Harmonic metrics on Higgs sheaves and uniformization of varieties of general type
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Abstract
We prove a criterion for the existence of harmonic metrics on Higgs bundles that are defined on smooth loci of klt varieties. As one application, we resolve the quasiétale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball by discrete, cocompact groups of automorphisms that act freely in codimension one. As a further application, we establish a nonabelian Hodge correspondence on smooth loci of klt varieties.
Mathematics Subject Classification
32Q30 14E20 14E30 53C071 Introduction
1.1 Main result of this paper
The core notion of nonabelian Hodge theory, as pioneered by Corlette, Donaldson, Hitchin, and Simpson, is certainly that of a harmonic bundle. Most (if not all) important results of this theory depend on existence results for harmonic metrics in bundles over projective manifolds, which are usually established using highly nontrivial analytic methods.
In view of the minimal model program, it is clear that these results should be studied in the more general context of varieties with terminal or canonical singularities or even for klt (= Kawamata log terminal) varieties. In this context, extending Simpson’s theory [37] from smooth projective manifolds to minimal models, the paper [9] established a natural equivalence between the category of local systems and the category of semistable, locally free Higgs sheaves with vanishing Chern classes on projective varieties with klt singularities
Theorem 1.1
[9, Thm. 1.1] Let X be a projective, klt variety. Then, there exists an equivalence between the category of local systems on X and the category of semistable, locally free Higgs sheaves with vanishing Chern classes on X. \(\square \)
However, for geometric applications we also need to understand (flat) locally free Higgs sheaves on the smooth locus \(X_{{{\,\mathrm{reg}\,}}} \) of a klt variety X which extend to X as reflexive Higgs sheaf rather than as a locally free Higgs sheaf. Thanks to the work of Simpson, JostZuo, Mochizuki and others, there is by now a developed theory of harmonic bundles for noncompact, quasiprojective manifolds \(X^{\circ }\), establishing the existence of a (essentially unique) harmonic metric on a given semisimple flat bundle \((E,\nabla _E)\) on \(X^{\circ }\). In particular, E inherits a holomorphic structure and a Higgs field. In this context, considering a compactification X of \(X^{\circ }\) by a simple normal crossing divisor, the notion of a tame and purely imaginary bundle (with respect to the compactification) play a decisive role. In case of a projective klt variety X with smooth locus \(X^{\circ } = X_{{{\,\mathrm{reg}\,}}}\), the situation simplifies to some extent since \(X {\setminus } X^{\circ }\) has codimension at least 2, once one is willing to pay the price that X is singular. Our point is that the singularities arising from the minimal model program are mild enough to still obtain a nonabelian Hodge theory on \(X_{{{\,\mathrm{reg}\,}}}\) that can be formulated in downtoearth terms. Our main result can be seen as an existence result for harmonic metrics on bundles that are defined on smooth loci of klt varieties.
Theorem 1.2
 (1.2.1)The Higgs sheaf \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) is (poly)stable with respect to H and the \(\mathbb {Q}\)Chern characters satisfy$$\begin{aligned} \widehat{ch}_1(\mathscr {E}_X)\cdot [H]^{n1} = 0 \quad \text {and}\quad \widehat{ch}_2(\mathscr {E}_X)\cdot [H]^{n 2} = 0. \end{aligned}$$
 (1.2.2)
The sheaf \(\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}\) is locally free, \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) is induced by a tame, purely imaginary harmonic bundle whose associated flat bundle is (semi)simple.
Remark 1.3
The symbols \(\widehat{ch}_\bullet \) that appear in Theorem 1.2 denote \(\mathbb {Q}\)Chern characters, which exist because klt varieties have quotient singularities in codimension two. We refer to Sect. 2.7 and [10, Sect. 1.7] for a discussion, and to [8, Sect. 3] for a proper definition.
We illustrate the usefulness of Theorem 1.2 with two applications.
1.2 Application: a nonabelian Hodge correspondence for local systems on \(X_{{{\,\mathrm{reg}\,}}}\)
The first application pertains to the nonabelian Hodge correspondence. As already mentioned, Theorem 1.1 extends the classic correspondence to projective varieties with klt singularities, in particular to minimal models of varieties of general type.
As one immediate consequence of Theorem 1.2, we find that our singular varieties exhibit two nonabelian Hodge correspondences, one linking local systems on the singular space X with locally free Higgs sheaves there, and one linking local systems on the smooth locus \(X_{{{\,\mathrm{reg}\,}}}\) with Higgs sheaves that may acquire certain singularities along the singular locus of X. One of the key features of the theory is that the two correspondences coincide after passing to a finite quasiétale cover \(Y \rightarrow X\). We refer to reader to Sect. 6 for precise formulations.
1.3 Application: quasiétale uniformisation
Again, in view of the minimal model program it is clear that these results should be studied in the more general context of varieties with terminal, canonical, or even klt singularities. In this context, the Miyaoka–Yau inequality has been generalised as follows; the formulation again uses \(\mathbb {Q}\)Chern classes as constructed in [8, Sect. 3].
Theorem 1.4
The reader is encouraged to also have a look at [14], where related inequalities for pairs are discussed.
Our new quasiétale uniformisation result for varieties of general type may then be formulated as follows.
Theorem 1.5
We refer to Sect. 2.2 for the definition of “quasiétale” and for further references.
Corollary 1.6
(Chern class equality forces quotient singularities) In the setting of Theorem 1.4, equality in the \(\mathbb {Q}\)Miyaoka–Yau Inequality implies that \(X_{{{\,\mathrm{can}\,}}}\) has at worst quotient singularities. \(\square \)
Remark 1.7
Theorem 1.5 was shown by the authors in [8] under the additional assumption that the variety X be nonsingular in codimension two; this technical condition allowed us to use a significantly simpler argument.
The canonical models that appear in Theorem 1.5 are themselves quotients of the unit ball. The following characterisation is a minor generalisation of [8, Thm. 1.3]; the proof given in [8, Sect. 9.1] applies nearly verbatim and is therefore omitted.
Theorem 1.8
 (1.8.1)
The space X is of the form \(\mathbb {B}^n/\widehat{\Gamma }\) for a discrete, cocompact subgroup \(\widehat{\Gamma } < {{\,\mathrm{Aut}\,}}_{\mathscr {O}}(\mathbb {B}^n)\) whose action on \(\mathbb {B}^n\) is fixedpoint free in codimension one.
 (1.8.2)
The space X is of the form Y / G, where Y is a smooth ball quotient, and G is a finite group of automorphisms of Y whose action is fixedpoint free in codimension one.
 (1.8.3)
The space X is projective and klt, the canonical divisor \(K_X\) is ample, and we have equality in (1.5.1). \(\square \)
The reader is referred to [8, Sect. 10] for a discussion of the expectations regarding quotients of the ball by properly discontinuous group actions having fixed points in codimension one.
1.4 Structure of the paper
Section 2 gathers notation, known results and global conventions that will be used throughout the paper.
1.4.1 Part I
Part I of this paper begins in Sect. 3 with a review of Mochizuki’s theory of tame and purely imaginary harmonic bundles on quasiprojective varieties and discusses them in the particular setting where the quasiprojective variety is the smooth locus of a klt variety. Section 4 briefly reviews the somewhat delicate notion of a Higgs sheaf on a singular space, and discusses stability of Higgs bundles that are defined on the the smooth locus of a klt variety only. The core of Part I is, however, Sect. 5 where the the central existence result for harmonic structures is shown.
1.4.2 Part II
Part II of this paper concerns applications. Section 6 shows in brief how existence of harmonic structures leads to nonabelian Hodge correspondences pertaining to local systems on the smooth locus of a klt space. Section 7 proves our main result on quasiétale uniformisation, Theorem 1.5. Section 8 studies singular ball quotients, asking what positivity one might expect in \(\Omega ^{[1]}_X\), and what hyperbolicity properties might hold in the underlying spaces.
2 Notation and elementary facts
2.1 Global conventions
Throughout the present paper, all varieties and schemes will be defined over the complex numbers. We follow the notation used in the standard reference books [15, 20], with the exception that klt pairs are assumed to have an effective boundary divisor, see Sect. 2.6 below.
A morphism of vector bundles is always assumed to have constant rank. Notation introduced in our previous papers [8, 9], will briefly be recalled before it is used.
Throughout the paper, we will freely switch between the algebraic and analytic context if no confusion is likely to arise; sheaves on quasiprojective varieties will always be algebraic.
2.2 Varieties, subsets, morphisms
In line with the notation used in [15], varieties are always assumed to be irreducible. The following will be used for notational convenience.
Notation 2.1
(Big and small subsets) Let X be a normal, quasiprojective variety. A closed subset \(Z \subset X\) is called small if \({{\,\mathrm{codim}\,}}_X Z \ge 2\). An open subset \(U \subseteq X\) is called big if \(X{ {\setminus }} U\) is small.
Galois morphisms appear prominently in the literature, but their precise definition is not consistent. We will use the following definition, which does not ask Galois morphisms to be étale.
