Titles and abstracts

Titles and Abstracts

Harold Blum (The University of Utah)

Title: Moduli of Fano varieties with complements

Abstract：While the theories of KSB 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 divisor, in which we consider all slc degenerations. In certain cases when X is a degeneration of P^2, this approach is successful. This is ongoing joint work with K. Ascher, D. Bejleri, K. DeVleming, G. Inchiostro, Y. Liu, X. Wang.

Paolo Cascini (Imperial College London)

Title: On the Minimal Model Program for foliations

Abstract：I will survive some recent results on the study of the birational geometry of foliations over complex projective varieties, focusing, in particular, on the case of algebraically integrable foliations.

Ivan Cheltsov (The University of Edinburgh)

Title: K-moduli of one-parameter families of smooth Fano threefolds

Abstract: I will report on a joint work (in progress) with Hamid Abban, Erroxe Etxabarri-Alberdi, Dongchen Jiao, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Elena Denisova, and Theo Papazachariou about the K-moduli of smooth Fano threefolds in deformation families 2-22, 2-24, 2-25, 3-12, 3-13, 4-13 (see https://www.fanography.info/ for the description of these families). These are all one-dimensional families of smooth Fano threefolds for which K-moduli exist. We know all smooth K-polystable Fano threefolds in these families. I will explain how to find their K-polystable singular limits.

Kristin deVleming (University of Massachusetts Amherst)

Title：A conjecture of Mori and families of plane curves

Abstract：Consider a smooth family of hypersurfaces of degree d in P^{n+1}. When is every smooth projective limit of this family also a hypersurface? While it is easy to construct examples of limits that are not hypersurfaces when the degree d is composite, Mori conjectured that, if d is prime and n>2, every smooth projective limit is indeed a hypersurface. However, there are counterexamples when n=1 or 2; for example, one can take a family of degree 5 plane curves and degenerate to a smooth hyperelliptic (non-planar) curve. In this talk, we will propose a

re-formulation of Mori's conjecture that explains the failure in low dimensions, provide results in dimension one, and discuss a general approach to the problem using moduli spaces of pairs. This is joint work with David Stapleton.

Osamu Fujino (Kyoto University)

Title: Vanishing theorems for projective morphisms between complex analytic spaces

Abstract: We explain some new vanishing theorems in a complex analytic setting.

We will use them for the study of the minimal model program for projective morphisms

between complex analytic spaces.

Angela Gibney (University of Pennsylvania)

Title: Vector bundles on the moduli space of curves from representations of VOAs

Abstract: Given any vertex operator algebra V, Zhu defined an associative algebra A(V), and showed that to any A(V)-module, one can associate an admissible V-module. This gives rise to a functor taking n-tuples of A(V)-modules to a sheaf of coinvariants (and its dual sheaf of conformal blocks) on the moduli space of stable n-pointed curves of genus g. If V is strongly rational (in which case A(V) is finite and semi-simple), much is known about these sheaves, including that they are coherent and satisfy a factorization property. Factorization ultimately allows one to show the sheaves are vector bundles with Chern classes in the tautological ring. In this talk I will describe a program in which we are aiming for analogous results after removing the assumption of rationality. As a first step, we replace the standard factorization formula with an inductive one that holds for sheaves defined by modules over any VOA of CFT-type. As an application, we show that if V is strongly finite, then sheaves of coinvariants and conformal blocks are coherent. This is a preliminary description of new and ongoing joint work with Krashen and Damiolini, extending work with Damiolini and Tarasca.

Antonella Grassi (Università di Bologna)

Title: Kodaira's birational classification of singular elliptic fibers on threefolds

Abstract: I will discuss a birational Kodaira's classifications for 3-dimensional elliptic fibrations and its applications. This work is in part motivated from questions from physics.

Shihoko Ishii (The University of Tokyo)

Title: Liftings of ideals in positive chatacteristic to characteristic 0

Abstract: We study a singularity of a pair, consisting of a smooth variety and a multi-ideal, defined over a field of positive characteristic. Singularities of a variety defined over a field of characteristic $0$ are extensively studied by making use of many good properties which are not available for the positive characteristic case. So we cannot expect the same results in positive characteristic by the same method as in characterisitc $0$. In my talk I will propose another way to prove the results which are known in characteristic $0$. The way is to construct a bridge between positive characteristic and characteristic $0$ by means of inversion of modulo $p$-reduction and discussion of arc spaces. I will introduce our expected conjecture and its consequences. Then, I will show my recent results towards the conjecture. Actually, for every ``multi-ideal" on a smooth variety over the base field of positive characteristic, there exists its lifting ``multi-fractional ideal" on a smooth variety over characteristic $0$ with the same mld (minimal log discrepancy). By this bridge, we can induce some results on mld for positive characteristic from those for characteristic $0$.

