Algebraic Geometry in East Asia

Online Seminar

This is a joint effort of many algebraic geometers in East Asia. We aim to create a platform for algebraic geometers and students for further interaction and cooperation.

The seminar will take place on Friday at GMT 1:00 or GMT 7:00, unless otherwise specified, in order to accommodate most participants in East Asia.

ageaseminar  A@T  gmail.com

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Last updated 2020/11/23

Upcoming Talks:

2021/1/22 GMT 7:00-8:00

Sho Tanimoto (Kumamoto University)

Title: Classifying sections of del Pezzo fibrations

Abstract:

Mori invented a technique called as Bend and Break lemma which claims that if we deform a curve with fixed points, then it breaks into the union of several curves such that some of them are rational. This technique has wide applications ranging from rationally connectedness of smooth Fano varieties, Cone theorem for smooth projective varieties, to boundedness of smooth Fano varieties. However, a priori there is no control on breaking curves so in particular, an outcome of Bend and Break could be a singular point of the moduli space of rational curves. With Brian Lehmann, we propose Movable Bend and Break conjecture which claims that a free rational curve of enough high degree can degenerate to the union of two free rational curves in the moduli space of stable maps, and we confirm this conjecture for sections of del Pezzo fibrations over an arbitrary smooth projective curve. In this talk I will explain some of ideas of the proof of MBB for del Pezzo fibrations as well as its applications to Batyrev¡¦s conjecture and Geometric Mann¡¦s conjecture. This is joint work with Brian Lehmann.

2021/1/22 GMT 8:15-9:15

Fei Hu

Title: Some comparison problems on correspondences

Abstract:

Although the transcendental part of Weil's cohomology theory remains mysterious, one may try to understand it by investigating the pullback actions of morphisms, or more generally, correspondences, on the cohomology group and its algebraic part.

Inspired by a result of Esnault and Srinivas on automorphisms of surfaces as well as recent advances in complex dynamics, Truong raised a question on the comparison of two dynamical degrees, which are defined using pullback actions of dynamical correspondences on cycle class groups and cohomology groups, respectively. An affirmative answer to his question would surprisingly imply Weil¡¦s Riemann hypothesis.

In this talk, I propose more general comparison problems on the norms and spectral radii of the pullback actions of certain correspondences (which are more natural in some sense). I will talk about their connections with Truong¡¦s dynamical degree comparison and the standard conjectures. Under certain technical assumption, some partial results will be given. I will also discuss some applications to Abelian varieties and surfaces. This talk is based on joint work with Tuyen Truong.

2020/12/4 GMT 1:00-2:00

Jakub Witaszek (University of Michigan)

Title: On the four-dimensional Minimal Model Program for singularities and families in positive characteristic

Abstract:

I will discuss new developments on the four-dimensional Minimal Model Program in positive characteristic. This is based on a joint work with Christopher Hacon.

2020/12/4 GMT 2:15-3:15

Yu-Wei Fan (UC Berkeley)

Title: Stokes matrices, surfaces, and points on spheres

Abstract:

Moduli spaces of points on n-spheres carry natural actions of braid groups. For n=0,1, and 3, we prove that these symmetries extend to actions of mapping class groups of positive genus surfaces, through exceptional isomorphisms with certain moduli of local systems. This relies on the existence of group structure for spheres in these dimensions. We also apply the exceptional isomorphisms to the study of Stokes matrices and exceptional collections of triangulated categories. Joint work with Junho Peter Whang.

 Date Time Speaker Title 2020/12/04 GMT 1 Jakub Witaszek(Michigan) 2020/12/04 GMT 2:15 Yu-Wei Fan (UC Berkeley) 2020/12/18 GMT 7 Chen Jiang (Fudan) 2020/12/18 GMT 8:15 Ya Deng (IHES) 2021/01/08 GMT 1 Junliang Shen (MIT) 2021/01/08 GMT 2:15 Yohsuke Matsuzawa (Brown Univ.) 2021/01/22 GMT 7 Sho Tanimoto (Kumamoto University) 2021/01/22 GMT 8:15 Fei Hu (University of Oslo) 2021/02/05 GMT 7 Lei Zhang (USTC) 2021/02/19 GMT 7 Joonyeong Won (KIAS) 2021/02/19 GMT 7 Soheyla Feyzbakhsh (Imperial College)

More Confirmed Speakers:

Organizers: Yujiro Kawamata (Tokyo), Xiaotao Sun (Tianjin), JongHae Keum (KIAS, Seoul), Jungkai Chen (NTU, Taipei), Conan Nai Chung Leung (CUHK, Hong Kong), Phung Ho Hai (VAST, Hanoi), De Qi Zhang (NUS, Singapore), Yusuke Nakamura (Tokyo), Baohua Fu (CAS, Beijing), Kiryong Chung (Kyungpook National Univ., Daegu), Hsueh-Yung Lin (IPMU, Tokyo)

Archive:

2020/08/07  GMT 8:00-9:00

Caucher Birkar (University of Cambridge)

Title: Geometry and moduli of polarised varieties

Abstract:

In this talk I will discuss projective varieties polarised by ample divisors (or more generally nef and big divisors) and outline some recent results about the geometry and moduli spaces of such varieties.

