Projectively flat KLT varieties

In the context of uniformisation problems, we study projective varieties with klt singularities whose cotangent sheaf admits a projectively flat structure over the smooth locus. Generalising work of Jahnke-Radloff, we show that torus quotients are the only klt varieties with semistable cotangent sheaf and extremal Chern classes. An analogous result for varieties with nef normalised cotangent sheaves follows.

1. Introduction 1.1. Projective manifolds with projectively flat cotangent bundle. -Let E be a locally free sheaf of rank r 1 on a complex, projective manifold X of dimension n 2. If E is semistable with respect to some ample divisor H ∈ Div(X), then the Bogomolov-Gieseker Inequality holds: In case of equality, the sheaf E is known to be projectively flat. We refer the reader to Section 3.1 for a brief discussion of projective flatness. Motivated by the structure theory of higher-dimensional projective manifolds, we are particularly interested in the case where E is the cotangent bundle Ω 1 X of an n-dimensional manifold X. In this setup, the equality case of (1.0.1) reads While semistability of Ω 1 X occurs in many relevant cases and has important geometric consequences, the Equality (1.0.2) poses severe restrictions on the geometry of X.
-In case K X is ample, Equality (1.0.2) will never hold, owing to the stronger Miyaoka-Yau inequality.
-If K X ≡ 0, then by Yau's theorem, X is an étale quotient of an Abelian variety.
-If X is Fano and Kähler-Einstein, again the equality (1.0.2) cannot occur, owing to the Chen-Oguie inequality.
In their remarkable paper [JR13], Jahnke and Radloff proved the following complete characterisation of manifolds with semistable cotangent bundle for which Equality (1.0.2) holds. Let X be a projective manifold of dimension n and assume that Ω 1 X is H-semistable for some ample line bundle H. If Equality (1.0.2) holds, then X is a finite étale quotient of a torus.
In particular, this deals with manifolds of intermediate Kodaira dimension, and moreover implies that in the Fano case Equality (1.0.2) can never happen, independent of the existence question for Kähler-Einstein metrics.
1.2. Main result of this paper. -We have learned from numerous previous results, including [GKP16,LT18,GKT18,GKPT20,GKPT19a,GKP20], that the natural context for uniformisation results is that of minimal model theory. The aim of this paper is therefore to generalise the theorem of Jahnke-Radloff to the case when X has klt singularities.
Theorem 1.2 (Characterisation of quasi-Abelian varieties). -Let X be a projective klt space of dimension n 2 and let H ∈ Div(X) be ample. Assume that Ω [1] X is semistable with respect to H and that its Q-Chern classes satisfy the equation Then, X is quasi-Abelian and has at worst quotient singularities.
In Theorem 1.2, we say that a normal projective variety X is quasi-Abelian if there exists a quasi-étale cover X → X from an Abelian variety X to X. The symbols c • denote Q-Chern classes on the klt variety X, as recalled in [GKPT19a,§3].
1.3. Normalised cotangent sheaves. -Work of Narasimhan, Seshadri and others, summarised for example in [JR13, Th. 1.1] and explained in detail by Nakayama in [Nak98, Th. A], can be used to reformulate Jahnke-Radloff's result in terms of positivity properties of natural tensor sheaves: a projective manifold X of dimension n is quasi-Abelian if and only if the normalised cotangent bundle, Sym n Ω 1 X ⊗O X (−K X ), is nef.
While the arguments presented by Nakayama use intersection theory computations on the total space of the projectivised cotangent bundle that cannot immediately be carried over to singular varieties, we are nevertheless able to obtain the analogous result in our setup.
Theorem 1.3. -Let X be a normal projective variety of dimension n 2. Assume that X is klt and that the reflexive normalised cotangent sheaf Sym n Ω 1 X ⊗ O X (−K X ) * * is nef. Then, X is quasi-Abelian.
Definition 2.1 recalls the meaning of nef for a sheaf that is not necessarily locally free.
1.4. Strategy of proof. -While the general strategy of proof is similar to the one employed by Jahnke and Radloff, our argument introduces a number of new tools; this includes a detailed analysis of sheaves that are projectively flat on the smooth a klt variety, especially as their behaviour near the singularities is concerned. On the one hand, these tools allow us to deal with the (serious) complications arising from the singularities. On the other hand, they enable us to streamline parts of Jahnke-Radloff's proof, thereby clarifying the underlying geometric principles.
As an intermediate step, we obtain the following result, which might be of independent interest to some readers. Acknowledgements. -We thank Indranil Biswas and Stefan Schröer for answering our questions. We thank the anonymous referee for an extremely helpful and detailed report that improved the presentation of this paper considerably. Section 4 has been re-written, strengthened and substantially shortened in response to the reviewer's comments. In particular, the final version of some of the results and proofs in Section 4.2 were suggested by the referee. 2. Conventions, notation, and variations of standard facts 2.1. Global conventions. -Throughout this paper, all schemes, varieties and morphisms will be defined over the complex number field. We follow the notation and conventions of Hartshorne's book [Har77]. In particular, varieties are always assumed to be irreducible. For all notation around Mori theory, such as klt spaces and klt pairs, we refer the reader to [KM98].

2.2.
Varieties and complex spaces. -In order to keep notation simple, we will sometimes, when there is no danger of confusion, not distinguish between algebraic varieties and their underlying complex spaces. Along these lines, if X is a quasi-projective complex variety, we write π 1 (X) for the fundamental group of the associated complex space.
2.3. Reflexive sheaves. -As in most other papers on the subject, we will frequently consider reflexive sheaves and take reflexive hulls. Given a normal, quasi-projective variety (or normal, irreducible complex space) X, we write Ω 2.4. Nef sheaves. -We recall the notion of a nef sheaf in brief and collect basic properties.
Definition 2.1 (Nef and ample sheaves, [Anc82]). -Let X be a normal, projective variety and let S = 0 be a non-trivial coherent sheaf on X. We call S ample/nef if the locally free sheaf O P(S ) (1) ∈ Pic(P(S )) is ample/nef.
We refer the reader to [Gro61] for the definition of P(S ), and to [Anc82, §2 & Th. 2.9] for a more detailed discussion of amplitude and for further references. We mention a few elementary facts without proof.
(2.2.1) A direct sum of sheaves on X is nef iff every summand is nef.
(2.2.3) A sheaf E is nef on X if and only if for every smooth curve C and every morphism γ : C → X, the pull-back γ * E is nef.
