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We show that generalized log canonical thresholds for complex analytic spaces satisfy the ACC (ascending chain condition) and we characterize the accumulation points.

In this paper, we prove a strengthening of the generic vanishing result in characteristic p>0given in Hacon and Patakfalvi (Am J Math 138(4):963–998, 2016). As a consequence of this result, we show that irreducible Θdivisors are strongly F-regular and we prove a related result for pluri-theta divisors.

Any two birational Mori fibre spaces are connected by a sequence of Sarkisov links.

Abstract.: Let X be a smooth complex projective algebraic variety of maximal Albanese dimension. We give a characterization of in terms of the set . An immediate consequence of this is that the Kodaira dimension is invariant under smooth deformations. We then study the pluricanonical maps . We prove that if X is of general type, is generically finite for and birational for . More generally, we show that for the image of is of dimension equal to and for , is the stable canonical map.

Abstract.: We prove that any smooth complex projective variety X with plurigenera P 1(X)=P 2(X)=1 and irregularity q(X)=dim(X) is birational to an abelian variety.

Abstract.: The purpose of this paper is to give two applications of Fourier transforms and generic vanishing theorems: – we give a cohomological characterization of principal polarizations– we prove that if X an abelian variety and a polarization of type (1, ...,1,2), then a general pair is log canonical.

The purpose of this paper is to show how generalizations of generic vanishing theorems to a${\mirrored Q}$-divisor setting can be used to study the geometric properties of pluritheta divisors on a principally polarized Abelian variety (PPAV for short).