Chenyang Xu: Moduli spaces of Fano varieties

Abstract: We will describe the purely algebraic construction of moduli spaces parametrizing Fano varieties with K-(semi,poly)stability, called K-moduli spaces, and its fundamental properties. As a by-product, it also completes the solution of Yau-Tian-Donaldson Conjecture to all Fano varieties case (including singular ones).

Lecture 1: we will discuss the background of K-stability and algebraic geometer’s gradually evolving understanding of the concept.

Lecture 2: we will discuss the construction of K-moduli spaces.

Lecture 3: we will focus on a new finite generation theorem we proved recently (joint with Yuchen Liu and Ziquan Zhuang), which completes the solution to several main questions in K-stability, including the compactedness of the K-moduli as well as the Yau-Tian-Donaldson Conjecture for general Fano varieties.