Title:
The algebraic geometry of the Putman-Wieland conjecture
Abstract:
Suppose we are given an unramified covering $\Sigma_{g'} \to \Sigma_g$ of topological
surfaces with $g \geq 3$. The Putman-Wieland conjecture predicts that the
action of the mapping class group of $Mod_{g,1}$ on $H_1(\Sigma_{g'})$ has no nonzero fixed vectors.
More generally, one might predict that this action has big monodromy in a
suitable sense.
The Putman-Wieland conjecture is closely related to Ivanov's conjecture, predicting that finite index
subgroups of the mapping class group (for $g \geq 3$) have finite abelianization.
We will discuss joint work with Daniel Litt and Will Sawin making progress toward
the Putman-Wieland conjecture, as well as several big monodromy variants.
The methods are Hodge theoretic in nature via studying the derivative of an
associated period map.