Definition 2.2
(Covers and covering maps, Galois morphisms) A cover or covering map is a finite, surjective morphism \(\gamma : X \rightarrow Y\) of normal, quasiprojective varieties. The covering map \(\gamma \) is called Galois if there exists a finite group \(G \subset {{\,\mathrm{Aut}\,}}(X)\) such that \(\gamma \) is isomorphic to the quotient map.
As pointed out in the introduction, quasiétale morphisms feature prominently in this paper. We recall the definition and refer the reader to [8, Def. 2.11] for a more detailed discussion.
Definition 2.3
(Quasiétale morphisms) A morphism \(f : X \rightarrow Y\) between normal varieties is called quasiétale if f is of relative dimension zero and étale in codimension one. In other words, f is quasiétale if \(\dim X = \dim Y\) and if there exists a closed, subset \(Z \subseteq X\) of codimension \({{\,\mathrm{codim}\,}}_X Z \ge 2\) such that \(f_{X{\setminus }Z} : X{\setminus }Z \rightarrow Y\) is étale.
2.3 Nef sheaves
While positivity notions for vector bundles are wellestablished in the literature, we will need these notions also for coherent sheaves.
Definition 2.4
(Nef and ample sheaves, [1]) Let X be a normal, projective variety and let \(\mathscr {S}\not = 0\) be a nontrivial coherent sheaf on X, not necessarily locally free. We call \(\mathscr {S}\)ample (resp. nef) if the locally free sheaf \(\mathscr {O}_{\mathbb {P}(\mathscr {S})}(1) \in {{\,\mathrm{Pic}\,}}(\mathbb {P}(\mathscr {S}))\) is ample (resp. nef).
We refer the reader to [12] for the definition of \(\mathbb {P}(\mathscr {S})\), and to [1, Sect. 2 and Thm. 2.9] for a more detailed discussion of amplitude and for further references. We mention a few elementary facts without proof.
Fact 2.5
 (2.5.1)
Ample sheaves on X are nef.
 (2.5.2)
A direct sum of sheaves on X if nef iff every summand is nef.
 (2.5.3)
Pullbacks and quotients of nef sheaves are nef.
 (2.5.4)
A sheaf \(\mathscr {E}\) is nef on X if and only if for every smooth curve C and every morphism \(\gamma : C \rightarrow X\), the pullback \(\gamma ^* \mathscr {E}\) is nef.\(\square \)
2.4 Connections on complex vector bundles
Connections on complex vector bundles play a prominent role in this paper. We recall two elementary facts that will become relevant later.
Fact 2.6
(Extension of bundles with connection) Let M be a \(\mathcal {C}^\infty \)manifold and \(M^{\circ } \subseteq M\) an open subset. Write \(\widehat{\pi }_1\) for the profinite completion of the fundamental group, and let \(\widehat{\rho } : \widehat{\pi }_1(M^{\circ }) \rightarrow \widehat{\pi }_1(M)\) be the natural morphism induced by the inclusion. If \(\widehat{\rho }\) is isomorphic, then any flat, complex bundle \((E^{\circ }, \nabla _{E^{\circ }})\) on \(M^{\circ }\) admits an extension to a flat, complex bundle \((E, \nabla _E)\) on M. The extension is unique up to canonical isomorphism. \(\square \)
If \(\gamma : X \rightarrow Y\) is a ramified Galois cover of complex manifolds, if E is a smooth, complex bundle over Y and h a smooth, Galoisinvariant Hermitian metric on \(\gamma ^*E\), it is generally not true that h comes from a smooth metric on Y. In contrast, the following result asserts that flat connections in \(\gamma ^*E\) do indeed descend once they are invariant. This is probably known to experts. The arXiv version of this paper includes a full proof.
Proposition 2.7
(Descent of Ginvariant, flat connections) Let \(\gamma : X \rightarrow Y\) be a Galois cover of complex manifolds, with Galois group G. Let \(E_Y\) be a smooth, complex bundle over Y and let \(\nabla _X\) be a flat, Ginvariant connection on \(E_X := \gamma ^*E_Y\). Then, there exists a flat connection \(\nabla _Y\) on \(E_Y\) such that \(\nabla _X = \gamma ^* \nabla _Y\). \(\square \)
2.5 Higgs sheaves
Let X be a normal variety or normal complex space. Following the notation introduced in [8, Def. 5.1], a Higgs sheaf is a pair \((\mathscr {E}, \theta )\) of a coherent sheaf \(\mathscr {E}\) of \(\mathscr {O}_X\)modules, together with an \(\mathscr {O}_X\)linear sheaf morphism \(\theta : \mathscr {E}\rightarrow \mathscr {E}\otimes \Omega ^{[1]}_X\), called Higgs field, such that the induced morphism \(\mathscr {E}\rightarrow \mathscr {E}\otimes \Omega ^{[2]}_X\) vanishes. We refer the reader to [8, Sect. 5] for related notions, including the definition of a Higgs Gsheaf, and slope stability of Higgs sheaves with respect to nef divisor classes.
2.5.1 Categories used in the nonabelian Hodge correspondence

the Higgs sheaf \((\mathscr {E}, \theta )\) is semistable with respect to H, and

the Chern characters of \(\mathscr {E}\) satisfy \(ch_1(\mathscr {E})\cdot [H]^{n1} = ch_2(\mathscr {E})\cdot [H]^{n2} = 0\).
2.6 KLT spaces and \(\mathbb {Q}\)Chern classes
A klt pair\((X,\Delta )\) consists of a normal variety X and a Weil \(\mathbb {Q}\)divisor \(\Delta = \sum _i a_i D_i\) with \(a_i \in \mathbb {Q}\cap (0,1)\) such that \(K_X + \Delta \) is \(\mathbb {Q}\)Cartier and such that \({{\,\mathrm{discrep}\,}}(X,\Delta ) > 1\), where the discrepancy \({{\,\mathrm{discrep}\,}}(X,\Delta )\) is defined in [20, Def. 2.28] using [20, Def. 2.25]. In contrast to [20, Def. 2.34] we do not allow noneffective boundary divisors \(\Delta \).
Definition 2.8
A normal, quasiprojective variety X is called klt space if there exists an effective \(\mathbb {Q}\)divisor \(\Delta \) that makes the pair \((X,\Delta )\) klt.
2.7 \(\mathbb {Q}\)Chern classes
Lemma 2.9
Remark 2.10
The sheaf \(\gamma ^{[*]} \mathscr {E}\) in Lemma 2.9 is usually not isomorphic to the pullback \(\gamma ^* \mathscr {E}\).
3 Part I. Existence of harmonic bundle structures
4 Harmonic bundles
Harmonic bundles are key tools in nonabelian Hodge theory that provide the link between flat structures and Higgs fields. We briefly recall the definition, explain relevant properties, and recall the notion of “tameness” that is used to establish a good theory in noncompact, compactifiable situations. Finally, we prove a boundedness result for Higgs bundles admitting a tame and purely imaginary harmonic structure.
Fact and Definition 3.1

A holomorphic vector bundle \((E, \bar{\partial })\) and a Hermitian metric h on E.

A Higgs field \(\theta : \mathscr {E}\rightarrow \mathscr {E}\otimes \Omega ^{1}_X\), where \(\mathscr {E}= \ker \bar{\partial }\) is the sheaf of holomorphic sections.
Notation 3.2
(Flat bundles and local systems associated with harmonic bundles) Given a harmonic bundle \(\mathbb {E}= (E, \bar{\partial }, \theta , h)\) as in Fact and Definition 3.1, we denote the associated flat bundle by \((E, \nabla _\mathbb {E})\) and write \(\mathsf{E}\in {{\,\mathrm{\mathsf LSys}\,}}_M\) for the local system.
Notation 3.3

Given a locally free sheaf \(\mathscr {E}\) on M with associated holomorphic bundle \((E, \bar{\partial })\), we say that \(\mathscr {E}\)admits a harmonic bundle structure if there exists a harmonic bundle of the form \((E, \bar{\partial }, \theta , h)\).

Given a locally free Higgs sheaf \((\mathscr {E}, \theta )\) on M with associated holomorphic bundle \((E, \bar{\partial })\), we say that \((\mathscr {E}, \theta )\)admits a harmonic bundle structure if there exists a harmonic bundle of the form \((E, \bar{\partial }, \theta , h)\).

We say that a flat bundle \((E,\nabla )\)admits a harmonic bundle structure if there exists a harmonic bundle \(\mathbb {E}= (E, \bar{\partial }, \theta , h)\) such that \(\nabla = \nabla _\mathbb {E}\).
Remark 3.4
In the setup of Notation 3.3, let N be a complex submanifold of M. Assume that \((\mathscr {E}, \theta )\) admits a harmonic bundle structure. Then, an easy local computation shows that the locally free Higgs sheaf \((\mathscr {E}, \theta )_N\) admits a harmonic bundle structure given by restriction.
4.1 Tame and purely imaginary bundles
In order to study Higgs bundles on quasiprojective, nonprojective varieties, we consider “tame” harmonic bundles. These are harmonic bundles on the complement of a divisor whose growth near the divisor is sufficiently controlled.
4.1.1 Basic definitions
The following is not Simpson’s original definition of “tameness”, but is equivalent to it.
