# MPS Conference on Higher Dimensional Geometry, August 22-26, 2022

Date & Time

Organizers:
Paolo Cascini, Imperial College
Ivan Cheltsov, University of Edinburgh
James McKernan, USCD
Chenyang Xu, Princeton University

Speakers:
Harold Blum, Stony Brook University
Lukas Braun, University of Freiburg
Giulio Codogni, University of Rome Tor Vergata
János Kollár, Princeton University
Eveline Legendre, Toulouse Mathematics Institute
Chi Li, Rutgers University
Jennifer Li, Princeton University
Yuchen Liu, Northwestern University
Nivedita Niswanathan, Loughborough University
Calum Spicer, King’s College London
Roberto Svaldi, EPFL
Chenyang Xu, Princeton University
Ziquan Zhuang, MIT

• Agenda

#### MONDAY, AUGUST 22nd

 9:30 AM János Kollár | Moduli of varieties of general type (1/3) 11:00 AM Chenyang Xu | Survey of local stability theory 1:30 PM Giulio Codogni | Ample cone of moduli spaces and Harder-Narsimhan filtration 2:30 PM Radu Laza| Higher Du Bois and higher rational singularities 4:00 PM Nivedita Niswanathan | On K-stability of some singular del Pezzo surfaces

#### TUESDAY, AUGUST 23rd

 9:30 AM János Kollár | Moduli of varieties of general type (2/3) 11:00 AM Yuechen Liu | Recent developments in K-stability (1/3) 1:30 PM Chi Li | Polarized Hodge structures for Clemens manifolds 2:30 PM Lukas Braun | Reductive quotients of klt varieties 4:00 PM Short Talks

#### WEDNESDAY, AUGUST 24th

 9:30 AM Roberto Svaldi | Minimal model program for foliated surfaces: a different approach 11:00 AM Calum Spicer | Minimal model program for foliated surfaces: a different approach

#### THURSDAY, AUGUST 25th

 9:30 AM János Kollár| Moduli of varieties of general type (3/3) 11:00 AM Yuchen Liu | Recent developments in K-stability (2/3) 1:30 PM Ziquan Zhuang | Stable degenerations of klt singularities 2:30 PM Jennifer Li | A cone conjecture for log Calabi-Yau surfaces 4:00 PM Problem Session

#### FRIDAY, AUGUST 26th

 9:30 AM Yuchen Liu | Recent developments in K-stability (3/3) 11:00 AM Harold Blum | Moduli of Fano varieties with complements
• Abstracts

Harold Blum
Moduli of Fano varieties with complements

While the theories of KSBA-stability and K-stability have been successful in constructing compact moduli spaces of canonically polarized varieties and Fano varieties, respectively, the case of K-trivial varieties remains less well understood. I will discuss a new approach to this problem in the case of K-trivial pairs (X,D), where X is a Fano variety and D is an anticanonical Q-divisor, in which we consider all slc degenerations. In the case when X is a degeneration of P^2, this approach is successful. This is joint work with K. Ascher, D. Bejleri, K. DeVleming, G. Inchiostro, Y. Liu, X. Wang.

Lukas Braun
Reductive quotients of klt varieties

In this talk, I will explain the proof of the recent result, obtained together with Daniel Greb, Kevin Langlois, and Joaquin Moraga,that reductive quotients of klt type varieties are of klt type. If time permits, I will also discuss several applications of our result, e.g. on quotients of Fano type varieties, good moduli spaces, and collapsing of homogeneous bundles.

Giulio Codogni
Ample cone of moduli spaces and Harder-Narsimhan filtration

I will present some results about the Harder-Narasimhan filtration of vector bundles associated to one-parameter families of KSB-stable and K-stable varieties. As main application, I will give a quantitative description of a portion of the ample cone of KSB moduli spaces. The talk is based on a joint work and a work in progress with L. Tasin and F. Viviani.

János Kollár
Moduli of varieties of general type

We discuss the moduli theory of varieties of general type, focusing on new results and open problems.

Higher Du Bois and higher rational singularities

Two fundamental classes of singularities are the rational singularities, and the Du Bois singularities. Recently, M. Mustață, M. Popa, M. Saito and their collaborators have introduced a natural generalization of the Du Bois singularities, the higher Du Bois singularities. In this talk, I will discuss the companion notion of higher rational singularities and establish some basic properties for both of these classes of singularities. I will conclude by explaining some important applications of these types of singularities to degenerations and deformations of algebraic varieties. This is joint work with R. Friedman.

