On the bicanonical map of irregular varieties
arXiv:0907.4363
Abstract
From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional generalization to this context of curves of genus 2: varieties birationally equivalent to the theta-divisor of an indecomposable principally polarized abelian variety. The proof is based on the (generalized) Fourier-Mukai transform.
20 pages. There was an inaccuracy in the proof of Theorem 4.13. We have fixed the problem giving a slightly different proof. The modifications only affect to Section 4 of the paper: there is a new definition and we have eliminated a Corollary, which now becomes useless (although it was correct)