Zsolt Patakfalvi
Title: Semi-positivity in positive characteristics
Abstract: Results of Griffiths, Fujita, Kawamata, Viehweg, Kollár, etc. stating semi-positivity of relative canonical bundles and of the pushforwards of their powers were crucial in the development of modern algebraic geometry. Most of these results required the characteristic zero assumption, partially due to the use of Hodge theory. In this talk I present semi-positivity results in positive characteristics. The main focus is moduli theoretic situations, in which the best known results in positive characteristics were for families of stable curves by Szpiro and Kollár and for families of K3 surfaces for Maulik. I present results for arbitrary fiber dimensions allowing sharply F-pure (char p equivalent of log canonical) singularities and semi-ample or ample canonical sheaves for the fibers. The proof avoids characteristic zero and in particular Hodge Theory. The main tool is a lifting theorem by Schwede in positive characteristics. I will also review briefly the history of the characteristic zero results and discuss some applications and motivations, most of which are characteristic independent as soon as one has the adequate semi-positivity statements in place.