Definition 3.5
Fact and Definition 3.6
The harmonic bundle \((E, \bar{\partial }_E, \theta , h)\) is called purely imaginary with respect to (M, D) if all eigenvalues of the residues of \(\theta _M\) along the irreducible components of D are purely imaginary for one (equivalently any) extension \((\mathscr {E}_M, \theta _M)\) of \((\mathscr {E}, \theta )\). \(\square \)
It is important to notice that the notion of tame purely imaginary bundles does not depend on the compactification.
Fact and Definition 3.7
(Tame and purely imaginary bundles on quasiprojective varieties, [27, Lem. 25.29] and [26, Cor. 8.7]) Let X be a smooth, quasiprojective variety and let \(\mathbb {E}:= (E, \bar{\partial }, \theta , h)\) be a harmonic bundle on \(X^{an}\). Let \(\overline{X}_1\) and \(\overline{X}_2\) be two smooth, projective compactifications of X such that \(D_\bullet := \overline{X}_\bullet {\setminus }X\) are snc divisors. Then, \(\mathbb {E}\) is tame and purely imaginary with respect to \((\overline{X}_1,D_1)\) if and only if \(\mathbb {E}\) is tame and purely imaginary with respect to \((\overline{X}_2, D_2)\). We can therefore speak about tame and purely imaginary harmonic bundles on \(X^{an}\). \(\square \)
Remark 3.8
(Automatic algebraicity I) In the setup of Fact and Definition 3.7, setting as usual \(\mathscr {E}:= \ker \bar{\partial }\), we may apply Serre’s GAGA to the extension of \(\mathscr {E}\) to a smooth projective simple normal crossings compactification of X (as in Definition 3.5) to see that the holomorphic Higgs sheaf \((\mathscr {E}, \theta )\) on \(X^{an}\) can be endowed with an algebraic structure.
Remark and Notation 3.9
(Uniqueness of the algebraic structure) We will often consider the setup of Fact and Definition 3.7 in a situation where the smooth variety X is a big open subset of a normal projective variety \(\overline{X}\). If \(\mathscr {E}\) is any locally free sheaf on \(X^{an}\) that admits a tame and purely imaginary harmonic bundle structure, then \(\mathscr {E}\) can be endowed with an algebraic structure and hence has a coherent extension to \(\overline{X}^{an}\). By [35, Thm. 1], the analytic sheaf \(\mathscr {E}\) will then have a unique reflexive extension to \(\overline{X}^{an}\). It then follows from GAGA that the induced algebraic structure on \(\mathscr {E}\) is unique up to isomorphism; the same holds for the Higgs field as well.
In this situation, we simply say that \(\mathscr {E}\) and \((\mathscr {E}, \theta )\) are algebraic, and freely switch between the analytic and the algebraic category if no confusion seems likely.
Remark 3.10
Flat sheaves admitting a tame and purely imaginary harmonic bundle structure are necessarily semisimple, see [27, Prop. 22.15].
Notation 3.11
Lemma 3.12
(Flat subsheaves in tame and purely imaginary harmonic bundles) Let X be a smooth, quasiprojective variety and let \(\mathbb {E}= (E,\bar{\partial }_E, \theta , h)\) be a tame and purely imaginary harmonic bundle on \(X^{an}\) with induced flat connection \(\nabla _\mathbb {E}\). If \(F \subseteq E\) is any complex subbundle that is invariant with respect to \(\nabla _\mathbb {E}\), then \(\bar{\partial }\) restricts to equip F with the structure of a Higgsinvariant, holomorphic subbundle of \((E, \bar{\partial })\).
Proof
It suffices to consider the case where F with its induced flat structure is irreducible. But there, the description of the tame and purely imaginary harmonic bundle \(\mathbb {E}\) in [25, Lem. A.13] immediately implies that the metric complement \(F^\perp \) of F is likewise invariant with respect to \(\nabla _\mathbb {E}\). The claim then follows from the description of the operators \(\bar{\partial }\) and \(\theta \) in terms of \(\nabla _\mathbb {E}\) and h, cf. [37, p. 13]. The arXiv version of this paper spells out all details. \(\square \)
4.1.2 Existence and uniqueness
If X is smooth and quasiprojective, then a result of JostZuo [17] implies that every semisimple flat bundle on X admits a tame and purely imaginary metric, which is essentially unique.^{2} We summarise the results relevant for us in the following theorem, see [25, Lem. A.13] and further references given there.
Theorem 3.13
(Existence and uniqueness of harmonic structures) Let X be a smooth quasiprojective variety. Then, every semisimple flat vector bundle \((E,\nabla _E)\) on X admits a tame, purely imaginary harmonic bundle structure \(\mathbb {E}= (E, \bar{\partial }, \theta , h)\). The metric h is unique up to flat automorphisms of E, and, as a consequence, the operators in the induced decomposition (3.1.1) are independent of the choice of such h. \(\square \)
The following consequences will be used later.
Corollary 3.14
(Higgs bundles determined by induced connection) Let X be a smooth, quasiprojective variety and let \(\mathbb {E}:= (E, \bar{\partial }_E, \theta _\mathscr {E}, h_E)\) and \(\mathbb {F}:= (F, \bar{\partial }_F, \theta _\mathscr {F}, h_F)\) be two tame, purely imaginary harmonic bundles on \(X^{(an)}\), with associated locally free sheaves \(\mathscr {E}\) and \(\mathscr {F}\). Assume that the flat bundles \((E, \nabla _\mathbb {E})\) and \((F, \nabla _\mathbb {F})\) are isomorphic. Then, also the corresponding holomorphic Higgs bundles are holomorphically isomorphic, \((\mathscr {E}, \theta _\mathscr {E}) \cong (\mathscr {F}, \theta _\mathscr {F})\).
Proof
By assumption, there exists a smooth isomorphism \(\Phi : E \rightarrow F\) such that \(\Phi ^*\nabla _{\mathbb {F}} = \nabla _\mathbb {E}\). The pullback \(\Phi ^*\mathbb {F}= (E, \Phi ^* \bar{\partial }_F, \Phi ^*\theta _\mathscr {F}, \Phi ^*h_F)\) will thus equip E with a second tame, purely imaginary harmonic bundle structure, whose associated flat connection \(\nabla _{\Phi ^*\mathbb {F}}\) equals \(\nabla _\mathbb {E}\). But then it follows from Theorem 3.13 that the differential operators in the two harmonic bundle structures agree: \(\bar{\partial }_E = \Phi ^* \bar{\partial }_F\) and \(\theta _\mathscr {E}= \Phi ^*\theta _\mathscr {F}\). In other words, \(\Phi \) is holomorphic and induces a holomorphic isomorphism of Higgs bundles. \(\square \)
Remark 3.15
(Automatic algebraicity II) In the setting of Corollary 3.14, recall from Remark 3.8 that the Higgs sheaves \((\mathscr {E}, \theta _\mathscr {E})\) and \((\mathscr {F}, \theta _\mathscr {F})\) are in fact algebraic. If X is isomorphic to a big open subset in a normal variety, then the holomorphic isomorphism given in Corollary 3.14 extends to an isomorphism between reflexive closures, and is therefore likewise algebraic.
Corollary 3.16
(Extension of harmonic bundles from hyperplanes) Let X be a normal, projective variety of dimension \(\dim X > 2\), and let \(H \in {{\,\mathrm{Div}\,}}(X)\) be ample. If \(m \gg 0\) is large enough and \(D \in m \cdot H\) is general, then D is normal, \(D_{{{\,\mathrm{reg}\,}}} = D \cap X_{{{\,\mathrm{reg}\,}}}\), and the restriction map \({{\,\mathrm{\mathsf TPIHiggs}\,}}_{X_{{{\,\mathrm{reg}\,}}}} \rightarrow {{\,\mathrm{\mathsf TPIHiggs}\,}}_{D_{{{\,\mathrm{reg}\,}}}}\) is surjective.
Proof
4.2 TPIHarmonic bundles on klt spaces
Let X be a projective klt space. Using the nonabelian Hodge correspondence for locally free Higgs sheaves on klt spaces, [9, Thm. 3.4], we will show boundedness of the family \({{\,\mathrm{\mathsf TPIHiggs}\,}}_{X_{{{\,\mathrm{reg}\,}}}}\). As a consequence, we will obtain in Corollary 3.20 a criterion for the existence of harmonic structures: a given Higgs bundle on \(X_{{{\,\mathrm{reg}\,}}}\) admits a harmonic structure if and only if its restriction to the smooth locus of a general hypersurface does. The proof uses the existence, for every klt space X, of a “maximally quasiétale cover”. This is a quasiétale cover \(\gamma : Y \rightarrow X\) such that the natural map of étale fundamental groups, \(\widehat{\pi }_1(Y_{{{\,\mathrm{reg}\,}}}) \rightarrow \widehat{\pi }_1(Y)\), is isomorphic. The existence of such a cover was established in [7, Thm. 1.5].