2020/08/14 GMT 7:00-8:00

Yukinobu Toda (Kavli IPMU) (video)

Title: On d-critical birational geometry and categorical DT theories

Abstract:

In this talk, I will explain an idea of d-critical birational geometry, which deals with certain "virtual" birational maps among schemes with d-critical structures. One of the motivations of this new framework is to categorify wall-crossing formulas of Donaldson-Thomas invariants. I will propose an analogue of D/K equivalence conjecture in d-critical birational geometry, which should lead to a categorification of wall-crossing formulas of DT invariants.

The main result in this talk is to realize the above story for local surfaces. I will show the window theorem for categorical DT theories on local surfaces, which is used to categorify wall-crossing invariance of genus zero GV invariants, MNOP/PT correspondence, etc.

2020/08/14 GMT 8:15-9:15

Huai-Liang Chang (HKUST) (video)

Title: BCOV Feynman structure for Gromov Witten invariants

Abstract:

Gromov Witten invariants Fg encodes the numbers of genus g curves in Calabi Yau threefolds. They play a role in enumerative geometry and are not easy to be determined.

In 1993 Bershadsky, Cecotti, Ooguri, Vafa exhibited a hidden "Feynman structure" governing all Fg¡¦s at once. Their argument was via path integral while its counterpart in mathematics had been missing for decades.

In 2018, considering the moduli of a special kind of algebro geometric objects, "Mixed Spin P fields", is developed and provides the wanted "Feynman structure". In this talk we will see genuine ideas behind these features.

2020/08/28  GMT 7:00-8:00

Xun Yu (Tianjin University) (video)

Title: Automorphism groups of smooth hypersurfaces

Abstract:

I will discuss automorphism groups of smooth hypersurfaces in the projective space and explain an approach to classify automorphism groups of smooth quintic threefolds and smooth cubic threefolds. This talk is based on my joint works with Professor Keiji Oguiso and Li Wei.

2020/08/28 GMT 8:15-9:15

Tuyen Trung Truong (University of Oslo) (video)

Title: Rationality of quotients of Abelian varieties and computer algebra

Abstract:

This talk concerns the question of what variety of the form X/G, where X is an Abelian variety and G a finite subgroup of Aut(X), is rational. It is motivated by some interesting geometric and dynamical system questions. Most of the work so far concerns the case where X is of the form E^m where E is an elliptic curve, and G is a finite subgroup of Aut(E). I will review the current known results and approaches, and explain why it could be necessary to use computer algebra to resolve the question, and a brief discussion on works in this direction (including my ongoing joint work with Keiji Oguiso).

2020/09/11 GMT 2:15-3:15

Jeongseok Oh (KIAS) (video)

Title: Counting sheaves on Calabi-Yau 4-folds

Abstract:

We define a localised Euler class for isotropic sections, and isotropic cones, in SO(N) bundles. We use this to give an algebraic definition of Borisov-Joyce sheaf counting invariants on Calabi-Yau 4-folds. When a torus acts, we prove a localisation result. This talk is based on the joint work with Richard. P. Thomas.

2020/09/25  GMT 7:00-8:00

Jie Liu (Chinese Academy of Sciences) (video)

Title: Strictly nef subsheaves in tangent bundle

Abstract:

Since the seminal works of Mori and Siu-Yau on the solutions to Hartshorne conjecture and Frankel conjecture, it becomes apparent that the positivity of the tangent bundle of a complex projective manifold carries important geometric information. In this talk, we will discuss the structure of projective manifolds whose tangent bundle contains a locally free strictly nef subsheaf and present a new characterisation of projective spaces. This is a joint work with Wenhao Ou (AMSS) and Xiaokui Yang (YMSC).

2020/09/25  GMT 8:30-9:30

Junyan Cao (Université Côte d'Azur) (video)

Title: On the Ohsawa-Takegoshi extension theorem

Abstract:

Since it was established, the Ohsawa-Takegoshi extension theorem turned out to be a fundamental tool in complex geometry. We establish a new extension result for twisted canonical forms defined on a hypersurface with simple normal crossings of a projective manifold with a control on its L2 norme. It is a joint work with Mihai Păun.

