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. 

 

To receive the announcement and the Zoom password, please send an empty mail with the title "Subscription" to the following address:

ageaseminar  A@T  gmail.com

 

This is a mirror site of

https://sites.google.com/ncts.ntu.edu.tw/agea-seminar

Last updated 2021/10/12


Upcoming Talks:

TBA

 

Date

Time

Speaker

Title

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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:

2021/10/08 GMT 7:00-8:00

Speaker: Christian Schnell (Stony Brook University) (video)

Title: Finiteness for self-dual classes in variations of Hodge structure

Abstract:

I will talk about a new finiteness theorem for variations of Hodge structure. It is a generalization of the Cattani-Deligne-Kaplan theorem from Hodge classes to so-called self-dual (and anti-self-dual) classes. For example, among integral cohomology classes of degree 4, those of type (4,0) + (2,2) + (0,4) are self-dual, and those of type (3,1) + (1,3) are anti-self-dual. The result is suggested by considerations in theoretical physics, and the proof uses o-minimality and the definability of period mappings. This is joint work with Benjamin Bakker, Thomas Grimm, and Jacob Tsimerman.

2021/10/08 GMT 8:15-9:15

Speaker: Nguyen-Bac Dang (Université Paris-Saclay) (video)

Title: Spectral interpretations of dynamical degrees

Abstract:

This talk is based on a joint work with Charles Favre. I will explain how one can control the degree of the iterates of rational maps in arbitrary dimension by applying method from functional analysis. Namely, we endow some particular norms on the space of b-divisors and on the spaces of b-classes and study the eigenvalues of the pullback operator induced by a rational map.

 

2021/9/24 GMT 7:00-8:00

Speaker: Kenta Hashizume (University of Tokyo) (video)

Title: Adjunction and inversion of adjunction

Abstract:

Finding a relation between singularities of a variety and singularities of subvarietes is a natural problem. An answer to the problem, called adjunction and inversion of adjunction for log canonical pairs, plays a critical role in the recent developments of the birational geometry. In this talk, I will introduce a generalization of the result, that is, adjunction and inversion of adjunction for normal pairs. This is a joint work with Osamu Fujino.

 

2021/9/24 GMT 8:15-9:15

Speaker: Takuzo Okada (Saga University) (video)

Title: Birational geometry of sextic double solids with cA points

Abstract:

A sextic double solid is a Fano 3-fold which is a double cover of the projective 3-space branched along a sextic surface. Iskovskikh proved that a smooth sextic double solid is birationally superrigid, that is, it does not admit a non-biregular birational map to a Mori fiber space. Later on Cheltsov and Park showed that the same conclusion holds for sextic double solids with ordinary double points. In this talk I will explain birational (non-)superrigidity of sextic double solids with cA points. This talk is based on a joint work with Krylov and Paemurru.

 

2021/09/10 GMT 1:00-2:00

Speaker: Kyoung-Seog Lee (Miami University) (video)

Title: Derived categories and motives of moduli spaces of vector bundles on curves

Abstract: Derived categories and motives are important invariants of algebraic varieties invented by Grothendieck and his collaborators around 1960s. In 2005, Orlov conjectured that they will be closely related and now there are several evidences supporting his conjecture. On the other hand, moduli spaces of vector bundles on curves provide attractive and important examples of algebraic varieties and there have been intensive works studying them. In this talk, I will discuss derived categories and motives of moduli spaces of vector bundles on curves. This talk is based on several joint works with I. Biswas, T. Gomez, H.-B. Moon and M. S. Narasimhan

 

2021/09/10 GMT 2:15-3:15

Speaker: Insong Choe (Konkuk University) (video)

Title: Symplectic and orthogonal Hecke curves

Abstract: A Hecke curve is a rational curve on the moduli space $SU_C(r,d)$ of vector bundles over an algebraic curve, constructed by using the Hecke transformation. The Hecke curves played an important role in Jun-Muk Hwang's works on the geometry of $SU_C(r,d)$. Later, Xiaotao Sun proved that they have the minimal degree among the rational curves passing through a general point. We construct rational curves on the moduli spaces of symplectic and orthogonal bundles by using symplecitic/orthogonal version of Hecke transformation. It turns out that the symplectic Hecke curves are special kind of Hecke curves, while the orthogonal Hecke curves have degree $2d$, where $d$ is the degree of Hecke curves. Also we show that those curves have the minimal degree among the rational curves passing through a general point. This is a joint work with Kiryong Chung and Sanghyeon Lee.