Proposition 3.17
Remark and Notation 3.18
Proof of Proposition 3.17
Write \((\mathscr {F}_{Y^{\circ }}, \theta _{\mathscr {F}_{Y^{\circ }}}) := \delta ^* (\mathscr {E}_{X^{\circ }}, \theta _{\mathscr {E}_{X^{\circ }}})\) and choose a tame, purely imaginary harmonic bundle structure \(\mathbb {E}_{X^{\circ }}\) for \((\mathscr {E}_{X^{\circ }}, \theta _{\mathscr {E}_{X^{\circ }}})\). Recall from [27, Lem. 25.29] that the pullback \(\mathbb {F}_{Y^{\circ }} := \delta ^* \mathbb {E}_{X^{\circ }}\) is a tame and purely imaginary harmonic bundle structure for \((\mathscr {F}_{Y^{\circ }}, \theta _{\mathscr {F}_{Y^{\circ }}})\). The induced local system \(\mathsf{F}_{Y^{\circ }}\) is semisimple by [27, Prop. 22.15]. The assumption that Y is maximally quasiétale implies that \(\mathsf{F}_{Y^{\circ }}\) extends from \(Y^{\circ }\) to a semisimple local system \(\mathsf{F}_Y \in {{\,\mathrm{\mathsf LSys}\,}}_Y\) that is defined on all of Y, see [13, Thm. 1.2b] or [7, Sect. 8.1]. In particular, the nonabelian Hodge correspondence for klt spaces, [9, Thm. 3.4], applies to yield a locally free Higgs sheaf \((\mathscr {F}_Y, \theta _{\mathscr {F}_Y}) := \eta _Y(\mathsf{F}_Y) \in {{\,\mathrm{\mathsf Higgs}\,}}_Y\). More is true: We have seen in [9, Prop. 3.11] that \((\mathscr {F}_Y, \theta _{\mathscr {F}_Y})_{Y^{\circ }}\) admits a tame, purely imaginary harmonic bundle structure \(\mathbb {F}'_{Y^{\circ }}\) whose associated local system \(\mathsf{F}'_{Y^{\circ }}\) is isomorphic to \(\mathsf{F}_{Y^{\circ }}\). Corollary 3.14 thus yields the desired isomorphism (3.17.1). \(\square \)
Corollary 3.19
Proof
The following corollary is now a direct consequence of Remark 3.4, of the boundedness result above, of Corollary 3.16, and of the iterated Bertinitype theorem for bounded families, [9, Prop. 7.3]. We emphasise that it contains a criterion for a reflexive Higgs sheaf to be locally free.
Corollary 3.20
Proof
The conditions on the singularities of the hypersurface D stated above are met by a Zariskiopen subset of any basepoint free linear system owing to Seidenberg’s theorem [2, Thm. 1.7.1] and [20, Lem. 5.17].
5 Higgs bundles and Higgs sheaves
To prepare for the proof of the uniformisation result in the Part II of this paper and to fix notation, we briefly recall the notion of \(\mathbb {Q}\)varieties in the specific case of surfaces; details concerning this notion can be found in [8, Part I]. Section 4.2 discusses stability notions for Higgs sheaves that are defined only on the smooth locus of a projective klt space.
5.1 Higgs sheaves on \(\mathbb {Q}\)surfaces
For surfaces with quotient singularities, we show how a Higgs \(\mathbb {Q}\)sheaf can be constructed from a Higgs sheaf that is defined on the smooth locus of the underlying surface; the theory can certainly be developed to cover more general cases, but we restrict ourselves to the material necessary for the arguments in the proof of our main result. For further material, the reader is referred to [8, Sect. 5.5], where the notion of Higgs \(\mathbb {Q}\)sheaves is defined in general.
Construction 4.1
Construction 4.2
(Higgs \(\mathbb {Q}\)bundle from Higgs bundle on \(S_{{{\,\mathrm{reg}\,}}}\)). In the setup of Construction 4.1, assume that \(S_{{{\,\mathrm{reg}\,}}}\) is equipped with a locally free Higgs sheaf \((\mathscr {E}_{S_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{S_{{{\,\mathrm{reg}\,}}}}})\). We will denote the reflexive extension of \(\mathscr {E}_{S_{{{\,\mathrm{reg}\,}}}}\) to S by \(\mathscr {E}_S\). Slightly generalising [8, Constructions 3.8 and 5.15], one constructs a locally free Higgs \(\mathbb {Q}\)sheaf \((\mathscr {E}_S, \theta _{\mathscr {E}_S})^{[\mathbb {Q}]}\) on \(S^\mathbb {Q}\), given by the collection \((\mathscr {E}_{S_\alpha }, \theta _{\mathscr {E}_{S_\alpha }})\) of locally free Higgs sheaves on the \(S_\alpha \) that can be obtained by extending the Higgs bundles \((p_\alpha _{p_\alpha ^{1}(S_{{{\,\mathrm{reg}\,}}})})^{*}(\mathscr {E}_{S_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{S_{{{\,\mathrm{reg}\,}}}}})\) to a Higgs bundle on \(S_\alpha \) using the Riemann Extension Theorem.
More precisely, we set \(S_\alpha ^{\circ } = S_\alpha {{\setminus }} p_\alpha ^{1}(S_{{{\,\mathrm{sing}\,}}})\). Each \(\mathscr {E}_{S_\alpha }^{\circ } : = \big ( p_\alpha _{p_\alpha ^{1}(S_{{{\,\mathrm{reg}\,}}})} \big )^*(\mathscr {E})\) extends to a locally free sheaf \(\mathscr {E}_{S_\alpha }\) on \(S_\alpha \). In addition, as the induced Higgs fields on \(\mathscr {E}_{S_\alpha }^{\circ }\) are sections of \({{\,\mathrm{\mathscr {E} \textit{nd}}\,}}(\mathscr {E}_{S_\alpha }^{\circ }) \otimes \Omega ^{1}_{S_\alpha ^{\circ }}\), they extend to sections of \({{\,\mathrm{\mathscr {E} \textit{nd}}\,}}(\mathscr {E}_{S_\alpha }) \otimes \Omega ^{1}_{S_\alpha } \). We denote these extended sections by \(\theta _{\mathscr {E}_{S_\alpha }}\). Now, the pullbacks \(q_\alpha ^*(\mathscr {E}_{S_\alpha }, \theta _{\mathscr {E}_{S_\alpha }})\) are locally free Higgs sheaves on \(\widehat{S}_\alpha \) that glue to give a locally free Higgs Gsheaf \((\mathscr {E}_{\widehat{S}}, \theta _{\mathscr {E}_{\widehat{S}}})\) on \(\widehat{S}\). The construction works without change for sheaves without (or with the trivial) Higgs field.
Remark 4.3
In Construction 4.2, we have \(\mathscr {E}_{\widehat{S}} \cong \gamma ^{[*]} \mathscr {E}_S\) and therefore \(\gamma _*(\mathscr {E}_{\widehat{S}})^G \cong \mathscr {E}_S\).
Construction 4.4
(Pullback out of a \(\mathbb {Q}\)structure) In the setting of Construction 4.2, consider a Gequivariant resolution of singularities, \(\pi : \widetilde{S} \rightarrow \widehat{S}\). Given an arbitrary Higgs sheaf \((\mathscr {F},\tau )\) on \(\widehat{S}\), there is generally no way to define a Higgs field on the pullback \(\pi ^* \mathscr {F}\) even in cases where \(\mathscr {F}\) is locally free. In our special setting, however, recall from [8, Lem. 5.17] that there exists a Ginvariant Higgs field on the Gsheaf \(\mathscr {E}_{\widetilde{S}} := \pi ^* \mathscr {E}_{\widehat{S}}\) that agrees with the pullback of \(\theta _{\mathscr {E}_{\widehat{S}}}\) wherever \(\widehat{S}\) is smooth.
5.2 Stability and polystability
Stability properties of Higgs sheaves on singular, projective varieties are defined and discussed in detail in [8, Sect. 5.6]. In the situation at hand, it makes sense to generalise this to the case where a Higgs sheaf is defined on the smooth locus only. We recall [9, Def. 2.19] in our setting.
Definition 4.5
The stability notion of Definition 4.5 is compatible with the existing notions. The proof of the following fact is elementary and therefore omitted.
Fact 4.6
 (4.6.1)
The sheaf \((\mathscr {E}_X, \theta _{\mathscr {E}_X})\) is stable with respect to H.
 (4.6.2)
The sheaf \((\mathscr {E}_X, \theta _{\mathscr {E}_X})_{X_{{{\,\mathrm{reg}\,}}}}\) is stable with respect to H.
The following lemma discusses behaviour of stability when taking tensor products, and shows polystability for Higgs bundles in \({{\,\mathrm{\mathsf TPIHiggs}\,}}\).
Lemma 4.7
 (4.7.1)
The Higgs bundle \((\mathscr {F}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {F}_{X_{{{\,\mathrm{reg}\,}}}}})\) is in \({{\,\mathrm{\mathsf TPIHiggs}\,}}_{X_{{{\,\mathrm{reg}\,}}}}\).
 (4.7.2)
The Higgs bundle \((\mathscr {F}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {F}_{X_{{{\,\mathrm{reg}\,}}}}})\) is a tensor product of two Higgs bundles on \(X_{{{\,\mathrm{reg}\,}}}\) that are stable with respect to H.
