Varieties with Ample Tangent Sheaves

TL;DR

This paper proves that any normal projective variety over complex numbers with an ample tangent sheaf must be isomorphic to complex projective space, extending Mori's theorem.

Contribution

It generalizes Mori's theorem from smooth projective manifolds to normal projective varieties with ample tangent sheaves.

Findings

Normal projective varieties with ample tangent sheaf are isomorphic to complex projective space.

Extends the classification of varieties with ample tangent bundles to a broader class.

Confirms the uniqueness of projective space in this context.

Abstract

This paper generalises Mori's famous theorem about "Projective manifolds with ample tangent bundles" to normal projective varieties in the following way: A normal projective variety over $\mathbb{C}$ with ample tangent sheaf is isomorphic to the complex projective space.

Equations

$${\mathcal{T}}_X\to {\pi }_{\ast }{\mathcal{T}}_{\hat{X}}.$$

$$X\simeq {\mathbb{P}}_n.$$

$$f:{\pi }^{\ast }{\mathcal{T}}_X\to {\mathcal{T}}_{\hat{X}}$$

$${\pi }^{\ast }{\mathcal{T}}_X$$

$$\tilde{C}$$

$$\overline{{\nu }^{\ast }f}:\mathcal{A}\text{lhook}\text{joinrel}\text{relbar}\text{joinrel}\to {\nu }^{\ast }{\mathcal{T}}_{\hat{X}}$$

Full text

A normal projective variety over $\mathbb{C}$ with ample tangent sheaf is isomorphic to the complex projective space.

1. Introduction

In this paper we give a proof for the following theorem.

Main Theorem**.**

A normal projective variety over $\mathbb{C}$ with ample tangent sheaf is isomorphic to the projective space.

We work over the field of complex numbers $\mathbb{C}$. Besides that restriction, the theorem is a generalisation to singular varieties of Mori’s famous result.

Theorem** ([Mor79]).**

An $n$-dimensional projective manifold $X$ over an algebraically closed field $\mathbb{K}$ with ample tangent bundle is isomorphic to the projective space $\mathbb{P}_{\mathbb{K}}^{n}$.

Mori’s work has been generalised over the years in various ways, for example by Andreatta and Wiśniewski [AW01]: For $X$ being $\mathbb{P}_{n}$ it suffices that $\mathcal{T}_{X}$ contains an ample subbundle. This has been altered by Aprodu, Kebekus and Peternell [AKP08, Section 4]. They add the assumption that $X$ has Picard number 1, but an ample subsheaf (not necessarily locally free) of $\mathcal{T}_{X}$ then induces $X\simeq\mathbb{P}_{n}$. Generalising those results, Liu [Liu16] recently showed that $X$ is already the projective space if $\mathcal{T}_{X}$ contains an ample subsheaf (again not necessarily locally free). Kebekus [Keb02] even characterises $\mathbb{P}_{n}$ only by using the anticanonical degree of all rational curves being greater than $n$. All these efforts, besides Ballico’s article [Bal93], keep the preliminary that $X$ is smooth. Ballico’s paper on the other hand treats mainly positive characteristic, as he requires the tangent sheaf to be locally free. Which, the Zariski-Lipman conjecture suggests, is most likely never the case over the complex numbers, if $X$ is singular.

Outline of our proof

We consider a special desingularisation $\hat{X}$ of the given variety $X$ of dimension $\geq 2$ (normal curves are smooth) and prove that $\hat{X}$ is the projective space. As $\mathbb{P}_{n}$ is minimal, $X$ itself is already the projective space. To show that $\hat{X}$ is the projective space, we combine two strong results.

First, we relate $\mathcal{T}_{X}$ to $\mathcal{T}_{\hat{X}}$: For a suitable desingularisation $\pi\colon\hat{X}\rightarrow X$, there is a morphism $f\colon\pi^{*}\mathcal{T}_{X}\rightarrow\mathcal{T}_{\hat{X}}$ that is an isomorphism outside $\pi^{-1}(\operatorname{Sing}(X))$ (Theorem 3.2).