Chi Li
Polarized Hodge structures for Clemens manifolds

A conifold transition is a geometric transformation that is used to connect different moduli spaces of Calabi-Yau threefolds. Let X be a projective Calabi-Yau threefold. A conifold transition first contracts X along disjoint rational curves with normal bundles of type (-1,-1), and then smooths the resulting singular complex space Z to a new compact complex manifold Y. Such Y is called a Clemens manifold and can be non-Kahler. We prove that any small smoothing Y of Z satisfies ddbar-lemma. We also show that the resulting pure Hodge structure of weight three is polarized by the cup product. This answers some questions of R. Friedman. The proof uses the theory of limiting mixed Hodge structures and basic linear algebra.

Jennifer Li
A cone conjecture for log Calabi-Yau surfaces

In 1993, Morrison conjectured that the automorphism group of a Calabi-Yau 3-fold acts on its nef cone with a rational polyhedral fundamental domain. In this talk, I will discuss a version of this conjecture for log Calabi-Yau surfaces. In particular, for a generic log Calabi-Yau surface with singular boundary, the monodromy group acts on the nef effective cone with a rational polyhedral fundamental domain. In addition, the automorphism group of the unique surface with a split mixed Hodge structure in each deformation type acts on the nef effective cone with a rational polyhedral fundamental domain.

Yuchen Liu (3 lectures)
Recent developments in K-stability

K-stability was introduced by Tian and Donaldson to characterize the solution of the Kahler-Einstein problem on Fano varieties. In the past decade, a purely algebraic geometric study of K-stability has prospered, based on the birational classification theory of varieties centered around the minimal model program. As one of the most important consequences, the K-moduli theory for Fano varieties has been established using purely algebraic methods. In this lecture series, we will give an overview of the recent developments in the algebraic theory of K-stability. In the first part, we will discuss Fujita-Li’s valuative criterion. In the second part, we will discuss the construction of K-moduli spaces from purely algebraic methods. In the third part, we will discuss explicit K-moduli spaces of log Fano varieties and their wall-crossing.

Roberto Svaldi, Calum Spicer
Minimal model program for foliated surfaces: a different approach.

The birational classification of foliated surface is pretty much complete, thanks to the work of Brunella, Mendes, McQuillan. In recent joint work we explore a new approach to studying the singularities and the minimal model program for foliated surfaces inspired by the work of Pereira-Svaldi. The basic idea is rather simple: rather than just considering the canonical divisor $$K{\mathcal F}$$ of a foliation $$\mathcal F$$ (the classic analogue of the canonical divisor in the foliated world) together with the linear system $$|mK\mathcal{F}|$$, $$m \in \mathbb N$$, one can consider perturbed divisors $$K_{\mathcal F}+\epsilon K_X$$, $$\epsilon>0$$ and linear systems of the form $$|nKX + mK{\mathcal F}|$$, $$n,m >0$$. Those perturbed divisors (and the related linear systems) encode a lot of the positivity features that classically the canonical divisor (and pluricanonical forms) on a projective variety display and that do not necessarily hold for $$K_\mathcal{F}$$ alone. The price to pay for working with these divisors is to define a new category of singularities for foliated varieties. We will introduce these new singularities and try to explain how they behave via examples in the 1st talk. The 2nd talk will instead be devoted to discussing new results and applications for this class of divisors, discussing new results on the boundedness of surface foliations, and applications of these results to some classical problems in foliation theory, for instance, on the problem of bounding the degree of orbits of vector fields in the complex plane.

Nivedita Niswanathan
On K-stability of some singular del Pezzo surfaces

There has been a lot of development recently in understanding the existence of Kahler-Einstein metrics on Fano manifolds due to the Yau-Tian-Donaldson conjecture, which gives us a way of looking at this problem in terms of the notion of K-stability. In particular, this problem is solved in totality for smooth del Pezzo surfaces by Tian. For del Pezzo surfaces with quotient singularities, there are partial results. In this talk, we will consider singular del Pezzo surfaces which are quasi-smooth, well-formed hypersurfaces in weighted projective space, and understand what we can say about their K-stability. This is ongoing joint work with In-Kyun Kim and Joonyeong Won.

Chenyang Xu
Survey of local stability theory

A few years ago, Chi Li introduced the concept of normalized volume function for any klt singularity and proposed to study the minimizing valuation. Together with later work of Li-Xu, this initiated the foundation of local K-stability theory, centered around the Stable Degeneration Conjecture. In the past few years, a lot of interests were attracted to the Stable Degeneration Conjecture, and it was completely settled recently (for the last step, see Ziquan Zhuang’s lecture in this conference). In this talk, I will survey the local stability theory. The lecture can be regarded as the pretalk for Zhuang’s lecture.

