibly after passing to a quasi-étale cover, that the algebraic holonomy group $H_{x}({\mathcal{F}})$ is connected.

Thus it remains to show that all reflexive symmetric powers $S^{[l]}{\mathcal{F}}$ are $H-$stable for some ample divisor $H$. By Lemma 2.14 it is sufficient to show that the algebraic holonomy group of ${\mathcal{F}}$ is $\mbox{SL}(F_{x})$ or $\mbox{Sp}(F_{x})$. We can now follow the proof of [GGK17, Thm.12.15]: by [GGK17, Prop.12.14], it suffices to that $S^{[l]}{\mathcal{F}}$ is indecomposable for some $l\geq 2$. Now observe that the (differential-geometric) Bochner principle [GGK17, Thm. 8.1] also applies to the direct factors of the tangent sheaf $T_{X}$. Thus any direct summand of $S^{}{\mathcal{F}}$ would create a $G$-invariant subspace of the $G$-representation $S^{2}F_{x}$. However, for $G=SU(F_{x})$ and $G=Sp(F_{x})$, the induced representation on the second symmetric power is irreducible [Wey49], [FH91, Sect.24.1 and 24.2]. Thus $S^{}{\mathcal{F}}$ is indecomposable and so are all the sheaves $S^{[m]}{\mathcal{F}}$, completing the proof of Theorem 1.5.

B. Proof of Theorem 1.6

The claim is invariant under quasi-étale covers, so by [GGK17, Thm. E], possibly after a quasi-étale cover, the variety $X$ is Calabi-Yau or irreducible symplectic. By [GGK17, Prop.12.14, Thm.12.15] all the symmetric powers $S^{[l]}T_{X}$ are $H$-stable.

Arguing by contradiction suppose that $T_{X}$ is pseudoeffective. Hence by Theorem 1.1 we have $c_{2}(X)\cdot H^{n-2}=0$. But then $X$ is a quasi-étale quotient of a torus by [GKP16a, Thm.1.17], contradicting our assumption that $T_{X}$ is strongly stable. By duality all the symmetric powers $S^{[l]}\Omega_{X}^{}$ are $H$-stable, so the proof for $\Omega_{X}^{}$ is analogous.

Almost nef sheaves

We start with some technical preparation.

Then ${\mathcal{E}}$ is numerically flat.

By the classical Bogomolov-Gieseker inequality, [Miy87, 4.7], we have $c_{2}({\mathcal{E}})\geq 0$. A theorem of Simpson [Sim92, Cor.3.10] states moreover that $c_{2}({\mathcal{E}})=0$ if and only if ${\mathcal{E}}$ is numerically flat. Thus it suffices to show that $c_{2}({\mathcal{E}})\leq 0.$

On the other hand since $c_{1}({\mathcal{E}})=0$ the Leray-Hirsch relation

simplifies to $\zeta^{r}=-\pi^{*}c_{2}({\mathcal{E}})\zeta^{r-2}$. Consequently,

Since $a>0$ we arrive at $c_{2}({\mathcal{E}})\leq 0$. ∎

Lemma. Let $S$ be a smooth projective surface, and let ${\mathcal{E}}$ be a locally free sheaf of rank $r\geq 2$ on $S$. Let $(C_{i})_{i\in I}$ be an at most countable collection of irreducible curves in $S$.

Let $H$ be an ample divisor on $S$. Then for every $m\gg 0$ there exists a divisor $D\in|\zeta+m(r-1)\pi^{*}H|$ with the following properties:

Let $\{p_{1},\ldots,p_{k}\}\subset S$ be the non-flat locus of $\pi|_{D}:D\rightarrow S$. Then $p_{j}\not\in C_{i}$ for every $j\in\{1,\ldots,k\}$ and $i\in I$

For every $i\in I$ the restriction of the tautological class $\zeta|_{D\cap\pi^{-1}(C_{i})}$ is nef.

Lemma. Let $Z$ be a projective variety of dimension $d$, and let ${\mathcal{E}}$ be a locally free sheaf of rank $r$ on $Z$. Let $V\subset H^{0}(Z,{\mathcal{E}})$ be a linear subspace such that ${\mathcal{E}}$ is generated by the $V$, i.e. we have a surjective evaluation morphism

a) If $r>d$, then a general choice of $r-d$ elements of $V$ defines a subbundle

b) If $r\geq d$, then a general choice of $r-d+1$ elements of $V$ defines an injective morphism

that is a subbundle in the complement of finitely many points.

It is well-known [Har77, II,Ex.8.2] that if $r>d$, then a general section does not vanish, so a) follows by induction on $r-d$. It is also well-known that if $r=d$, then a general section vanishes only in finitely many points, so b) follows from a) and induction. ∎

For $m\gg 0$ we know by Serre’s theorem that the sheaf ${\mathcal{E}}^{*}\otimes{\mathcal{O}}_{S}(mH)$ is globally generated; we denote by $V$ the space of global sections.

For every $i\in I$, the restricted vector bundle $({\mathcal{E}}^{*}\otimes{\mathcal{O}}_{S}(mH))|_{C_{i}}$ is generated by global sections of $V_{i}:=V|_{C_{i}}$ (i.e. those global sections of $({\mathcal{E}}^{*}\otimes{\mathcal{O}}_{S}(mH))|_{C_{i}}$ that lift to global sections on $S$). By Lemma 5.3 for a general choice of elements

Since there are only countably many curves $C_{i}$ we can thus fix very general sections

Moreover by part b) of Lemma 5.3 the map is injective in the complement of finitely many points $p_{1},\ldots,p_{k}$. Dualising and tensoring with ${\mathcal{O}}_{S}(mH)$ we obtain a morphism

that is surjective in the complement of finitely many points and the restriction

Let $H$ be a very ample line bundle on $X$.

Step 1. The case $\dim X=2$. Then by assumption $X$ is smooth, hence the reflexive sheaf ${\mathcal{E}}$ is locally free. By Proposition 5.1 it is sufficient to find a divisor $D\in|\zeta+l\pi^{*}H|$ such that $\zeta^{r}\cdot D\geq 0$.

Using that almost nefness and numerically flatness of vector bundles are invariant under a birational morphism, the following variant of Theorem 1.8 follows by passing to a desingularisation.

Corollary. Let $X$ be a normal projective variety, and let ${\mathcal{E}}$ be a locally free sheaf on $X$ such that $c_{1}({\mathcal{E}})=0.$ If ${\mathcal{E}}$ is almost nef, then ${\mathcal{E}}$ is numerically flat.

References