2020/10/9 GMT 7:00-8:00

Frank Gounelas (Göttingen University) (video)

Title: Curves on K3 surfaces

Abstract:

I will survey the recent completion (joint with Chen-Liedtke) of the remaining cases of the conjecture that a projective K3 surface contains infinitely many rational curves. As a consequence of this along with the Bogomolov-Miyaoka-Yau inequality and the deformation theory of stable maps, I will explain (joint with Chen) how in characteristic zero one can deduce the existence of infinitely many curves of any geometric genus moving with maximal variation in moduli on a K3 surface. In particular this leads to an algebraic proof of a theorem of Kobayashi on vanishing of global symmetric differentials and applications to 0-cycles.

2020/10/9 GMT 8:15-9:15

Toshiyuki Katsura (University of Tokyo) (video)

Title: Counting Richelot isogenies of superspecial curves of genus 2

Abstract:

Recently, supersingular elliptic curve isogeny cryptography has been extended to the genus-2 case by using superspecial curves of genus 2 and their Richelot isogeny graphs. In view of this situation, we examine the structure of Richelot isogenies of superspecial curves of genus 2, and give a characterization of decomposed Richelot isogenies. We also give a concrete formula of the number of such decomposed Richelot isogenies up to isomorphism between superspecial principally polarized abelian surfaces. This is a joint work with Katsuyuki Takashima (Mitsubishi Electic Co.).

2020/10/23  GMT 7:00-8:00

Takahiro Shibata (National University of Singapore) (video)

Title: Invariant subvarieties with small dynamical degree

Abstract:

Given a self-morphism on an algebraic variety, we can consider various dynamical problems on it. Motivated by an arithmetic-dynamical problem, we consider invariant subvarieties whose first dynamical degree is less than that of the ambient variety. We give an estimate of the number of them in certain cases.

2020/10/23  GMT 8:15-9:15

Sheng Meng (KIAS) (video)

Title: Dynamical equivariant minimal model program

Abstract:

I will describe the minimal model program (MMP) in the study of complex dynamics and how MMP can be applied to many conjectures with dynamical or arithmetical flavours. Several open questions will also be proposed in this talk.

2020/11/06  GMT 1:00-2:00

Hong Duc Nguyen (Thang Long University) (video)

Title: Cohomology of contact loci

Abstract:

We construct a spectral sequence converging to the cohomology with compact support of the $m$-th contact locus of a complex polynomial.  The first page is explicitly described in terms of a log resolution and coincides with the first page of McLean's spectral sequence converging to the Floer cohomology of the $m$-th iterate of the monodromy, when the polynomial has an isolated singularity. Inspired by this connection we conjecture that the Floer cohomology of the $m$-th iterate of the monodromy of $f$ is isomorphic to the compactly supported cohomology of the $m$-th contact locus of $f$, and that this isomorphism comes from an isomorphism of McLean spectral sequence with ours.

2020/11/06  GMT 1:00-2:00

Qizheng Yin (Peking University) (video)

Title: The Chow ring of Hilb(K3) revisited

Abstract:

The Chow ring of hyper-Kähler varieties should enjoy similar properties as the Chow ring of abelian varieties. In particular, a Beauville type decomposition is believed (by Beauville himself) to exist for all hyper-Kähler varieties. In this talk, we discuss a general approach towards the Beauville type decomposition of the Chow ring. We carry it out explicitly for the Hilbert scheme of points of K3 surfaces, and prove the multiplicativity of the resulting decomposition. Joint work with Andrei Negut and Georg Oberdieck.

2020/11/20  GMT 7:00-8:00

Jinhyung Park (Sogang University) (video)

Title: A Castelnuovo-Mumford regularity bound for threefolds with mild singularities

Abstract:

The Eisenbud-Goto regularity conjecture says that the Castelnuovo-Mumford regularity of an embedded projective variety is bounded above by degree - codimension +1, but McCullough-Peeva recently constructed highly singular counterexamples to the conjecture. It is natural to make a precise distinction between mildly singular varieties satisfying the regularity conjecture and highly singular varieties not satisfying the regularity conjecture. In this talk, we consider the threefold case. We prove that every projective threefold with rational singularities has a nice regularity bound, which is slightly weaker than the conjectured bound, and we show that every normal projective threefold with Cohen-Macaulay Du Bois singularities in codimension two satisfies the regularity conjecture. The codimension two case is particularly interesting because one of the counterexamples to the regularity conjecture appears in this case. This is joint work with Wenbo Niu.

2020/11/20  GMT 7:00-8:00

Evgeny Shinder (University of Sheffield) (video)

Title: Semiorthogonal decompositions for singular varieties

Abstract:

I will explain a semiorthogonal decomposition for derived categories of singular projective varieties into derived categories of finite-dimensional algebras, due to Professor Kawamata, generalizing the concept of an exceptional collection in the smooth case. I will present known constructions of these for nodal curves (Burban), torsion-free toric surfaces (Karmazyn-Kuznetsov-Shinder) and two nodal threefolds (Kawamata). Finally, I will explain obstructions coming from the K_{-1} group, and how it translates to maximal nonfactoriality in the nodal threefold case. This is joint work with M. Kalck and N. Pavic.

