Christian Schnell : On the canonical extension of a local system

Posted by , part of the Algebraic Geometry Seminar.

At

Oct. 11, 2007, 4 p.m.

In

SEO 636

Abstract

In any family of projective varieties, the integral cohomology groups of the smooth fibers form a local system over the base $B$. In general, this local system is defined only over an open subset of $B$, because some fibers may be singular, and its behavior near the ``boundary'' contains information about the original family. Associated to the local system, there is also a vector bundle with a flat connection. In 1970, Deligne showed in a more general setting how to construct a ``canonical extension'' for this type of bundle, using the monodromy of the local system, and his construction has since played a role in Hodge theory. In the talk, we answer the natural question of what happens to the local system itself at the boundary.