Abstract |
A general theme in geometry is the classification of
algebraic/differential geometric structures which satisfy a positivity
property. I will describe an "asymptotic" version of this theme
based on joint work with Cheltsov, Martinez-Garcia, and Zhang. On the
algebraic side, we introduce the class of asymptotically log Fano
varieties and state a classification theorem in dimension 2,
generalizing the classical efforts of the 19th century Italian school.
The novelty here is the use of a convex optimization theorem that
reduce the asymptotic positivity to determining intersection
properties of high-dimensional convex bodies. On the differential
side, I will give a conjectural picture for existence of singular
Kahler-Einstein metrics, explain progress towards this conjecture, and
relations to singular Kahler-Ricci solitons. Time permitting, I will
also explain some conjectures and results about the "small angle
limit" when the angle tends to zero, making ties to non-compact
Calabi-Yau fibrations and steady Ricci solitons. The talk should be
accessible to non-experts.
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