Ziquan Zhuang
Stable degenerations of klt singularities

Several years ago, Chi Li introduced the normalized volumes of valuations in his work on K-stability. The stable degeneration conjecture, due to Li and Xu, predicts a local stability theory of klt singularities through the minimizers of the the normalized volume functions. I’ll talk about the recent solution of this conjecture, focusing on the finite generation property of valuations. Based on joint work with Chenyang Xu.

#### SHORT TALKS

Elena Denisova
On K-stability of P3 blown up along the disjoint union of a twisted cubic curve and a line.

We discuss how to find all K-polystable smooth Fano threefolds that can be obtained as blowup of P^3 along the disjoint union of a twisted cubic curve and a line.

Jacob Keller
K-stability of moduli of bundles on curves.

Moduli spaces of vector bundles on curves with fixed determinants are Fano varieties that play an important role in algebraic geometry as well as other areas of mathematics and physics. This talk will outline an approach to proving these Fano varieties are K-stable, using toric degenerations and the Luna slice theorem for stacks. The K-stability of these spaces has the consequence that there exist components of the K-polystable moduli space which are birational to Mg and are therefore of general type.

Yujie Luo
On Shokurov’s conjecture on $$(\epsilon,n)$$-complements for rationally connected threefolds

We show the existence of $$(\epsilon,n)$$-complements for rationally connected Calabi-Yau threefolds. As a corollary, we show that the set of rationally connected threefold $$X$$ which has an $$(\epsilon,\mathbb{R}r)$$-complement is bounded in codimension one. This is joint work with Guodu Chen and Jingjun Han.

Lisa Marquand
Symplectic birational involutions of manifolds of OG10 type.

A big open problem surrounding hyperkähler manifolds is the construction of new examples: currently there are only 4 known deformation types. One approach is to consider finite symplectic group actions of a known hyperkähler manifold, and study the symplectic resolution (if it exists) of the fixed locus. In this talk, we will obtain a partial classification of birational symplectic involutions of manifolds of OG10 type. We do this from two vantage points: firstly relating to automorphisms of the Leech lattice, and secondly we relate to automorphisms of cubic fourfolds. More specifically, we compute the algebraic sublattice of the middle cohomology of a cubic fourfold with a certain involution explicitly. This has several consequences with regard to cubic fourfolds; namely we exhibit a 10-dimensional family of rational cubic fourfolds.

Erik Paemurru
Parameter space of divisorial contractions

In the literature, local analytic types of 3-dimensional divisorial contractions with centre a point have been almost classified. In some cases, a local analytic divisorial contraction can correspond to uncountably many global algebraic divisorial contractions. In this case, we show how to construct a parameter space of global algebraic divisorial contractions.

Theo Papazachariou
K-moduli for log Fano complete intersections

An important category of geometric objects in algebraic geometry is smooth Fano varieties. These have been classified in 1, 10 and 105 families in dimensions 1, 2 and 3 respectively, while in higher dimensions the number of Fano families is yet unknown. An important problem is compactifying these families into moduli spaces via K-stability. In this talk, I will describe the compactification of the family of Fano threefolds, which is obtained by blowing up the projective space along a complete intersection of two quadrics which is an elliptic curve, into a K-moduli space using Geometric Invariant Theory (GIT). A more interesting setting occurs in the case of pairs of varieties and a hyperplane section where the K-moduli compactifications tessellate depending on a parameter. In this case it has been shown recently that the K-moduli decompose into a wall-chamber decomposition depending on a parameter, but wall-crossing phenomena are still difficult to describe explicitly. Using GIT, I will describe an explicit example of wall-crossing in the K-moduli spaces, where both variety and divisor differ in the deformation families before and after the wall, given by log pairs of Fano surfaces of degree 4 and a hyperplane section.

Lu Qi
Convexity of volumes of filtrations on local rings

I will talk about a convexity property of volumes of filtrations in the local setting, which has applications in the theory of volumes of valuations and K-stability. Moreover, we find a construction which can be used to generalize some classical results in commutative algebra. This is based on joint work with Harold Blum and Yuchen Liu.

Arman Sarikyan
On the rationality of Fano-Enriques threefolds with terminal cyclic quotient singularities.

A three-dimensional non-Gorenstein Fano variety with at most canonical singularities is called a Fano-Enriques threefold if it contains an ample Cartier divisor that is an Enriques surface with at most canonical singularities. There is no complete classification of Fano-Enriques threefolds yet. However, L. Bayle has classified Fano-Enriques threefolds with terminal cyclic quotient singularities in terms of their canonical coverings, which are smooth Fano threefolds in this case. The rationality of Fano-Enriques threefolds is an open classical problem that goes back to the works of G. Fano and F. Enriques. In this talk we will discuss the rationality of Fano-Enriques threefolds with terminal cyclic quotient singularities.