Irregular varieites with geometric genus one, theta divisors, and fake tori

TL;DR

This paper investigates the structure of compact Kähler manifolds with geometric genus one, revealing conditions under which they map onto theta divisors or are fibered over genus two curves, with implications for their Hodge numbers.

Contribution

It establishes new geometric characterizations of Kähler manifolds with geometric genus one, including their mapping properties and topological classifications, especially relating to theta divisors and fiber structures.

Findings

Manifolds with non-surjective Albanese map map onto ample divisors in abelian varieties.

Under certain topology conditions, these manifolds are fibered over genus two curves.

Identifies conditions under which such manifolds have Hodge numbers matching abelian varieties.

Abstract

We study the Albanese image of a compact K\"ahler manifold whose geometric genus is one. We prove that if the Albanese map is not surjective, then the manifold maps surjectively onto an ample divisor in some abelian variety, and in many cases the ample divisor is a theta divisor. With a further natural assumption on the topology of the manifold, we prove that the manifold is an algebraic fiber space over a genus two curve. Finally we apply these results to study the geometry of a compact K\"ahler manifold which has the same Hodge numbers as those of an abelian variety of the same dimension.