# MPS Conference on Higher Dimensional Geometry, October 2022

Date & Time

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

• Meeting Report

Yuri Tschinkel opened the conference by presenting his recent joint results with Brendan Hassett and Andrew Kresch on equivariant birational geometry. Then, Brendan Hassett presented his joint results with Yuri Tschinkel about rational points on derived equivalent K3-surfaces and related equivariant problem about a cyclic group action on derived equivalent K3 surfaces. After this, Jun-Muk Hwang gave a well-structured talk about his recent work on the space of lines covering a smooth hypersurface in the complex projective space. Giulia Saccà gave a last talk of the first day of the conference about antisymplectic involutions of projective hyperkahler manifolds. After Giulia’s talk, we had an excellent poster session with poster presentations by Anna Abasheva (Columbia University), Jennifer Li (Princeton), Lisa Marquand (Stonybrook), Lu Qi (Princeton), Chengxi Wang (UCLA) and Kaiqi Yang (NYU).

During the second day of the conference, we had talks by Yujiro Kawamata, Christopher Hacon, Alena Pirutka, Dan Abramovich and Paul Hacking. Kawamata spoke about results on deformation theory over non-commutative base. Hacon spoke about his recent joint results with his former PhD student Omprokash Das about the existence of minimal models for Kahler varieties. Pirutka spoke about her joint results with Francois Charles on the uniform bounds for the torsion part of the Chow group of codimension two cycles of algebraic varieties in flat families. Abramovich reported on the recent work of his current PhD students Veronica Arena and Stephen Obinna. Then Hacking reported on the work in progress with his current PhD student Cristian Rodriguez about mirror symmetry for (singular) Fano 3-folds.

During the third day of the conference Cinzia Casagrande, Lawrence Ein and Ekaterina Amerik spoke about Fano manifolds with Lefschetz defect three, degree of irrationality of generic complete intersections, algebraically coisotropic submanifolds of holomorphic symplectic manifolds, respectively. During the fourth day of the conference Carolina Araujo, Tommaso de Fernex, Jungkai Chen, Kento Fujita, Karl Schwede spoke about birational geometry of Calabi-Yau pairs, geometry of arc spaces, classification of birational maps between threefold, K-stability of smooth Fano threefolds, étale fundamental groups, respectively.

During the final day of the conference, we had three very good talks by Mattias Jonsson, Jorge Pereira and Valery Alexeev. Jonsson presented his recent joint results with Sebastien Boucksom on K-stability of polarized algebraic varieties, and gave a consistence overview of the state of art of this field of algebraic geometry. Then Jorge Pereira reported on his joint results with Stephane Druel, Brent Pym and Frederic Touzet about holomorphic Poisson manifolds. Finally, Valery Alexeev presented his recent joint results with Philip Engel about compactifications of moduli spaces of K3 surfaces with a nonsymplectic involution.

We received very positive feedback about the conference. It provided an opportunity for many people to interact with each other in person for the whole week, which will undoubtedly result in many new collaborations.

• Agenda

#### MONDAY, OCTOBER 24

 9:30 AM Yuri Tschinkel | Equivariant birational geometry 11:00 AM Brendan Hassett | Derived equivalence, rational points, and automorphisms of K3 surfaces 1:00 PM Jun-Muk Hwang | Natural distributions on the spaces of lines covering smooth hypersurfaces 2:30 PM Giulia Saccà 4:00 PM Poster Session

#### TUESDAY, OCTOBER 25

 9:30 AM Yujiro Kawamata | Deformations over non-commutative base 11:00 AM Christopher Hacon 1:00 PM Alena Pirutka 2:30 PM Dan Abramovich | The Chow ring of a weighted projective bundle and of a weighted blowup 4:00 PM Paul Hacking | Mirror symmetry for Q-Fano 3-folds

#### WEDNESDAY, OCTOBER 26

 9:30 AM Cinzia Casagrande | Fano manifolds with Lefschetz defect 3 11:00 AM Lawrence Ein 1:00 PM Katya Amerik | On algebraically coisotropic submanifolds

#### THURSDAY, OCTOBER 27

 9:30 AM Carolina Araujo | Birational geometry of Calabi-Yau pairs 11:00 AM Tommaso de Fernex 1:00 PM Jungkai Chen 2:30 PM Kento Fujita | The Calabi problem for Fano threefolds 4:00 PM Karl Schwede | Perfectoid signature and an application to étale fundamental groups

