Description:

Title: Algebraicity of Shafarevich maps and the Shafarevich conjecture

Abstract: For a normal complex algebraic variety X equipped with a complex representation V of its fundamental group, a Shafarevich map f:X->Y is a map which contracts precisely those algebraic subvarieties on which V has finite monodromy. Such maps were constructed for projective X by Eyssidieux, and recently have been constructed analytically in the quasiprojective case by Brunebarbe and Deng--Yamanoi, in both cases using techniques from non-abelian Hodge theory. In joint work with Y. Brunebarbe and J. Tsimerman, we show that these maps are algebraic. This is a generalization of the Griffiths conjecture on the algebraicity of images of period maps, and the proof critically uses o-minimal GAGA. We will also explain how the same techniques can be used to prove the Shafarevich conjecture in the "linear case", which puts strong restrictions on the complex analytic varieties that arise as universal covers of algebraic varieties admitting linear representations of their fundamental groups.