On generic nefness of tangent sheaves
Abstract.
We show that the tangent bundle of a projective manifold with nef anticanonical class is generically nef. That is, its restriction to a curve cut out by general sufficiently ample divisors is a nef vector bundle. As a consequence, the second Chern class of such a manifold has nonnegative intersections with ample divisors. We also investigate under which conditions these positivities are strict.
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Contents
 I Positivity of tangent sheaves
 1 Slope semistability and HarderNarasimhan filtrations
 2 Foliations and relative tangent sheaves
 3 Proof of Theorem 7
 4 Proofs of Theorem 4, 3 and 5
 5 An orbifold version of generic nefness
 II Discussion on equality conditions
 6 Proof of Theorem 9
 7 Equality conditions of Miyaoka inequality
 8 Irregularity of projective varieties
 9 Proof of Theorem 10
Introduction
From the viewpoint of the minimal model program, complex projective manifolds should be birationally classified according to the sign of the canonical class . It is natural to ask how far we can lift the positivity (or the negativity) of to the cotangent sheaf . The following two theorems were due to Miyaoka (See [Miy87, Corollary 6.4]).
Theorem 1.
Let be a complex projective manifold such that is pseudoeffective. Then the sheaf is generically nef. That is, is a nef vector bundle for any MehtaRamanathangeneral curve .
Theorem 2.
Let be a complex projective manifold of dimension such that is nef. Then for any ample divisors , we have
where stands for the second Chern class.
A MehtaRamanathangeneral curve is the complete intersection of sufficiently ample divisors in general positions, where is the dimension of . In this note, we are interested in complex projective varieties with nef anticanonical class . These varieties have been studied by many mathematicians, e.g. DemaillyPeternellSchneider ([DPS93],[DPS94], [DPS96], [DPS01]), Zhang ([Zha96], [Zha05]), Păun ([Pău97], [Pău12]), Peternell ([Pet12]), CampanaDemaillyPeternell ([CDP15]), CaoHöring ([CH17]), Cao ([Cao16])… Based on their works, we prove the following theorem.
Theorem 3.
Let be a complex projective variety of dimension at least with factorial log canonical singularities. Assume that that is nef. Then the reflexive tangent sheaf is generically nef. That is, is a nef vector bundle for any MehtaRamanathangeneral curve .
The statement above was conjecture by Peternell in [Pet12, Conjecture 1.5]. If we assume that is the intersection of sufficiently ample divisors of the same class, then Cao (see [Cao13, Theorem 1.2]) and Guenancia (see [Gue16, Theorem C]) proved the nefness of independently by analytic methods. Our approach is more algebraic though.
We note that Theorem 3 does not hold if we only assume that is pseudoeffective. The following example was due to Demailly, Peternell and Schneider (see [DPS01, Example 4.14]). Let be a curve of genus and let be a line bundle on of degree smaller than . Let and let be the natural projection. Then, on the one hand, is effective. On the other hand, we have a natural surjective morphism . Since the is ample, the vector bundle is not nef if is a general very ample divisor.
We also remark that Theorem 3 does not hold if we replace MehtaRamanathangeneral curves by movable curves. For example, Boucksom, Demailly, Păun and Peternell showed that if is a projective K3surface or a projective CalabiYau threefold, then there is a dominant family of curves such that is not nef for general (see [BDPP13, Theorem 7.7]).
For movable curves classes, we prove the following theorem, which implies Theorem 3 by MehtaRamanathan theorem (see [MR82, Theorem 6.1]).
Theorem 4.
Let be a complex projective variety with factorial log canonical singularities. Assume that is nef. Let be a movable class of curves. Then for any nonzero torsionfree quotient sheaf of , we have , where stands for the first Chern class.
As a corollary, we obtain the following theorem, which was proved by Xie in the case of smooth threefolds (see [Xie04, Theorem 1.2]).
Theorem 5.
Let be a normal complex projective variety of dimension with nef anticanonical class . Assume that has factorial log canonical singularities and is smooth in codimension . Then for any nef divisors , we have
There are two main ingredients for the proof of Theorem 4. The first one is the following theorem (see also Theorem 1) by Campana and Păun on algebraicity of foliations (See [CP15a, Theorem 1.1]).
Theorem 6.
Let be a projective manifold. Let be any movable curve class. Let be a foliation. Assume that the slope, with respect to , of any nonzero torsionfree quotient of is strictly positive. Then has algebraic leaves. That is, is the foliation induced by some rational dominant map . Moreover, general leaves of are rationally connected.
Another crucial theorem is the following one, which is a refined version of a theorem of Chen and Zhang (see [CZ13, Main Theorem]).
Theorem 7.
Let be a projective factorial log canonical pair with nef. Let be a rational dominant map with . Let be the foliation induced by . Then is pseudoeffective, where is the vertical part of over , and is the canonical class of .
Let us sketch the proof of the Theorem 4. Our idea is inspired by Peternell’s proof in the case of rational surfaces (see [Pet12, Theorem 5.9]). We assume by contradiction that there is some movable class and some torsionfree quotient such that . Then we can find a suitable subsheaf, namely , in the HarderNarasimhan semistable filtration of such that is negative, and that is a foliation on . In particular,
Moreover, by using Theorem 6, we can show that is induced by a rational dominant map . Then from Theorem 7, we obtain that
This is a contradiction.
An orbifold version of Theorem 1 was established by Campana and Păun (see [CP15b, Theorem 2.1]): if is a projective factorial log canonical pair with pseudoeffective, then the orbifold cotangent sheaf is generically nef for any adapted Kawamata finite cover . By using the orbifold version of Theorem 6 of Campana and Păun (see [CP15a, Theorem 1.4]), we can also deduce the following orbifold version of Theorem 3.
Theorem 8.
Let be a complex projective factorial log canonical pair such that is nef. Then the orbifold tangent sheaf is generically nef for any adapted Kawamata finite cover .
It is natural to ask under which conditions the positivities in the theorems above are strict. In the second part of the paper, we prove the the following two theorems.
Theorem 9.
Let be a smooth projective complex manifold of dimension with nef anticanonical class . Then the following properties are equivalent:

