Speaker: Ben Bakker (UIC)

Title: The linear Shafarevich conjecture for quasiprojective varieties (part 1)

Abstract:

Shafarevich asked whether the universal cover of a smooth projective variety X is always holomorphically convex, meaning it admits a proper map to a Stein space. This was proven in the linear case---namely when X admits an almost faithful representation of its fundamental group---by Eyssidieux--Katzarkov--Pantev--Ramachandran using techniques from non-abelian Hodge theory. In joint work with Y. Brunebarbe and J. Tsimerman, we prove a version of the linear Shafarevich conjecture for quasiprojective varieties. The proof relies on a number of recent advances in non-abelian Hodge theory in the non-proper case.

In the first talk I will outline the general strategy and explain why non-abelian Hodge theory naturally shows up in the context of Shafarevich's question. In the second talk I will provide some details of the proof, including the role played by the twistor geometry of the stack of local systems and the algebraic integrability of Katzarkov--Zuo foliations. As a bonus, I will also explain how these techniques prove the algebraicity of Shafarevich morphisms, which generalizes Griffiths' conjecture on the algebraicity of the images of period maps to arbitrary local systems.