2.5. Flat sheaves. -One key notion in our argument is that of a flat sheaf. We briefly recall the definition. -If X is any normal, irreducible complex space and F is any locally free coherent sheaf on X, we call F flat if it is defined by a (finite-dimensional) complex representation of the fundamental group π 1 (X). A locally free coherent sheaf on a quasi-projective variety is called flat if the associated analytic sheaf on the underlying complex space is flat.
Remark 2.4 (Simple properties of flat sheaves). -Tensor powers, duals, symmetric products and wedge products of locally free, flat sheaves are locally free and flat. The pull-back of a locally free, flat sheaf under an arbitrary morphism is locally free and flat. If F is a locally free and flat sheaf on a normal, irreducible complex space X, then there exists a description of F in terms of a trivialising covering and transition functions where all transitions functions are constant.
The following lemma is a direct consequence of [Kob87, Chap. II, Prop. 3.1]. We leave the details to the reader. 2.6. Covering maps and quasi-étale morphisms. -A cover or covering map is a finite, surjective morphism γ : X → Y of normal, quasi-projective varieties (or normal, irreducible complex spaces). The covering map γ is called Galois if there exists a finite group G ⊂ Aut(X) such that γ is isomorphic to the quotient map.
A morphism f : X → Y between normal varieties (or normal, irreducible complex spaces) is called quasi-étale if f is of relative dimension zero and étale in codimension one. In other words, f is quasi-étale if dim X = dim Y and if there exists a closed, subset Z ⊆ X of codimension codim X Z 2 such that f | X Z : X Z → Y is étale.

2.7.
Maximally quasi-étale spaces. -Let X be a normal, quasi-projective variety (or a normal, irreducible complex space). We say that X is maximally quasi-étale if the natural push-forward map of fundamental groups, induces an isomorphism between the profinite completions, π 1 (X reg ) ∼ = π 1 (X).
Remark 2.8 (Extension of representations and flat sheaves). -Assume that X is a normal, irreducible complex space that is maximally quasi-étale. Using Malcev's theorem, it has been shown in [GKP16, §8.1] that any representation ρ : π 1 (X reg ) → GL(N, C) factorises via the fundamental group of X, In particular, it follows that any flat bundle on X reg extends to a flat bundle on X.
2.8. Local fundamental groups of contraction morphisms. -The behaviour of the fundamental group under extremal contractions and resolutions of singularities has been studied by Takayama in a series of papers. The following result is due to him.
Proposition 2.9 (Local fundamental groups of contraction morphisms) Let X be a projective klt variety and f : X → Y the contraction of a K X -negative extremal ray. If y ∈ Y is any point, then there exists a neighbourhood U = U (y) ⊆ Y , open in the analytic topology, such that f −1 (U ) is connected and simply connected. Takayama  2.9. Abundance. -We will later use the following special case of the abundance conjecture. Even though the proof uses only standard results from the literature, we found it worth to include it here in full, for the reader's convenience and for later referencing. Then, X is a good minimal model. In other words, K X is semiample.
Proof. -Blowing up X in the indeterminacy locus, we obtain a diagram where π is isomorphic around the general fibre X y ∼ = X y of f or f , respectively. Next, consider the relative Iitaka fibration for π * K X on X/Y . As discussed in [BC15,§2.4], this is given by the (standard) Iitaka fibration for a line bundle of the form L := π * K X + f * (sufficiently ample). Since sufficiently high powers of L have no base points along the general fibre X y , we may replace X by a further blow-up and obtain a diagram as follows, where Z is smooth and where L is numerically trivial on the general fibre X z of X over Z. The numerical dimension ν(X) and the Kodaira dimension κ(X) are then estimated in terms of X y and dim Y as follows, It follows that κ(X) = ν(X), and hence by [KMM87, Cor. 6-1-13] that K X is semiample, as desired. Lemma 2.11 (Negativity Lemma I).
-Let f : X → Y be a surjective morphism of normal, projective varieties. If E ∈ Q Div(X) is non-zero and effective with codim Y f (supp E) 2, then E is not nef.
Proof. -Choose a very ample line bundle L on X and a general tuple of sections This is a normal subvariety of X. The restricted morphism f | S is surjective and generically finite, and E ∩ S is a non-zero, effective, Q-Cartier Q-divisor on S. The Stein factorisation of f | S is then birational and contracts E ∩ S. The negativity lemma, [KM98, Lem. 3.39(1)] with −B = E ∩ S, then applies to conclude that E ∩ S (and therefore E) are non-nef.
Lemma 2.12 (Negativity Lemma II). -Let f : X → Y be a surjective morphism of normal, projective varieties, where Y is a curve. If E ∈ Q Div(X) is non-zero, effective, and maps to a point in Y , then either κ(E) = 1 or E is not nef.
Proof. -Write y := f (E). If supp(E) equals the set-theoretic fibre f −1 (y), then κ(E) = 1. We will therefore assume that supp(E) is a proper subset of f −1 (y). Choose a very ample line bundle L on X and a general tuple of sections (H 1 , . . . , H dim X−2 ) ∈ |L | ×(dim X−dim Y ) . Set S := H 1 ∩ · · · ∩ H dim X−2 . This is a normal surface in X. The divisor E ∩ S is a non-zero, effective, Q-Cartier Q-divisor on S, supported on a proper subset of (f | S ) −1 (y). Zariski's Lemma, [BHPVdV04, Chap. III, Lem. 8.2], will then show that (E ∩ S) 2 < 0. It follows that E ∩ S (and therefore E) cannot be nef. Definition 2.13 (Abelian group scheme). -An Abelian group scheme over a base B is a smooth, proper morphism a : A → B between smooth varieties such that every fibre of a is an Abelian variety, and such that there exists a section B → A.
Definition 2.14 (Generically large fundamental group, [Kol93, Defn. 6.1]) Let X be a normal, projective variety and let Y ⊂ X be a closed subvariety. We say that X has a generically large fundamental group on Y if for all very general points y ∈ Y and for every closed and positive-dimensional subvariety y ∈ Z ⊂ Y with normalisation Z, the image of the natural morphism π 1 Z → π 1 (X) is infinite.
The generalisation of Kollár's result is then formulated as follows.
Proposition 2.15 (Characterisation of étale quotients of Abelian group schemes) Let f : X → Y be a surjective morphism with connected fibres between normal, projective varieties. Assume that there exists ∆ ∈ Q Div(X) such that (X, ∆) is klt. Let y ∈ Y be a very general point with fibre X y and assume that the following holds.
