Chern class inequalities for nonuniruled projective varieties

It is known that projective minimal models satisfy the celebrated Miyaoka–Yau inequalities. In this article, we extend these inequalities to the set of all smooth, projective, and nonuniruled varieties.


INTRODUCTION
Interconnections between topological, analytic, and algebraic structures of compact complex varieties is a central theme in various branches of geometry and topology.Many of the classical results in this area use characteristic classes, in particular Chern classes to describe such connections.Apart from the celebrated Hirzebruch-Riemann-Roch theorems, prominent examples were found by Bogomolov [3] and -in a different direction -Yau [40], as a consequence of his (and Aubin's) solution to Calabi's conjecture.More precisely, he established that an -dimensional compact Kähler manifold (, ) with  1 () < 0 satisfies the inequality In a more general setting, using his generic semipositivity result, Miyaoka [28] showed that any minimal variety †  satisfies the inequality for every ample divisor  ⊂ .The combination of the two inequalities ((1.0.1) and (1.0.2)) and their analogs are nowadays referred to as the Miyaoka-Yau inequalities.
The purpose of this article is to establish that, as long as  is not covered by rational curves, it satisfies a Chern class inequality generalizing (1.0.2).Throughout this paper, all varieties will be over ℂ. Theorem 1.1.Let  be a smooth, projective, and nonuniruled variety of dimension , and  any ample divisor.There is a decomposition   =  +  into ℚ-divisors, with  ⋅  ⋅  −2 = 0, such that (1.1.1) Moreover, we have  = 0, when   is nef.
A few remarks about the statement of Theorem 1.1 are as follows.First, we note that it is well known that, for a nonuniruled variety, Chern class inequalities of the form (1.1.1)cannot be gleaned from the ones for its minimal models, when they exist.Second, as is evident from the statement of the theorem, the quantity on the right-hand side of (1.1.1)(which we may think of as an error term) is forced on us by the negative part of Zariski decomposition.More precisely, given a complete intersection surface  ⊂  defined by very general members of very ample linear systems, the divisor  is an extension of the negative part of the Zariski decomposition for   |  (see Section 3 for the details).This extension is in the sense of the Noether-Lefschetz-type theorems (Proposition 3.1).
Theorem 1.1 is a special case of the following more general result that we obtain in this article, which is, in fact, needed for the proof of Theorem 1.1.Theorem 1.2.Let (, ) be a log-smooth pair of dimension , with  being a rational divisor.Assume that  is an ample divisor.If   +  is pseudoeffective, then there is a decomposition   +  =  + , that is -orthogonal in the sense that  ⋅  ⋅  −2 = 0, and for which the inequality holds.Furthermore, when   +  is nef, we have  = 0.
The Chern classes ĉ (⋅) in Theorem 1.2 are in the sense of orbifolds (see (2.10.1)), and when  is reduced, they coincide with the usual notion of Chern classes.

General strategy of the proof
For simplicity, we will focus mostly on Theorem 1.1; the case where  = 0.
As was observed by Miyaoka [28] and later on Simpson [33], it is sometimes possible to use the Bogomolov inequality [3] to establish Miyaoka-Yau inequalities.But the Bogomolov inequality is generally valid when the polarization is defined by ample or nef divisors, which is applicable -for the purpose of Miyaoka-Yau inequalities -when the variety is minimal.But for a general nonuniruled variety , no such polarization exists.On the other hand, we show in the current article that, thanks to the result of Boucksom-Demailly-Păun-Peternell [1], after cutting down by hyperplanes, the above divisor  defines a so-called movable cycle; a potentially natural choice for a polarization.But in general there is no topological Bogomolov-type inequality for sheaves that are semistable with respect to a movable class .At best, assuming that ℰ is locally free, one can use a Gauduchon metric   constructed in [6,Append.],with  −1  ≡ , and Li-Yau's result on the existence of Hermitian-Einstein metrics [24] to establish the Bogomolov inequality with respect to  −2  .But since  −2