Secondly, we use a corollary given by Cho, Miyaoka and Shepherd-Barron [CMSB02, Corollary 0.4 (11)] that Kebekus [Keb02] later proved directly (although he claims a weaker result): A uniruled manifold $\hat{X}$ is isomorphic to the projective space, if the anticanonical degree $-K_{\hat{X}}.\hat{C}$ is greater or equal $n+1$ for all rational curves $\hat{C}$ through a general point $p$. The uniruledness of $\hat{X}$ follows from the negativity of $K_{\hat{X}}$ and the anticanonical degree is calculated using the splitting of $\mathcal{T}_{\hat{X}}|_{\hat{C}}$ on the normalisation of $\hat{C}$ (Lemma 3.3). Hence $\hat{X}\simeq\mathbb{P}_{n}\simeq X$.

2. Preliminaries

Let us first recall the definition of the tangent sheaf for a proper variety, as it is a central term in this paper.

Definition 2.1** (tangent sheaf).**

Let $X$ be a algebraic variety, then its tangent sheaf $\mathcal{T}_{X}\coloneqq\operatorname{{\mathcal{H}}\mathit{om}}(\Omega_{X}^{1},\mathcal{O}_{X})$ is the dual of the cotangent sheaf.

We want to work on a desingularisation $\hat{X}$ of the normal variety $X$, so we have to connect $\mathcal{T}_{X}$ with $\mathcal{T}_{\hat{X}}$:

Theorem 2.2**.**

Let $X$ be a normal projective variety with tangent sheaf $\mathcal{T}_{X}$. Then there is a desingularisation $\pi\colon\hat{X}\rightarrow X$ and an $\mathcal{O}_{X}$-module isomorphism

[TABLE]

Proof.

Graf and Kovács [GK14, Theorem 4.2] state that there is a resolution $\pi\colon\hat{X}\rightarrow X$ such that $\pi_{*}\mathcal{T}_{\hat{X}}$ is reflexive. The sheaves $\mathcal{T}_{X}$ and $\pi_{*}\mathcal{T}_{\hat{X}}$ are reflexive, $X$ is normal and $\pi$ is an isomorphism outside the preimage of a set of codimension $2$. Thus we obtain an isomorphism $\mathcal{T}_{X}\rightarrow\pi_{*}\mathcal{T}_{\hat{X}}$. ∎

Remark*.*

For a more thorough understanding of the map $\mathcal{T}_{X}\rightarrow\pi_{*}\mathcal{T}_{\hat{X}}$ and the resolution $\pi$, see the paper of Greb, Kebekus and Kovács [GKK10, Section 4].

The most cited definition for ample sheaves is in Ancona’s paper [Anc82]. He defines ampleness and provides some equivalent characterisations, but gives very few properties. Kubota [Kub70] on the other hand works over graded $\mathcal{O}_{X}$-modules and gives some properties, but does not use the most modern language.

So we recall a definition and the most important properties we use throughout this work.

Definition 2.3** (ample sheaf).**

Let $X$ be a proper algebraic variety and $\mathcal{E}$ a coherent sheaf on $X$. Then we say $\mathcal{E}$ is ample if for every coherent sheaf $\mathcal{F}$ on $X$ there exists an $n=n(\mathcal{F})$ such that $\mathcal{F}\otimes S^{m}\mathcal{E}$ is globally generated for $m\geq n$.

Remark*.*

Other characterisations of ampleness can be found in [Anc82]. Note that an ample sheaf, unlike an ample vector bundle, on a proper variety $X$ does not yield that its support is projective, but only Moishezon [GPR94, Remark p. 244].