-The fibre X y has a finite étale cover that is birational to an Abelian variety.
-The variety X has a generically large fundamental group on X y .
Then, there exists an étale cover γ X : X → X such that the fibration a : A → Y obtained as the Stein factorisation of (f • γ X ) : X → Y is birational to an Abelian group scheme over a proper base. Proof.
-If X and Y are smooth, this is [Kol95, Th. 6.3]. If not, choose ∆ ∈ Q Div(X) such that (X, ∆) is klt. We will consider resolutions of the singularities and construct a diagram of surjective morphisms between normal, projective varieties, where the morphism q a is birational to an Abelian group scheme over a proper base. For this, begin by choosing appropriate resolution morphisms π X and π Y to construct the right square in (2.15.1). Once this is done, choose a very general point y ∈ Y with very general image point y := π Y ( y). Using a result of Campana, [Cam91, Prop. 1.3], we see that the variety X has a generically large fundamental group on X y . Kollár's result therefore applies to f and gives the left square.
There is more: using the assumption that (X, ∆) is klt, Takayama has shown in [Tak03, Th. 1.1] that the natural push-forward map (π X ) * : π 1 X → π 1 (X) is isomorphic. The étale morphism γ X therefore comes from an étale cover of X, say γ X : X → X. Stein factorising π X •γ X and π Y •γ Y , we will therefore obtain a diagram of the following form, This map makes (2.15.2) commute and is birational to q a, which is in turn birational to an Abelian group scheme. To end the proof, it will therefore suffice to show that a is actually a morphism. This is not so hard: if x ∈ X is any point with preimage Z : The Rigidity Lemma in the form of [Deb01, Lem. 1.15] will then show that a is well-defined at x.

Projective flatness
3.1. Projectively flat bundles and sheaves. -Projective flatness is the core notion of this paper. We will only recall the most relevant definition here and refer the reader to [GKP20, §3] for a detailed discussion of projectively flat bundles and sheaves.
Definition 3.1 (Projectively flat bundles and sheaves on complex spaces) Let X be a normal and irreducible complex space, let r ∈ N be any number and let P → X be a P r -bundle. We call the bundle P → X (holomorphically) projectively flat if there exists a representation of the fundamental group, ρ : π 1 (X) → PGL(r + 1, C), and an isomorphism of complex spaces over X where X is the universal cover of X and where the action π 1 (X) X × P r is the diagonal action. A locally free sheaf F of O X -modules is called (holomorphically) projectively flat if the associated bundle P(F ) is projectively flat.
Definition 3.2 (Projectively flat bundles and sheaves on complex varieties) Let X be a connected, complex, quasi-projective variety and let r ∈ N be any number. A P r -bundle P → X is called projectively flat if the associated analytic bundle P (an) → X (an) is projectively flat. Ditto for coherent sheaves.
3.2. Projectively flat bundles induced by a solvable representation. -One way to analyse a projectively flat sheaf E on a manifold or variety X is to look at the Albanese map X → Alb(X), and to try and understand the restriction E | F to a general fibre. The key insight here is that the Albanese map is a Shafarevich map for the commutator subgroup, [Kol93,Prop. 4.3]. This allows to relate the geometry of the Albanese map to the group theory of the representation that defines the projectively flat structure on E . The following proposition, which is a variation of material that appears in [JR13], is key to our analysis.

Proposition 3.3 (Projectively flat bundles induced by a solvable representation)
Let X be a normal, irreducible complex space that is maximally quasi-étale. Let F be a rank-n reflexive sheaf on X such that F | Xreg is locally free and projectively flat, so that P(F | Xreg ) is isomorphic to a projective bundle P ρ , for a representation ρ : π 1 (X reg ) → PGL(n, C). If the image of ρ is contained in a connected, solvable, algebraic subgroup G ⊂ PGL(n, C), then there exists an isomorphism where A is Weil divisorial (1) and where F is locally free and admits a filtration such that the following holds.
Remark 3.4. -In the setting of Proposition 3.3, note that there exists an isomorphism of Weil divisorial sheaves, det F ∼ = A [⊗n] ⊗ det F . Recall from Remark 2.4 that the invertible factor det F is flat and hence by Lemma 2.5 numerically trivial.
Proof of Proposition 3.3. -To begin, we claim that the projective representation ρ lifts to a linear representation, which we will denote by σ. To formulate our result more precisely, write B(n, C) ⊂ GL(n, C) for the subgroup of upper-triangular matrices and B 1 (n, C) ⊂ B(n, C) for the subgroup of upper-triangular matrices whose top-left entry equals one. In other words, With this notation, we claim there exists a connected linear group G ⊆ B 1 (n, C) ⊂ GL(n, C) such that the natural projection π : GL(n, C) → PGL(n, C) induces an isomorphism between G and G. To this end, let G ⊂ SL(n, C) be the maximal connected subgroup of π −1 (G) ∩ SL(n, C). The restricted morphism π| G : G → G is then finite and surjective, and induces an isomorphism of the Lie algebras Lie( G ) and Lie(G). In particular, G is solvable. Therefore, choosing an appropriate basis of C n , Lie's Theorem allows us to assume without loss of generality that G is contained in the subgroup B(n, C) of upper-triangular matrices. Finally, set G : The obvious fact that any invertible upper triangular matrix A ∈ G ⊆ B(n, C) has a unique multiple λ · A that lies in B 1 (n, C) implies that the restriction π| G : G → G is then isomorphic, as desired. Remark 2.8 and the assumption that X is maximally quasi-étale allow us to identify representations of π 1 (X) and π 1 (X reg ). With these identifications made, the following commutative diagram summarises the situation, If F denotes the locally free, flat sheaf on X associated with σ, then the factorisation of σ via the group B 1 (n, C) implies that F admits a filtration as in (3.3.2) that satisfies conditions 3.3.3 and 3.3.4. By construction, we have P(F | Xreg ) ∼ = P(F | Xreg ), and so there exists an invertible sheaf A reg on X reg such that F | Xreg ∼ = F | Xreg ⊗A reg . A presentation of F as in Equation (3.3.1) therefore exists by taking A := ι * A reg , where ι : X reg → X is the inclusion.

