# 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

• Agenda

#### MONDAY, OCTOBER 24

 8:30 AM CHECK-IN & BREAKFAST 9:30 AM Yuri Tschinkel | Equivariant birational geometry 10:30 AM BREAK 11:00 AM Brendan Hassett | Derived equivalence, rational points, and automorphisms of K3 surfaces 12:00 PM LUNCH 1:00 PM Jun-Muk Hwang | Natural distributions on the spaces of lines covering smooth hypersurfaces 2:00 PM BREAK 2:30 PM Giulia Saccà 3:30 PM BREAK 4:00 PM Poster Session 5:00 PM DAY ONE CONCLUDES

#### TUESDAY, OCTOBER 25

 8:30 AM CHECK-IN & BREAKFAST 9:30 AM Yujiro Kawamata | Deformations over non-commutative base 10:30 AM BREAK 11:00 AM Christopher Hacon 12:00 PM LUNCH 1:00 PM Alena Pirutka 2:00 PM BREAK 2:30 PM Dan Abramovich | The Chow ring of a weighted projective bundle and of a weighted blowup 3:30 PM BREAK 4:00 PM Paul Hacking | Mirror symmetry for Q-Fano 3-folds 5:00 PM DAY TWO CONCLUDES

#### WEDNESDAY, OCTOBER 26

 8:30 AM CHECK-IN & BREAKFAST 9:30 AM Cinzia Casagrande | Fano manifolds with Lefschetz defect 3 10:30 AM BREAK 11:00 AM Lawrence Ein 12:00 PM LUNCH 1:00 PM Katya Amerik | On algebraically coisotropic submanifolds 2:00 PM DAY THREE CONCLUDES

#### THURSDAY, OCTOBER 27

 8:30 AM CHECK-IN & BREAKFAST 9:30 AM Carolina Araujo | Birational geometry of Calabi-Yau pairs 10:30 AM BREAK 11:00 AM Tommaso de Fernex 12:00 PM LUNCH 1:00 PM Jungkai Chen 2:00 PM BREAK 2:30 PM Kento Fujita | The Calabi problem for Fano threefolds 3:30 PM BREAK 4:00 PM Karl Schwede | Perfectoid signature and an application to étale fundamental groups 5:00 PM DAY FOUR CONCLUDES

#### FRIDAY, OCTOBER 28

 8:30 AM CHECK-IN & BREAKFAST 9:30 AM Mattias Jonsson | Divisorial stability: openness and cscK metrics 10:30 AM BREAK 11:00 AM Jorge Pereira 12:00 PM LUNCH 1:00 PM Valery Alexeev | Mirror symmetric compactifications of moduli spaces of K3 surfaces with a nonsymplectic involution 2:00 PM MEETING CONCLUDES
• Abstracts

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

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

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

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).

• Accommodations

Group A – PIs & Speakers
Business-class airfare for flights over 5 hours
Hotel accommodations for up to 3 nights
Reimbursement of Local Expenses

Group B – Out-of-town Participants
Economy Airfare
Hotel Accommodations for up to 3 nights
Reimbursement of Local Expenses

Group C – Local Participants
No funding provided besides hosted conference meals.

Group D – Remote Participants
A Zoom link will be provided.

Personal Car
For participants in Groups A & B driving to Manhattan, the James NoMad hotel offers valet parking. Please note there are no in-and-out privileges when using the hotel’s garage; therefore, participants are encouraged to walk or take public transportation to the Simons Foundation.

• Hotel

Participants in Groups A & B who require accommodations are hosted by the foundation for a maximum of six nights at The James NoMad Hotel. Any additional nights are at the attendee’s own expense. To arrange accommodations, please register at the link above.

22 E 29th St
New York, NY 10016
(between 28th and 29th Streets)

• COVID-19 Policy

ALL in-person meeting attendees must be vaccinated against the COVID-19 virus with a World Health Organization approved vaccine, be beyond the 14-day inoculation period of their final dose, and provide proof of vaccination upon arrival to the conference. Acceptable vaccines can be found at the bottom of this page on the WHO’s site.

• Reimbursement and Travel Policy

Any expenses not directly paid for by the Simons Foundation are subject to reimbursement based on the foundation’s travel policy. An email will be sent within a week following the conclusion of the meeting with further instructions on submitting your expenses via the foundation’s web-based expense reimbursement platform.
Receipts are required for any expenses over \$25 USD and are due within 30 days of the conclusion of the meeting. Should you have any questions, please contact Emily Klein.

• Contacts

Registration and Travel Assistance
Ovation Travel Group
sfnevents@ovationtravel.com
(917) 408-8384 (24-Hours)
www.ovationtravel.com

Meeting Questions and Assistance
Emily Klein
Event Coordinator, MPS, Simons Foundation
eklein@simonsfoundation.org
(646) 751-1262

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