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We prove the nonvanishing theorem for $3$ -folds over an algebraically closed field $k$ of characteristic $p\gt 5$ .

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., Comment: v1: this article supersedes arXiv:1302.1655; v2: slightly more general results, simplified topological arguments in Part II of the paper, improved exposition; v3: improved exposition, minor changes as requested by referee; to appear in Duke Mathematical Journal

We prove a 1999 conjecture of Veys, which says that the opposite of the log canonical threshold is the only possible pole of maximal order of Denef and Loeser's motivic zeta function associated with a germ of a regular function on a smooth variety over a field of characteristic zero. We apply similar methods to study the weight function on the Berkovich skeleton associated with a degeneration of Calabi-Yau varieties. Our results suggest that the weight function induces a flow on the non-archimedean analytification of the degeneration towards the Kontsevich-Soibelman skeleton., Comment: to appear in Duke Mathematical Journal

We give a new and self-contained proof of the finite generation of adjoint rings with big boundaries. As a consequence, we show that the canonical ring of a smooth projective variety is finitely generated., to appear in Duke Math. J

We prove that the ascending chain condition (ACC) for log canonical (lc) thresholds in dimension $d$ , existence of log flips in dimension $d$ , and the log minimal model program (LMMP) in dimension $d{-}1$ imply termination of any sequence of log flips starting with a $d$ -dimensional effective lc pair and also imply termination of flops in dimension $d$ . In particular, the latter terminations in dimension $4$ follow from the Alexeev-Borisov conjecture in dimension $3$

We correct a sign error in the published version of the paper.

We construct a compactification M_d of the moduli space of plane curves of degree d. We regard a plane curve C as a surface-divisor pair (P^2,C) and define M_d as a moduli space of pairs (X,D) where X is a degeneration of the plane. We show that, if d is not divisible by 3, the stack M_d is smooth and the degenerate surfaces X can be described explicitly., 46 pages. Final version to be published in Duke Mathematical Journal

For $Y,Y^+$ three-dimensional smooth varieties related by a flop, Bondal and Orlov conjectured that the derived categories $D^b({\rm coh}(Y))$ and $D^b({\rm coh}(Y^+))$ are equivalent. This conjecture was recently proved by Bridgeland. Our aim in this paper is to give a partially new proof of Bridgeland's result using noncommutative rings. The new proof also covers some mild singular and higher-dimensional situations (including those occuring in the recent paper by Chen [11]).

Consider the Fulton-MacPherson configuration space of $n$ points on $\P^1$, which is isomorphic to a certain moduli space of stable maps to $\P^1$. We compute the cone of effective ${\mathfrak S}_n$-invariant divisors on this space. This yields a geometric interpretation of known asymptotic formulas for the number of integral points of bounded height on compactifications of $\SL_2$ in the space of binary forms of degree $n\ge 3$., Comment: 26 pages, LaTeX