is rationally connected;

is generically ample, that is, is an ample vector bundle for any MehtaRamanathangeneral curve .
The next theorem was pointed out to the author by Junyan Cao, and our proof borrows the idea from [Cao13].
Theorem 10.
Let be a smooth complex projective manifold of dimension with nef anticanonical class . Then the following properties are equivalent:

there are ample divisors such that

there is a finite étale cover such that is either isomorphic to an abelian variety or isomorphic to a bundle over an abelian variety.
This theorem answers a question of Yau (see [Yau93, Problem 66]) in the projective case. We note that one could not expect the bundle to be trivial, see for example [DPS94, Example 3.5].
Throughout this paper, we will work over , the field of complex numbers.
Acknowledgment. The author would like to express his gratitude to Stéphane Druel and Burt Totaro for reading the preliminary version of this paper and warm encouragement. He is grateful to Junyan Cao for pointing out the application Theorem 10 to him. He would also like to thank Jun Li, Chen Jiang, Claire Voisin, Yuan Wang and Jian Xiao for general discussions.
Part I Positivity of tangent sheaves
1. Slope semistability and HarderNarasimhan filtrations
In this section, we will study some properties on HarderNarasimhan semistable filtrations. Let be a normal projective variety and let be a movable curve class. Assume that either is factorial or is the class of a complete intersection of basepointfree divisors. Then for any torsionfree coherent sheaf with positive rank on , the slope of with respect to is the number
We would like to refer to [GKP14, Appendix A] for more details on slope notions on singular spaces. The maximal slope is defined as follows,
The supremum is in fact a maximum (see e.g. [GKP14, Proposition A.2]). The sheaf is called semistable (or just semistable if there is no ambiguity) if . If is not semistable, then there is a unique maximal subsheaf of such that . This is called the maximal destabilizing subsheaf and is automatically semistable.
There is always a unique filtration, called the HarderNarasimhan semistable filtration, of saturated subsheaves,
such that is the maximal destabilizing subsheaf of for all and that the sequence is strictly decreasing.
The minimal slope (see e.g. [CP15a, Definition 2.3]) is defined as follows,
This infimum is also a minimum. Indeed, we have , where is the saturated subsheaf defined in the HarderNarasimhan semistable filtration above (see [CP11, Proposition 1.3]).
We will use the following lemma in the proof of Theorem 3.
Lemma 1.
Let be a normal projective factorial variety and let be a movable curve class in . Let be a torsionfree sheaf on such that and . Let
with be the HarderNarasimhan semistable filtration with respect to . Then there is some such that and .
Proof.
We have . Let be the smallest integer in such that . Since , we know that . We consider the following exact sequence
By the definition of , we have . Thus . ∎
2. Foliations and relative tangent sheaves
Let be a normal variety of dimension at least and let be the reflexive tangent sheaf. A foliation on is a nonzero saturated subsheaf of which is closed under Lie brackets. The canonical class of is a Weil divisor such that
where is the reflexive hull of the top wedge product of . We say that has algebraic leaves if the dimension of the Zariski closure of a general leaf of is equal to the rank of .
Typical examples of foliations are relative tangent sheaves as follows. Consider a rational dominant map between normal varieties. Assume that . Let be the smooth locus of . Let be a nonempty smooth open subset of such that is regular and . The relative tangent sheaf of is defined as the kernel of the natural morphism
There is a unique saturated subsheaf of the reflexive tangent sheaf such that . We call the relative tangent sheaf of . It is a foliation on . We note that a foliation on has algebraic leaves if and only if it is induced by some rational dominant map as above (see e.g. [AD13, Lemma 3.2]).