𝐺
is not closed, this would not yield a topological inequality.However, thanks to a fundamental result of Langer [21,Thm. 3.4], semistability with respect to a certain subset of movable classes does lead to the classical Bogomolov inequality.Having this important fact in mind, we use the definition of the Zariski decomposition to show that the intersection of  with  −2 belongs to this smaller subset, as long as  is of general type, which is the content of in Proposition 3.8.With this observation, and using further properties of , we then show that, thanks to Campana-Păun's result on positivity properties of the (log-)cotangent sheaf with respect to movable cycles [7], much of Miyaoka's original approach can then be adapted to establish (1.1.1).
In the more general setting of nonuniruled varieties, that is, when   is pseudoeffective [1], given an ample divisor  and any  ∈ ℕ, we consider the pair (, 1   ), which is now of loggeneral type.Here, the log-version of Theorem 1.1 is needed, forcing us to resort to orbifolds (in the sense of Campana) and their Chern classes as was defined in [14, Sect.2], following [30].With the inequality (1.2.1) at hand, one can then extract the inequality (1.1.1)through a limiting process, which is reminiscent of [14], but employed for somewhat different reasons.

Related results
Chern class inequalities for surfaces and their connection to the Zariski decomposition were first studied by Miyaoka in [27] and later on by Wahl [39], Megyesi [25], Langer [20], and others.In higher dimensions, when   +  is movable and dim  = 3, the inequality (1.2.1) is established in [32] for (mildly) singular pairs.Under the assumption that   +  is nef and big, such Chern class inequalities have a rich history and were discovered by Kobayashi [19], Tsuji [37], and Tian [36], to name a few.More recently, and in a more general setting, they have been studied in joint papers with Greb-Kebekus-Peternell [12] and with Guenancia [14].Further results have been established by Deng [8] and Hai-Schreieder [16].Finally, we note that the methods that we use in this article show that coefficient of  in (1.2.1) can be sharpened.In Section 4, we make some predictions about possibly optimal versions of Theorem 1.2.

Cones of curves and divisors and stability notions
Assuming that  is projective, let NE 1 () ℚ ⊂ N 1 () ℚ denote the convex cone of classes of effective ℚ-divisors and set NE 1 () ℚ to be its closure, called the pseudoeffective cone.
We define the notion of slope stability in the following general setting.
Through the Harder-Narasimhan filtration, semistable sheaves form the building blocks of coherent, torsion-free sheaves.But to ensure the existence of such (unique) filtrations, we generally need more assumptions on  in Definition 2.2.In this article, we require the existence of Harder-Narasimhan filtration under the assumption that  ∈  1 () ℚ is movable.Thankfully, such filtrations are known to exist for such classes [11,Sect.2].
As discussed in the introduction, for semistability to lead to a suitable Bogomolov inequality, we need to work with a smaller set of 1-cycles than those in Mov 1 ().To do so, we use the following definition.
Theorem 2.5 [21,Thm. 3.4].For any torsion-free sheaf ℱ of rank  on , satisfying the inequality Here  1 (ℱ) is thought of as the dual of its image under the cycle map, represented by the reflexivization of det(ℱ) (similarly for  1 (ℱ ′ )).

Orbifold sheaves and Chern classes
We follow the definitions and constructions of [14, Sects.2,3] in the generally simpler context of log-smooth pairs.We refer to [4,17,35] and [5] for more examples and details on pairs and associated notions of adapted morphisms.
A pair (, ) consists of a variety  and  = ∑   ⋅   ∈ Div() ℚ , with for some   ,   ∈ ℕ.A pair (, ) is said to be log-smooth, if  is smooth and  has simple normal crossing support.We say that (, ) is (quasi-)projective, if  is so.We now recall a few basic notions regarding morphisms, sheaves, and Chern classes encoding the fractional part of  in (, ).Definition 2.7 (Adapted morphisms).Given a quasi-projective pair (, ), a finite, Galois, and surjective morphism  ∶  →  of schemes is called adapted (to (, )), if the following conditions are satisfied.
(2.7.2) For every   , with   ≠ 1, there are   ∈ ℕ and a reduced divisor 3) The morphism  is étale at every generic point of ⌊⌋.
Furthermore, if   = 1, for all , we say that  is strictly adapted.
Definition 2.10 (Orbifold cotangent sheaf).In the setting of Notation 2.9, we define the orbifold cotangent sheaf Ω 1 (,,) of (, ) with respect to  by the kernel of the morphism which is naturally defined using the residue map.

