On a conjecture of Beltrametti and Sommese

TL;DR

This paper proves a weak version of a conjecture relating to the existence of global sections of certain divisors on projective manifolds, with specific results in three dimensions confirming a stronger non-vanishing conjecture and applications to Seshadri constants.

Contribution

It establishes a weak version of Beltrametti and Sommese's conjecture in all dimensions and confirms a stronger non-vanishing conjecture in three dimensions, with applications.

Findings

Proved a weak version of the conjecture in arbitrary dimension.

Confirmed the non-vanishing conjecture in dimension three.

Applied results to Seshadri constants.

Abstract

Let X be a projective manifold of dimension n. Beltrametti and Sommese conjectured that if A is an ample divisor such that $K_X+(n-1)A$ is nef, then $K_X+(n-1)A$ has non-zero global sections. We prove a weak version of this conjecture in arbitrary dimension. In dimension three, we prove the stronger non-vanishing conjecture of Ambro, Ionescu and Kawamata and give an application to Seshadri constants.

Equations

$$H^0(X,{\mathcal{O}}_X(K_X+(n-1)A))\ne 0.$$

$$H^0(X,{\mathcal{O}}_X(K_X+jA))\ne 0.$$

$$H^0(X,{\mathcal{O}}_X(K_X+(n-1)A))\ne 0.$$

$$\chi (X,{\mathcal{O}}_X(K_X+tA))=h^0(X,{\mathcal{O}}_X(K_X+tA))\forall t\in \mathbb{N},$$

$$\chi (X,{\mathcal{O}}_X(K_X+tA))=(-1{)}^n\chi (X,{\mathcal{O}}_X(-tA))$$

$$\chi (X,{\mathcal{O}}_X(-tA))=[ch(-tA)\cdot td(T_X){]}_n,$$

$$ch(-tA)=\sum _{k=0}^n\frac{(-tA{)}^k}{k!}$$

$$td(T_X)=1-\frac{1}{2}{K}_X+\frac{1}{12}(K_X^2+c_2(X))+\dots +\chi (X,{\mathcal{O}}_X)$$

$$\frac{A^n}{n!}{t}^n+\frac{A^{n-1}\cdot K_X}{2(n-1)!}{t}^{n-1}+\frac{A^{n-2}\cdot (K_X^2+c_2(X))}{12(n-2)!}{t}^{n-2}+\dots +(-1{)}^n\chi (X,{\mathcal{O}}_X).$$

$$K_{X/Y}+(n-m)A=(n-m)p_Y^{\ast }{A}_Y-p_{{\mathbb{P}}^{n-m}}^{\ast }H$$

$$H^0(X,{\mathcal{O}}_X(K_X+A))\ne 0.$$

$$H^0(X,{\mathcal{O}}_X(-K_X))=H^0(X,{\mathcal{O}}_X(K_X+2(-K_X)))=0.$$

$$\epsilon (K_X+A,x)\ge 1$$

$$\epsilon (L,x)\ge 1$$

$$K_{X/Y}:=K_X-{\phi }^{\ast }{K}_Y$$

$$\mathrm{Pic}(X{)}^k\times A_k(X)\to \mathbb{Z},(D_1,\dots ,D_k,[Z])\mapsto D_1\cdots D_k\cdot [Z]$$

$$D_1\cdots D_k\cdot [Z]=:D_1\cdots D_k\cdot Z.$$

$$\alpha \cdot C\ge 0$$

$$\mathcal{F}<\delta >$$

$$D_1\cap \dots \cap D_{n-1}$$

$$c_1({\mathcal{O}}_{\mathbb{P}(\mathcal{F}{|}_C)}(1))+{\pi }^{\ast }\delta$$

$$D\cdot H_1\cdots H_{n-1}\ge 0$$

$$(K_{X^{\prime }}+{\mu }^{\ast }A)\cdot C=(K_X+A)\cdot C.$$

$$D_1\cap \dots \cap D_{n-2}$$

$$H_1\cdots H_{n-2}\cdot c_2(E<\delta >)\ge 0,$$

$$D_1\cdots D_{n-2}\cdot c_2(E<\delta >)=c_2(E<\delta >{|}_S)\ge 0.$$

$$H_1\cdots H_{n-2}\cdot c_2(X)\ge -H_1\cdots H_{n-2}\cdot (\frac{n-1}{n}{K}_X\cdot D+\frac{n-1}{2n}{D}^2),$$

$$c_1(E<\delta >)=c_1(E)+r\delta$$

$$c_2(E<\delta >)=c_2(E)+(r-1)c_1(E)\cdot \delta +\frac{r(r-1)}{2}{\delta }^2.$$

Full text

1. Introduction

1.A. The main result

The aim of this paper is to study the following effective non-vanishing conjecture, due to Beltrametti and Sommese [BS95, Conj. 7.2.7].

