Departmental Colloquium

Title

Moduli theory of Fano varieties

Abstract

The construction of moduli spaces for curves marked a significant milestone in modern algebraic geometry. For higher-dimensional varieties, the moduli theory of those with negative curvature has been studied for several decades and known to link 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 to characterize the existence of Kähler-Einstein metrics.

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