2025 Simons Collaboration on Moduli of Varieties Annual Meeting

The 2025 annual meeting for the Simons Collaboration on the Moduli of Varieties was held from Thursday, October 2nd, to Friday, October 3rd, 2025, at the Simons Foundation in New York. The conference focused on higher dimensional geometry and featured eight distinguished speakers: Ivan Cheltsov (University of Edinburgh), Kristin DeVleming (University of California, San Diego), Janos Kollár (Princeton University), James McKernan (University of California, San Diego), Mircea Mustaţă (University of Michigan, Ann Arbor), Alena Pirutka (New York University), Chenyang Xu (Princeton University), and Ziquan Zhuang (Johns Hopkins University). Notably, six of the eight speakers, Cheltsov, Kollár, McKernan, Mustaţă, Xu, and Zhuang, are collaboration principal investigators.

In addition to the speakers, approximately 100 mathematicians from leading research universities worldwide participated. The research level was exceptionally high, with extensive interaction among participants. Throughout coffee and lunch breaks, lively discussions between speakers and attendees continued beyond the formal talks.

The meeting opened with an outstanding talk by János Kollár titled “Abelian Fibrations.” He presented recent breakthrough results on sections of Jacobian fibrations (joint work with Giulia Saccà, another Principal Investigator of the grant) and higher direct images of dualizing sheaves (joint work with Sándor Kovács, who also attended the conference). In the first part, Kollár described new topological obstructions to extending sections of Lagrangian fibrations, exemplified by the universal Jacobian over lines in a linear system on a smooth projective surface with Picard rank 1. When the surface is a K3 surface, this corresponds to a Lagrangian fibration on a smooth hyperkähler manifold. Kollár and Saccà identified a topological obstruction contradicting recent results by Bogomolov, Kamenova, and Verbitsky. The second part focused on flat morphisms between varieties with rational singularities, proving that higher direct images of the structure sheaf are locally free — a far-reaching generalization of Kollár’s classical theorems from 1986. This was a highly insightful and well-received talk.

Kristin DeVleming (University of California, San Diego) delivered the second talk, “The Hassett–Keel Program in Genus 4.” She reported on joint results with Kenneth Ascher (a conference participant), Yuchen Liu (invited but unable to attend due to caregiving responsibilities), and Xiaowei Wang (a participant) regarding the modular interpretation of steps in the log minimal model program for the Deligne–Mumford moduli space of stable genus 4 curves. She began by reviewing earlier work by Sebastian Casalaina–Martin, Radu Laza, and Jarod Alper, who were also present. DeVleming then explained constructions involving the hyperelliptic flip via stackified Chow quotients, wall-crossing for moduli spaces of pairs in the contexts of K-stability and KSBA stability, and the recent moduli spaces of boundary polarized Calabi–Yau surface pairs. This was an energetic and engaging presentation.

After a scientifically rich lunch on Thursday, Alena Pirutka (New York University) gave the third talk, “On Rationality Over the Reals.” She discussed rationality of algebraic varieties over a field k, focusing on the subtle distinction between being rational over k versus geometrically rational. While rationality is well-understood for dimension <= 2, it remains open for geometrically rational three-dimensional varieties. Pirutka summarized joint work with Olivier Benoist (École Normale Supérieure, Paris) on the rationality of such varieties defined over the real numbers or more general real closed fields, particularly those admitting conic or quadric surface bundle structures with connected real loci and vanishing intermediate Jacobian obstructions. Using unramified cohomology and birational rigidity techniques, she presented both positive and negative results on rationality. The talk was notably well organized and insightful. Mircea Mustaţă presented the fourth talk, "Minimal Exponents of Hypersurface Singularities." Reporting on joint work with Qianyu Chen (postdoctoral researcher funded by the grant), Mustaţă introduced a novel description of the minimal exponent of hypersurface singularities via higher direct images of suitably twisted sheaves of logarithmic forms on log resolutions. This generalizes the classical description by Morihiko Saito in terms of microlocal multiplier ideal sheaves (also called higher multiplier ideal sheaves). This was a technically dense but highly informative talk. James McKernan, director of the grant, concluded Thursday's sessions with a talk on "Forgetful Functors." He recalled the 2023 work of Kollár, Lieblich, Olsson, and Sawin on Torelli-type theorems in the Zariski topology for varieties of dimension >= 2 over arbitrary fields. Their results showed that for proper normal varieties over uncountable algebraically closed fields of characteristic zero, the Zariski topological space determines the linear equivalence of divisors, and thus uniquely identifies the underlying scheme. McKernan presented a generalization for varieties defined over countable fields, taking surfaces over the algebraic closure of the rational numbers as a model case. The talk was rich in innovative ideas.