1.1**.**

Conjecture.* Let $X$ be a projective manifold of dimension $n$, and let $A$ be an ample Cartier divisor such that $K_{X}+(n-1)A$ is nef. Then we have*

[TABLE]

By the classification of Fujita and Ionescu [Ion86, Fuj87] the adjoint divisor $K_{X}+(n-1)A$ is nef unless we are in a very special situation ($X$ is a projective space, quadric etc.), so the conjecture applies to adjoint linear systems on almost every variety. If $X$ is a surface it is an immediate consequence of the Riemann-Roch formula and classical results on surfaces, but in higher dimension the situation is much more complicated. Conjecture 1.1 and its (conjectural) generalisation due to Ambro [Amb99], Ionescu [Cet93] and Kawamata [Kaw00] have been studied by several authors during the last years [Kaw00], [CCZ05], [Xie05], [Fuk06], [Fuk07], [Bro09], [BH08]. We prove a weak version of the Beltrametti-Sommese conjecture in arbitrary dimension:

1.2**.**

Theorem.* Let $X$ be a normal, projective variety of dimension $n\geq 2$ with at most rational singularities, and let $A$ be a nef and big Cartier divisor on $X$ such that $K_{X}+(n-1)A$ is generically nef (cf. Definition 2.6). Then there exists a $j\in\{1,\ldots,n-1\}$ such that*

[TABLE]

In particular if $A$ is effective, then

[TABLE]

If $X$ has irrational singularities, the statement still holds unless $(X,A)$ is birationally a scroll (cf. Definition 1.3) over a curve of positive genus.

Note that the conclusion of our theorem is a priori111It is an open problem due to Tsuji [MKe04] whether $h^{0}(X,{\mathcal{O}}_{X}(K_{X}+jA))\leq h^{0}(X,{\mathcal{O}}_{X}(K_{X}+(j+1)A))$ holds without assuming $A$ effective. weaker than Conjecture 1.1, but it should be equally useful for applications.

1.B. The technique

Let $X$ be a projective manifold of dimension $n$, and let $A$ be a nef and big Cartier divisor on $X$. By the Kawamata-Viehweg vanishing theorem one has

[TABLE]

so the non-vanishing problem reduces to studying the Hilbert polynomial $\chi(X,{\mathcal{O}}_{X}(K_{X}+tA))$. By Serre duality

[TABLE]

is a polynomial of degree $n$ in $t$ which can be computed by the Riemann-Roch formula

[TABLE]

where $[\ ]_{n}$ denotes the component of degree $n$ in $A(X)\otimes\mathbb{Q}$. Using the formulae

[TABLE]

for the Chern character and

[TABLE]

for the Todd class of $T_{X}$, we see that $\chi(X,{\mathcal{O}}_{X}(K_{X}+tA))$ equals

[TABLE]

The idea of the proof of Theorem 1.2 is now as follows: we argue by contradiction and suppose that for all $j\in\{1,\ldots,n-1\}$ we have $h^{0}(X,{\mathcal{O}}_{X}(K_{X}+jA))=0$. Thus $1,\ldots,n-1$ are roots of the Hilbert polynomial $\chi(X,{\mathcal{O}}_{X}(K_{X}+tA))$. It is an undergraduate exercise to translate the assumption into equations involving the coefficients of the Hilbert polynomial above (cf. Lemma 4.1), but it significantly simplifies the problem by reducing it to controlling the characteristic classes $K_{X},c_{2}(X)$ and $\chi(X,{\mathcal{O}}_{X})$. Somewhat surprisingly this immediately allows us to deal with the case where $X$ is rationally connected (so $\chi(X,{\mathcal{O}}_{X})=1$). If $X$ is a minimal model Miyaoka’s theorem tells us that the second Chern class $c_{2}(X)$ is pseudoeffective which is largely sufficient to conclude. More generally if $X$ is not uniruled we still know that $\Omega_{X}$ is generically nef, so a twisted version of Miyaoka’s statement [Fuk05, Thm.2.1], [BH08, Cor.3.13] still allows us to control the second Chern class. The most delicate case is thus when $X$ is uniruled but not rationally connected. In particular the case of birational scrolls will need some additional effort.

