On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluri- canonical 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 pa- per 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 gen- eralization to this context of curves of genus 2: varieties birationally equivalent to the theta-divisor of an indecomposable principally polar- ized abelian variety. The proof is based on the (generalized) Fourier- Mukai transform.