Orbifold Chern classes
Let  ∶  → (, ) be a strictly adapted morphism for a log-smooth pair (, ).Assume that  is smooth and set  ∶= Gal(∕).Given a coherent -sheaf ℰ on , we have   (ℰ) ∈   ()  .Here   (⋅) denotes the th Chern class and   ()  the group of -invariant, -cocycles in .We define the th orbifold Chern class of ℰ by where   is the natural map   ∶   ()  ⊗ ℚ →  − () ⊗ ℚ defined by the composition of cap product with [] and pushforward.With the above definition, when  is projective, ĉ (ℰ) defines a multilinear form on  1 () − ℚ .Furthermore, with  being flat, from [30,Thm. 3.1], it follows that  is in fact a group isomorphism.Thus, similar to [30] (or [14, Append.]),we can use this isomorphism to equip  * () ⊗ ℚ with a ring structure compatible with that of  * ()  ⊗ ℚ.In this way, products of orbifold Chern classes can also be consistently defined in  * () ⊗ ℚ.
One can check that for -sheaves on , defined by pullback of sheaves on , the above notion of orbifold Chern classes is consistent with the projection formula, when applicable.

CONSTRUCTING MOVABLE CYCLES IN 𝑩 + 𝑯 VIA ZARISKI DECOMPOSITION
Our main goal in this section is to use Zariski decomposition on certain complete-intersection surfaces to construct global moving cycles in  +  , which is the content of Proposition 3.8.which is naturally defined by using adjunction.By the Kodaira vanishing  1 (  + ) = 0, the induced exact cohomology sequence partially reads Claim 3.2.We have ℎ 2,0 () < ℎ 2,0 (), if and only if  is a strict inclusion.
For Item (3.1.2),we will keep the notations for the proof of Item (3.1.1).Again, since N 1 () ≅  1 () (see, e.g., [22,Ex. 3.1.29]),it suffices to prove N 1 () ≅ N 1 ( NL ).As  NL is reduced, we have a commutative diagram of long exact cohomology sequences arising from the two exponential sequences on  and  NL .In particular, we have Now, with the vertical arrow on the left being an isomorphism by Item (3.1.1)and the one on the right being an injection by the Lefschetz hyperplane theorem ([22,Thm. 3.1.17]or [38, 2.3.2]), the isomorphism Pic() → Pic( NL ) descends to an isomorphism (Pic()∕ ≡) → (Pic( NL )∕ ≡), as required.□ Before stating the application of Proposition 3.1 that we need, we briefly review Nakayama's -decomposition.

MIYAOKA-YAU-TYPE INEQUALITIES FOR NONUNIRULED VARIETIES
We now proceed to the proof of Theorem 1.2.

4.1
The general-type case For the rest of this subsection, we will closely follow the arguments of [28,Prop. 7.1], adapting them to our setting by using the results in Section 3.
Proof of Claim 4.2.As ( ∏ −2 =1 ||)∖ 0  consists of a union of only countable number of closed subsets, by a successive application of the Lefschetz hyperplane theorem, for a general member of ||, items (4.2.1) and (4.2.2) are guaranteed to hold (note that ℰ 1 is reflexive and thus locally free in codimension 2).Same is true for Item (4.2.3) by the following observation: after removing a closed subscheme of , the surjection in (4.1.1)defines  * , locally analytically, as a sum of rank one foliations (trivially integrable).Therefore, by choosing Ŝ transversal to the associated leaves, and using Nakayama's lemma, we can ensure that Ω Case I: rank(ℰ 1 ) = 2 Using the injection (4.2.4)