Varieties with vanishing holomorphic Euler characteristic

TL;DR

This paper investigates complex projective varieties with maximal Albanese dimension and vanishing holomorphic Euler characteristic, classifying examples in dimension 3 and proposing a general structural conjecture.

Contribution

It proves that such varieties have at least three simple factors in their Albanese variety and classifies all examples in dimension 3, also proposing a conjecture for higher dimensions.

Findings

Albanese variety has at least three simple factors.

In dimension 3, classified all such varieties up to covers.

Proposed a conjecture on the structure in all dimensions.

Abstract

We study smooth complex projective varieties $X$ of maximal Albanese dimension and of general type satisfying with vanishing holomorphic Euler characteristic. We prove that the Albanese variety of $X$ has at least three simple factors. Examples were constructed by Ein and Lazarsfeld, and we prove that in dimension 3, these examples are (up to abelian \'etale covers) the only ones. By results of Ueno, another source of examples is provided by varieties $X$ of maximal Albanese dimension and of general type with $p_g(X)=1$. Examples were constructed by Chen and Hacon, and again, we prove that in dimension 3, these examples are (up to abelian \'etale covers) the only ones. We also formulate a conjecture on the general structure of these varieties in all dimensions.