The following properties can be found in Debarre’s paper [Deb06, Section 2] or the proof in the vector bundle case (as in [Laz04]) carries over to coherent sheaves:

Proposition 2.4**.**

Let $X$ and $Y$ be normal projective varieties, $f\colon Y\rightarrow X$ a finite morphism, $\mathcal{E}$, $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ sheaves of $\mathcal{O}_{X}$-modules and $\mathcal{E}$ ample, then

1. (1)

$f^{*}\mathcal{E}$* is ample (in particular restrictions of ample sheaves are ample)* 2. (2)

every quotient of $\mathcal{E}$ is ample 3. (3)

$\mathcal{E}_{1}\oplus\mathcal{E}_{2}$* is ample if and only if $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ are both ample*

Proposition 2.5** ([Laz04, 6.4.17]).**

Let $C$ be a smooth curve and ${\mathcal{E}}$ and ${\mathcal{F}}$ vector bundles on $C$. If ${\mathcal{E}}$ is ample and there is a homomorphism ${\mathcal{E}}\rightarrow{\mathcal{F}}$, surjective outside of finitely many points, then ${\mathcal{F}}$ is ample.

We need one further result which is, besides Theorem 2.2, the main ingredient for our result:

Theorem 2.6** ([CMSB02, Corollary 0.4 (11)]).**

A uniruled projective complex manifold $X$ of dimension $n$ with a dense open subspace $U$ such that for all $p\in U$ and all rational curves $C$ through $p$ the inequality $-K_{X}.C\geq n+1$ holds, is isomorphic to $\mathbb{P}_{n}$.

3. Projective varieties with ample tangent sheaves

Now we get to the main result of the paper:

Theorem 3.1**.**

Let $X$ be a normal projective variety over $\mathbb{C}$ of dimension $n$ with ample tangent sheaf $\mathcal{T}_{X}$, then

[TABLE]

Before proving the main theorem we have to adapt the results given in Section 2.

Theorem 3.2**.**

Let $X$ be a normal projective variety, then there is a desingularisation $\pi\colon\hat{X}\rightarrow X$ and an $\mathcal{O}_{\hat{X}}$-module homomorphism

[TABLE]

that is an isomorphism outside $\pi^{-1}(\operatorname{Sing}(X))$.

Proof.

Using Theorem 2.2, we obtain an isomorphism $\mathcal{T}_{X}\rightarrow\pi_{*}\mathcal{T}_{\hat{X}}$ for a suitable resolution $\pi\colon\hat{X}\rightarrow X$. The map $\pi$ is an isomorphism outside $\pi^{-1}(\operatorname{Sing}(X))$ (one has to retrace the resolution guaranteed by [GK14, Theorem 4.2] to [Kol07, Theorem 3.45] for this property). Pulling back $\mathcal{T}_{X}\rightarrow\pi_{*}\mathcal{T}_{\hat{X}}$ and using the natural morphism $c\colon\pi^{*}\pi_{*}\mathcal{T}_{\hat{X}}\rightarrow\mathcal{T}_{\hat{X}}$, there is the diagram

[TABLE]

Considering the maps $g$ and $c$, it is easy to check that they, and therefore $f$, are isomorphisms outside $\pi^{-1}(\operatorname{Sing}(X))$. ∎

Remark*.*

The editor pointed out to the author that Kawamata [Kaw85, p. 14] made use of the map $f$ as well.

Lemma 3.3**.**

Let $X$ be a normal projective variety of dimension $n$ with ample tangent sheaf $\mathcal{T}_{X}$ and $C\subset X$ a closed curve that intersects $\operatorname{Sing}(X)$ in at most finitely many points. Let $\pi\colon\hat{X}\rightarrow X$ be a desingularisation as in Theorem 3.2, $\hat{C}$ the strict transform of $C$ and $\eta\colon\tilde{C}\rightarrow\hat{C}$ the normalisation of $\hat{C}$. Accordingly, there is the following commutative diagram:

[TABLE]

Then $\nu^{*}\mathcal{T}_{\hat{X}}$ is an ample vector bundle and the anticanonical degree $-K_{\hat{X}}.\hat{C}$ is positive. If $\hat{C}$ is a rational curve, $-K_{\hat{X}}.\hat{C}\geq n+1$.