 

2021/09/03 GMT 1:00-2:00

Speaker: Jingjun Han (Johns Hopkins University) (video)

Title: Shokurov's conjecture on conic bundles with canonical singularities

Abstract: A conic bundle is a contraction $X\to Z$ between normal varieties of relative dimension $1$ such that the anit-canonical divisor is relatively ample. In this talk, I will prove a conjecture of Shokurov which predicts that, if $X\to Z$ is a conic bundle such that $X$ has canonical singularities, then base variety $Z$ is always $\frac{1}{2}$-lc, and the multiplicities of the fibers over codimension $1$ points are bounded from above by $2$. Both values $\frac{1}{2}$ and $2$ are sharp. This is a joint work with Chen Jiang and Yujie Luo.

 

2021/09/03 GMT 2:15-3:15

Speaker: Jia Jia (National University of Singapore) (video)

Title: Surjective Endomorphisms of Affine and Projective Surfaces.

Abstract: In this talk, we will give structure theorems of finite surjective endomorphisms of smooth affine surfaces and normal projective surfaces. Combining with some local dynamics and known results, we will talk about their applications to Zariski Dense Orbit and Kawaguchi-Silverman Conjectures. These are joint work with Takahiro Shibata, Junyi Xie and De-Qi Zhang.

 

2021/08/13  GMT 7:00-8:00

Speaker: Jihao Liu (University of Utah) (video)

Title: Minimal model program for generalized lc pairs

Abstract: The theory of generalized pairs was introduced by C. Birkar and D.-Q. Zhang in order to tackle the effective Iitaka fibration conjecture, and has proven to be a powerful tool in birational geometry. It has recently become apparent that the minimal model program for generalized pairs is closely related to the minimal model program for usual pairs and varieties. A folklore conjecture proposed by J. Han and Z. Li and recently re-emphasized by Birkar asks whether we can always run the minimal model program for generalized pairs with at worst generalized lc singularities. In this talk, we will confirm this conjecture by proving the cone theorem, contraction theorem, and the existence of flips for generalized lc pairs. As an immediate consequence, we will complete the minimal model program for generalized lc pairs in dimension <=3 and the pseudo-effective case in dimension 4. This is joint work with C. D. Hacon

 

2021/08/13  GMT 8:15-9:15

Speaker: Thomas Krämer (Humboldt-Universität zu Berlin) (video)

Title: Big Tannaka groups on abelian varieties

Abstract: Lawrence and Sawin have shown that up to translation, any abelian variety over a number field contains only finitely many smooth ample hypersurfaces with given fundamental class and good reduction outside a given finite set of primes. A key ingredient in their proof is that certain Tannaka groups attached to smooth hypersurfaces are big. In the talk I will give a general introduction to Tannaka groups of perverse sheaves on abelian varieties and explain how to determine them for subvarieties of higher codimension (this is work in progress with Ariyan Javanpeykar, Christian Lehn and Marco Maculan).

 

2021/07/30  GMT 1:00-2:00

Speaker: Hsian-Hua Tseng (Ohio State University) (video)

 

Title: Relative Gromov-​Witten theory without log geometry

 

Abstract: 

We describe a new Gromov-​Witten theory of a space relative to a simple normal-​crossing divisor constructed using multi-​root stacks. 

 

2021/07/30  GMT 2:15-3:15

Speaker: Shusuke Otabe (Tokyo Denki University)

 

Title: Universal triviality of the Chow group of zero-cycles and unramified logarithmic Hodge-Witt cohomology

 

Abstract:

Auel-Bigazzi-Böhning-Graf von Bothmer proved that if a proper smooth variety over a field has universally trivial Chow group of zero-cycles, then its cohomological Brauer group is trivial as well. Binda-Rülling-Saito recently prove that the same conclusion is true for all reciprocity sheaves. For example, unramified logarithmic Hodge-Witt cohomology has the structure of reciprocity sheaf. In this talk, I will discuss another proof of the triviality of the unramified cohomology, where the key ingredient is a certain kind of moving lemma. This is a joint work with Wataru Kai and Takao Yamazaki. 