The following theorem is a singular version of Theorem 6.
Theorem 1.
Let be a projective normal factorial variety. Let be a movable curve class. Assume that is a saturated subsheaf of such that

,

Then is a foliation and has algebraic leaves. Moreover, general leaves of are rationally connected.
Proof.
Let be a resolution of singularities, and let be the numerical pullback such that
for any divisor class on (see [GKP14, Construction A.15]). Since is movable, so is by [GKP14, Lemma A.17]. If and are torsionfree sheaves on and respectively such that is isomorphic to in codimension , then
Therefore, if be the saturated subsheaf of induced by , then
and
Hence is a foliation and has algebraic leaves by [CP15a, Theorem 1.4]. Moreover, general leaves of are rationally connected. The theorem then follows from the property that . ∎
3. Proof of Theorem 7
In this section, we will prove Theorem 7. It follows from the following theorem, which is a special case of a theorem of Druel (see [Dru15, Proposition 4.1]).
Theorem 1.
Let be a surjective morphism between normal projective factorial varieties. Let be an effective divisor in . Assume that the pair is log canonical, where is a general fiber of . If there is some positive integer such that is Cartier and that , then is pseudoeffective, where is the foliation induced by .
Proof of Theorem 7.
There is a log resolution of such that the induced map is a morphism. By blowing up and if necessary, we may assume that is smooth. We write
where and are effective divisors without common components. Moreover has exceptional support. Since is log canonical, so is the pair .
\xymatrix
Z \ar[d]^g \ar[r]^π & X
Y &
Let be an ample divisor in and let be a rational number. Then is ample. We can then choose a smooth irreducible divisor
such that is a log canonical. We have
Let . Then is effective, and we have
(*) 
Let be a general fiber of . We claim that has nonzero global sections for large enough and sufficiently divisible integer . Indeed, by (*), we have
The righthandside above is a big divisor. Hence has nonzero global sections for large enough and sufficiently divisible integer .
By Theorem 1, we obtain that is pseudoeffective, where is the foliation induced by . Hence
is pseudoeffective.
Since this is true for arbitrary , we obtain that
is pseudoeffective. Thus
is pseudoeffective. ∎
4. Proofs of Theorem 4, 3 and 5
Proof of Theorem 4.
Assume the opposite. Then . In particular, is not semistable. Let
with be the HarderNarasimhan semistable filtration. Then by Lemma 1, there is some such that and . We have the following inequality,
Hence is a foliation and has algebraic leaves by Theorem 1. We denote by . Then
for .
Since has algebraic leaves, there is a rational dominant map such that is induced by . Moreover, since is a nonzero proper subsheaf, we have . Hence Theorem 7 shows that
This is a contradiction. ∎
5. An orbifold version of generic nefness
In this section, we will prove Theorem 8. We would like to refer to [CP15a, Section 5] for detailed notions of orbifolds. Let be a projective factorial log canonical pair of dimension . Let be a Kawamata finite cover adapted to . Let be ample divisors in . Then the orbifold cotangent sheaf (respectively the orbifold tangent sheaf ) is said to be generically semipositive with respect to if for any nonzero torsionfree quotient of (respectively of ), we have
We say that (respectively ) is generically nef if it is generically semipositive with respect to any ample divisors in . In particular, we note that if is an integral divisor and if is the identity map, then generic nefness is the same as generic nefness by MehtaRamanathan theorem (see [MR82, Theorem 6.1]).
In order to prove Theorem 8, we will need the following version of [CP15a, Theorem 1.4] for singular spaces.
Theorem 1.
Let be a projective factorial log canonical pair. Let be a finite cover adapted to . Let be very ample divisors in and let be the class of . Assume that there is a saturated subsheaf of such that

