| Description |
Vanishing theorems play a paramount role in modern birational geometry. Over elds of characteristic zero, the classical vanishing theorems of Kodaira, Nakano and Kawamata- Viehweg show that under positivity assumptions on a line bundle L, one can guarantee that the higher cohomology groups Hi(X; !X L) of a smooth projective variety are zero. If the positivity assumptions are dropped (namely, if the line bundles involved have zero rst Chern class), a celebrated theorem of Green and Lazarsfeld shows that one can still obtain vanishing of certain cohomology groups for an open subset of topologically trivial line bundles. Generic vanishing theorems have proved to be remarkably useful, and among their many applications, one can highlight Ein and Lazarsfeld's results on the birational geometry of irregular varieties or the study of pluricanonical maps of varieties of maximal Albanese dimension. Over elds of positive characteristic, a theorem of Hacon and Kovacs shows that the obvious extension of Green and Lazarsfeld's generic vanishing theorem is false. Notwithstanding this, recent work of Hacon and Patakfalvi provides a generic vanishing statement in positive characteristic which, albeit necessarily weaker, is strong enough to prove positive characteristic versions of Kawamata's celebrated characterization of abelian varieties. The objective of this dissertation is twofold. In the rst place, we generalize Hacon and Patakfalvi's generic vanishing theorem in the context of Pareschi and Popa's generic vanishing theory. Once the generic vanishing groundwork is laid down, we apply it to derive relevant geometric consequences. Concretely, we prove some results of Ein and Lazarsfeld on the geometry of irregular varieties and on the singularities of Theta divisors and we also prove some statements that might shed some light on the problem of the rationality of the tricanonical map of varieties of general type and of maximal Albanese dimension in positive characteristic. |