1.3**.**

Definition.* Let $X$ be a normal, projective variety, and let $A$ be a nef and big Cartier divisor on $X$. We say that $(X,A)$ is birationally a scroll if there exists a birational morphism $\mu:X^{\prime}\rightarrow X$ from a projective manifold $X^{\prime}$ and a fibration $\varphi:X^{\prime}\rightarrow Y$ onto a projective manifold $Y$ such that the general fibre $F$ admits a birational morphism $\tau:F\rightarrow\mathbb{P}^{n-m}$ and ${\mathcal{O}}_{F}(\mu^{*}A)\simeq\tau^{*}{\mathcal{O}}_{\mathbb{P}^{n-m}}(1)$.*

Using the foliated Mori theory due to Miyaoka and Bogomolov-McQuillan we prove the following:

1.4**.**

Theorem.* Let $X$ be a normal, projective variety of dimension $n$. Let $A$ be a nef and big Cartier divisor on $X$. If $(X,A)$ is not birationally a scroll, then $\Omega_{X}<A>$ is generically nef.*

This theorem can be seen as a foliated version of the well-known statement that if $X$ is a projective manifold and $A$ is ample, then $K_{X}+nA$ is nef unless $X\simeq\mathbb{P}^{n}$ and $A\simeq{\mathcal{O}}_{\mathbb{P}^{n}}(1)$. Note that if $(X,A)$ is birationally a scroll, then $\Omega_{X}<A>$ is not generically nef even if we assume that $\det(\Omega_{X}<A>)$ is generically nef. Indeed for $n\geq 2$, set $X:=Y\times\mathbb{P}^{n-m}$ where $Y$ is a projective manifold of dimension $1\leq m\leq n-1$ with nef canonical divisor. Let $A_{Y}$ be an ample Cartier divisor on $Y$ and $H$ be the hyperplane divisor on $\mathbb{P}^{n-m}$, then $A:=p_{Y}^{*}A_{Y}+p_{\mathbb{P}^{n-m}}^{*}H$ is ample and $K_{X}+nA$ is nef. Nevertheless the twisted bundle $\Omega_{X}<A>$ is not generically nef: this would imply that $\Omega_{X/Y}<A>$ is generically nef, yet even its determinant

[TABLE]

is not generically nef.

1.C. Generalisations and applications

In dimension three, the techniques developed for the proof of Theorem 1.2 give an affirmative answer for the stronger non-vanishing conjecture of Ambro, Ionescu and Kawamata which so far is only known in rather special cases.

1.5**.**

Theorem.* Let $X$ be a normal, projective threefold with at most $\mathbb{Q}$-factorial canonical singularities, and let $A$ be a nef and big Cartier divisor on $X$ such that $K_{X}+A$ is nef. Then we have*

[TABLE]

Note that while $K_{X}$ is only supposed to be $\mathbb{Q}$-Cartier, it is crucial for our proof to suppose that $A$ is Cartier. In fact the statement is false if $A$ is a Weil divisor which is merely $\mathbb{Q}$-Cartier: Iano-Fletcher has constructed an example [IF00, Ex.16.1] of a $\mathbb{Q}$-Fano threefold $X$ of index $30$ with terminal singularities such that

[TABLE]

As A. Broustet pointed out to me, a desingularisation of the threefold in the Iano-Fletcher example gives an example of a smooth projective threefold $X$ and $A$ a big Cartier divisor on $X$ such that $K_{X}+A$ is pseudoeffective but not effective. Thus our statement is almost optimal. A direct consequence of Theorem 1.5 is the following special case of Lazarsfeld’s conjecture on Seshadri constants (cf. [Bro09, Lemme 4.11]).

1.6**.**

Theorem.* Let $X$ be a normal, projective threefold with at most $\mathbb{Q}$-factorial canonical singularities, and let $A$ be a nef and big Cartier divisor on $X$ such that $K_{X}+A$ is nef and big. Then we have*

[TABLE]

for every $x\in X$ sufficiently general.

In particular if the anticanonical divisor of $X$ is nef and $L$ is a nef and big Cartier divisor on $X$, then

[TABLE]

for every $x\in X$ sufficiently general.

Acknowledgements. I would like to thank A. Broustet, S. Boucksom and T. de Fernex for discussions on various questions related to this paper. I would like to thank the referee for pointing out a number of rather serious mistakes in the initial version of this paper.

2. Notation and basic material

We work over the complex numbers, topological notions always refer to the Zariski topology. For general definitions we refer to Hartshorne’s book [Har77]. We will frequently use standard terminology and results of the minimal model program (MMP) as explained in [KM98] or [Deb01].