Proof.

The choice of $\pi$ yields the map $f\colon\pi^{*}\mathcal{T}_{X}\rightarrow\mathcal{T}_{\hat{X}}$. Pulling back $f$ via $\nu$ and dividing out the kernel gives

[TABLE]

with $\mathcal{A}\coloneqq\faktor{\nu^{*}\pi^{*}\mathcal{T}_{X}}{\ker(\nu^{*}f)}$. The sheaf $\mathcal{A}$ is ample, since $\mathcal{T}_{X}$ is ample, $\pi\circ\nu$ is finite and quotients of ample sheaves are ample again. Moreover $\mathcal{A}$ is locally free of rank $n$ because it is a torsion-free sheaf on a smooth curve, $\pi\circ\nu$ is an isomorphism outside of finitely many points and $\ker(\nu^{*}f)$ is supported on only finitely many points. Using Proposition 2.5, we deduce that $\nu^{*}\mathcal{T}_{\hat{X}}$ is an ample vector bundle. Because $-K_{\hat{X}}.\hat{C}=\deg\nu^{*}\mathcal{T}_{\hat{X}}$, the anticanonical degree is certainly positive. Since $\nu^{*}\mathcal{T}_{\hat{X}}$ splits on $\mathbb{P}_{1}$ and a direct sum of ample vector bundles is ample only if all summands are ample, we obtain $\nu^{*}\mathcal{T}_{\hat{X}}\simeq\bigoplus_{i=1}^{n}\mathcal{O}_{\mathbb{P}_{1}}(a_{i})$ with $a_{i}\geq 1$ for all $i$. The dual of the homomorphism $\nu^{*}\Omega_{\hat{X}}^{1}\rightarrow\Omega_{\tilde{C}}^{1}$ is a non-trivial map ${\mathcal{T}_{\mathbb{P}_{1}}\simeq\mathcal{O}_{\mathbb{P}_{1}}(2)\rightarrow\nu^{*}\mathcal{T}_{\hat{X}}}$. Thus $a_{i}\geq 2$ for at least one $i$ and we can conclude $-K_{\hat{X}}.\hat{C}=\sum_{i=1}^{n}a_{i}\geq n+1$. ∎

Now we use Lemma 3.3 to show that the assumptions of Theorem 2.6 are fulfilled for $\hat{X}$ and hence $X$ is isomorphic to $\mathbb{P}_{n}$.

Proof of Theorem 3.1.

Normal curves are smooth, so we can assume that $n\geq 2$. Let $\pi\colon\hat{X}\rightarrow X$ be a desingularisation as in Lemma 3.3 and let $p\in\hat{X}\setminus\pi^{-1}(\operatorname{Sing}(X))$ be any general point outside the exceptional locus.

Since $\hat{X}$ is projective, there is an irreducible curve $\hat{C}$ through $p$. As $\hat{C}$ is the strict transform of a closed curve $C\subset X$, $K_{\hat{X}}.\hat{C}<0$ according to Lemma 3.3. Therefore $\hat{X}$ is uniruled by [MM86, Theorem 1].

Any rational curve $\hat{C}\subset\hat{X}$ containing $p$ projects to a curve $C$ on $X$. The curve $C$ meets $\operatorname{Sing}(X)$ in at most finitely many points, thus Lemma 3.3 applies and we have the assumptions of Theorem 2.6 fulfilled. So $\hat{X}$ is isomorphic to the projective space $\mathbb{P}_{n}$. Hence $X\simeq\mathbb{P}_{n}$ too. ∎

Acknowledgement*.*

I want to thank Prof. Dr. Thomas Peternell for his guidance and support and Dr. Patrick Graf for his advice in many occasions, proofreading and especially for hinting me towards [GK14, Theorem 4.2]. In addition I thank Andreas Demleitner and Dr. Florian Schrack for proofreading, their advice and countless conversations.