#### FRIDAY, OCTOBER 28

 9:30 AM Mattias Jonsson | Divisorial stability: openness and cscK metrics 11:00 AM Jorge Pereira 1:00 PM Valery Alexeev | Mirror symmetric compactifications of moduli spaces of K3 surfaces with a nonsymplectic involution
• Abstracts & Slides

Dan Abramovich
The Chow ring of a weighted projective bundle and of a weighted blowup

Abstract: This is a report on work of Brown PhD students Veronica Arena and Stephen Obinna.
The Chow groups of a blowup of a smooth variety along a smooth subvariety is described in Fulton’s book using Grothendieck’s “key formula”, involving the Chow groups of the blown up variety, the center of blowup, and the Chern classes of its normal bundle. If interested in weighted blowups, one expects everything to generalize directly. This is in hindsight correct, except that at every turn there is an interesting and delightful surprise, shedding light on the original formulas for usual blowups, especially when one wants to pin down the integral Chow ring of a stack theoretic weighted blowup.

Valery Alexeev
Mirror symmetric compactifications of moduli spaces of K3 surfaces with a nonsymplectic involution

Abstract: There are 75 moduli spaces F_S of K3 surfaces with a nonsymplectic involution. We give a detailed description of Kulikov models for each of them. In the 50 cases when the fixed locus of the involution has a component C of genus g>1, we identify normalizations of the KSBA compactifications of F_S, using the stable pairs (X,\epsilon C), with explicit semitoroidal compactifications of F_S. This is a joint work with Philip Engel.

Ekaterina Amerik
On algebraically coisotropic submanifolds

Abstract: This is a joint work with F. Campana. Recall that a submanifold $$X$$ in a holomorphic symplectic manifold $$M$$ is said to be coisotropic if the corank of the restriction of the holomorphic symplectic form $$s$$ is maximal possible, that is equal to the codimension of $$X$$. In particular a hypersurface is always coisotropic. The kernel of the restriction of $$s$$ defines a foliation on $$X$$; if it is a fibration, $$X$$ is said to be algebraically coisotropic. A few years ago we proved that a non-uniruled algebraically coisotropic hypersurface $$X\subset M$$ is a finite etale quotient of $$C\times Y\subset S\times Y$$, where $$C\subset S$$ is a curve in a holomorphic symplectic surface, and $$Y$$ is arbitrary holomorphic symplectic. We prove some partial results on the higher-codimensional analogue of this, with emphasis on the (easy) abelian case. The key point, like in our earlier work, is the isotriviality of the fibration.

Carolina Araujo
Birational geometry of Calabi-Yau pairs
View Slides (PDF)

Abstract: Consider the following problem, posed by Gizatullin: “Which automorphisms of a smooth quartic K3 surface in $$\mathbb{P}^3$$ are induced by Cremona transformations of the ambient space?” When $$S\subset \mathbb{P}^3$$ is a smooth quartic surface, the pair $$(\mathbb{P}^3,S)$$ is an example of a Calabi-Yau pair, that is, a mildly singular pair $$(X,D)$$ consisting of a normal projective variety $$X$$ and an effective Weil divisor $$D$$ on $$X$$ such that $$K_X+D\sim 0$$. In this talk, I will explain a general framework to study the birational geometry of Calabi-Yau pairs. This is a joint work with Alessio Corti and Alex Massarenti.

Cinzia Casagrande
Fano manifolds with Lefschetz defect 3

Abstract: We will talk about a structure result for some (smooth, complex) Fano varieties X, which depends on the Lefschetz defect delta(X), an invariant of X defined as follows. Consider a prime divisor D in X and the restriction r:H^2(X,R)->H^2(D,R). Then delta(X) is the maximal dimension of ker(r), where D varies among all prime divisors in X. If delta(X)>3, then X is isomorphic to a product SxT, where S is a surface. When delta(X)=3, X does not need to be a product, but we will see that it still has a very explicit structure. More precisely, there exists a smooth Fano variety T with dim T=dim X-2 such that X is obtained from T with two possible explicit constructions; in both cases there is a P^2-bundle Z over T such that X is the blow-up of Z along three pairwise disjoint smooth, irreducible, codimension 2 subvarieties. This structure theorem allows to complete the classification of Fano 4-folds with Lefschetz defect at least 3. This is a joint work with Eleonora Romano and Saverio Secci.