A variety is an integral scheme of finite type over $\mathbb{C}$, a manifold is a smooth variety. A fibration is a proper, surjective morphism $\varphi:X\rightarrow Y$ between normal varieties such that $\dim X>\dim Y$ and $\varphi_{*}{\mathcal{O}}_{X}\simeq{\mathcal{O}}_{Y}$, that is all the fibres are connected. Fibres are always scheme-theoretic fibres. Points are always supposed to be closed.

Let $X$ be a normal variety. The singular locus of $X$ has codimension at least two, so we have an isomorphism $Cl(X_{\operatorname*{nons}})\rightarrow Cl(X)$. We define the canonical divisor $K_{X}\in Cl(X)$ as the image of $\det T_{X_{\operatorname*{nons}}}$. Let $\varphi:X\rightarrow Y$ be a fibration between projective manifolds. We set

[TABLE]

for the relative canonical divisor.

A property (smoothness, local freeness, etc.) depending on a point $x\in X$ holds in codimension $k$, if there exists a closed subset $Z\subset X$ of codimension bigger than $k$ such that the property holds for every $x\in X\setminus Z$.

Let $X$ be a normal, projective variety. A $\mathbb{Q}$-divisor will always be a $\mathbb{Q}$-Weil divisor, not necessarily $\mathbb{Q}$-Cartier. We will frequently use that on a normal variety $X$, there is a bijection between Weil divisors $D$ and reflexive sheaves of rank one ${\mathcal{O}}_{X}(D)$ [Rei80, App., Thm. 3].

Let $X$ be a normal, projective variety. For every $k\in\{0,\ldots,\dim X\}$ we denote by $A_{k}(X)$ the group of $k$-dimensional cycles modulo rational equivalence, and by $\operatorname*{Pic}(X)$ the group of isomorphism classes of line bundles. We denote by

[TABLE]

the intersection product as defined in [Ful84, Ch.2]. More generally if we consider Cartier divisors and cycles with coefficients in $\mathbb{Q}$, we get a pairing with values in $\mathbb{Q}$ which we often abbreviate by

[TABLE]

Suppose now that $X$ is a normal, projective variety of dimension $n$ that is smooth in codimension two. Then we have an isomorphism $A_{n-2}(X_{\operatorname*{nons}})\rightarrow A_{n-2}(X)$, so if $E$ is a coherent sheaf on $X$, we define $c_{2}(E)\in A_{n-2}(X)$ as the image of $c_{2}(E|_{X_{\operatorname*{nons}}})$ under this isomorphism. In particular we define the second Chern class $c_{2}(X)$ as the image of $c_{2}(T_{X_{\operatorname*{nons}}})$.

We denote by $N^{1}(X)_{\mathbb{R}}$ the vector space of $\mathbb{R}$-Cartier divisors modulo numerical equivalence, and by $N_{1}(X)_{\mathbb{R}}$ its dual, the space of 1-cycles modulo numerical equivalence. A divisor class $\alpha\in N^{1}(X)_{\mathbb{R}}$ is pseudoeffective if it is in the closure of the cone of effective divisors in $N^{1}(X)_{\mathbb{R}}$. By [BDPP04] this is equivalent to

[TABLE]

for every $C$ a member of a covering family of curves for $X$.

Birationally, every projective manifold admits a fibration that separates the rationally connected part and the non-uniruled part: the MRC-fibration or rationally connected quotient:

2.1**.**

Theorem.* [Cam92], [GHS03], [KMM92]

Let $X$ be a uniruled, projective manifold. Then there exists a projective manifold $X^{\prime}$, a birational morphism $\mu:X^{\prime}\rightarrow X$ and a fibration $\varphi:X^{\prime}\rightarrow Y$ onto a projective manifold $Y$ such that the general fibre is rationally connected and the variety $Y$ is not uniruled.*

2.2**.**

Remarks.

1. a)

We call $Y$ the base of the MRC-fibration. This is a slight abuse of language since the MRC-fibration is only unique up to birational equivalence of fibrations (cf. [Cam04]). Since the dimension of $Y$ does not depend on the birational model, it still makes sense to speak of the dimension of the base of the MRC-fibration. 2. b)

If $X$ is a normal, projective variety, we define the MRC-fibration of $X$ to be the MRC-fibration of some desingularisation $X^{\prime}\rightarrow X$. We say that a normal variety is rationally connected if $X^{\prime}$ is rationally connected. Note that with this definition a cone over an elliptic curve is not rationally connected (it is merely rationally chain-connected). 3. c)

The MRC-fibration is almost regular, i.e. there exist open dense sets $X_{0}\subset X$ and $Y_{0}\subset Y$ such that the restriction of the rational map $\varphi:X\dashrightarrow Y$ to $X_{0}$ gives a regular (proper) fibration $\varphi|_{X_{0}}:X_{0}\rightarrow Y_{0}$. In particular we can see the general $\varphi$-fibre as a submanifold of $X$. Note also that if $Y$ has dimension one, the almost regular map $\varphi$ is regular.

