Given a quasi-projective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite etale covers from the smooth locus $X_{\mathrm{reg}}$ of $X$ to $X$ itself. A simplified version of our main results states that there exists a Galois cover $Y \rightarrow X$, ramified only over the singularities of $X$, such that the etale fundamental groups of $Y$ and of $Y_{\mathrm{reg}}$ agree. In particular, every etale cover of $Y_{\mathrm{reg}}$ extends to an etale cover of $Y$. 
As first major application, we show that every flat holomorphic bundle defined on $Y_{\mathrm{reg}}$ extends to a flat bundle that is defined on all of $Y$. As a consequence, we generalise a classical result of Yau to the singular case: every variety with at worst terminal singularities and with vanishing first and second Chern class is a finite quotient of an Abelian variety. As a further application, we verify a conjecture of Nakayama and Zhang describing the structure of varieties that admit polarised endomorphisms.

\'Etale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties

We investigate the holonomy group of singular Kahler-Einstein metrics on klt varieties with numerically trivial canonical divisor. Finiteness of the number of connected components, a Bochner principle for holomorphic tensors, and a connection between irreducibility of holonomy representations and stability of the tangent sheaf are established. As a consequence, known decompositions for tangent sheaves of varieties with trivial canonical divisor are refined. In particular, we show that up to finite quasi-etale covers, varieties with strongly stable tangent sheaf are either Calabi-Yau or irreducible holomorphic symplectic. These results form one building block for Horing-Peternell's recent proof of a singular version of the Beauville-Bogomolov Decomposition Theorem.

https://msp.org/gt/2019/23-4/gt-v23-n4-p08-s.pdf

Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups

We investigate under what conditions holomorphic forms defined on the regular locus of a reduced complex space extend to holomorphic (or logarithmic) forms on a resolution of singularities. We give a simple necessary and sufficient condition for this, whose proof relies on the Decomposition Theorem and Saito's theory of mixed Hodge modules. We use it to generalize the theorem of Greb-Kebekus-Kovacs-Peternell to complex spaces with rational singularities, and to prove the existence of a functorial pull-back for reflexive differentials on such spaces. We also use our methods to settle the "local vanishing conjecture" proposed by Mustaţa, Olano, and Popa.

Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities

Klt varieties with trivial canonical class -- Holonomy, differential forms, and fundamental groups

Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q > 1$. 
Suppose $f$ is separable and $X$ is $\mathbb{Q}$-Gorenstein and normal. We show that the anti-canonical divisor $-K_X$ is numerically equivalent to an effective $\mathbb{Q}$-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (Theorem C) and also covering singular varieties over an algebraically closed field of arbitrary characteristic. 
Suppose $f$ is separable and $X$ is normal. We show that the Albanese morphism of $X$ is an algebraic fibre space and $f$ induces polarized endomorphisms on the Albanese and also the Picard variety of $X$, and $K_X$ being pseudo-effective and $\mathbb{Q}$-Cartier means being a torsion $\mathbb{Q}$-divisor. 
Let $f^{Gal}:\overline{X}\to X$ be the Galois closure of $f$. We show that if $p>5$ and co-prime to $deg\, f^{Gal}$ then one can run the minimal model program (MMP) $f$-equivariantly, after replacing $f$ by a positive power, for a mildly singular threefold $X$ and reach a variety $Y$ with torsion canonical divisor (and also with $Y$ being a quasi-etale quotient of an abelian variety when $\dim(Y)\le 2$). Along the way, we show that a power of $f$ acts as a scalar multiplication on the Neron-Severi group of $X$ (modulo torsion) when $X$ is a smooth and rationally chain connected projective variety of dimension at most three.

