Syzygies, Pluricanonical Maps and the Birational Geometry of Irregular Varieties

Abstract

In this thesis we looked into three different problems which share, as a common factor, the exstensive use of the Fourier–Mukai transform as a research tool. In the first Part we investigated the syzygies of Kummer varieties (i.e. the quotient of an abelian variety X by the Z/2Z induced by the group operation), extending to higher syzygies results on projective normality and degree of equations of Sasaki ([S1]), Kempf ([K5]) and Khaled ([K6, K7]). The second Part of is dedicated to the study of pluricanonical linear systems on varieties of maximal Albanese dimension. More precisely, in Chapter 3 we prove that the 4-canonical map of a smooth variety of general type and maximal Albanese dimension is always birational into its image, the content of this section can also be found in [T2]. Chapter 4 is based on a joint work with Z. Jiang and M. Lahoz ([JLT]) in which we prove that, in any Kodaira dimension, the 4-canonical map of a smooth variety of maximal Albanese dimension induces the Iitaka fibration, while, in the case of varieties of general type, the 3-canonical map is sufficient (and hence the 3-canonical map of these varieties is always birational). We remark that these last results are both sharp. Finally, in the last part of this thesis we consider the problem of classification of varieties with small invariants. The final goal of our investigation is to provide a complete cohomological charaterization of products of theta divisors by proving that every smooth projective variety X, of maximal Albanese dimension, with Euler characteristic equal to 1, and whose Albanese image is not fibered by tori is birational to a product of theta divisors. Under these hypothesis we show that the Albanese map has degree one. Furthermore, we present a new characterization of -divisor in principally polarized abelian varieties.