The second day began with Chenyang Xu’s talk “Properness of K-moduli,” reporting joint work with Harold Blum (participant), Yuchen Liu, and Ziquan Zhuang. K-stability has enabled the development of moduli theory for Fano varieties and log Fano pairs, producing the K-moduli space, a projective good moduli space parametrizing K-polystable log Fano pairs with fixed invariants. Properness of this moduli space, proven by Liu, Xu, and Zhuang in 2022, is a key and challenging property. Xu explained a novel approach based on the relative stability threshold of families over discrete valuation rings and divisorial valuations, which, by iterated replacement of special fibers, yields a new, more direct proof of properness relying solely on birational geometry. The talk was elegant and insightful.

Ivan Cheltsov gave the second talk, “Finite Simple Subgroups of Real Cremona Groups,” focused on classifying finite simple (non-abelian) subgroups of the real three-dimensional Cremona group (the group of birational self-maps of real three-dimensional space). He reviewed the classification of finite subgroups in the complex one- and two-dimensional Cremona groups and highlighted results by Yuri Prokhorov on the complex three-dimensional case, where seven finite simple non-abelian subgroups appear. Cheltsov then presented joint ongoing work with Antoine Pinardin (his former PhD student and conference participant) and Prokhorov, proving that every finite simple non-abelian subgroup of the real three-dimensional Cremona group is isomorphic either to the simple group of order 60 or that of order 360. Despite the group-theoretic nature of these results, the talk was highly geometric and accessible.

The conference concluded with Ziquan Zhuang’s talk “Boundedness in General Type MMP,” reporting on recent joint work with Jingjun Han (Fudan University) and Lu Qi (East China Normal University, participant). The Minimal Model Program (MMP) seeks to construct birational models of projective varieties with minimal complexity. A fundamental open problem is the termination of the MMP — conjectured to occur after finitely many steps. Consequently, invariants of varieties should be bounded throughout any MMP sequence. Zhuang showed that local volume properties imply that, in any general type MMP sequence, the minimal log discrepancy of singularities takes only finitely many values, and the fibers of all extremal contractions and flips form a bounded family. A key component is analyzing local volume behavior within the MMP framework. The presentation was engaging and clear.

The 2025 Simons Collaboration on Moduli of Varieties Annual Meeting was a great success. The talks attracted many leading mathematicians who actively engaged in discussions during coffee and lunch breaks. These exchanges are expected to foster new collaborative projects. The organizers were very satisfied with the quality and depth of each presentation and appreciated the creative and stimulating atmosphere throughout the meeting.

Ivan Cheltsov
University of Edinburgh

Finite Simple Subgroups of Real Cremona Groups

In this talk, we describe all finite simple subgroups of the groups of birational transformations of the real projective plane and the real projective space. This is a joint work with Antoine Pinardin (Basel) and Yuri Prokhorov (Moscow).
 

Kristin DeVleming
University of California, San Diego

The Hassett-Keel Program in Genus 4

Studying the birational models arising from the MMP on M_g, the moduli space of genus g curves, and determining a modular interpretation of each model is known as the Hassett-Keel program. The first few steps are well-understood yet the program remains quite incomplete in general for any genus g > 3. In this talk, we will complete the Hassett-Keel program in genus 4. This is joint work with Kenneth Ascher, Yuchen Liu, and Xiaowei Wang.
 

János Kollár
Princeton University

Abelian Fibrations
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János Kollár will discuss recent results on Abelian fibrations, including untwisting, lifting of sections, and the relative Picard scheme.
 

James McKernan
University of California, San Diego

Kähler Versus Projective

It is a very natural question to understand the difference between Kähler varieties and projective varieties. On the other hand, there has been a lot of recent progress on the bimeromorphic geometry of Kähler varieties. In particular, the minimal model program exists for some threefolds and fourfolds, and we have abundance for threefolds. We review this progress and its relation to the difference between Kähler and projective varieties.
 

Mircea Mustata
University of Michigan

Minimal Exponents of Hypersurface Singularities

Mircea Mustata will give an introduction to the minimal exponent, an invariant of hypersurface singularities that measures rational singularities, and then discuss some recent results in this area.
 

Alena Pirutka
New York University

On Rationality Over the Reals

In this talk, we will discuss rationality questions for some threefolds defined over the reals and more general real closed fields, following joint works with O. Benoist, J.-L. Colliot-Thélène, and F. Scavia.
 

Chenyang Xu
Princeton University

Properness of K-moduli

We present a new proof of the properness of K-moduli spaces. While our approach still relies on the higher-rank finite generation theorem, it avoids the use of Halpern-Leistner’s Θ-stratification theory. Instead, we develop a purely birational method, rooted in a relative framework for K-stability, which provides a more direct geometric proof of properness.

This is based on joint with Harold Blum, Yuchen Liu, and Ziquan Zhuang.
 

Ziquan Zhuang
John Hopkins University

Boundedness in General Type MMP

Ziquan Zhuang will explain how local volumes (an invariant related to K-stability) can be used to show that in any general type MMP, the minimal log discrepancy of singularities takes only finitely many values, and the fibers of all flipping contractions and flips fall into finitely many deformation types. An application is the effective termination of fivefold general type MMP.

This is based on joint work with Jingjun Han, Jihao Liu, and Lu Qi.