# Summer Lectures in Geometry

### 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 2

For detailed overviews of each course see http://camgsd.ist.utl.pt/encontros/slg/.