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

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. $$\kappa(X)\ge\kappa(X_{\overline\eta})+\kappa(Z).$$