Varieties with splitting cotangent sheaves
One relevant case that keeps appearing in the discussion of varieties with projectively flat sheaf of reflexive differentials is the one where Ω [1] X decomposes as a direct sum of Weil divisorial sheaves. We discuss settings where there are local or global decompositions of this form. The results of this section will be used later in the proof of our main result, but might be of independent interest. In relation to [JR13], the local results are new, prompted by the appearance of singularities in our context, and the results on varieties with globally split cotangent sheaf allow us to replace some of the more involved arguments of loc. cit. by a clearer structure result.
X is projectively flat, for instance because the flatness criterion of [GKP20, Th. 1.6] applies, the local description of projectively flat sheaves, [GKP20, Prop. 3.11], can often be used to reduce to the case where Ω X decomposes locally. We will show that this already forces X to have quotient singularities. where L is reflexive of rank one. Then, X has only isolated, cyclic quotient singularities.
We prove Proposition 4.1 in Section 4.3.
-If X is a quasi-projective variety with klt singularities where Ω [1] X is of the form L ⊕ dim X for a Weil-divisorial sheaf L , then the proof of Proposition 4.1 applies verbatim to show that X is Zariski-locally a quotient of a smooth variety. We do not use this fact however.
X decomposes globally, for instance because it is projectively flat and X is known to be simply connected. This obviously puts strong conditions on the global geometry of X; see [Dru00] for further results in the smooth setup.
The following proposition, including the proof, is due to the referee. He also suggested Theorem 4.4, which is stronger than our original result that included a pseudoeffectivity assumption. Our previous arguments were of analytic nature and used a theorem of Demailly [Dem02].
For reasons that will become apparent in a second, we are particularly interested in the coordinate hyperplanes To see how the Z i come into play, restate the assumption as a decomposition of the tangent bundle: T U ∼ = A 1 ⊕ · · · ⊕ A n . The summands A • ∼ = L * are regular rank-one foliations on U , with normal bundles The assertion is thus shown.
Applying Proposition 4.3 to the smooth locus of a klt variety, we obtain the following characterisation of quasi-abelian varieties. Proof.
-We have seen in Proposition 4.3 that the Q-Cartier sheaves L are numerically trivial, and then so is ω X ∼ = L [⊗n] . Since X is klt, it follows from [Nak04, Prop. V.4.9] or [Amb05, Th. 4.2] that ω X and L are torsion. As a consequence, we find a finite quasi-étale cover γ : X → X such that the reflexive pull-back γ [ * ] L is trivial. The space X is klt and its tangent sheaf  . Recall that ρ decomposes into direct a sum of onedimensional representations. Identifying Z m with roots of unity, we can thus find linear coordinates on V such that multiplication with ξ ∈ Z m is given as An elementary computation in these coordinates shows that S • exists, and can in fact be chosen to be dense in S.
Returning to the general case where X is arbitrary, let s ∈ S be any point. By assumption, there exists an open neighbourhood U = U (x) and a cyclic cover γ : U → U , say with group G. Shrinking U and passing to a subgroup, if need be, we may assume without loss of generality that γ is totally branched over s, that is γ −1 (s) = { s}. The cyclic group thus acts on U and fixes s. Proof of Proposition 4.1. -As a first step in the proof of Proposition 4.1, we claim that L is Q-Cartier. In fact, taking determinants of both sides in (4.1.1), we obtain that L [⊗ dim X] ∼ = ω X , which is Q-Cartier by assumption. Choose a minimal number N ∈ N + such that L [⊗N ] is locally free. The reflexive tensor product Ω As a second step, we show that X has cyclic quotient singularities. In fact, given any x ∈ X, we find an open neighbourhood U = U (x) over which L [⊗N ] is trivial. Chose a trivialisation and let γ : U → U be the associated index-one cover, which is cyclic of order N . The complex space U has again klt singularities and γ [ * ] L ∼ = O U . In particular, it follows that Ω , then asserts that U is smooth. We conclude that X has cyclic quotient singularities only.
As a third and last step, we show that the singularities of X are isolated. Assume to the contrary and let S ⊆ X sing be a positive-dimensional, irreducible component. We have seen in Lemma 4.5 that there a non-empty, open subset S • ⊆ S such that every x ∈ S • admits an open neighbourhood U = U (x) ⊆ X of product form U = W × V , where W is smooth and V has an isolated quotient singularity. In particular, dim W = dim S and Ω V . In particular, the reflexive tensor power Ω Xreg is projectively flat, at least after going to a quasi-étale cover. This allows to apply following lemma in various settings that appear throughout the proof of Theorem 1.2.
Lemma 5.1 (Consequences of projectively flat cotangent bundle). -Let X be a smooth, quasi-projective variety of dimension n where Ω 1 X is projectively flat. (5.1.1) If K X is nef, then X does not contain complete rational curves. (5.1.2) Assume that there exists an immersion η : F → X where F is projective and smooth. If η * K X is nef and if K F ≡ 0, then F is quasi-Abelian.
Proof. -We prove the statements separately. Both proofs use the fact that every locally free, projectively flat sheaf on a simply connected space is isomorphic to a direct sum of the form L ⊕ • where L is invertible, cf. [GKP20, Prop. 3.11].
As to the first statement, let η : P 1 → X be a non-constant map. Since η * Ω 1 X is projectively flat, there is an integer m such that η * Ω 1 X ∼ = O P1 (m) ⊕n . The canonical morphism η * Ω 1 X → Ω 1 P1 then yields m −2, contradicting the nefness of η * K X . As to the second statement, applying the classic Decomposition Theorem [Bea83, Th. 1] and possibly passing to a finite étale cover, we may assume that F is of product form, F ∼ = A × Z, where A is Abelian (possibly a point) and where Z is simply connected with K Z = 0. We aim to prove that Z is a point, argue by contradiction and assume that m := dim Z is positive. Let ι : Z → F denote the inclusion morphism. The pull-back (η • ι) * Ω 1 X is projectively flat and hence of the form L ⊕n , for a suitable line bundle L ∈ Pic(Z). The obvious surjection induces a non-trivial map L ⊗m → ω Z ∼ = O Z . But then the assumption that η * K X and hence also L are nef shows that L ⊗m cannot be a proper ideal sheaf, so L ⊗m ∼ = O Z . It follows that either L is torsion or that H 0 Z, Ω 1 Z = 0, both contradicting the simple connectedness of Z.

5.2.
