Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties

Abstract

Part I. Preparations 8 3. Notation, conventions, and facts used in the proof 8 4. Chern classes on singular varieties 11 5. Bertini-type theorems for sheaves and their moduli 14 Part II. Quasi-étale covers of klt spaces 16 6. Proof of Theorems 2.1 and 1.1 16 7. Direct applications 18 8. Flat sheaves on klt base spaces 21 9. Varieties with vanishing Chern classes 23 10. Varieties admitting polarised endomorphisms 25 11. Examples, counterexamples, and sharpness of results 26 Appendices 29 Appendix A. Zariski's Main Theorem in the equivariant setting 29 Appendix B. Galois closure 31 References 31

Étale Fundamental Groups of Strongly $\boldsymbol{F}$-Regular Schemes

“…The expected general answer is that for more severe singularities on X, there are more obstructions. Analogous to [GKP13], we show that schemes with strongly F -regular singularities are mild in this sense.…”

“…Note that we do not require any quasi-projectivity hypothesis. We actually obtain a slightly stronger version, just as in [GKP13]. As a corollary, we also obtain a variant of [GKP13, Theorem 1.10], see Corollary 4.8, which says in particular that there exists an integer N > 0 such that for every Q-Cartier divisor D on X with index not divisible by p, N · D is in fact Cartier.…”

“…Just as in [GKP13], Corollary 4.8 implies that for any F -finite Noetherian integral strongly F -regular scheme, there exists an integer N > 0 such that if D is Z (p) -Cartier, then N · D is Cartier. The point is that by quasi-compactness, X can be covered by finitely many open X • satisfying Corollary 4.8.…”

“…In [Xu14], Chenyang Xu proved that the algebraic local fundamental group (the profinite completion of the topological fundamental group of the link) of a complex klt singularity is finite. This result was then used by Greb-Kebekus-Peternell [GKP13] to show that for a complex quasi-projective variety X with klt singularities, any sequence of finite quasi-étale (i.e.étale in codimension one) and generically Galois covers eventually becomesétale. In particular, for any quasi-projective klt variety X/C there exists a finite quasi-étale generically Galois cover ρ : X − → X such that any further quasi-étale cover Y − → X is actuallyétale.…”

Uniformisation of higher-dimensional minimal varieties

“…Our hope is that a comparison between the two proofs would prove useful in clarifying the main ideas and techniques behind both results. We have therefore chosen to present an outline of the proof following the strategy of [GKP16], even though this is covered in at least one other survey, [KP14, Section 9]. We remark that the case of canonical threefolds with vanishing Chern classes was achieved by Shepherd-Barron and Wilson in [SBW94].…”