2.A. $\mathbb{Q}$-twisted sheaves and generic nefness

We adapt the notion of $\mathbb{Q}$-twisted vector bundles [Laz04, Ch.6.2] to our setting.

2.3**.**

Definition.* [Miy87] Let $X$ be a normal, projective variety. A $\mathbb{Q}$-twisted sheaf*

[TABLE]

is an ordered pair consisting of a coherent sheaf ${\mathcal{F}}$ and a numerical equivalence class $\delta\in N^{1}(X)_{\mathbb{Q}}$. The $\mathbb{Q}$-twisted sheaf ${\mathcal{F}}<\delta>$ is torsion-free if ${\mathcal{F}}$ is torsion-free. If $A$ is $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on $X$ we write ${\mathcal{F}}<A>$ for the twist of ${\mathcal{F}}$ by the numerical class of $A$.

In the situations we are interested in ${\mathcal{F}}$ will either be a torsion-free sheaf on a normal variety or the sheaf of Kähler differentials of a normal variety.

2.4**.**

Definition.* Let $X$ be a normal, projective variety, and let $H_{1},\ldots,H_{n-1}$ be a collection of ample Cartier divisors. A MR-general curve $C\subset X$ is an intersection*

[TABLE]

for general $D_{j}\in|m_{j}H_{j}|$ where $m_{j}\gg 0$.

2.5**.**

Remark. The abbreviation MR stands of course for Mehta-Ramanathan, alluding to the well-known fact [MR82] that the Harder-Narasimhan filtration of a torsion-free sheaf commutes with restriction to a MR-general curve.

Let $X$ be a normal, projective variety, and let ${\mathcal{F}}$ be a coherent sheaf that is locally free in codimension one. A MR-general curve $C$ is contained in the open set where ${\mathcal{F}}$ is locally free. Thus ${\mathcal{F}}|_{C}:={\mathcal{F}}\otimes{\mathcal{O}}_{C}$ is a vector bundle and the following definition makes sense.

2.6**.**

Definition.* Let $X$ be a normal, projective variety of dimension $n$, and let ${\mathcal{F}}$ be a coherent sheaf on $X$ that is locally free in codimension one. The $\mathbb{Q}$-twisted sheaf ${\mathcal{F}}<\delta>$ is generically nef if its restriction to every MR-general curve $C$ is a nef $\mathbb{Q}$-vector bundle in the sense of [Laz04, Defn. 6.2.3], i.e.*

[TABLE]

is a nef class in $N^{1}(\mathbb{P}({\mathcal{F}}|_{C}))_{\mathbb{R}}$, where ${\mathcal{O}}_{\mathbb{P}({\mathcal{F}}|_{C})}(1)$ is the tautological line bundle on the projectivised vector bundle $\mathbb{P}({\mathcal{F}}|_{C})$.

A $\mathbb{Q}$-divisor $D$ on $X$ is generically nef if

[TABLE]

for any collection of ample Cartier divisors $H_{1},\ldots,H_{n-1}$.

2.7**.**

Remarks. a) An effective $\mathbb{Q}$-divisor $D$ is generically nef.

b) A $\mathbb{Q}$-divisor $D$ on $X$ is generically nef if for $m\gg 0$ sufficiently divisible the reflexive sheaf ${\mathcal{O}}_{X}(mD)$ is generically nef.

c) If $D$ is a pseudoeffective $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor, it is generically nef.

For lack of reference we collect some basic properties of generically nef sheaves. The proof is elementary and left to the reader.

2.8**.**

Lemma.**

1. a)

Let $X$ be a normal, projective variety of dimension $n$, and let ${\mathcal{F}}$ be a coherent sheaf on $X$ that is locally free in codimension one. The $\mathbb{Q}$-twisted sheaf ${\mathcal{F}}<\delta>$ is generically nef if and only if its bidual ${\mathcal{F}}^{**}<\delta>$ is generically nef. 2. b)

A $\mathbb{Q}$-divisor $D$ on a normal, projective variety $X$ is generically nef if and only if

[TABLE]

for any collection of nef Cartier divisors $H_{1},\ldots,H_{n-1}$. 3. c)

Let $\mu:X^{\prime}\rightarrow X$ be a birational morphism between normal varieties, and let $A$ be $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on $X$. Let $C\subset X$ be a MR-general curve222A MR-general curve does not meet the image of the exceptional locus, so we can consider it also as a curve in $X^{\prime}$., then

[TABLE]

If $K_{X}+A$ is not generically nef, then $K_{X^{\prime}}+\mu^{*}A$ is not generically nef.