Lena Ji (The University of Michigan)

Title: The Noether–Lefschetz theorem in arbitrary characteristic

Abstract：The classical Noether–Lefschetz theorem over the complex numbers states that the restriction map on divisor class groups from P^3 to a very general surface of degree at least 4 is an isomorphism. In this talk, we will show a Noether–Lefschetz result for varieties over fields of arbitrary characteristic. The proof uses the relative Jacobian of a curve fibration, and it also works for singular varieties (for Weil divisors).

Chen Jiang (Fudan University)

Title: Algebraic reverse Khovanskii--Teissier inequality via Okounkov bodies

Abstract: I will discuss the algebraic reverse Khovanskii--Teissier inequality. Namely, let $A, B, C$ be nef divisors on a projective variety of dimension $n$, then for any integer $1\leq k\leq n-1$,

$$(B^k\cdot A^{n-k})\cdot (A^k\cdot C^{n-k})\geq \frac{k!(n-k)!}{n!}(A^n)\cdot (B^k\cdot C^{n-k}).$$

The same inequality in the analytic setting was obtained by Lehmann and Xiao for compact K\"ahler manifolds using the Calabi--Yau theorem, while our approach is purely algebraic using (multipoint) Okounkov bodies. This is a joint work with Zhiyuan Li.

Yujiro Kawamata (The University of Tokyo)

Title: Deformations over non-commutative base

Abstract: If one allows the base of the deformations to be non-commutative, then there are more deformations than usual deformations. The deformations over commutative base can sometimes be regarded as the first order approximation of more general higher order deformations. Though the formal theories of deformations are parallel and the extension to the non-commutative case is simple, some new phenomena and invariants appear. I will explain these by some examples.

János Kollár (Princeton University)

Title: Moduli of varieties

Abstract: TBA.

Jihao Liu (Northwestern University)

Title：On Shokurov’s ACC conjecture for mlds in dimension 3

Abstract: We discuss recent progress on Shokurov’s ACC conjecture for mlds in dimension 3. Partially joint works with Jingjun Han, Yujie Luo, and Liudan Xiao.

Yuchen Liu (Northwestern University)

Title: ACC for local volumes and boundedness of singularities

Abstract: Kawamata log terminal (klt) singularities form an important class of singularities due to its fundamental roles in MMP, K\”ahler-Einstein geometry, and K-stability. Recently, Chi Li invented a new invariant called the local volume of a klt singularity which encodes lots of interesting geometric and topological information. In this talk, we will explore the relation between local volumes and certain boundedness condition of singularities related to the existence of \epsilon-plt blow-ups. As a main result, we show that the set of local volumes of klt singularities is discrete away from zero (resp. satisfies ACC) if the coefficient set is finite (resp. satisfies DCC) and the ambient spaces are analytically bounded. This is based on joint work with Jingjun Han and Lu Qi.

James M^cKernan (University of California San Diego)

Title: TBA

Abstract: TBA.

Joaquín Moraga (Princeton University)

Title: Coregularity of Fano varieties

Abstract: In this talk, we will introduce the coregularity of Fano varieties.

This invariant measures how large of a dual complex can we find among log Calabi-Yau structures on a Fano variety. The coregularity relates to log canonical thresholds, existence of complements, and the index of log Calabi-Yau pairs. In this talk, we will discuss some recent results about this invariant and other future directions. The results of this talk are joint work with Fernando Figueroa, Stefano Filipazzi, Mirko Mauri, and Junyao Peng.

Yusuke Nakamura (The University of Tokyo)

Title: Inversion of adjunction for quotient singularities

Abstract: In this talk, we will discuss the minimal log discrepancies of quotient singularities. I will show that the PIA (precise inversion of adjunction) conjecture holds for quotient singularities. The main tool of this talk involves the theory of the arc space of a quotient singularity established by Denef and Loeser. I will also explain some technical difficulties when dealing with non-linear group actions. This is joint work with Kohsuke Shibata.

