Iitaka's $C_{n,m}$ conjecture for 3-folds in positive characteristic

TL;DR

This paper proves the subadditivity of Kodaira dimensions for certain fibrations of smooth projective 3-folds over curves in positive characteristic, confirming Iitaka's $C_{n,m}$ conjecture in this setting.

Contribution

It establishes the Iitaka $C_{n,m}$ conjecture for smooth projective 3-folds over algebraically closed fields with characteristic greater than 5.

Findings

Subadditivity of Kodaira dimensions proven for specific 3-fold fibrations.

Validates Iitaka's $C_{n,m}$ conjecture in positive characteristic.

Applicable to smooth projective 3-folds over curves in characteristic p > 5.

Abstract

In this paper, we prove that for a fibration $f:X\to Z$ from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field $k$ with $\mathrm{char} k =p >5$, if the geometric generic fiber $X_{\overline\eta}$ is smooth, then subadditivity of Kodaira dimensions holds, i.e. $$\kappa(X)\ge\kappa(X_{\overline\eta})+\kappa(Z).$$