 

2021/07/16  GMT 7:00-8:00

Speaker: Qingyuan Jiang (University of Edinburgh) (video)

 

Title: On the derived categories of Quot schemes of locally free quotients

 

Abstract: 

  Quot schemes of locally free quotients of a given coherent sheaf, introduced by Grothendieck, are generalizations of projectivizations and Grassmannian bundles, and are closely related to degeneracy loci of maps between vector bundles. In this talk, we will discuss the structure of the derived categories of these Quot schemes in the case when the coherent sheaf has homological dimension $\le 1$. This framework not only allows us to relax the regularity conditions on various known formulae -- such as the ones for blowups, Cayley's trick, standard flips, projectivizations, and Grassmannain flips, but it also leads us to many new phenomena such as virtual flips, and blowup formulae for blowups along determinantal subschemes of codimension $\le 4$. We will illustrate the idea of proof in concrete cases, and if time allowed, we will also discuss the applications to the case of moduli of linear series on curves, and Brill-Noether theory for moduli of stable objects in K3 categories.

 

2021/07/16  GMT 8:15-9:15

Speaker: Le Quy Thuong (Vietnam National University) (video)

 

Title: The ACVF theory and motivic Milnor fibers

 

Abstract:

  In this talk, I review recent studies on the theory of algebraically closed value fields of equal characteristic zero (ACVF theory) developed by Hrushovski-Kazhdan and Hrushovski-Loeser. More precisely, I consider a concrete Grothendieck ring of definable subsets in the VF-sort and prove the structure theorem of this ring which can be presented via materials from extended residue field sort and value group sort. One can construct a ring homomorphism HL from this ring to the Grothendieck ring of algebraic varieties, from which the motivic Milnor fiber can be described in terms of a certain definable subset in VF-sort. As applications, I sketch proofs of the integral identity conjecture and the motivic Thom-Sebastiani theorem using HL, as well as mention the recent work of Fichou-Yin in the same topic.

 

2021/07/02  GMT 7:00-8:00

Speaker: Han-Bom Moon (Fordham University, New York) (video)

Title: Point configurations, phylogenetic trees, and dissimilarity vectors

Abstract:

In 2004 Pachter and Speyer introduced the dissimilarity maps for phylogenetic trees and asked two important questions about their relationship with tropical Grassmannian. Multiple authors answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. In this talk, we present a weighted variant of the dissimilarity map and show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter-Speyer envisioned. This tropical variety has a geometric interpretation in terms of point configurations on rational normal curves. This is joint work with Alessio Caminata, Noah Giansiracusa, and Luca Schaffler.

 

2021/07/02  GMT 8:15-9:15

Speaker: Yifei Chen (Chinese Academy of Sciences) (video)

Title: Jordan property of automorphism groups of surfaces of positive characteristic

Abstract:

A classical theorem of C. Jordan asserts the general linear group G over a field of characteristic zero is Jordan. That is, any finite subgroup of G contains a normal abelian subgroup of index at most J, where J is an integer only depends on the group G. J.-P. Serre proved that the same property holds for the Cremona group of rank 2. In this talk, we will discuss Jordan property for automorphism groups of surfaces of positive characteristic. This is a joint work with C. Shramov.

 

2021/06/18  GMT 7:00-8:00

Speaker: Mingshuo Zhou (Tianjin University) (video)

Title: Moduli space of parabolic bundles over a curve

Abstract:

In this talk, we will review a program (by Narasinhan-Ramadas and Sun) on the proof of Verlinde formula by using degeneration of moduli space of parabolic bundles over a curve. We will also show how the degeneration argument can be used to prove F-splitting of moduli space of parabolic bundles (for generic choice of parabolic points) over a generic curve in positive charactersitic. This is a joint work with Professor Xiaotao Sun. 

 

2021/06/18  GMT 8:15-9:15

Speaker: Zhi Jiang (Shanghai Center for Mathematical Sciences) (video)

Title: On syzygies of homogeneous varieties

Abstract:

We discuss some recent progress on syzygies of ample line bundles on homogeneous varieties, including abelian varieties and rational homogeneous varieties.

 

2021/5/28 GMT 7:00-8:00

Zhiyu Tian (BICMR-Beijing University) (video)

Title: Some conjectures about Kato homology of rationally connected varieties and KLT singularities

Abstract:

A natural question about zero cycles on a variety defied over an arithmetically interesting field is the injectivity/surjectivity of the cycle class map. This leads to the study of a Gersten type complex defined by Bloch-Ogus and Kato. I will present some conjectures about this complex for rationally connected varieties and Kawamata log terminal (KLT) singularities. I will also present some evidence for the conjectures, and explain how they fit into a variety of conjectures about the stability phenomenon observed in topology and number theory.