Proof
To begin, we remark that \((\mathscr {F}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {F}_{X_{{{\,\mathrm{reg}\,}}}}})\) is semistable. In fact, Remark 3.4 and the restriction theorem for semistable Higgs sheaves on \(X_{{{\,\mathrm{reg}\,}}}\), [9, Thm. 6.1], respectively, allows us to restrict ourselves to the case where \(X = X_{{{\,\mathrm{reg}\,}}}\) is a smooth projective curve. There, the result is classically known in either of the two cases.
The proof of polystability proceeds by induction on the dimension on X. If \(\dim X = 1\), then X is a smooth projective curve and the result is classically known in either case. As for the inductive step, assume that the result was known for all varieties of dimension less than \(\dim X\), and assume that there exists a saturated, Higgsinvariant subsheaf \(\mathscr {A}_{X_{{{\,\mathrm{reg}\,}}}} \subsetneq \mathscr {F}_{X_{{{\,\mathrm{reg}\,}}}}\) whose slope equals that of \(\mathscr {F}_{X_{{{\,\mathrm{reg}\,}}}}\) and that is stable with respect to H. We need to show that \(\mathscr {A}_{X_{{{\,\mathrm{reg}\,}}}}\) is a direct summand. More precisely, we need to find a morphism of Higgs sheaves, \(\mathscr {F}_{X_{{{\,\mathrm{reg}\,}}}} \rightarrow \mathscr {A}_{X_{{{\,\mathrm{reg}\,}}}}\), that is a projection onto \(\mathscr {A}_{X_{{{\,\mathrm{reg}\,}}}}\).
 (4.7.3)
The subsheaf \(\mathscr {A}_{X_{{{\,\mathrm{reg}\,}}}}_{Y_{{{\,\mathrm{reg}\,}}}} \subsetneq \mathscr {F}_{X_{{{\,\mathrm{reg}\,}}}}_{Y_{{{\,\mathrm{reg}\,}}}}\) Higgsinvariant and stable. This is the Restriction Theorem for stable Higgs sheaves on \(X_{{{\,\mathrm{reg}\,}}}\), [9, Thm. 6.1].
 (4.7.4)
The restriction \({{\,\mathrm{Hom}\,}}(\mathscr {F}_{X_{{{\,\mathrm{reg}\,}}}}, \mathscr {A}_{X_{{{\,\mathrm{reg}\,}}}} ) \rightarrow {{\,\mathrm{Hom}\,}}(\mathscr {F}_{X_{{{\,\mathrm{reg}\,}}}}_{Y_{{{\,\mathrm{reg}\,}}}}, \mathscr {A}_{X_{{{\,\mathrm{reg}\,}}}}_{Y_{{{\,\mathrm{reg}\,}}}} )\) is bijective. This follows from standard arguments, compare [9, Prop. 7.3 and proof].
6 Existence of harmonic structures
The following is the main result in Part I of the present paper. It generalises earlier results obtained in [7, 23, 28, 34].
Theorem 5.1
 (5.1.1)The Higgs sheaf \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) is (poly)stable with respect to H and the \(\mathbb {Q}\)Chern characters satisfy$$\begin{aligned} \widehat{ch}_1(\mathscr {E}_X)\cdot [H]^{n1} = 0 \quad \text {and}\quad \widehat{ch}_2(\mathscr {E}_X)\cdot [H]^{n 2} = 0. \end{aligned}$$
 (5.1.2)
The sheaf \(\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}\) is locally free and \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) is induced by a tame, purely imaginary harmonic bundle whose associated flat bundle is (semi)simple.
6.1 Proof of Theorem 5.1
The two implications are proven separately.
6.2 Implication (5.1.1) \(\Rightarrow \) (5.1.2)
Suppose we already know that \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) admits a tame and purely imaginary harmonic bundle structure \(\mathbb {E}_{X_{{{\,\mathrm{reg}\,}}}}\). Then the induced flat bundle \((E_{X_{{{\,\mathrm{reg}\,}}}}, \nabla _{\mathbb {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) is always semisimple, see Remark 3.10. If \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) is even stable, then Lemma 3.12 implies that \((E_{X_{{{\,\mathrm{reg}\,}}}}, \nabla _{\mathbb {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) is simple. It therefore remains to establish a tame, purely imaginary harmonic bundle structure. The proof is rather long, and therefore subdivided into six steps.
6.3 Implication (5.1.1) \(\Rightarrow \) (5.1.2), Step 1: Reduction to stable sheaves on surfaces
Standard arguments involving the Bogomolov–Gieseker inequality show that it suffices to consider the stable case only. We refer to [6, Step 2 in proof of Thm. 6.2] for details.
Assumption w.l.o.g. 5.2
The Higgs sheaf \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) is stable with respect to H.
 (5.3.1)
The intersection \(S := D_1 \cap \cdots \cap D_{n2}\) is irreducible and normal. Moreover, S is a klt space and \(S_{{{\,\mathrm{reg}\,}}} = S \cap X_{{{\,\mathrm{reg}\,}}}\). This is Seidenberg’s theorem [2, Thm. 1.7.1] and [20, Lem. 5.17].
 (5.3.2)
The restricted sheaf \(\mathscr {E}_S := \mathscr {E}_X_S\) is reflexive, hence locally free on \(S_{{{\,\mathrm{reg}\,}}}\), and satisfies \(\widehat{ch}_2(\mathscr {E}_S) = 0\) as well as \(\widehat{c}_1(\mathscr {E}_Y)\cdot [H_S] = 0\). This is [16, Cor. 1.1.14] and [8, Thm. 3.13].
 (5.3.3)
The locally free Higgs sheaf \((\mathscr {E}_{S_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{S_{{{\,\mathrm{reg}\,}}}}}) := (\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})_{S_{{{\,\mathrm{reg}\,}}}}\) is stable with respect to \(H_S\). This is the Restriction Theorem [9, Thm. 6.1].
 (5.3.4)
The reflexive Higgs sheaf \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) is locally free and admits a tame and purely imaginary harmonic bundle structure if and only if the locally free Higgs sheaf \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})_{S_{{{\,\mathrm{reg}\,}}}}\) admits a tame and purely imaginary harmonic bundle structure. This is Corollary 3.20.
Assumption w.l.o.g. 5.3
The dimension of X is equal to two.
6.4 Implication (5.1.1) \(\Rightarrow \) (5.1.2), Step 2: The \(\mathbb {Q}\)structure
Claim 5.4
(Stability and Chern classes of \(\mathscr {E}_{\widetilde{X}}\)) The Higgs Gbundle \((\mathscr {E}_{\widetilde{X}}, \theta _{\mathscr {E}_{\widetilde{X}}})\) is Gstable with respect to a Ginvariant ample divisor \(\widetilde{H} \in {{\,\mathrm{Div}\,}}(\widetilde{X})\). The Chern classes \(c_i(\mathscr {E}_{\widetilde{X}}) \in H^{2i} \bigl (\widetilde{X},\, \mathbb {R}\bigr )\) vanish.
Proof of Claim 5.4
The first claim is proven as in the proof of [8, Prop. 6.2], using the locally free Higgs Gsheaf \((\mathscr {E}_{\widehat{X}}, \theta _{\mathscr {E}_{\widehat{X}}})\) on \(\widehat{X}\) obtained in Construction 4.2. For the second claim, observe that the maximally destabilising subsheaf of the Higgs bundle^{3}\((\mathscr {E}_{\widetilde{X}}, \theta _{\mathscr {E}_{\widetilde{X}}})\) with respect to \(\widetilde{H}\) is automatically Ginvariant. In particular, it follows from Gstability that \((\mathscr {E}_{\widetilde{X}}, \theta _{\mathscr {E}_{\widetilde{X}}})\) is at least semistable with respect to \(\widetilde{H}\). Moreover, from (5.3.2), from [8, Thm. 3.13], and from the functorial properties of Chern classes [4, Thm. 3.2(d)] we conclude that \(c_1(\mathscr {E}_{\widetilde{X}}) \cdot \widetilde{H} = ch_2(\mathscr {E}_{\widetilde{X}})= 0\). Vanishing of the \(c_i\) hence follows from the Hodge index theorem and the BogomolovGieseker inequality as in the first paragraph of [6, Sect. 6.3]. \(\square \) (Claim 5.4)
For the remainder of the proof, we fix one Ginvariant ample divisor \(\widetilde{H} \in {{\,\mathrm{Div}\,}}(\widetilde{X})\) as in Claim 5.4 and use this divisor to equip \(\widetilde{X}\) with a Ginvariant Kähler metric \(\omega _{\widetilde{H}}\). We denote the holomorphic vector bundle associated with \(\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}\) by \((E_{X_{{{\,\mathrm{reg}\,}}}}, \bar{\partial }_{E_{X_{{{\,\mathrm{reg}\,}}}}})\). We use similar notation also for other bundles, including \((E_{\widetilde{X}}, \bar{\partial }_{E_{\widetilde{X}}})\).
6.4.1 Open sets
6.4.2 Orbifold charts
6.5 Implication (5.1.1) \(\Rightarrow \) (5.1.2), Step 3: Simplification
The following claim allows us to concentrate on the big, open set \(X^{\circ } \subseteq X_{{{\,\mathrm{reg}\,}}}\), simplifying notation substantially.
