On Ueno's Conjecture K

TL;DR

This paper investigates the relationship between the Kodaira dimension of a smooth complex projective variety with Kodaira dimension zero and the Kodaira dimension of its general fiber under the Albanese map, establishing an upper bound.

Contribution

It proves that for such varieties, the Kodaira dimension of a general fiber is at most the dimension of the space of holomorphic 1-forms.

Findings

The Kodaira dimension of a general fiber is bounded above by h^0(Ω^1_X).

The result links the geometry of the variety to its Albanese fiber structure.

Abstract

We show that if $X$ is a smooth complex projective variety with Kodaira dimension $0$ then the Kodaira dimension of a general fiber of its Albanese map is at most $h^0(\Omega ^1 _X)$.