Chenyang Xu Distinguished Lecturer
Moduli Theory of Fano Varieties
Chenyang Xu
Department of Mathematics
Princeton University
Thursday, March 13, 2025
4:00 pm
120 Snow Hall
Reception at 3:30 pm, 406 Snow
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.
Chenyang Xu is professor of mathematics at Princeton University since 2020. A leader in Algebraic Geometry with a focus on birational geometry, Professor Xu completed his PhD at Princeton in 2008 under the supervision of Professor János Kollár. Xu was a CLE Moore Instructor at MIT from 2008-2011, after which he took positions as an assistant professor at the University of Utah before joining Peking University as a research fellow and then a professor.
In 2016, he was announced as a winner of the ICTP Ramanujan Prize for that year, "in recognition of Xu's outstanding works in algebraic geometry, notably in the area of birational geometry, including works both on log canonical pairs and on Q-Fano varieties, and on the topology of singularities and their dual complexes."
He is one of five winners of the 2019 New Horizons Prize for Early-Career Achievement in Mathematics, associated with the Breakthrough Prize in Mathematics for his research in the minimal model program and applications to the moduli of algebraic varieties.
He was elected as a Fellow of the American Mathematical Society in the 2020 Class, for "contributions to algebraic geometry, in particular the minimal model program and the K-stability of Fano varieties". In 2021, he received the Cole Prize in Algebra from the AMS.
Xu has made fundamental contributions in algebraic geometry, particularly in the area of birational geometry and on the topology of singularities and their dual complexes.
More specifically, jointly with C. Hacon and J. McKernan, Xu has developed the theory on boundedness of log general type pairs. One major application of the theory is proving the finiteness of automorphisms of a general type variety, advancing Hurwitz's century old work on curves and G. Xiao’s work on surfaces in the early 80s. Another major accomplishment of the theory is the final resolution of Shokurov’s ACC conjecture. The theory has also extended the work of Deligne-Mumford on stable curves to all dimensions; In a joint work with Chi Li, Xu has established a procedure using the Minimal Model Program to study the K-stability of Fano varieties, reducing K-stability issues to problems on special test configurations; Xu has also proved, jointly with Hacon, the existence of pl-flips in three dimensions in characteristic p, with prime p>5, generalizing a result of S. Mori in characteristic 0; Jointly with Kollar, Xu has developed a theory of studying dual complexes using Minimal Model Program. More specifically, they investigated the dual complex of a log Calabi-Yau pair and proved the finiteness result of its fundamental group, solving a conjecture by Kontsevich-Soilberman for up to four dimensions.
Xu has developed an impressively wide range of innovative techniques to tackle a broad spectrum of geometric problems in algebraic geometry and beyond, solving a series of fundamental problems.