Zsolt Patakfalvi
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Title:  Semi-positivity in positive characteristics
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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.