 

2021/5/28 GMT 8:15-9:15

Joao Pedro dos Santos (Universite de Paris) (video) 

Title: Group schemes from ODEs defined over a discrete valuation ring. 

Abstract:

Differential Galois theory has the objective to study linear ODEs (or connections) with the help of algebraic groups. Roughly and explicitly, to a matrix $A\in \mathrm{Mat}_n( \mathbb C(x) )$ and a differential system $y'=Ay$, we associate a subgroup of $GL_n(\mathbb C)$, the differential Galois group, whose function is to measure the complexity of the solutions. There are three paths to this theory: Picard-Vessiot extensions, monodromy representations and Tannakian categories. 

If instead of working with complex coefficients we deal with a discrete valuation ring $R$, the construction of the differential Galois groups are less obvious and the theory of groups gives place to that of group schemes. This puts forward the Tannakian approach and relevant concepts from algebraic geometry like formal group schemes and blowups. In this talk, I shall explain how to associate to these differential equations certain flat $R$-group schemes, what properties these may have--what to expect from a group having a generically faithful representation which becomes trivial under specialisation?--and how to compute with the help of the analytic method of monodromy. The talk is a horizontal report on several works done in collaboration with P.H.Hai and his students N.D.Duong and P.T.Tam over the past years. 

 

2021/5/14 GMT 6:00-7:00

Yuji Odaka (Kyoto University) (video)

Title: On (various) geometric compactifications of moduli of K3 surfaces 

Abstract:

What we mean by “geometric compactifications” in the title is it still parametrizes “geometric objects” at the boundary. In algebraic geometry, it is natural to expect degenerate varieties as such objects. For the moduli of polarized K3 surfaces (or K-trivial varieties in general) case,  it is natural to expect slc and K-trivial degenerations, but there are many such compactifications for a fixed moduli component, showing flexibility / ambiguity / difficulty of the problem. This talk is planned to mainly focus the following. In K3 surfaces (and hyperKahler varieties), there is a canonical geometric compactification whose boundary and parametrized objects are Not varieties but tropical geometric or with more PL flavor. This is ongoing joint work with Y.Oshima (cf., arXiv:1810.07685, 2010.00416). 

In general, there is a canonical PARTIAL compactification (quasi-projective variety) of moduli of polarized K-trivial varieties (essentially due to Birkar and Zhang), as a completion with respect to the Weil-Petersson metric. This is characterized by K-stability. 

 

2021/5/14 GMT 7:15-8:15

YongJoo Shin (Chungnam National University) (video)

Title: Complex minimal surfaces of general type with pg= 0 and K2 = 7 via bidouble covers

Abstract:

Let S be a minimal surface of general type with pg(S) = 0 and K2S = 7 over the field of complex numbers. Inoue firstly constructed such surfaces S described as Galois Z2×Z2-covers over the four-noda cubic surface. Chen later found different surfaces S constructed as Galois Z2×Z2-covers over six nodal del Pezzo surfaces of degree one. In this talk we construct a two-dimensional family of surfaces S different from ones by Inoue and Chen. The construction uses Galois Z2×Z2-covers over rational surfaces with Picard number three, with eight nodes and with two elliptic fibrations. This is a joint work with Yifan Chen.

 

2021/4/30 GMT 7:00-8:00

Yi Gu (Suzhou University) (video)

Title: On the equivariant automorphism group of surface fibrations

Abstract:

Let f:X→C be a relatively minimal surface fibration with smooth generic fibre. We will discuss the finiteness of its equivariant automorphism group, which is the group of pairs {ÎAut(X)´Aut(C)|} with natural group law. We will give a complete classification of those surface fibrations with infinite equivariant automorphism group in any characteristic. As an application, we will show how this classification can be used to study the bounded subgroup property and the Jordan property for automorphism group of algebraic surfaces.

