Felix Klein Lectures

Hodge theory and o-minimality

Benjamin Bakker (Georgia)

 

Dates:

Introductory lecture: Thursday 16 May 10:15 am 

Venue: MPI, Vivatsgasse 7, Bonn

Monday, 20 May 2:15-3:45 pm

Tuesday 21 May 2:15-3:45 pm

Thursday 23 May 10:15-11:45 am

Friday 24 May 2:15-3:45 pm

Venue:
Lipschitz hall, Endenicher Allee 60, Bonn 

 

Abstract:

The cohomology groups of complex algebraic varieties come equipped with a powerful invariant called a Hodge structure. Going back to the foundational work of Griffiths, Hodge theory has found many important applications to algebraic and arithmetic geometry, but its intrinsically analytic nature often leads to complications.

Recent joint work with Y. Brunebarbe, B. Klingler, and J. Tsimerman has shown that in fact many Hodge-theoretic constructions can be carried out in an intermediate geometric category, and o-minimality provides the crucial tameness hypothesis to make this precise. A salient feature of the o-minimal category is that it allows for the local flexibility of the analytic category while preserving the global behavior of the algebraic category.

The goal of these lectures is to give a survey of recent advances in Hodge theory via o-minimal techniques aimed at non-experts. We will start by introducing the basic notions of o-minimal geometry with a view towards algebraization theorems. Among the Hodge-theoretic applications, we will easily recover a theorem of Cattani--Deligne--Kaplan on the algebraicity of Hodge loci and describe a proof of a conjecture of Griffiths on the quasiprojectivity of images of period maps.