Claim 5.5
To prove Theorem 5.1, it suffices to show that the Higgs bundle \((\mathscr {E}_{X^{\circ }}, \theta _{\mathscr {E}_{X^{\circ }}})\) admits a tame and purely imaginary harmonic bundle structure.
6.6 Implication (5.1.1) \(\Rightarrow \) (5.1.2), Step 4: Ginvariant harmonic structure on \({\varvec{E}}_{{\tilde{\varvec{X}}}}\)
To start the core of the argument, we will now show the existence of a Ginvariant harmonic structure on \((\mathscr {E}_{\widetilde{X}}, \theta _{\mathscr {E}_{\widetilde{X}}})\).
Claim 5.6
The locally free Higgs Gsheaf \((\mathscr {E}_{\widetilde{X}}, \theta _{\mathscr {E}_{\widetilde{X}}})\) admits a Ginvariant harmonic bundle structure \(\mathbb {E}_{\widetilde{X}} := (E_{\widetilde{X}}, \bar{\partial }_{E_{\widetilde{X}}}, \theta _{\mathscr {E}_{\widetilde{X}}}, h_{E_{\widetilde{X}}})\).
Proof of Claim 5.6
Claim 5.7
If \(F_{\widetilde{X}}\) is a Ginvariant flat subbundle of \(E_{\widetilde{X}}\), then either \(F_{\widetilde{X}} = 0\) or \(F_{\widetilde{X}} = E_{\widetilde{X}}\).
Proof of Claim 5.7
We have seen in Lemma 3.12 that \(F_{\widetilde{X}}\) is a holomorphic subbundle of \((E_{\widetilde{X}}, \bar{\partial }_{E_{\widetilde{X}}})\) and yields a Higgsinvariant, locally free subsheaf \(\mathscr {F}_{\widetilde{X}}\) of \((\mathscr {E}_{\widetilde{X}}, \theta _{\mathscr {E}_{\widetilde{X}}})\). The assumption that \(F_{\widetilde{X}}\) is a Ginvariant subbundle of \(E_{\widetilde{X}}\) guarantees that \(\mathscr {F}_{\widetilde{X}}\) is a Higgsinvariant Gsubsheaf of the Higgs Gsheaf \((\mathscr {E}_{\widetilde{X}}, \theta _{\mathscr {E}_{\widetilde{X}}})\). Using that \(F_{\widetilde{X}}\) is invariant with respect to the flat connection \(\nabla _{\mathbb {E}_{\widetilde{X}}}\), we obtain that \(F_{\widetilde{X}}\) is again flat, so that all its Chern classes vanish. Claim 5.7 thus follows from Claim 5.4 above. \(\square \) (Claim 5.7)
Claim 5.8
If \(F_{\widetilde{X}^{\circ }}\) is a Ginvariant flat subbundle of \((E_{\widetilde{X}^{\circ }}, \nabla _{\mathbb {E}_{\widetilde{X}}}_{\widetilde{X}^{\circ }})\), then \(F_{\widetilde{X}^{\circ }}\) extends to a Ginvariant flat subbundle of \(E_{\widetilde{X}}\).
Proof of Claim 5.8
The question is local over the \(X_\alpha \). More precisely, using the notation introduced in Step 2, it suffices to show that for every index \(\alpha \), the restricted bundle \(F_{\widetilde{X}^{\circ }_\alpha } := F_{\widetilde{X}^{\circ }}_{\widetilde{X}^{\circ }_\alpha }\) on \(\widetilde{X}^{\circ }_\alpha \) extends to a subbundle \(F_{\widetilde{X}_\alpha } \subseteq E_{\widetilde{X}_\alpha }\) on \(\widetilde{X}_\alpha \) that is invariant with respect to \(\nabla _{\mathbb {E}_{\widetilde{X}}}\). The Ginvariance follows then automatically from density of \(\widetilde{X}^{\circ }_\alpha \subseteq \widetilde{X}_\alpha \).
Recall from Constructions 4.2 and 4.4 that \(E_{\widetilde{X}^{\circ }_\alpha }\) is a pullback from \(X^{\circ }_\alpha \), say \(E_{\widetilde{X}^{\circ }_\alpha } \cong (\Pi ^{\circ }_\alpha )^* E_{X^{\circ }_\alpha }\). We claim that both the connection \(\nabla _{\mathbb {E}_{\widetilde{X}}}_{\widetilde{X}^{\circ }}\) and the subbundle \(F_{\widetilde{X}^{\circ }_\alpha }\) descend to \(X^{\circ }_\alpha \), too. Indeed, since \(\nabla _{\mathbb {E}_{\widetilde{X}}}\) is invariant under G, and hence also invariant under the Galois group \(H_\alpha = {\text {Gal}}(\Pi ^{\circ }_\alpha ) \subseteq G\), we may apply Proposition 2.7 to show the existence of a connection \(\nabla _{E_{X^{\circ }_\alpha }}\) on \(E_{X^{\circ }_\alpha }\) such that \(\nabla _{\mathbb {E}_{\widetilde{X}}}_{\widetilde{X}^{\circ }} = (\Pi ^{\circ }_\alpha )^* \nabla _{E_{X^{\circ }_\alpha }}\). Moreover, it follows from [8, Prop. 2.16] applied to the associated locally free sheaves of \(\mathscr {O}_{\widetilde{X}^{\circ }_\alpha }\)modules that there exists a subbundle \(F_{X^{\circ }_\alpha } \subseteq E_{X^{\circ }_\alpha }\) with \(F_{\widetilde{X}^{\circ }_\alpha } = (\Pi ^{\circ }_\alpha )^* F_{X^{\circ }_\alpha }\). The subbundle \(F_{X^{\circ }_\alpha }\) is then clearly invariant with respect to the connection \(\nabla _{E_{X^{\circ }_\alpha }}\).
As \(X^{\circ }_\alpha \) is a big open subset of the smooth, quasiprojective surface \(X_\alpha \), the natural morphism of fundamental groups, \(\pi _1(X^{\circ }_\alpha ) \rightarrow \pi _1(X_\alpha )\), is isomorphic. Fact 2.6 therefore asserts that the subbundle \(F_{X^{\circ }_\alpha } \subseteq E_{X^{\circ }_\alpha }\) extends from \(X^{\circ }_\alpha \) to a \(\nabla _{E_{X^{\circ }_\alpha }}\)invariant subbundle \(F_{X^{\circ }_\alpha } \subseteq E_{X^{\circ }_\alpha }\) that exists on all of \(X_\alpha \). Pulling back, we define the desired extension of \(F_{\widetilde{X}^{\circ }_\alpha }\) as \(F_{\widetilde{X}_\alpha } := (\Pi ^{\circ }_\alpha )^* F_{X^{\circ }_\alpha }\). \(\square \) (Claim 5.8)
Combining Claims 5.7 and 5.8, we arrive at the following result.
Construction 5.9
If \(F_{\widetilde{X}^{\circ }}\) is a Ginvariant subbundle of \(E_{\widetilde{X}^{\circ }}\) that is also invariant with respect to \(\nabla _{\mathbb {E}_{\widetilde{X}}}_{\widetilde{X}^{\circ }}\), then \(F_{\widetilde{X}^{\circ }} = 0\) or \(F_{\widetilde{X}^{\circ }} = E_{\widetilde{X}^{\circ }}\). \(\square \) (Consequence 5.9)
6.7 Implication (5.1.1) \(\Rightarrow \) (5.1.2), Step 5: Harmonic structure on \(E_{X^{\circ }}\)
Eventually, we would like to show that the metric \(h_{E_{\widetilde{X}}}\) descends to a smooth Hermitian metric on \(E_{X^{\circ }}\). However, owing to branching of the map \(\gamma \) over the smooth part of \(X_{{{\,\mathrm{reg}\,}}}\), it is not clear from the outset whether the natural stratified \(\mathcal {C}^\infty \)structure on the quotient \(\widetilde{X}^{\circ }/G\) will coincide with the \(\mathcal {C}^\infty \)structure induced by the complex structure on \(X^{\circ }\). Rather than showing descent of the metric directly, we will first discuss the flat structure on \(E^{\circ }\) and construct a metric from there.
6.8 Implication (5.1.1) \(\Rightarrow \) (5.1.2), Step 6: Comparison
There are now two tame and purely imaginary, Ginvariant harmonic bundle structures on \(E_{\widetilde{X}^{\circ }}\). First, the restriction \(\mathbb {E}_{\widetilde{X}}_{\widetilde{X}^{\circ }}\), which is obviously tame and purely imaginary. The associated connection \(\nabla _{\mathbb {E}_{\widetilde{X}}}_{\widetilde{X}^{\circ }}\) on \(E_{\widetilde{X}^{\circ }}\) is semisimple by Remark 3.10. Second, by [27, Lem. 25.29] the pullback of the harmonic structure \((\psi ^{\circ })^* \mathbb {E}'_{X^{\circ }}\) on \(E_{X^{\circ }}\) is also tame and purely imaginary. The associated connection on \(E_{\widetilde{X}^{\circ }}\) is the pullback of \(\nabla _{E_{X^{\circ }}}\), and hence likewise equal to \(\nabla _{\mathbb {E}_{\widetilde{X}}}_{\widetilde{X}^{\circ }}\).
