School of Mathematical Sciences

Department Colloquium

Schreiber 006, 12:15

Stanford University

Abstract:The Uniformization Theorem implies that any compact Riemann surface has a constant curvature metric with the sign of the curvature determined by the genus. Kahler-Einstein (KE) metrics are a natural generalization of constant curvature metrics, and the search for such metrics has a long and rich history, going back to Schouten, Kahler (30's), Calabi (50's), Aubin, Yau (70's) and Tian (90's), among others. Yet, despite much progress, a complete picture is available only in complex dimension 2.

In contrast to such smooth KE metrics, in the mid 90's Tian conjectured the existence of KE metrics with conical singularities along a divisor (i.e., for which the manifold is `bent' at some angle along a complex hypersurface), motivated by applications to algebraic geometry and Calabi-Yau manifolds. More recently, Donaldson suggested a program for constructing smooth KE metrics of positive curvature out of such singular ones, and put forward several influential conjectures.

In this talk we will try to give an introduction to Kahler-Einstein geometry and briefly describe some recent work mostly joint with R. Mazzeo that resolves some of these conjectures. It follows that many algebraic varieties that may not admit smooth KE metrics (e.g., Fano or minimal varieties) nevertheless admit KE metrics bent along a divisor.

Coffee will be served at 12:00 before the lecture

at Schreiber building 006