Proof of Theorem 1.2. -We maintain notation and assumptions of Theorem 1.2 throughout the present Section 5.2. The proof follows the strategy of [JR13] closely, but has to overcome a fair number of technical difficulties arising from the presence of singularities. It is fairly long and therefore subdivided into numerous steps. The main idea is to show abundance for X, and then to analyse the Iitaka fibration. Abundance is shown in Steps 4, 5 and 6, with an inductive argument using repeated covers, fibrations, and restrictions to fibres.
-Steps 1-3 set the stage, prove minimality of X and put limits on its possible numerical dimension and Kodaira dimension.
-Step 4 fibres X using a Shafarevich map construction. The general fibre is called F .
-Step 5, which is the longest step of this proof, considers a suitable cover of F of F and takes a general fibre of its Albanese map. An analysis of this fibre will then give abundance for F .
-Step 6 uses abundance for F to prove abundance for X.
-Steps 7-8 end the proof by showing that the Iitaka fibration of a suitable cover of X is birational (and then isomorphic) to an Abelian group scheme over a proper base.
Step 1: Simplification. -Since assumption and conclusion of Theorem 1.2 are stable under quasi-étale covers, we may apply [GKP16, Th. 1.5], pass to a maximal quasi-étale cover and assume that the following holds in addition.
With this assumption in place, the criterion for projective flatness, [GKP20, Th. 1.6], implies that Ω 1 Xreg is projectively flat. As we have seen, this has a number of interesting consequences. Notation 5.5. -We chose a projectively flat bundle P → X as in Consequence 5.3 and maintain this choice throughout. We write τ : π 1 (X) → PGL(n, C) for the representation that defines the projectively flat structure on P.
If K X is numerically trivial, then the Chern class equality (1.2.1) reduces to c 2 (X) · [H] n−2 = 0, and we may apply [LT18, Th. 1.2] to conclude that X is quasi-Abelian. This allows to make the following assumption.
Assumption w.l.o.g. 5.6 (K X ≡ 0). -The canonical class K X is not numerically trivial.
Step 2: Minimality of X. -Next, we show that the variety X is minimal.
Claim 5.7 (Minimality of X). -The canonical divisor K X is nef.
Proof of Claim 5.7. -Choose an integer m such that ω [⊗m] X is invertible. We argue by contradiction and assume that there exists a contraction of a K X -negative extremal ray. Choosing a resolution of X, we find a sequence of morphisms If y ∈ Y is any point, Takayama's result on the local fundamental groups of φ, Proposition 2.9, allows to find a neighbourhood U = U (y) ⊆ Y (an) , open in the analytic topology, such that V := φ −1 (U ) ⊆ X (an) is connected and simply connected. Using simple connectedness, the local description of projectively flat sheaves, [GKP20, Prop. 3.11], provides us with a reflexive, rank-one coherent analytic sheaf L on V and an isomorphism Ω V , intersects all curves trivially, as this will contradict negativity of the contracted extremal ray. To this end, write V := π −1 (V ) and consider the following commutative diagram, (5.7.1) X is reflexive, the morphism α is injective by [ST71, Th. 1.14] and [BS76, Cor. II.3.15]; see also [BS76, Th. II.3.6]. By commutativity of (5.7.1), the Chern class c 1 (π * L [⊗n·m] ) of the pullback via π vanishes. The claim follows from the projection formula for intersection numbers.
Step 3: Kodaira dimension. -We aim to show that the minimal variety X is quasi-Abelian, which implies in particular that κ(X) = 0. As a first step, we show that X is at least not of general type.
Claim 5.8. -The variety X is not of general type.
Proof of Claim 5.8. -Suppose to the contrary that X is of general type. Using that X is also minimal, recall from [GKPT19a, Th. 1.1] that the Chern classes of X satisfy a Q-Miyaoka-Yau inequality, On the other hand, we assume that X ) has vanishing Q-Chern classes with respect any ample bundle, and hence also with respect to any nef bundle. We obtain that But we know that K X is big and nef, so that [K X ] n > 0. Putting (in)equalities (5.8.1) and (5.8.2) together hence produces a contradiction. (Claim 5.8) Step 4: The Shafarevich map. -Recall Notation 5.5 and consider the representation τ : π 1 (X) → PGL(n, C) that defines the projectively flat structure on P. Write for the algebraic Zariski closure of its image. This is a linear algebraic group, which has finitely many components. Applying Selberg's Lemma (2) and passing to an appropriate étale cover of X, we may hence assume without loss of generality that the following holds.
Assumption w.l.o.g. 5.9 (Connectivity and torsion-freeness). -The linear algebraic group G X is connected. Writing Rad(G X ) G X for the solvable radical of G X , (2) see [Alp87] for a brief account of this classic result.
the image of the composed group homomorphism to the linear algebraic group G X Rad(G X ) is torsion-free, If G X reduces to a point, we have Ω We may then apply Theorem 4.4 to conclude that X is quasi-Abelian. As a consequence, we may assume without loss of generality that the following holds.
Assumption w.l.o.g. 5.10 (Representation has infinite image). -The group G X is of positive dimension.
We now follow the strategy of [JR13] and use Assumption 5.10 to consider the Shafarevich map for the solvable radical of G X . We refer the reader to [Kol95,§3] for more on Shafarevich maps.
Construction 5.11. -Consider the composed morphism (5.9.1) of Assumption 5.9, let K π 1 (X) be its kernel and consider an associated K-Shafarevich map, sha K (X) : The following two properties of sha K (X) will be most relevant in the sequel.
has finite image.
Claim 5.14 (The base of the Shafarevich map). -The (smooth) variety Sha K (X) is of general type.
Proof of Claim 5.14. -Choose a resolution π : X → X, recall from [Tak03, Th. 1.1] that the natural push-forward map π * : π 1 ( X) → π 1 (X) is isomorphic and observe that the composed map sha K (X) • π : X Sha K (X) is therefore a Shafarevich map for K π 1 ( X). Using the assumption that the image of the composed morphism (5.9.1) is torsion-free, the claim then follows from [CCE15, Th. 1].
(Claim 5.14) Claims 5.8 and 5.14 together imply in particular that the fibres of sha K (X) are positive-dimensional. These will be investigated next.
Step 5: Fibres of the Shafarevich map. -Throughout the present step, choose a general fibre F ⊂ X of sha K (X). There are two cases to consider: either Sha K (X) is a point and F = X is potentially singular, or F is a proper subvariety of X, and then it avoids the (finitely many!) singularities of X, cf. Consequence 5.4, and must hence be smooth. Either way, we will show in this step that F is quasi-Abelian. The proof is somewhat long and therefore divided into sub-steps.
