Weak positivity theorem and Frobenius stable canonical rings of geometric generic fibers

TL;DR

This paper proves a weak positivity theorem in positive characteristic under certain conditions on the canonical rings of geometric generic fibers, leading to new cases of subadditivity of Kodaira dimensions.

Contribution

It introduces a weak positivity theorem in positive characteristic assuming finitely generated canonical rings and large Frobenius stable canonical rings.

Findings

Established weak positivity theorem in positive characteristic.

Applied the theorem to prove subadditivity of Kodaira dimensions in new cases.

Connected properties of canonical rings with geometric generic fibers.

Abstract

In this paper, we prove the weak positivity theorem in positive characteristic when the canonical ring of the geometric generic fiber $F$ is finitely generated and the Frobenius stable canonical ring of $F$ is large enough. As its application, we show the subadditivity of Kodaira dimensions in some new cases.