is invariant, where is the Galois group of .

,

Then the saturation of in defines an algebraic foliation on . Moreover, is the saturation of in .
Proof.
Let be a log resolution of which is an isomorphism over the smooth locus of . Let be the normalization of . Then the natural morphism is an adapted finite cover of , where . Let , and .
\xymatrix
Z’ \ar[d]_π’ \ar[r] & Z \ar[d]^π
X’ \ar[r]^r & X
There is a unique invariant saturated subsheaf of such that is isomorphic to . Let be the class of . We have
Hence by [CP15a, Theorem 1.4], the saturation of in defines an algebraic foliation on . Moreover, is the saturation of in (see [CP15a, Corollary 5.10]). Let be the saturation of the natural image of in . Then is an algebraic foliation and is the saturation of in . ∎
Now we will prove Theorem 8.
Proof of Theorem 8.
The proof is similar to the one of Theorem 3. Assume the opposite. Then there are very ample divisors such that is not generically semipositive with respect to . Let be the class of .
By applying Lemma 1 to , we can find a saturated subsheaf of such that and that . From the uniqueness of HarderNarasimhan filtration, we know that , as a component in the HarderNarasimhan filtration, is invariant under the Galois group of . Moreover, as in the proof of Theorem 4, we have
By Theorem 1, the saturation of in defines an algebraic foliation on . Assume that is the relative tangent sheaf of some dominant rational map .
Part II Discussion on equality conditions
6. Proof of Theorem 9
We will prove Theorem 9 in this section. We will first prove the following lemma.
Lemma 1.
Let be a smooth projective variety and let be a cycle of pure dimension . Assume that for any ample divisors , we have . Then the following two conditions are equivalent:

there are ample divisors such that ;