Polarized endomorphisms of normal projective threefolds in arbitrary characteristic

We prove that for a compact Kahler threefold with canonical singularities and vanishing first Chern class, the projective fibres are dense in the semiuniversal deformation space. This implies that every Kahler threefold of Kodaira dimension zero admits small projective deformations after a suitable bimeromorphic modification. As a corollary, we see that the fundamental group of any Kahler threefold is a quotient of an extension of fundamental groups of projective manifolds, up to subgroups of finite index. 
In the course of the proof, we show that for a canonical threefold with $c_1 = 0$, the Albanese map decomposes as a product after a finite etale base change. This generalizes a result of Kawamata, valid in all dimensions, to the Kahler case. Furthermore we generalize a Hodge-theoretic criterion for algebraic approximability, due to Green and Voisin, to quotients of a manifold by a finite group.

Algebraic approximation of K\"ahler threefolds of Kodaira dimension zero

Theorem 2 (Bogomolov–Sommese vanishing on lc C-pairs). Let (X,D′) be a complex projective log canonical pair, and let D ≤ D′ be a divisor such that (X,D) is a C-pair. If A ⊂ Sym C Ω p X(logD) is a Weil divisorial subsheaf, then κC(A ) ≤ p. A C-pair is a pair (X,D) where all the coefficients of D are of the form 1 − 1/n for n ∈ N ∪ {∞}. This notion was introduced by Campana under the name orbifoldes geometriques. The C-Kodaira dimension κC of a Weil divisorial sheaf of differential forms on (X,D) is a natural generalization of the Kodaira dimension of a line bundle, which takes into account the fractional part of D.

/pdf/bogomolov-sommese-vanishing-on-log-canonical-pairs-1qfhfc9fhe.pdf

Bogomolov-Sommese vanishing on log canonical pairs

Given a logarithmic $1$-form on the snc locus of a log canonical surface pair $(X, D)$ over a perfect field of characteristic $p \ge 7$, we show that it extends with at worst logarithmic poles to any resolution of singularities. We also prove the analogous statement for regular differential forms, under an additional tameness hypothesis. In addition, residue and restriction sequences for tamely dlt pairs are established. 
We give a number of examples showing that our results are sharp in the surface case, and that they fail in higher dimensions. On the other hand, our techniques yield a new proof of the characteristic zero Logarithmic Extension Theorem in any dimension.

Differential forms on log canonical spaces in positive characteristic

Let (X, D) be a projective log canonical pair. We show that for any natural number p, the sheaf (Omega_X^p(log D))^** of reflexive logarithmic p-forms does not contain a Weil divisorial subsheaf whose Kodaira-Iitaka dimension exceeds p. This generalizes a classical theorem of Bogomolov and Sommese. 
In fact, we prove a more general version of this result which also deals with the orbifoldes g\'eom\'etriques introduced by Campana. The main ingredients to the proof are the extension theorem of Greb-Kebekus-Kov\'acs-Peternell, a new version of the Negativity lemma, the Minimal Model Program, and a residue map for symmetric differentials on dlt pairs. 
We also give an example showing that the statement cannot be generalized to spaces with Du Bois singularities. As an application, we give a Kodaira-Akizuki-Nakano-type vanishing result for log canonical pairs which holds for reflexive as well as for K\"ahler differentials.

We investigate properties of potentially Du Bois singularities, that is, those that occur on the underlying space of a Du Bois pair. We show that a normal variety $X$ with potentially Du Bois singularities and Cartier canonical divisor $K_X$ is necessarily log canonical, and hence Du Bois. As an immediate corollary, we obtain the Lipman-Zariski conjecture for varieties with potentially Du Bois singularities. 
We also show that for a normal surface singularity, the notions of Du Bois and potentially Du Bois singularities coincide. In contrast, we give an example showing that in dimension at least three, a normal potentially Du Bois singularity $x \in X$ need not be Du Bois even if one assumes the canonical divisor $K_X$ to be $\mathbb{Q}$-Cartier.

Potentially Du Bois spaces

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Papers

Algebraic approximation of K\"ahler threefolds of Kodaira dimension zero

Bogomolov-Sommese vanishing on log canonical pairs

Differential forms on log canonical spaces in positive characteristic

Bogomolov-Sommese vanishing on log canonical pairs

Potentially Du Bois spaces