Seminar aimed at a general audience by Cristiano Spotti (University of Cambridge)

Title: Compact moduli spaces of Kähler-Einstein Fano varieties

2016.04.01 | Jane Jamshidi

Date Mon 11 Apr
Time 10:15 11:15
Location 1531-113 (Aud. D1)

An important problem at the interface between Differential and Algebraic Geometry consists in finding "canonical" Riemannian metrics on complex algebraic manifolds. Kähler-Einstein (KE) metrics are the natural generalization of metrics with constant Gauss curvature on Riemann surfaces. The case of KE metrics with zero or negative Einstein constant has been understood by the works of Yau and Aubin in the late 70s, while the "positive case" has been recently solved by Chen, Donaldson and Sun, who proved that the existence of a KE metric with positive Einstein constant is equivalent to the purely algebro-geometric notion of K-stability of the underlying complex (Fano) manifold (an Hitchin-Kobayashi correspondence for varieties).

       In this talk, I will focus on how the "spaces" of all such KE Fano manifolds (KE/K-moduli spaces) look like and how to naturally compactifying them by adding certain singular Fano varieties, thanks to the extension of the existence theory to the singular setting. This moduli compactification can be seen as a generalization of the Deligne-Mumford stable curves compactification (and its higher dimensional KSBA analogue), even if, at present, its construction is based on highly transcendental (not algebraic) techniques. In complex dimension greater than or equal to three there are no non-trivial known examples of such compact KE/K-moduli spaces, but, in contrast, I will show how in two dimension (del Pezzo case) the picture is now very explicit.

 

NOTE: This seminar is aimed at a general audience of mathematicians.