Let $X$ be a projective variety that is smooth in codimension two, and let $D$ be a $\mathbb{Q}$-divisor on $X$. Let $S\subset X$ be a surface that is a complete intersection of general very ample divisors. Then $S$ is not contained $\mbox{Supp}(D)$, so the restriction $D|_{S}$ is well-defined. Moreover $S$ is smooth, so $D|_{S}$ is $\mathbb{Q}$-Cartier and the following definition makes sense.

2.9**.**

Definition.* Let $X$ be a normal, projective variety of dimension $n$ that is smooth in codimension two, and let $D$ be a $\mathbb{Q}$-divisor on $X$. We say that $D$ is nef in codimension one if for every collection $H_{1},\ldots,H_{n-2}$ of ample Cartier divisors and $S\subset X$ a complete intersection*

[TABLE]

of general $D_{j}\in|m_{j}H_{j}|$ where $m_{j}\gg 0$, the restriction $D|_{S}$ is nef.

2.10**.**

Lemma.333This statement seems now a bit too optimistic to me, see http://math.unice.fr/h̃oering/articles/remark-bs-conjecture.pdf for a correction.* Let $X$ be a normal, projective variety of dimension $n\geq 2$ that is smooth in codimension two. Let $E$ be a reflexive sheaf over $X$ such that $\det E$ is $\mathbb{Q}$-Cartier, and $\delta$ a numerical equivalence class in $N^{1}(X)_{\mathbb{Q}}$. If $E<\delta>$ is generically nef and $c_{1}(E<\delta>)$ is nef in codimension one, then*

[TABLE]

where $H_{1},\ldots,H_{n-2}$ is a collection of ample Cartier divisors on $X$.

Recall that the isomorphism $A_{n-2}(X_{\operatorname*{nons}})\rightarrow A_{n-2}(X)$ allows to define the second Chern class $c_{2}(E<\delta>)$.

Proof.

By linearity of the intersection form it is sufficient to prove that if $S$ is a complete intersection cut out by general elements $D_{j}\in|m_{j}H_{j}|$ for $m_{j}\gg 0$, then

[TABLE]

Since $X$ is smooth in codimension two, the surface $S$ is smooth. The reflexive sheaf $E$ being locally free in codimension two, the restriction $E|_{S}$ is a vector bundle. Moreover $E<\delta>|_{S}$ is generically nef and $c_{1}(E<\delta>|_{S})$ is nef. We conclude with [LM97, Thm. 8’]. ∎

2.11**.**

Corollary.444This statement seems now a bit too optimistic to me, see http://math.unice.fr/h̃oering/articles/remark-bs-conjecture.pdf for a correction.*

Let $X$ be a normal, projective variety of dimension $n\geq 2$ that is smooth in codimension two. Let $D$ be a nef $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on $X$ such that $\Omega_{X}<\frac{1}{n}D>$ is generically nef and $K_{X}+D$ is nef in codimension one. Then we have*

[TABLE]

where $H_{1},\ldots,H_{n-2}$ is any collection of nef Cartier divisors on $X$.

We recall that the usual formulas for tensor products of vector bundles extends to $\mathbb{Q}$-vector bundles [Laz04, Ch.6.2, Ch.8.1]: let $X$ be a normal, projective variety that is smooth in codimension two, and let $E$ be a coherent sheaf of rank $r$ over $X$. If $\delta\in N^{1}(X)_{\mathbb{Q}}$ is a numerical class, then

[TABLE]

[TABLE]

Proof of Corollary 2.11.

By the linearity of the intersection form, it is sufficient to show the statement in the case where the Cartier divisors $H_{i}$ are ample. Moreover by Lemma 2.8,a) the sheaf $\Omega_{X}<\frac{1}{n}D>$ is generically nef if and only its bidual $\Omega_{X}^{**}<\frac{1}{n}D>$ is generically nef. Since $X$ is smooth in codimension two, we have $c_{2}(\Omega_{X})=c_{2}(\Omega_{X}^{**})$. Therefore Lemma 2.10 applies and yields

[TABLE]

Since by Formula (3)

[TABLE]

we get

[TABLE]

∎

2.B. Some technical lemmas

The following lemma shows that we can reduce the non-vanishing problem to non-singular varieties.

