Monday October 24th to Friday October 28th 2022

Organisers:

Paolo Cascini, Ivan Cheltsov, James McKernan and Chenyang Xu

Speakers:

1. Valery Alexeev

2. Ekaterina Amerik

3. Dan Abramovich

4. Carolina Araujo

5. Cinzia Casagrande

6. Jungkai Chen

7. Lawrence Ein

8. Tommaso de Fernex

9. Kento Fujita

10. Christopher Hacon

11. Paul Hacking

12. Brendan Hassett

13. Jun-Muk Hwang

14. Mattias Jonsson

15. Yujiro Kawamata

16. Jorge Pereira

17. Alena Pirutka

18. Giulia Saccà

19. Karl Schwede

20. Yuri Tschinkel

Schedule

Monday, October 24th

9:30 - 10:30 Yuri Tschinkel

11:00 - 12:00 Brendan Hassett

1:00 - 2:00 Jun-Muk Hwang

2:30 - 3:30 Giulia Saccà

4:00 - 5:00 Poster Session

Tuesday, October 25th

9:30 - 10:30 Yujiro Kawamata

11:00 - 12:00 Christopher Hacon

1:00 - 2:00 Alena Pirutka

2:30 - 3:30 Dan Abramovich

4:00 - 5:00 Paul Hacking

Wednesday, October 26th

9:30 - 10:30 Cinzia Casagrande

11:00 - 12:00 Lawrence Ein

1:00 - 2:00 Ekaterina Amerik

Thursday, October 27th

9:30 - 10:30 Carolina Araujo

11:00 - 12:00 Tommaso de Fernex

1:30 - 2:30 Jungkai Chen

2:30 - 3:30 Kento Fujita

4:00 - 5:00 Karl Schwede

Friday, October 28th

9:30 - 10:30 Mattias Jonsson

11:00 - 12:00 Jorge Pereira

1:00 - 2:00 Valery Alexeev

Titles and 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.

Jungkai Chen - Toward the classification and factorization of threefold birational maps.

Abstract: In minimal model program, divisorial contractions, flips and flops are considered to be the elementary maps. In dimension three, divisorial contractions to points are well-understood and can be realized as weighted under certain embeddings. In this talk, we will present the recent developments toward the classification of threefold divisorial contractions to curves. We will provide ouline our proposed program and demonstrate some new examples. If time permitted, we will talk about their factorization as well.

The talk is based on the joint work in progress with Hsin-Ku Chen and Jheng-Jie Chen.

Tommaso de Fernex - Geometry of arc spaces.

Abstract: The arc space of a variety is an infinite dimensional scheme whose geometric structure captures, in a way that is not yet fully understood, certain features of the singularities of the variety. Focusing on its local rings and invariants of these rings such as embedding dimension and codimension, we explore the local structure of arc spaces. Arc spaces are defined as the inverse limit of jet schemes and are often studied via their truncation maps; here we will present an alternative approach using projections to infinite dimensional affine spaces and finiteness properties of arc fibers. The talk is based on joint work with Christopher Chiu and Roi Docampo.

Lawrence Ein - Meaures of irrationality of generic hypersurfaces and complete intersections

Abstract: We’ll discuss some recent work on the irrationality of generic hypersurfaces and complete intersections.

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.

Christopher Hacon - Recent progress in the Kahler minimal model program

Abstract: In this talk we will discuss some recent results and challenges encountered trying to extend the minimal model program to the context of Kahler varieties.

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.

Jorge Pereira - A global splitting theorem for holomorphic Poisson manifolds.

Abstract: Any germ of Poisson manifold splits as the product of a germ of symplectic manifold and of a germ of Poisson with vanishing Poisson bracket (Weinstein splitting).

I will discuss a global version of this splitting for holomorphic Poisson structures on compact Kähler manifolds admitting a compact leaf with finite

fundamental group. Based on joint work with Stéphane Druel, Brent Pym, and Frédéric Touzet.

Alena Pirutka - Variation of CH2 over number fields.

Abstract: Let X be a smooth projective variety over a number field. Following Bass conjecture, it is believed that the Chow groups CH^i(X) are of finite type. In particular, this would imply that their torsion subgroups are finite.

For cycles of codimension two, and for X satisfying H^2(X,O_X)=0, the finiteness of the torsion subgroup was established by Colliot-Thélène and Raskind. In this talk we will discuss uniform bounds for the torsion subgroup in the Chow group of cycles of codimension two for families of varieties over number fields.

This is joint work with F. Charles.

Giulia Saccà - TBA

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).