Mathematical Research Letters

Volume 25 (2018)

Number 3

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

Pages: 783 – 802



Sho Ejiri (Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, Japan)

Lei Zhang (School of Mathematical Science, University of Science and Technology of China, Hefei, China)


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 $\mathrm{char}k = p \gt 5$, if the geometric generic fiber $X_{\overline{\eta}}$ is smooth, then subadditivity of Kodaira dimensions holds, i.e.\[\kappa (X) \geq \kappa (X_{\overline{\eta}}) + \kappa (Z) \; \textrm{.}\]

Received 16 April 2016

Accepted 2 March 2017

Published 3 August 2018