 

2021/4/30 GMT 8:15-9:15

Takehiko Yasuda (Osaka University) (video)

Title: On the isomorphism problem of projective schemes

Abstract:

I will talk about the isomorphism problem of projective schemes; is it algorithmically decidable whether or not two given projective (or, more generally, quasi-projective) schemes, say over an algebraic closure of Q, are isomorphic? I will explain that it is indeed decidable for the following classes of schemes: (1) one-dimensional projective schemes, (2) one-dimensional reduced quasi-projective schemes, (3) smooth projective varieties with either the canonical divisor or the anti-canonical divisor being big, and (4) K3 surfaces with finite automorphism group. Our main strategy is to compute Iso schemes for finitely many Hilbert polynomials. I will also discuss related decidability problems concerning positivity properties (such as ample, nef and big) of line bundles.

 

2021/4/16 GMT 1:00-2:00

Kuan-Wen Lai (University of Massachusetts Amherst) (video)

Title: On the irrationality of moduli spaces of K3 surfaces

Abstract:

As for moduli spaces of curves, the moduli space of polarized K3 surfaces of genus g is of general type and thus is irrational for g sufficiently large. In this work, we estimate how the irrationality grows with g in terms of the measure introduced by Moh and Heinzer. We proved that the growth is bounded by a polynomial in g of degree 15 and, for three sets of infinitely many genera, the bounds can be refined to polynomials of degree 10. These results are built upon the modularity of the generating series of these moduli spaces in certain ambient spaces, and also built upon the existence of Hodge theoretically associated cubic fourfolds, Gushel–Mukai fourfolds, and hyperkähler fourfolds. This is a collaboration with Daniele Agostini and Ignacio Barros (arXiv:2011.11025).

 

2021/4/16 GMT 2:15-3:15

Yu-Shen Lin (Boston University) (video)

Title: Special Lagrangian Fibrations in Log Calabi-Yau Surfaces and Mirror Symmetry

Abstract:

Strominger-Yau-Zaslow conjecture predicts that the Calabi-Yau manifolds admit special Lagrangian fibrations and the mirror can be constructed via the dual torus fibration. The conjecture has been the guiding principle for mirror symmetry while the original conjecture has little progress. In this talk, I will prove that the SYZ fibration exists in certain log Calabi-Yau surfaces and their mirrors indeed admit the dual torus fibration under suitable mirror maps. The result is an interplay between geometric analysis and complex algebraic geometry. The talk is based on joint works with T. Collins and A. Jacob. 

 

2021/4/2 GMT 7:00-8:00

Weizhe Zheng (Morningside Center of Mathematics) (video)

Title: Ultraproduct cohomology and the decomposition theorem

Abstract:

Ultraproducts of étale cohomology provide a large family of Weil cohomology theories for algebraic varieties. Their properties are closely related to questions of l-independence and torsion-freeness of l-adic cohomology. I will present recent progress in ultraproduct cohomology with coefficients, such as the decomposition theorem. This talk is based on joint work with Anna Cadoret.

 

2021/4/2 GMT 8:15-9:15

Kestutis Cesnavicius (Universite Paris Sud) (video)

Title: Grothendieck--Serre in the quasi--split unramified case

Abstract:

The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. To overcome obstacles that have so far kept the mixed characteristic case out of reach, we adapt Artin's construction of "good neighborhoods" to the setting where the base is a discrete valuation ring, build equivariant compactifications of tori over higher dimensional bases, and study the geometry of the affine Grassmannian in bad characteristics.

 

2021/3/19 GMT 1:00-2:00

Zhiyuan Li (Shanghai Center for Mathematical Sciences) (video)

Title: Twisted derived equivalence for abelian surfaces

Abstract:

Over complex numbers, the famous global Torelli theorem for K3 surfaces says that two integral Hodge isometric K3 surfaces are isomorphic. Recently, Huybrechts has shown that two rational Hodge isometric K3 surfaces are twisted derived equivalent. This is called the twisted derived Torelli theorem for K3. Natural questions arise for abelian varieties. In this talk, I will talk about the twisted derived equivalence for abelian surfaces, including the twisted derived Torelli theorem for abelian surfaces (over all fields) and its applications. This is a joint work with Haitao Zou.

 

2021/3/19 GMT 2:15-3:15

Michael Kemeny (University of Wisconsin-Madison) (video)

Title: Universal Secant Bundles and Syzygies 

Abstract:

We describe a universal approach to the secant bundle construction of syzygies provided by Ein and Lazarsfeld. As an application, we obtain a quick proof of Green's Conjecture on the shape of the equations of general canonical curves. Furthermore, we will explain how the same technique resolves a conjecture of von Bothmer and Schreyer on Geometric Syzygies of canonical curves.

