Colloquium Abstracts Spring 2025

Abstracts will be posted here for the colloquium talks when they are available.

Chenyang Xu

Princeton University
March 13, 2025

Moduli Theory of Fano Varieties

Moduli spaces, which parametrize classes of geometric objects, are central to algebraic geometry. The construction of moduli spaces for negatively curved curves by Mumford marked a significant milestone in modern moduli theory. For higher-dimensional varieties, the moduli theory of those with negative curvature, developed by Kollár and others, has long been linked to the minimal model program.

However, constructing moduli spaces for positively curved varieties, called Fano varieties, remained a challenge and an open question for higher-dimensional geometers. Recent breakthroughs have revealed that the key lies in the concept of K-stability—a notion introduced in complex geometry by Tian, Donaldson, and others to characterize the existence of Kähler-Einstein metrics.

In this lecture, I will discuss the development of these ideas, highlighting the interplay between algebraic and complex geometry, and the role of K-stabiity in establishing a moduli theory for Fano varieties.