Piotr Achinger: On Deligne's monodromy finiteness theorem
Abstract: Let S be a smooth complex algebraic variety with a base point 0. If
X/S is a family of smooth projective families, one obtains the
associated monodromy representation pi_1(S, 0)->GL(H^*(X_0, Q)) of the
topological fundamental group of S on the rational cohomology of X_0.
Let us say that a representation pi_1(S, 0) -> GL_n(Q) "comes from
geometry" if it is a direct summand of such a monodromy
representation. Representations coming from geometry are quite
special: they factor through GL_n(Z) up to conjugation and the
corresponding local system on S underlies a polarizable variation of
Hodge structures. Inspired by a theorem of Faltings, Deligne proved in
the mid-80s that there are only finitely many such representations of
rank n for fixed S and n. I will explain the beautifully simple proof
of this result, based on Griffiths' study of the geometry of period
domains. I will also mention the very recent results of Litt which in
particular imply a suitable analog for the etale fundamental group