Kento Fujita
The Calabi problem for Fano threefolds

Abstract: There are 105 irreducible families of smooth Fano threefolds, which have been classified by Iskovskikh, Mori and Mukai. For each family, we determine whether its general member admits a Kaehler-Einstein metric or not.
This is a joint work with Carolina Araujo, Ana-Maria Castravet, Ivan Cheltsov, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Suess and Nivedita Viswanathan.

Paul Hacking
Mirror symmetry for Q-Fano 3-folds

Abstract: This is a report on work in progress with my student Cristian Rodriguez. The mirror of a Q-Fano 3-fold of Picard rank 1 is a rigid K3 fibration over A^1 such that the total space is log Calabi-Yau and some power of the monodromy at infinity is maximally unipotent. We will explain this assertion in terms of the Strominger–Yau–Zaslow and homological mirror symmetry conjectures, and describe the correspondence explicitly for hypersurfaces in weighted projective space. The singularities of the K3 fibration are related to the Kuznetsov decomposition of the derived category of the Q-Fano via homological mirror symmetry.

Brendan Hassett
Derived equivalence, rational points, and automorphisms of K3 surfaces
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Abstract: Given K3 surfaces that are derived equivalent over a field k, how are their k-rational points related? We consider this question over k=C((t)), especially for isotrivial families, where we show that the existence of rational points is a derived invariant. This program naturally leads to questions on cyclic group actions on K3 surfaces under various equivalence relations. (Joint with Tschinkel).

Jun-Muk Hwang
Natural distributions on the spaces of lines covering smooth hypersurfaces

Abstract: The space of minimal rational curves on a uniruled projective manifold has a natural distribution. The growth vector of this distribution is its simplest numerical invariant, but often not easy to determine. As an example, we consider the case of the space of lines covering a smooth hypersurface in the complex projective space. We discuss a joint work with Qifeng Li, where this growth vector is determined for a general hypersurface of dimension 5 and degree 4.

Mattias Jonsson
Divisorial stability: openness and cscK metrics
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Abstract: A version of the Yau–Tian–Donaldson conjecture states that a polarized complex manifold admits a constant scalar curvature Kähler (cscK) metric in the given cohomology class iff it is a stable in a suitable sense. Chi Li defined a stability notion using filtrations on the section ring, and proved that this notion implies the existence of a cscK metric. I will report on joint work with Boucksom, where we show that Li’s notion is equivalent to a notion that we call divisorial stability, and which is defined in terms of finite subsets of divisorial valuations. This notion has the advantage of being defined for arbitrary ample numerical classes, and we show that divisorial stability is an open condition on the ample cone.

Yujiro Kawamata
Deformations over non-commutative base

Abstract: We consider deformations over non-commutative base space instead of the usual commutative base. Then there are more deformations which give more information. NC deformation theory works for sheaves on varieties as well as varieties themselves. NC deformations of flopping curves on 3-folds considered by Donovan-Wemyss give Gopakumar-Vafa invariants. NC deformations on surfaces with quotient singularities give Hacking’s vector bundles under Koll\’ar-Shepherd-Barron’s Q-Gorenstein smoothing.

Karl Schwede
Perfectoid signature and an application to étale fundamental groups

Abstract: In characteristic p > 0 commutative algebra, the F-signature measures how close a strongly F-regular ring is from being non-singular.Here F-regular singularities are a characteristic p > 0 analog of klt singularities. In this talk, using the perfectoidization of Bhatt-Scholze, we will introduce a mixed characteristic analog of F-signature. As an application, we show it can be used to provide an explicit upper bound on the size of the étale fundamental group of the regular locus of a BCM-regular singularities (related to results of Xu, Braun, Carvajal-Rojas, Tucker and others in characteristic zero and characteristic p). BCM-regular singularities can be thought of as a mixed characteristic analog of klt and F-regular singularities. This is joint work with Hanlin Cai, Seungsu Lee, Linquan Ma and Kevin Tucker.

Yuri Tschinkel
Equivariant birational geometry

Abstract: I will present some new results and constructions in higher-dimensional equivariant birational geometry (joint with B. Hassett and A. Kresch).

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