The main aim of the lecture is to survey results on known
restrictions on invariants of algebraic varieties and curves on such
varieties coming from the study of inequalities between Chern
classes of vector bundles. One of such restrictions is given by the
Bogomolov--Miyaoka--Yau inequality, which bounds the topological
Euler characteristic of a complex projective
surface. Generalizations of this inequality were used by
F. Hirzebruch to bound possible configurations of line arrangements
on a complex projective plane and by Y. Miyaoka to bound genus of
algebraic curves on surfaces.
I will focus on recent progress on analogous questions in positive
characteristic p, where all such results were classically known to
fail. We will show that, similarly to Deligne--Ilussie's work on
vanishing theorems, all such examples are related to problems with
lifting modulo \(p^2\).
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