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24/07/2012, 14:00 — 15:00 — Room P3.10, Mathematics Building

Bruno Oliveira, *University of Miami and New York University*

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Symmetric differentials and fundamental group

The relationship between the algebra of symmetric differentials
(sections of the symmetric powers of the holomorphic cotangent
bundle) and the topology of a projective manifold is still
considered quite mysterious. In general this relationship will be
quite loose, since for example it is known that there are manifolds
with the same topology but diametrically distinct spaces of
symmetric differentials (e.g. no symmetric differentials and
asymptotically as many as possible). On the other hand, it is
expected that properties of the fundamental group to be reflected
on the space of symmetric differentials.
The goal of these lectures is to show that there is a class of
symmetric differentials that is quite topological in nature. This
class constitutes an extension to all degrees of the class of
closed symmetric differentials of degree 1 (i.e. closed holomorphic
1-forms) which are well known to reflect topological properties. We
will: discuss primarily the case of degree 2; describe examples;
connect to the theory of foliations and fibrations; and show that
the presence of closed symmetric differentials imply that the
fundamental group has to be infinite.

Session 1