 

2021/3/5 GMT 7:00-8:00

Junyi Xie (CNRS Rennes)

Title: Some boundedness problems in Cremona group

Abstract:

This talk is based on a work with Cantat and Deserti. According to the degree sequence, there are 4 types (elliptic, Jonquieres, Halphen and Loxodromic) of elements f in Bir(P^2).  For a fixed degree d>=1, we study the set of these 4 types of elements of degree d. We show that for Halphen twists and Loxodromic transformations, such sets are constructible. This statement is not true for elliptic and Jonquieres elements.We also show that for a Jonquieres or Halphen twist f of degree d, the degree of the unique f-invariant pencil is bounded by a constant depending on d. This result may be considered as a positive answer to the Poincare problem of bounding the degree of first integrals,but for birational twists instead of algebraic foliations. As a consequence of this, we show that for two Halphen twists f and g, if they are conjugate in Bir(f), then they are conjugate by some element of degree bounded by a constant depending on deg(f)+deg(g). This statement is not true for Jonquieres twists. 

 

2021/3/5 GMT 8:15-9:15

Guolei Zhong (National University of Singapore)

Title: Fano threefolds and fourfolds admitting non-isomorphic endomorphisms.

Abstract: 

In this talk, we first show that a smooth Fano threefold X admits a non-isomorphic surjective endomorphism if and only if X is either toric or a product of a smooth rational curve and a del Pezzo surface. Second, we show that a smooth Fano fourfold Y with a conic bundle structure is toric if and only if Y admits an amplified endomorphism. The first part is a joint work with Sheng Meng and De-Qi Zhang, and the second part is a joint work with Jia jia.

 

2021/2/19 GMT 7:00-8:00     

Joonyeong Won (KIAS) (video)

Title: Sasaki-Einstein and Kähler-Einstein metric on 5-manifolds and weighted hypersurfaces

Abstract: 

By developing the method introduced by Kobayashi in 1960's, Boyer, Galicki and Kollár found many examples of simply connected Sasaki- Einstein 5-manifolds. For such examples they verified existence of orbifold Kähler-Einstein metrics on various log del Pezzo surfaces, in particular weighted log del Pezzo hypersurfaces. We discuss about recent progresses  of the existence problem of Sasaki -Einstein  and Kähler-Einstein metric on 5-manifold and weighted del Pezzo hypersurfaces respectively. 

 

2021/2/19 GMT 8:15-9:15     

Soheyla Feyzbakhsh (Imperial College) (video)

Title: An application of a Bogomolov-Gieseker type inequality to counting invariants

Abstract: 

In this talk, I will work on a smooth projective threefold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such as the projective space P^3 or the quintic threefold. I will show certain moduli spaces of 2-dimensional torsion sheaves on X are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in X. When X is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. This is joint work with Richard Thomas.

 

2021/2/5 GMT 1:00-2:00 

Lei Zhang (USTC) (video)

Title: Counterexample to Fujita conjecture in positive characteristic

Abstract: 

Fujita conjecture was proposed over complex numbers, which predicts that for a smooth projective variety X and an ample line bundle L on X, K_X + (dim X+1)L is base point free and K_X + nL is very ample if n > dim X+1. Joint with Yi Gu, Yongming Zhang, we find counterexamples to this elegant conjecture in positive characteristic. These examples stem from Raynaud’s surfaces. I will first report some related results on this topic and explain the construction and the proof. 

 

2021/1/22 GMT 7:00-8:00     

Sho Tanimoto (Kumamoto University) (video)

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 (University of Oslo) (video)

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. 

 

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. 

 

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—Witten 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

 

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. 

 

2020/11/20 GMT 7:00-8:00   

Jinhyung Park (Sogang University)

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 8:15-9:15   

Evgeny Shinder (University of Sheffield)

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.

 

2020/11/06 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 2:15-3:15   

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.

 

2020/10/23 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.

 

2020/10/09 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.

 

2020/10/09 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.).

 

2020/09/25 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 L^2 norme. It is a joint work with Mihai Păun.

 

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.

 

2020/08/28 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).

 

2020/08/14 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.

 

2020/08/07 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.

 

 

Sponsors: National Center for Theoretical Sciences