Numerical Kodaira dimension of algebraic fiber spaces in positive characteristic

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.

arXiv comment25 pages, v2: added an example which shows that the main theorem fails if the geometric generic fiber is not smooth. v3: Theorems 1.5 and 5.1 (Iitaka's inequality when the base space is of general type and the canonical divisor of the total space is relatively semi-ample) added