Notation 5.15. -Given an étale cover γ : F → F , consider the composed group morphism and let G γ ⊆ PGL(n, C) be the Zariski closure of its image.
Remark 5.16. -Given an étale cover γ : F → F , the group G γ is an algebraic subgroup of PGL(n, C) with finitely many connected components. Fact 5.13 implies that the maximal connected subgroup of G γ is solvable.
Step 5-1: Irregularity. -To begin our analysis of the fibres of the Shafarevich map, we show that the fibres have positive irregularity, at least once we pass to a suitable étale cover. This will later allow to analyse the fibres by means of their Albanese map.
Claim 5.17. -There exists an étale cover γ : Proof of Claim 5.17. -Passing to a first étale cover of F , we may assume without loss of generality that G γ is connected. If the composed map (5.15.1) has infinite image, then by Remark 5.16 this defines a solvable representation of π 1 F with infinite image. It follows that H 1 F , Z has positive rank and consequently, that dim C H 1 F , C > 0. The description of the natural Hodge structure on this space, [Sch16, Th. 1], then implies that h 0 F , Ω [1] F > 0, as desired. It therefore remains to consider the case where the composed map (5.15.1) has finite image. By Assumption 5.10, this implies that F is a proper subvariety of X, entirely contained in X reg . We may then choose γ such that the composed map (5.15.1) is trivial, implying as before that the pull-back of the projectively flat sheaf Ω X is a direct sum of the form, γ * Ω 1 X ∼ = L ⊕n , for some L ∈ Pic( F ). The pull-back of the conormal bundle sequence, which is non-trivial for reasons of dimension, will then read as follows, such that all sheaves F i are locally free and flat with rank(F i ) = i and such that the sheaf F 1 is trivial, F 1 ∼ = O X . In particular, there exists an inclusion X . The pull-back of the normal bundle sequence (which is trivial if F = X) reads Step 5-2: Numerical dimension and Kodaira dimension. -Claim 5.17 has fairly direct consequences for the Kodaira dimension of the fibres, which will turn out to be either zero or one, the second case disappearing eventually.
Claim 5.19 (Bounding the Kodaira dimension from below). -Maintaining Notation 5.18, if π : F → F is any resolution of singularities, then the Kodaira dimension of F is non-negative. In particular, it follows that κ( F ) 0.
Proof of Claim 5.19. -The assumption that h 0 F , Ω [1] F > 0, together with the Isomorphism (5.18.1) and the Filtration (5.18.2) allows to find an index i such that Writing L i := F i F i−1 and recalling that L i is invertible and flat, hence numerically trivial by Lemma 2.5, Equation (5.19.1) implies in particular that the sheaf also has a non-trivial section. In this setting, the extension theorem for differential forms, [GKKP11,Th. 4.3] yields a non-trivial morphism π [ * ] ω F → ω F and hence a non-trivial section of ω F ⊗ π * L , where again π * L ∈ Pic( F ) is numerically trivial. F . If this morphism is constantly zero, then A maps into the trivial sheaf γ * N * F/X , the nef sheaf A must thus be trivial, and the claim is shown. We will therefore continue this proof under the assumption that (5.20.2) is an embedding.
Next, choose a general tuple (H 1 , . . . , H m−2 ) ∈ |H| × · · · × |H| and consider the associated complete intersection surface S := H 1 ∩ · · · ∩ H m−2 . Then, S has klt singularities (3) and A | S is reflexive. The kernel of the restriction morphism for reflexive differentials is so in particular anti-ample. But since the nef sheaf A never maps to an anti-ample, we find that the composed sheaf morphism Proof of Claim 5.21. -Given that γ : F → F is étale, Claim 5.20 immediately implies that ν(F ) 1.
gives the two rightmost inequalities in (5.21.1), and so it remains to show that κ(F ) 0. But we already know from Claim 5.19 that κ( F ) 0, so that there exists a number p ∈ N and a non-trivial section σ ∈ H 0 F , ω ⊗p F . But then is a non-trivial Galois-invariant pluri-form on F , which hence descends to a non-trivial pluri-form on F .
(Claim 5.21) (3) The surface S will in fact almost always be smooth, except perhaps when X = F and dim X = 2.
J.É.P. -M., 2021, tome 8 Step 5-3: Fibres of the Albanese map. -We pointed out above that the Albanese map of F is not trivial, at least once we pass to a suitable cover. Using the fact that the Albanese yields a Shafarevich map for the commutator subgroup of the fundamental group, there is much that we can say about its fibres, which will eventually turn out to be abundant.
Construction 5.22. -Maintaining Notation 5.18, consider the Albanese map alb( F ) : F → Alb( F ) and recall from Claim 5.17 and Notation 5.18 that this map is non-trivial, which is to say that dim Alb( F ) > 0. Let F 1 F be a general fibre component of alb( F ). This might be a point if F is of maximal Albanese dimension. Since F has at worst isolated singularities, F 1 is necessarily smooth. Now, we use the fact that Albanese maps are Shafarevich maps by recalling from [Kol95, §0.1.3] that the push-forward morphism of Abelianised fundamental groups, has finite image. We can thus find an étale cover γ 1 : F 1 → F 1 such that the composed push-forward is trivial. The following diagram summarises the morphisms that we have discussed so far, Observing that img(η) X reg , we consider the exact sequence of differentials, Claim 5.23 (Numerical invariants of F 1 and F 1 ). -Assume the setting of Construction 5.22. Then, 0 κ(F 1 ) ν(F 1 ) 1 and 0 κ( F 1 ) ν( F 1 ) 1.
Proof of Claim 5.23. -In order to lighten notation a little bit, we will suppress ι 1 in the following discussion. Given that the morphism γ 1 is étale, it suffices to consider the variety F 1 only. We will also assume that F 1 is not a point, or else there is little to show, cf. Claim 5.21. On the one hand, using that img(η) X reg , we obtain a description of the canonical bundle, ω F1 Observing that γ * 1 A is invertible and that γ * 1 F is flat, this gives a numerical equiva- Proof of Claim 5.24. -As before, we consider the variety F 1 only and assume that it is not a point. We also consider the following commutative diagram of group morphisms, The representation τ •η * is solvable because it factors via the solvable group G γ . Better still, the representation τ • η * is in fact trivial, or else there would be a non-trivial Abelian subrepresentation, contradicting triviality of the Abelianised push-forward (ι 1 • γ 1 ) (ab) * . As a consequence, we find that η * Ω 1 X is isomorphic to a direct sum of line bundles, Recalling from Sequence (5.22.1) that η * Ω 1 X contains the trivial, non-zero subbundle Ω 1 F1/X , we find that L admits a section. But then η * Ω 1 X is generically generated, and so is its quotient Ω 1 F1 . By [Fuj13, Rem. 2.2], the manifold F 1 is then of maximal Albanese dimension. But then [Fuj13,Th. 4.2] implies that the nef canonical bundle K F1 is semiample, as claimed.