for any ample divisors , we have .
Proof.
It is enough to prove that (1) implies (2). Let be ample divisors such that , and let be any ample divisors. We need to prove that .
Let be a natural number such that is still an ample divisor. Then we have
Thus By repeating this procedure more times, we can obtain that ∎
Now we can prove Theorem 9.
Proof of Theorem 9.
The case when is trivial. We assume from now on that . First we assume that is rationally connected. Assume by contradiction that is not generically ample. Then there is a MehtaRamanathangeneral curve such that is nef but not ample. By [Har71, Theorem 2.4], there is a nonzero quotient bundle of of degree zero. Thus, by MehtaRamanathan theorem (see [MR82, Theorem 6.1]), we have , where is the class of . This implies that there is a surjective morphism such that is a nonzero torsionfree sheaf and that Since a quotient bundle of a nef bundle is still nef, a quotient torsionfree sheaf of a generically nef sheaf is also generically nef. Hence is generically nef and for any ample divisors . Thus by Lemma 1, for any ample divisors . This shows that is numerically zero. Since is rationally connected, it is simply connected. Therefore, we have The injective morphism then induces a nonzero morphism , where is the rank of . This is a contradiction, since by [Kol96, Corollary IV.3.8].
Now we assume that is not rationally connected. Then, by [Zha05, Corollary 1], there is a dominant rational map such that is smooth with Kodaira dimension and that the general fibers of are proper and rationally connected. Since is not rationally connected, has positive dimension . There is some positive integer such that . Thus Let be a MehtaRamanathangeneral curve and let be the class of . Then by [CP11, Corollary 5.11]. This implies that . Hence is not ample by MehtaRamanathan theorem (see [MR82, Theorem 6.1]). ∎
Remark 2.
Theorem 9 does not hold without assuming the smoothness of . For example, let be the group and let be an elliptic curve with an action of such that . We also endow with the action of by antipodes. Then acts on the product diagonally and the quotient is étale in codimension . In particular, has canonical singularities and is nef. In addition, as in [GKP14, Remark and Question 3.8], we have This implies that for any ample class .
7. Equality conditions of Miyaoka inequality
We recall that a nonzero torsionfree sheaf on a projective manifold of dimension is said to be generically semipositive for some ample divisors if for each nef divisor , we have , where is the class of . We remark that if is generically nef, then it is generically semipositive for any ample divisors .
In [Miy87, Theorem 6.1], Miyaoka proved that if is nef and if is generically semipositive, then . As a consequence, if is nef and if is generically nef, then for any ample divisors .
In this section, we will study the equality conditions of these inequalities. We will assume that is generically nef and that is nef. By Lemma 1, the equality holds for some ample divisors if and only if for any ample divisors . Thus, in order to study the equality conditions, we may assume that for some ample divisor .
Our idea is to look into the details in Miyaoka’s proof, and study every inequality inside. We will discuss following the numerical dimension of . Recall that the numerical dimension of a nef divisor is the largest integer such that .
Preparatory Lemmas
We will first collect some useful elementary results for this section.
Lemma 1.
Let be a nondegenerated symmetric bilinear form of signature on a real vector space . Let be two vectors. Assume that , and . Then and are linear dependent.
Proof.
If , then is definite and . We assume then that . By Sylvester theorem, there is an orthogonal basis of such that and that for all .
Let and be the coordinates of and respectively. Then by assumption, we have
Thus we have
From the equality condition of Cauchy inequality, this implies that and are linearly dependent. We then deduce that and are linearly dependent. ∎
Lemma 2.
Let be a smooth projective variety of dimension . Let be a torsionfree sheaf on . Then for any ample divisors , we have
Moreover, the equality holds if and only if is locally free in codimension .
Proof.
If , then the lemma follows from [Meg92, Lemma 10.9]. We assume then that . We may assume that are effective sufficiently ample divisors in general positions. Let be their intersection. Since is smooth, there is a finite free resolution of as follows,
Since is in general position, we may assume that is still torsionfree, and that
is again a free resolution. Hence
By the same argument, we may assume that is still reflexive and is isomorphic to . Moreover, we may also assume that
By [Meg92, Lemma 10.9], we have
and the equality holds if and only if is locally free. Since is in general position, the sheaf is locally free if and only is locally free in codimension . This completes the proof of the lemma. ∎
Case of
We will consider the case when has numerical dimension at least , and will show that is an extension of a torsionfree sheaf with numerically trivial first Chern class by an invertible sheaf. To this end, we only need the weaker condition that is generically semipositive for some ample divisors . We first prove the following lemma.
Lemma 3.
Let be a smooth projective variety of dimension . Let be a nonzero torsionfree sheaf on which is generically semipositive for some ample divisors . Assume that is nef and
Let be an exact sequence of nonzero torsionfree sheaves. If , then is generically semipositive, and
Proof.
Since , we have for any saturated subsheaf of