Semi-positivity in positive characteristics

Zsolt Patakfalvi

Published 2012-08-27, updated 2012-10-10Version 4

Let $f : (X, \Delta) \to Y$ be a flat, projective family of sharply $F$-pure, log-canonically polarized pairs over an algebraically closed field of characteristic $p >0$ such that $p \nmid \ind(K_{X/Y} + \Delta)$. We show that $K_{X/Y} + \Delta$ is nef and that $f_* (\sO_X(m (K_{X/Y} + \Delta)))$ is a nef vector bundle for $m \gg 0$ and divisible enough. Some of the results also extend to non log-canonically polarized pairs. The main motivation of the above results is projectivity of proper subspaces of the moduli space of stable pairs in positive characteristics. Other applications are Kodaira vanishing free, algebraic proofs of corresponding positivity results in characteristic zero, and special cases of subadditivity of Kodaira-dimension in positive characteristics.