(Claim 5.24) Step 5-4: Abundance and description of F . -In this step, we finally prove abundance for F and describe its Iitaka fibration. According to Claim 5.21, there are only two cases, κ(F ) = 0 and κ(F ) = 1, which we consider separately.
Proof of Claim 5.25. -To keep this proof readable, we consider three cases separately.
(5.25.1) The divisor K F is numerically trivial and F = X.
(5.25.2) The divisor K F is numerically trivial and F X. (5.25. 3) The divisor K F is not numerically trivial.
Case 5.25.1 is easiest. If K X is numerically trivial, then Equation (1.2.1) guarantees that X · [H] n−2 = 0. Then, X is quasi-Abelian by [LT18, Th. 1.2] and we are done. In Case 5.25.2, apply Item 5.1.2 of Lemma 5.1 to the inclusion F ⊂ X reg to find again that F is quasi-Abelian.
We consider Case 5.25.3 for the remainder of the proof, with the aim of producing a contradiction. To begin, we consider a strong log resolution of F and the Stein factorisation of the associated Albanese morphism. We obtain a commutative diagram as follows Observe that the image F 1 = π( F 1 ) is a connected component of a general fibre of alb F , which avoids the (finitely many) singularities of F . The restriction π| F1 : F 1 → F 1 is thus isomorphic, and Claim 5.24 applies to show that F 1 is a good minimal model. More is true. Since κ(F ) = 0 by assumption, we deduce from Claim 5.19 that κ( F ) = 0. Hence it follows from [Lai11,Th. 4.2] that F has a good minimal model F min , which has terminal singularities and numerically trivial canonical class.
We use the existence of F min to describe K F in more detail. To this end, resolve the singularities of the rational map F min F and obtain morphisms as follows, The fact that F min has terminal singularities implies that K Y ≡ E, where E ∈ Q Div(Y ) is effective with π 2 -exceptional support. Recalling that every fibre of π 2 is rationally connected by [HM07, Cor. 1.5], we find that every component of supp E is uniruled. The canonical class K F is then numerically equivalent to E := (π 1 ) * E. The Q-divisor E is effective, nef and not trivial (because K F is not numerically trivial), and its components are again uniruled. But the Albanese map alb F contracts all rational curves in supp E! We claim that the image set alb F (supp E) cannot contain any isolated points.
-If dim alb F ( F ) 2, this follows from nefness of E ≡ K F = γ * K X and from the Negativity Lemma 2.11.
-If dim alb F ( F ) = 1, this follows from nefness of E ≡ K F = γ * K X , from the assumption that 0 = κ(F ) = κ( F ) and from the Negativity Lemma 2.12.
Either way, we find that E contains rational curves that stay away from the finitely many singular points of F . Since K X is nef, this contradicts Item 5.1.1 of Lemma 5.1.
(Claim 5.25) Step 6: Abundance for X, description of the Iitaka fibration. -We have seen in the previous step that the general fibres of the Shafarevich map are abundant. As we will see now, this implies Abundance for X.
Claim 5.27 (Abundance for X). -The canonical bundle K X is semiample.
Proof of Claim 5.27. -If Sha K (X) is a point, then F = X and we have seen in Claim 5.21 that 0 κ(F ) = κ(X) 1. We know that abundance holds because the two cases have been described in Claims 5.25 and 5.26, respectively. If Sha K (X) is positive-dimensional, then we can apply Proposition 2.10 to the Shafarevich map.
-Assumption 2.10.1 is satisfied by Claim 5.14 -Assumption 2.10.2 is satisfied as nefness of K X has been shown in Claim 5. Step 7: Abelian group schemes. -Following the ideas of Jahnke-Radloff, [JR13], we use Kollár's characterisation of étale quotients of Abelian group schemes to show that the Iitaka fibration of a suitable étale cover of X is birational to an Abelian group scheme.
Claim 5.30. -There exists an étale cover X → X whose Iitaka fibration is birational to an Abelian group scheme over a smooth projective base admitting a level three structure. More precisely, there exists a commutative diagram where S is smooth and projective, and α is a family of polarised Abelian varieties admitting a level three structure.
Proof of Claim 5.30. -We construct from right to left a diagram as follows, where q a has connected fibres, where α and q α are Abelian group schemes and where α has the additional structure of a family of Abelian varieties with level three structure. To begin, we claim that the variety X has generically large fundamental group along the general fibre X y of the Iitaka fibration. In case where X is smooth, this claim has been shown in [JR13,Prop. 5.1]. We leave it to the reader to check that in our case, where X has only finitely many singularities, the proof of [JR13, Prop. 5.1] still applies verbatim. Given that the fibres of the Iitaka fibration are quasi-Abelian, Proposition 2.15 will therefore apply to yield the right and middle square of Diagram (5.30.2) -strictly speaking, Proposition 2.15 gives ρ only as a rational map, but we can always blow up to make it a morphism, and then pull back q A. Once this is done, we can find a suitable étale cover S → q S such that the pull-back A := q A × q S S is an Abelian group scheme with level three structure, and a result of Grothendieck allows to equip A/S with a polarisation, cf. [Ray70, Th. XI.1.4]. For later reference, we note that the following property holds by construction.
(5.30.3) If s ∈ S is any general point, with image y ∈ Iit(X), then the induced map between fibres, A s X y is birational.
We still need to construct X. We begin its construction with the observation that K where β * H is ample on Y ; this shows that q a is the Iitaka fibration for q X and allows to apply Ambro's canonical bundle formula, Remark 5.29, which shows that q Y is the underlying space of a klt pair. We are interested in this, because [Tak03, Th. 1.1] then asserts that π 1 q S = π 1 q Y . As a consequence, we find that the Stein factorisation of the morphism S → q Y factors via an étale cover of q Y and therefore gives a diagram, where X := q X × q Y Y and where the map φ is induced by the universal property of the fibre product. Observe, exactly as in (5.30.4), that the natural projection map X → Y is the Iitaka fibration of X, iit( X) : X → Iit( X) = Y . Assertion 5.30.3 implies that φ is birational.
