Basic results on irregular varieties via Fourier-Mukai methods

Summary

Recently Fourier-Mukai methods have proved to be a valuable tool in the study of the geometry of irregular varieties. The purpose of this paper is to illustrate these ideas by revisiting some basic results. In particular, we show a simpler proof of the Chen-Hacon birational characterization of abelian varieties. We also provide a treatment, along the same lines, of previous work of Ein and Lazarsfeld. We complete the exposition by revisiting further results on theta divisors. Two preliminary sections of background material are included.

In recent years the systematic use of the classical Fourier-Mukai transform between dual abelian varieties, and of related integral transforms, has proved to be a valuable tool for investigating the geometry of irregular varieties. An especially interesting point is the interplay between vanishing notions naturally arising in the Fourier-Mukai context, as weak index theorems, and the generic vanishing theorems of Green and Lazarsfeld. This naturally leads to the notion of generic vanishing sheaves (GV-sheaves for short). The purpose of this paper is to exemplify these ideas by revisiting some basic results.

To be precise, we focus on the theorem of Chen and Hacon [2001a] characterizing (birationally) abelian varieties by means of the conditions q (X) = dim X and; this is stated as Lemma 4.2 below. We show that the Fourier-Mukai/Generic Vanishing package, in combination with Kollár's theorems on higher direct images of canonical bundles, produces a surprisingly quick and transparent proof of this result. Along the way, we provide a unified Fourier-Mukai treatment of most of the results of [Ein and Lazarsfeld 1997], where both the original and the present proof of the Chen-Hacon theorem find their roots. We complete the exposition with a refinement of Hacon's cohomological characterization of desingularizations of theta divisors, as it fits well in the same framework.

Although many of the results treated here have led to further developments (see, for example, [Chen and Hacon 2002; Hacon and Pardini 2002; Jiang 2011; Debarre and Hacon 2007]), we have not attempted to recover the latter with the present approach. However, we hope that the point of view illustrated in this paper will be useful in the further study of irregular varieties with low invariants.