Jihun Park (Pohang University of Science and Technology)

Title: Sasaki-Einstein 5-manifolds

Abstract: A numerous number of closed simply connected Sasaki-Einstein manifolds, in particular 5-manifolds, have been found based on the method introduced by Shoshichi Kobayashi, and developed by Charles Boyer, Krzysztof Galicki, and János Kollár. In this talk, we briefly explain how to find closed simply connected 5-manifolds that allow Sasaki-Einstein metrics. By using methods in K-stability, we newly verify that several closed simply connected 5-manifolds allow Sasaki-Einstein structures. We then list closed simply connected 5-manifolds that are known so far to admit Sasaki-Einstein structures.

Yuri Prokhorov (Steklov Mathematical Institute)

Title: Singular Del Pezzo varieties

Abstract: A del Pezzo variety $X$ is a Fano variety whose anticanonical class has the form

\[-K_X=(n-1)A,\]

where $A$ is an ample line bundle and $n$ is the dimension of $X$. This is a higher dimensional analog notion of del Pezzo surfaces. I am going to discuss biregular and birational classifications of del Pezzo varieties admitting terminal singularities. The talk is based on a joint work with Alexander Kuznetsov (in preparation).

Miles Reid (University of Warwick)

Title: Godeaux surfaces in mixed chacteristic

Abstract: It is known that Godeaux's construction of surfaces with chi = 1, K^2 = 1 as the quotient of a quintic surface by an action of the cyclic group of order 5 can be modified to work in chacteristic 5, with any of the possible group schemes of order 5. These cases can all be put together into a single deformation family in mixed characteristic, and a similar construction also produces nonsingular Calabi-Yau 3-folds with polarisation of degree A^3 = 1 and Pic^0 containing any of the group schemes ZZ/5 or mu_5 or alpha_5. For more, see

https://homepages.warwick.ac.uk/~masda/TOp/.

The aim of the talk is to do the same for Godeaux surfaces and CY 3-folds with 3-torsion.

Giulia Saccà (Columbia University)

Title: Moduli spaces on K3 categories are Irreducible Symplectic Varieties

Abstract: Recent developments by Druel, Greb-Guenancia-Kebekus, Horing-Peternell have led to the formulation of a decomposition theorem for singular (klt) projective varieties with numerical trivial canonical class. Irreducible symplectic varieties are one the building blocks provided by this theorem, and the singular analogue of irreducible hyper-Kahler manifolds. In this talk I will show that moduli spaces of Bridgeland stable objects on the Kuznetsov component of a cubic fourfold with respect to a generic stability condition are always projective irreducible symplectic varieties. I will rely on the recent work of Bayer-Lahoz-Macri-Neuer-Perry-Stellari, which, ending a long series of

results by several authors, proved the analogue statement in the smooth case.

Claire Voisin (Le Centre national de la recherche scientifique)

Title: A topological characterization of hyper-Kähler fourfolds of Hilb2(K3) type

Abstract: There are two known deformations types of hyper-Kähler (HK) fourfolds, namely Hilb2(K3) (Beauville, Fujiki) and the generalized Kummer variety K_2(A) (Beauville). It is however still unknown whether there are other topological types or deformation types of HK fourfolds. Some strong restrictions on the Betti numbers of HK fourfolds are known by work of Beauville, S. Salamon, Verbitsky and Guan. In this talk, I will sketch the proof of the following:

Theorem. A hyper-Kähler fourfold X is a deformation of Hilb2(K3) if and only if it has two integral degree 2 cohomology classes satisfying the conditions l4=0, m4=0, l^2m2=2. In particular, a HK fourfold which is homeomorphic to Hilb2(K3) is a deformation of Hilb2(K3).

This is joint work with Debarre, Huybrechts and Macrì.

Ziquan Zhuang (Massachusetts Institute of Technology)

Title: Boundedness of singularities and minimal log discrepancies of Kollár components

Abstract: Several years ago, Chi Li introduced the local volume of a klt singularity in his work on K-stability. The local-global analogy between klt singularities and Fano varieties, together with recent study in K-stability lead to the conjecture that klt singularities whose local volumes are bounded away from zero are bounded up to special degeneration. In this talk, I will discuss some recent work on this conjecture through the minimal log discrepancies of Kollár components.

Vyacheslav Shokurov (Johns Hopkins University)

Title: Boundedness and finiteness

Abstract: Recollection and reflection on minimal models.