(Claim 5.30) Claim 5.31. -There exists an étale cover X → X whose Iitaka fibration is birational to an Abelian group scheme over Iit( X) with level three structure. More precisely, there exists a commutative diagram where α is a family of polarised Abelian varieties with level three structure.
Proof of Claim 5.31. -Consider Diagram (5.30.1) and recall that polarised Abelian varieties with level three structure admit a fine moduli space A 3 with universal family U 3 → A 3 , and that moreover A 3 does not contain any rational curve [Kol93, Lem. 5.9.3]. On the other hand, while observing Remark 5.29 recall from [HM07, Cor. 1.5] that the fibres of the morphism ψ are rationally chain connected. It follows that the moduli map S → A 3 factors via ψ to give a morphism Iit( X) → A 3 . To conclude, set A := U 3 × A3 Iit( X).
(Claim 5.31) Step 8: End of proof. -We end the proof by showing that X itself is an Abelian group scheme over a smooth base with ample canonical bundle.
Claim 5.32. -In the setting of Claim 5.31, the rational map δ := φ −1 : X A is a morphism.
Proof of Claim 5.32. -Given that X is klt and that the fibres of α do not contain any rational curves, the claim follows from [HM07, Cor. 1.6].
(Claim 5.32) Claim 5.33. -The base variety Iit( X) is smooth. In particular, A is smooth.
Proof of Claim 5.33. -Assume not. Let s ∈ Iit( X) sing be any singular point, and let a ∈ α −1 (s) be a general point of the fibre. Note that the morphism α is smooth, so a will be a singular point of A. On the other hand, note that the general choice of a guarantees that δ −1 (a) does not contain any of the (finitely many) singular points of X. It follows that δ resolves the singularity at a, and that the fibre δ −1 (a) is thus necessarily positive-dimensional.
There is more that we can say. Recalling from Remark 5.29 that Iit(X), ∆ is klt for a certain Q-divisor ∆ it follows from smoothness of α that (A, α * ∆) is klt as well. But then δ −1 (a) iit( X) −1 (s) ∩ X reg is covered by rational curves, violating Item 5.1.1 of Lemma 5.1 from above.
(Claim 5.33) Claim 5.34. -The morphism δ is an isomorphism and X is therefore smooth.
Proof of Claim 5.34. -If a ∈ A is any (smooth!) point where δ is not isomorphic, then δ −1 (a) is positive-dimensional, and every section of K X will necessarily vanish along δ −1 (a). That contradicts the semiampleness of K X , Claim 5.27. (Claim 5.34) In the current, now nonsingular setup, the following has already been observed in [JR13, Proof of Lem. 5.1].
Claim 5.35. -The canonical bundle of Iit( X) is ample.
Proof of Claim 5.35. -The smooth variety Iit( X) is of general type, while the existence of σ and minimality of X and hence X together imply that its canonical bundle is nef, hence abundant. Fibres of the Iitaka fibration for Iit( X) are covered by rational curves, which do not exist by the existence of σ and the fact that X (as an étale cover of X) does not contain rational curves by Item 5.1.1 of Lemma 5.1. Now that we know that X is an Abelian group scheme over a smooth projective base with ample canonical bundle, the final step of the argument of Jahnke-Radloff, [JR13, Th. 6.1], applies to conclude also our proof.
(Theorem 1.2) 6. Proof of Theorem 1.3 Step 1: Preparations. -In the setup of Theorem 1.3, we set Let H be a very ample divisor on X and let (D 1 , . . . , D n−1 ) ∈ |H| ×(n−1) be a general tuple of divisors, with associated general complete intersection curve C := D 1 ∩ · · · ∩ D n−1 . The curve C is then smooth and entirely contained in X reg . The restriction F | C is locally free and nef, with c 1 (F | C ) = 0. It follows that F | C is semistable and therefore that F is semistable with respect to H. But then, Ω [1] X will likewise be semistable with respect to H. Step 2: Local freeness of F . -Next, choose a resolution of singularities π : X → X such that the quotient sheaf F := π * (F ) tor is locally free; such a resolution exists by [Ros68,Th. 3.5]. Recall from Fact 2.2 that since F is nef, then so are π * (F ), F and det F . There is more that we can say. Since the determinant of F is trivial by construction, det F ∼ = O X , we may write where E i are π-exceptional and a i ∈ Z.
The divisor a i · E i is nef. But then the Negativity Lemma, [KM98, Lem. 3.39(1)], asserts that a i 0 for all i, which of course means that all a i = 0. It follows that det F = O X and thus that F is numerically flat. This has two consequences.
-First, it follows from the descent theorem for vector bundles on resolutions of klt spaces, [GKPT19b, Th. 1.2], that F is locally free, numerically flat, and that F = π * (F ).
Step 3: Singularities of X.
-If X → X is any maximally quasi-étale cover, then local freeness of F immediately implies local freeness and nefness for the reflexive normalised cotangent sheaf of X. The covering space X therefore reproduces the assumptions of Theorem 1.3, which allows us to assume without loss of generality that X is itself maximally quasi-étale. Together with projective flatness of Ω 1 Xreg , this assumption allows to apply the local description of projectively flat sheaves found in [GKP20, Prop. 3.11]: every singular point x ∈ X (an) admit a neighbourhood U , open in the analytic topology, such that Ω [1] U L ⊕n U with some Weil-divisorial sheaf L U on U . It follows from Proposition 4.1 that X has at worst isolated singularities.
The claim thus follows from the semistability of Ω 1 X and Theorem 1.2.
Step 5: End of proof in case where X is a surface. -It remains to consider the case where dim X = 2, so that X is a surface with klt quotient singularities. We have seen above that Sym [2] Ω 1 X = F ⊗ O X (K X ), where F is a locally free numerically flat sheaf of rank three. The second Q-Chern class is therefore computed as On the other hand, the standard formula for the second Chern class of a symmetric product gives c 2 Sym [2] Ω [1] X = 2 · [K X ] 2 + 4 · c 2 (X). Comparing these two equations, we find 1 4 ·[K X ] 2 = c 2 (X). Thanks to the semistability of Ω [1] X , we may again apply Theorem 1.2 and end the proof.