On the bicanonical map of irregular varieties

TL;DR

This paper investigates the bicanonical map of irregular varieties with maximal Albanese dimension, showing that only certain higher-dimensional analogs of genus 2 curves have non-birational bicanonical maps, using Fourier-Mukai techniques.

Contribution

It characterizes primitive irregular varieties of maximal Albanese dimension with non-birational bicanonical maps, extending the understanding of pluricanonical maps in higher dimensions.

Findings

Only varieties birational to theta-divisors of indecomposable principally polarized abelian varieties have non-birational bicanonical maps.

The result generalizes the genus 2 curve case to higher dimensions.

Fourier-Mukai transform is a key tool in the proof.

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.