Syzygies, Pluricanonical Maps, and the Birational Geometry of Varieties of Maximal Albanese Dimension

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 research tool. In the first Part we investigated the syzygies of Kummer varieties (i.e. quotients of abelian varieties by the $\mathbb{Z}/2\Z$-action induced by the group operation), extending to higher syzygies results on projective normality and degree of equations of Sasaki, Kempf, and Khaled. The second Part of this Thesis (partially written in collaboration with Z.Jiang and M. Lahoz) is dedicated to the study of pluricanonical linear systems on varieties of maximal Albanese dimension. 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 $\Theta$-divisor in principally polarized abelian varieties.