2.12**.**

Lemma.* Let $X$ be a normal, projective variety of dimension $n$, and let $A$ be a Cartier divisor on $X$. Let $\nu:X^{\prime}\rightarrow X$ be a desingularisation. Then for all $j\in\mathbb{Z}$ we have an inclusion:*

[TABLE]

Proof.

Since $A$ is Cartier, the projection formula yields

[TABLE]

Note that since $\nu_{*}{\mathcal{O}}_{X^{\prime}}(K_{X^{\prime}})\otimes{\mathcal{O}}_{X}(jA)$ is torsion-free we have an inclusion

[TABLE]

Moreover for any reflexive sheaf ${\mathcal{F}}$ on a normal variety we have

[TABLE]

where $j:X_{\operatorname*{nons}}\hookrightarrow X$ is the inclusion. Thus $(\nu_{*}{\mathcal{O}}_{X^{\prime}}(K_{X^{\prime}})\otimes{\mathcal{O}}_{X}(jA))^{**}$ and ${\mathcal{O}}_{X}(K_{X}+jA)$ are isomorphic since they coincide on $X_{\operatorname*{nons}}$ and we get an inclusion

[TABLE]

∎

2.13**.**

Proposition.* Let $X$ be a normal, projective variety of dimension $n$, and let $A$ be a nef and big Cartier divisor on $X$. Then the following holds:*

1. a)

There exists a $j\in\{1,\ldots,n+1\}$ such that $H^{0}(X,{\mathcal{O}}_{X}(K_{X}+jA))\neq 0$. In particular the divisor $K_{X}+(n+1)A$ is generically nef. 2. b)

If $(K_{X}+nA)\cdot A^{n-1}\geq 0$, there exists a $j\in\{1,\ldots,n\}$ such that $H^{0}(X,{\mathcal{O}}_{X}(K_{X}+jA))\neq 0$. 3. c)

If $(K_{X}+nA)\cdot A^{n-1}<0$, there exists a birational morphism $\tau:X\rightarrow\mathbb{P}^{n}$ such that ${\mathcal{O}}_{X}(A)\simeq\tau^{*}{\mathcal{O}}_{\mathbb{P}^{n}}(1)$.

Proof.

By Lemma 2.12 statement a) follows from [Laz04, Prop.9.4.23].

b)+c) Let $\nu:X^{\prime}\rightarrow X$ be a desingularisation. We have $\nu_{*}K_{X^{\prime}}=K_{X}$, so by the projection formula

[TABLE]

Thus the condition lifts to $X^{\prime}$ and we conclude by Lemma 2.12 and [Fuj89, Thm.2.2]. ∎

The following basic fact is well-known to experts. For the convenience of the reader we include a proof.

2.14**.**

Lemma.* Let $\varphi:X\rightarrow Y$ be a fibration between projective manifolds $X$ and $Y$, and let $A$ nef and big $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on $X$. Suppose that for a general fibre $F$ one has*

[TABLE]

where $D$ is a Cartier divisor on $F$ such that $D\sim_{\mathbb{Q}}K_{F}+A|_{F}$. Then $K_{X/Y}+A$ is pseudoeffective.

Proof.

It is sufficient to show that $m(K_{X/Y}+A)$ is pseudoeffective for $m\gg 0$ sufficiently divisible so we choose $m\in\mathbb{N}$ such that $mA$ is Cartier and $H^{0}(F,{\mathcal{O}}_{F}(mK_{F}+mA|_{F}))\neq 0$. Hence the direct image sheaf $\varphi_{*}({\mathcal{O}}_{X}(m(K_{X/Y}+A)))$ is not zero. The Cartier divisor $mA$ is nef and big, so it follows from [Vie95, Ch.2], [Cam04, Thm.4.13], [BP08, Thm.0.2] that $\varphi_{*}({\mathcal{O}}_{X}(m(K_{X/Y}+A)))$ is weakly positive in the sense of Viehweg. Since ${\mathcal{O}}_{X}(m(K_{X/Y}+A))$ has rank one, the canonical morphism

[TABLE]

is generically surjective, so ${\mathcal{O}}_{X}(m(K_{X/Y}+A))$ is also weakly positive. Thus the divisor $m(K_{X/Y}+A)$ is pseudoeffective. ∎

If $A$ is a Cartier divisor we can combine Lemma 2.14 and Proposition 2.13 to obtain:

