Birational positivity in dimension 4 (with an appendix by Fr\'ed\'eric Campana)

TL;DR

This paper establishes bounds on the Kodaira dimension of subsheaves of differential forms for certain 4-dimensional varieties, leading to finiteness results for their fundamental groups under specific conditions.

Contribution

It proves a new bound on the Kodaira dimension of subsheaves of differential forms for non-singular projective varieties of dimension up to 4 with non-negative Kodaira dimension, extending positivity results.

Findings

Bound on Kodaira dimension of subsheaves of $\

Finiteness of the fundamental group for certain varieties with vanishing Kodaira dimension.

Abstract

In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of $\Omega^p$ is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an $X$ provided that $X$ has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.