Numerical Kodaira dimension of algebraic fiber spaces in positive characteristic

TL;DR

This paper extends key inequalities and theorems related to the numerical Kodaira dimension to positive characteristic, proving Iitaka's conjecture in certain cases.

Contribution

It establishes positive characteristic analogs of important theorems and proves Iitaka's conjecture when the base is of general type and the total space's canonical divisor is semi-ample.

Findings

Proved a positive characteristic version of Nakayama's inequality.

Established variants of global generation and weak positivity theorems in positive characteristic.

Confirmed Iitaka's conjecture under specified conditions.

Abstract

In this paper, we prove a positive characteristic analog of Nakayama's inequality on the numerical Kodaira dimension of algebraic fiber spaces when the generic fibers have nef canonical divisors. To this end, we establish variants of Popa and Schnell's global generation theorem, Viehweg's weak positivity theorem and Fujino's global generation theorem in positive characteristic. As a byproduct, we show that Iitaka's conjecture holds true in positive characteristic when the base space is of general type and the canonical divisor of the total space is relatively semi-ample.