6.9 Implication (5.1.2) \(\Rightarrow \) (5.1.1)
First, remark that in the semisimple case the implication (5.1.2) \(\Rightarrow \) (5.1.1) is immediate consequence of Lemma 4.7, Proposition 3.17, [9, Thm. 3.10], and standard calculus of \(\mathbb {Q}\)Chern classes; this is explained in detail in [8, Sect. 3.8].
7 Part II. Applications
8 Nonabelian Hodge correspondences for smooth loci
The existence result for tame and purely imaginary harmonic bundles, Theorem 5.1, yields a nonabelian Hodge correspondence that relates semisimple local systems on the smooth locus of a klt space to polystable Higgs bundles on that locus. As in Simpson’s work, this correspondence extends to a correspondence for arbitrary local systems.
8.1 Nonabelian Hodge correspondence for polystable bundles
Before formulating the nonabelian Hodge correspondence for polystable bundles in Theorem 6.3 below, we need to specify the appropriate category of bundles. The following definition will be used.
Definition 6.1
Let X be a projective klt space X and \(\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}\) be a locally free sheaf on \(X_{{{\,\mathrm{reg}\,}}}\), whose extension to a reflexive sheaf on X is denoted by \(\mathscr {E}_X\). If \(H \in {{\,\mathrm{Div}\,}}(X)\) is ample, we say that \(\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}\)has vanishing\(\mathbb {Q}\)Chern classes with respect toH if \(\widehat{ch}_1(\mathscr {E}_X)\cdot [H]^{n1} = 0\) and \(\widehat{ch}_2(\mathscr {E}_X)\cdot [H]^{n2} = 0\).
As one immediate consequence of the existence result for harmonic structures, Theorem 5.1, we see that a Higgs bundle on the smooth locus of a projective klt space is polystable and has vanishing \(\mathbb {Q}\)Chern classes after cutting down with respect to one ample class, iff the same holds for any other ample class. This gives rise to the following fact, which we use to define the relevant category of bundles.
Fact and Definition 6.2
 (6.2.1)
There exists an ample \(H \in {{\,\mathrm{Div}\,}}(X)\), such that \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) is polystable and has vanishing \(\mathbb {Q}\)Chern classes with respect to H.
 (6.2.2)
For any ample \(H \in {{\,\mathrm{Div}\,}}(X)\), the Higgs bundle \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) is polystable and has vanishing \(\mathbb {Q}\)Chern classes with respect to H.
The nonabelian Hodge correspondence for polystable bundles, which is a direct analogue of [37, Cor. 1.3], is now formulated as follows.
Theorem 6.3
(Nonabelian Hodge correspondence for \({{\,\mathrm{\mathsf pHiggs}\,}}_{X_{{{\,\mathrm{reg}\,}}}}\)) Let X be a projective klt space. Then, there exists an equivalence between the category \({{\,\mathrm{\mathsf pHiggs}\,}}_{X_{{{\,\mathrm{reg}\,}}}}\) and the category \({{\,\mathrm{\mathsf sLSys}\,}}_{X_{{{\,\mathrm{reg}\,}}}}\) of semisimple local systems on \(X_{{{\,\mathrm{reg}\,}}}\).
For \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}}) \in {{\,\mathrm{\mathsf pHiggs}\,}}_{X_{{{\,\mathrm{reg}\,}}}}\) that are restrictions of polystable Higgs bundles on X with vanishing Chern classes, and for local systems \(\mathsf{E}_{X_{{{\,\mathrm{reg}\,}}}} \in {{\,\mathrm{\mathsf sLSys}\,}}_{X_{{{\,\mathrm{reg}\,}}}}\) that are restrictions of local systems on X, the correspondence is compatible with the global nonabelian Hodge Correspondence for projective klt spaces found in [9, Sect. 3].
Sketch of proof
Starting with a semisimple local system \(\mathsf{E}_{X_{{{\,\mathrm{reg}\,}}}}\) on \(X_{{{\,\mathrm{reg}\,}}}\), let \((E_{X_{{{\,\mathrm{reg}\,}}}}, \nabla _{E_{X_{{{\,\mathrm{reg}\,}}}}})\) be an associated flat bundle. By Theorem 3.13, this bundle admits a tame and purely imaginary harmonic metric that is unique up to flat automorphisms; these preserve the induced decomposition (3.1.1). The uniquely determined associated Higgs bundle \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{X_{{{\,\mathrm{reg}\,}}}})\) carries a unique algebraic structure by Remark and Notation 3.9 and is moreover polystable by Lemma 4.7. Its \(\mathbb {Q}\)Chern classes vanish by Proposition 3.17. Assigning \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{X_{{{\,\mathrm{reg}\,}}}})\) to \((E_{X_{{{\,\mathrm{reg}\,}}}}, \nabla _{E_{X_{{{\,\mathrm{reg}\,}}}}})\) defines a functor \(\eta _{X_{{{\,\mathrm{reg}\,}}}}\) from \({{\,\mathrm{\mathsf sLSys}\,}}_{X_{{{\,\mathrm{reg}\,}}}}\) to \({{\,\mathrm{\mathsf pHiggs}\,}}_{X_{{{\,\mathrm{reg}\,}}}}\). Compatibility with the global nonabelian Hodge Correspondence for projective klt spaces follows from the construction in [9]; cf. especially [9, Prop. 3.10].
We claim that the functor \(\eta _{X_{{{\,\mathrm{reg}\,}}}}\) is an equivalence of categories; for that, we need to check that it is full, faithful and essentially surjective. While essential surjectivity quickly follows from Theorem 5.1, we need to argue a bit more to establish the other two conditions. To this end, let \(\gamma : Y \rightarrow X\) be a maximally quasiétale cover, as provided by [7, Thm. 1.5] and as used in the first part of Sect. 3.2. Denote the Galois group of \(\gamma \) by G. By the defining property of the maximally quasiétale cover, the pullback \(\gamma ^*(\mathsf{E}_{X_{{{\,\mathrm{reg}\,}}}})\) extends to a Gequivariant local system \(\mathsf{E}_Y\) on Y. The global nonabelian Hodge Correspondence for projective klt spaces then assigns a Gequivariant, locally free, polystable Higgs bundle \((\mathscr {E}_Y, \theta _{Y}) = \eta _Y(\gamma ^*(\mathsf{E}_{X_{{{\,\mathrm{reg}\,}}}}))\) to \(\gamma ^*(\mathsf{E}_{X_{{{\,\mathrm{reg}\,}}}})\), which is seen to coincide with \(\gamma ^*(\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{X_{{{\,\mathrm{reg}\,}}}}) = \gamma ^*(\eta _{X_{{{\,\mathrm{reg}\,}}}}(\mathsf{E}_{X_{{{\,\mathrm{reg}\,}}}}))\) wherever this makes sense, see Proposition 3.17 as well as Remark and Notation 3.18 and also again [9, Prop. 3.10]. The fact that \(\eta _Y\) is an equivalence of categories now quickly implies that \(\eta _{X_{{{\,\mathrm{reg}\,}}}}\) is full and faithful as well. \(\square \)
Remark 6.4
It follows from the preceding proof and from Proposition 3.17 that on a maximally quasiétale cover Y, the nonabelian Hodge correspondence for \({{\,\mathrm{\mathsf pHiggs}\,}}_{Y_{{{\,\mathrm{reg}\,}}}}\) coincides with the nonabelian Hodge Correspondence for the projective klt space Y when we apply the natural restriction functors on both sides of the correspondence.
8.2 Nonabelian Hodge correspondence for semistable bundles
In direct analogy to Simpson’s work, Theorem 6.3 extends to give an equivalence between the category of flat bundles and arbitrary local systems on \(X_{{{\,\mathrm{reg}\,}}}\). The (fairly standard) proof requires a version of the nonabelian Hodge correspondence for a maximally quasiétale cover [9, Thm. 3.4 and the discussion after Prop. 3.11], Theorem 6.3 above, and the formalities of differential graded categories (DGCs) established in [37, Sect. 3]. The details are left to the reader.
Fact and Definition 6.5
 (6.5.1)
There exists an ample \(H \in {{\,\mathrm{Div}\,}}(X)\), such that \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) is semistable and has vanishing \(\mathbb {Q}\)Chern classes with respect to H.
 (6.5.2)
For any ample \(H \in {{\,\mathrm{Div}\,}}(X)\), the Higgs bundle \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}})\) is semistable and has vanishing \(\mathbb {Q}\)Chern classes with respect to H.
In analogy to [37, Cor. 3.10], the nonabelian Hodge correspondence for semistable bundles now reads as follows.