2.15**.**

Proposition.* Let $X$ be a projective manifold of dimension $n$. Let $\mu:X^{\prime}\rightarrow X$ and $\varphi:X^{\prime}\rightarrow Y$ be a model of the MRC-fibration (cf. Theorem 2.1), and denote by $m$ the dimension of $Y$. Let $A$ be a nef and big Cartier divisor on $X$. Then*

[TABLE]

is pseudoeffective. If

[TABLE]

is not pseudoeffective, the general $\varphi$-fibre $F$ admits a birational morphism $\tau:F\rightarrow\mathbb{P}^{n-m}$ such that ${\mathcal{O}}_{F}(A)\simeq\tau^{*}{\mathcal{O}}_{\mathbb{P}^{n-m}}(1)$.

In particular $K_{X}+(n-m+1)A$ is pseudoeffective. If $K_{X}+(n-m)A$ is not pseudoeffective, the manifold $F$ admits a birational morphism $\tau:F\rightarrow\mathbb{P}^{n-m}$ such that ${\mathcal{O}}_{F}(A)\simeq\tau^{*}{\mathcal{O}}_{\mathbb{P}^{n-m}}(1)$.

The second statement of the proposition is a consequence of the first and the fundamental result due to Boucksom, Demailly, Păun and Peternell [BDPP04, Cor.0.3] on the pseudoeffectiveness of the canonical bundle of a non-uniruled, projective manifold.

3. The cotangent sheaf of uniruled varieties

Theorem 1.4 will be a consequence of the following statement.

3.1**.**

Theorem.* Let $X$ be a normal, projective variety of dimension $n$. Let $A$ be a nef and big $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on $X$. Then the $\mathbb{Q}$-twisted sheaf $\Omega_{X}<A>$ is generically nef (cf. Definition 2.6) unless there exists a birational morphism $\mu:X^{\prime}\rightarrow X$ from a projective manifold $X^{\prime}$ and a fibration $\varphi:X^{\prime}\rightarrow Y$ onto a projective manifold $Y$ of dimension $m<n$ such that the general fibre $F$ is rationally connected and*

[TABLE]

where $D$ is any Cartier divisor on $F$ such that $D\sim_{\mathbb{Q}}K_{F}+j\mu^{*}A|_{F}$ with $j\in[0,n-m]\cap\mathbb{Q}$.

Let $\varphi:X\rightarrow Y$ be a fibration between projective manifolds, and let $\Omega_{X}\rightarrow\Omega_{X/Y}\rightarrow 0$ be the canonical map between the sheaves of Kähler differentials. We define the relative tangent sheaf $T_{X/Y}$ to be the saturation of

[TABLE]

in $T_{X}$, and $\det T_{X/Y}$ the divisor corresponding to its determinant. The main difficulty of the proof is that in general the relative canonical bundle of a fibration does not coincide with the dual of $\det T_{X/Y}$. We overcome this difficulty by making an appropriate base change.

Proof of Theorem 3.1.

Let us assume that $\Omega_{X}<A>$ is not generically nef. We fix $L_{1},\ldots,L_{n-1}$ ample Cartier divisors on $X$ such that $\Omega_{X}<A>$ is not generically nef with respect to $L_{1},\ldots,L_{n-1}$. Let

[TABLE]

be a MR-general curve where $D_{i}\in|m_{i}L_{i}|$ general and $m_{i}\gg 0$ such that $\Omega_{X}<A>|_{C}$ is not nef. If ${\mathcal{F}}<A>$ is a non-zero torsion-free $\mathbb{Q}$-twisted sheaf on $X$, we define the slope

[TABLE]

By Equation (2) one has

[TABLE]

By definition the $\mathbb{Q}$-twisted sheaf ${\mathcal{F}}<A>$ is semistable if for every non-zero torsion-free subsheaf ${\mathcal{E}}\subset{\mathcal{F}}$, we have $\mu({\mathcal{E}}<A>)\leq\mu({\mathcal{F}}<A>)$.

Denote by $T_{X}:=\Omega_{X}^{*}$ the tangent sheaf of $X$, and let

[TABLE]

be the Harder-Narasimhan filtration of $T_{X}$ with respect to $L_{1},\ldots,L_{n-1}$. Then for $i=1,\ldots,r$, the graded pieces ${\mathcal{G}}_{i}:={\mathcal{F}}_{i}/{\mathcal{F}}_{i-1}$ are semistable torsion-free sheaves and if $\mu({\mathcal{G}}_{i})$ denotes the slope, we have a strictly decreasing sequence

[TABLE]

Since twisting with a $\mathbb{Q}$