2020/12/4 GMT 1:00-2:00

Jakub Witaszek (University of Michigan) (video)

Title: On the four-dimensional Minimal Model Program for singularities and families in positive characteristic

Abstract:

I will discuss new developments on the four-dimensional Minimal Model Program in positive characteristic. This is based on a joint work with Christopher Hacon.

2020/12/4 GMT 2:15-3:15

Yu-Wei Fan (UC Berkeley) (video)

Title: Stokes matrices, surfaces, and points on spheres

Abstract:

Moduli spaces of points on n-spheres carry natural actions of braid groups. For n=0,1, and 3, we prove that these symmetries extend to actions of mapping class groups of positive genus surfaces, through exceptional isomorphisms with certain moduli of local systems. This relies on the existence of group structure for spheres in these dimensions. We also apply the exceptional isomorphisms to the study of Stokes matrices and exceptional collections of triangulated categories. Joint work with Junho Peter Whang.

2020/12/18 GMT 7:00-8:00

Chen Jiang (Fudan University) (video)

Title: Positivity in hyperk\"{a}hler manifolds via Rozansky-Witten theory

Abstract:

For a hyperk\"{a}hler manifold $X$ of dimension $2n$, Huybrechts showed that there are constants $a_0, a_2, \dots, a_{2n}$ such that

$$\chi(L) =\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q_X(c_1(L))^{i}$$

for any line bundle $L$ on $X$, where $q_X$ is the Beauville-Bogomolov-Fujiki quadratic form of $X$. Here the polynomial

$\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q^{i}$ is called the Riemann--Roch polynomial of $X$.

In this talk, I will discuss recent progress on the positivity of coefficients of the Riemann--Roch polynomial and also positivity of Todd classes. Such positivity results follows from a Lefschetz-type decomposition of the root of Todd genus via the Rozansky¡XWitten theory, following the ideas of Hitchin, Sawon, and Nieper-Wißkirchen.

2020/12/18 GMT 8:15-9:15

Ya Deng (IHES) (video)

Title: Big Picard theorem for varieties admitting a variation of Hodge structures

Abstract:

In 1972, A. Borel proved generalized big Picard theorem for any hermitian locally symmetric variety $X$: any holomorphic map from the punctured disk to $X$ extends to a holomorphic map of the disk into any projective compactification of $X$. In particular any analytic map from a quasi-projective variety to $X$ is algebraic. Period domains, introduced by Griffiths in 1969, are classifying spaces for Hodge structures. They are transcendental generalizations of hermitian locally symmetric varieties. In this talk, I will present a generalized big Picard theorem for period domains, which extends the recent work by Bakker-Brunebarbe-Tsimerman.

2021/1/8 GMT 1:00-2:00

Junliang Shen (MIT) (video)

Title: Intersection cohomology of the moduli of of 1-dimensional sheaves and the moduli of Higgs bundles

Abstract:

In general, the topology of the moduli space of semistable sheaves on an algebraic variety relies heavily on the choice of the Euler characteristic of the sheaves. We show a striking phenomenon that, for the moduli of 1-dimensional semistable sheaves on a toric del Pezzo surface (e.g. P^2) or the moduli of semistable Higgs bundles with respect to a divisor of degree > 2g-2 on a curve, the intersection cohomology (together with the perverse and the Hodge filtrations) of the moduli space is independent of the choice of the Euler characteristic. This confirms a conjecture of Bousseau for P^2, and proves a conjecture of Toda in the case of certain local Calabi-Yau 3-folds. In the proof, a generalized version of Ngô's support theorem plays a crucial role. Based on joint with Davesh Maulik.

2021/1/8 GMT 2:15-3:15

Yohsuke Matsuzawa (Brown University) (video)

Title: Vojta's conjecture and arithmetic dynamics

Abstract:

I will discuss applications of Vojta's conjecture to some problems in arithmetic dynamics, concerning the growth of sizes of coordinates of orbits, greatest common divisors among coordinates, and prime factors of coordinates. These problems can be restated and generalized in terms of (local/global) height functions, and I proved estimates on asymptotic behavior of height functions along orbits assuming Vojta's conjecture. One of the key inputs is an asymptotic estimate of log canonical thresholds of (X, f^{-n}(Y)), where f : X->X is a self-morphism and Y is a closed subscheme of X.

As corollaries, I showed that Vojta's conjecture implies Dynamical Lang-Siegel conjecture for projective spaces (the sizes of coordinates grow in the same speed),and existence of primitive prime divisors in higher dimensional setting.

Sponsors: National Center for Theoretical Sciences