Theorem 6.6
(Nonabelian Hodge correspondence for \({{\,\mathrm{\mathsf Higgs}\,}}_{X_{{{\,\mathrm{reg}\,}}}}\)). Let X be a projective klt space. Then, there exists equivalence between the category \({{\,\mathrm{\mathsf Higgs}\,}}_{X_{{{\,\mathrm{reg}\,}}}}\) and the category \({{\,\mathrm{\mathsf LSys}\,}}_{x_{{{\,\mathrm{reg}\,}}}}\) of local systems on \(X_{{{\,\mathrm{reg}\,}}}\).
For \((\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}, \theta _{\mathscr {E}_{X_{{{\,\mathrm{reg}\,}}}}}) \in {{\,\mathrm{\mathsf Higgs}\,}}_{X_{{{\,\mathrm{reg}\,}}}}\) that are restrictions of bundles \((\mathscr {E}_X, \theta _{\mathscr {E}_X}) \in {{\,\mathrm{\mathsf Higgs}\,}}_X\) and for local systems \(\mathsf{E}_{X_{{{\,\mathrm{reg}\,}}}} \in {{\,\mathrm{\mathsf LSys}\,}}_{X_{{{\,\mathrm{reg}\,}}}}\) that are restrictions of local systems \(\mathsf{E}_X \in {{\,\mathrm{\mathsf LSys}\,}}_X\), the correspondence is compatible with the global nonabelian Hodge Correspondence for projective klt spaces found in [9, Sect. 3]. \(\square \)
A statement similar to Remark 6.4 continues to hold for the nonabelian Hodge correspondence for semistable bundles on a maximally quasiétale cover.
9 Proof of Theorem 1.5, uniformisation for minimal varieties
In this section, we prove Theorem 1.5. The strategy of proof in principle follows [8, Prop. 8.2 and 8.3]. The main new difficulty stems from the fact that a general complete intersection surface in a klt surface need not be smooth, but might have finite quotient singularities. We maintain notation and assumptions of Theorem 1.5 throughout.
9.1 Step 1: Reduction steps
Let \(\pi : X \rightarrow X_{{{\,\mathrm{can}\,}}}\) be the birational crepant morphism to the canonical model \(X_{{{\,\mathrm{can}\,}}}\), which is also klt, and whose canonical divisor \(K_{X_{{{\,\mathrm{can}\,}}}}\) is ample, cf. [20, Thm. 3.3].
Claim 7.1
We have the inequality \(\widehat{c}_2(\mathscr {T}_{X_{{{\,\mathrm{can}\,}}}})\cdot [K_{X_{{{\,\mathrm{can}\,}}}}]^{n2} \le \widehat{c}_2(\mathscr {T}_X) \cdot [K_X]^{n2}\).
Proof
As a direct consequence of Claim 7.1 and of the fact that \(\pi \) is crepant, we see that equality holds in the \(\mathbb {Q}\)Miyaoka–Yau inequality for \(X_{{{\,\mathrm{can}\,}}}\) as well. The variety \(X_{{{\,\mathrm{can}\,}}}\) therefore reproduces the assumptions made in Theorem 1.5, and we may assume for the remainder of the present proof that the divisor \(K_X\) is ample.
Likewise, if \(\gamma : Y \rightarrow X\) is any quasiétale cover, recall from [20, Prop. 5.20] that Y is again klt. We have remarked in [8, Lem. 3.16] that equality holds in the \(\mathbb {Q}\)Miyaoka–Yau inequality for Y, too. Replacing X by a suitable maximally quasiétale cover, [7, Thm. 1.5], we will therefore assume from now on that X is maximally quasiétale. Our aim is now to show that X is smooth. Once this is established, the main claim will follow from classical uniformisation results of Yau for smooth projective varieties.
9.2 Step 2: End of proof
10 Positivity in the sheaf of reflexive differentials
Given a singular ball quotient X, we ask for positivity in the sheaf \(\Omega ^{[1]}_X\) of reflexive differentials. In other words, we would like to answer the following question.
Question 8.1
(Positivity for singular ball quotients) Given a morphism \(f: Y \rightarrow X\) of projective varieties where X has canonical or klt singularities and where Y is a smooth ball quotient of general type, is \(\Omega ^{[1]}_X\) positive in a suitable sense?
The relevance of Question 8.1 is illustrated by Proposition 8.2 below, which relates positivity to nonexistence of rational curves. Moreover, it is motivated by the hyperbolicity statement [8, Cor. 1.4 and Sect. 9.3]. The answer to Question 8.1 turns out to be surprisingly delicate. One the one hand, we show in Sect. 8.2 that sufficiently high symmetric powers \({{\,\mathrm{Sym}\,}}^{[m]} \Omega ^{[1]}_X\) are always ample in the sense of Definition 2.4. On the other hand, Sect. 8.3 shows by way of example that even nefness of \(\Omega ^{[1]}_X\) fails in general. In this sense, we have no satisfactory answer to Question 8.1 at present.
10.1 Consequences of positivity
As pointed out above, positivity in the singular sheaf \(\Omega ^{[1]}_X\) directly relates to hyperbolicity properties of the underlying variety.
Proposition 8.2
 (8.2.1)
If \(\Omega ^{[1]}_X\) is nef, then X does not contain rational curves.
 (8.2.2)
If \(\Omega ^{[1]}_X\) is ample, then X does not contain any curve whose normalisation is of genus one.
The proof of Proposition 8.2 uses the fact that there exists a functorial pullback functor for reflexive differential forms on klt spaces that agrees with the standard pullback of Kähler differentials wherever that makes sense. We refer to [18, Sect. 5] for a precise reference, and to [10, Sect. 3] for an overview.
Lemma 8.3
(Pullback of reflexive differential is generically surjective) Let X be a quasiprojective klt space and \(Y \subseteq X\) be a smooth subvariety, with inclusion \(\iota : Y \rightarrow X\). Then, the pullback map \(d\iota : \iota ^*\Omega ^{[1]}_X \rightarrow \Omega ^{1}_Y\) is generically surjective.
Proof
Proof of Proposition 8.2
10.2 Positivity of symmetric differentials
As pointed out in the introduction, we show that the sheaves of reflexive symmetric differentials of sufficiently high degree are always ample on singular ball quotients.
Proposition 8.4
 (8.4.1)
The sheaf \(f^{[*]} \Omega ^{[1]}_X\) is ample.
 (8.4.2)
The sheaf \({{\,\mathrm{Sym}\,}}^{[m]} \Omega ^{[1]}_X\) is ample for \(m \gg 0\) sufficiently divisible.
Proof
To prove (8.4.1), recall from [18, Thm. 1.3] or [21, Thm. 14.1] that there exists a pullback morphism for reflexive differential forms, \(df: f^{[*]} \Omega ^{[1]}_X \rightarrow \Omega ^{1}_Y\), which is an isomorphism on the big open set where f is étale. Since both sheaves are reflexive, df is actually an isomorphism, and the first assertion follows.
10.3 Failure of positivity in general
In spite of the positivity result established in Proposition 8.4, Question 8.1 has a negative answer in general. First examples already exist in dimension two.
Example 8.5
(A klt ball quotient whose reflexive cotangent sheaf is not ample) The example given in [8, Sect. 9.4] shows that there exists a klt ball quotient surface S that is covered by curves whose normalisations are elliptic. As a consequence of Proposition 8.2 above, \(\Omega ^{[1]}_S\) is not ample.
Example 8.6
(A canonical ball quotient whose reflexive cotangent sheaf is not nef) Recall from [3] that there exists a fake projective plane Y that admits an automorphism \(\sigma \) of order three. Recall from [29, p. 233] that Y is a smooth ball quotient. The quotient surface \(X := Y/\langle \sigma \rangle \) has been studied by Keum. He proves in [19, Prop. 3.1] that X has exactly three singular points, which are canonical of type \(A_2\), and that \(K_X\) is Cartier with \([K_X]^{2} = 3\). With this description of X at hand, the following Proposition 8.7 shows that \(\Omega ^{[1]}_X\) cannot possibly be nef.
Proposition 8.7
Let X be a projective surface with nontrivial canonical singularities where \(\Omega ^{[1]}_X\) is nef. If \(\,[K_X]^{2} = 3\), then X has exactly one singular point.
Proof
Claim 8.8
The sheaf \(\mathscr {E}\) is nef. In particular, \(c_1(\mathscr {E})^{2} \ge 0\).
Proof of Claim 8.8
Footnotes
 1.
In local coordinates, if \(\theta = \sum _k \theta _k\, d z_k\), then \(\theta ^h = \sum _k \theta _k^{*}\, d \bar{z}_k\), where \(\theta _k^*\) is the adjoint of \(\theta _k\) with respect to the metric h.
 2.
See the argument in [27, Thm. 25.28].
 3.
Here, we view \((\mathscr {E}_{\widetilde{X}}, \theta _{\mathscr {E}_{\widetilde{X}}})\) as a Higgsbundle without its structure as a Gsheaf
 4.
For instance, this is the case when Y is a smooth ball quotient.
Notes
Acknowledgements
We would like to thank numerous colleagues for discussions, including Daniel Barlet, Oliver Bräunling, Philippe Eyssidieux, Jochen Heinloth, Andreas Höring, Annette Huber, Shane Kelly, JongHae Keum, Adrian Langer and Jörg Schürmann. We also thank the anonymous referee for helpful suggestions for improvement.
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