### AGEA Seminar

##### Online

This is a mirror page of AGEA Seminar. If you want to participate in the seminar, please email to register to AGEAseminar@gmail.comVideos for previous talks are available at the page https://sites.google.com/ncts.ntu.edu.tw/agea-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.

Organizing Committee:
Jungkai Chen (NCTS, Taiwan)
Meng Chen (Fudan, China)
Kiryong Chung (Kyungpook National Univ., Korea)
Baohua Fu (Chinese Academy of Science, China)
Yoshinori Gongyo (Tokyo, Japan)
Phung Ho Hai (VAST, Vietnam)
Yujiro Kawamata (Tokyo, Japan)
JongHae Keum (KIAS, Korea)
Conan Leung (CUHK, Hong Kong)
Hsueh-Yung Lin (NTU, Taiwan)
Yusuke Nakamura (Tokyo, Japan)
Xiaotao Sun (Tianjing Univ., China)
De-Qi Zhang (NUS, Singapore)

Other upcoming talks (To be confirmed by speakers):
Date                   Time                 Speaker
2022/02/11  GMT 7:00-8:00   Hsin-Ku Chen (NTU)
2022/02/11  GMT 8:15-9:15   Iacopo Brivio (NCTS)

2022/02/25  GMT 7:00-8:00   Quoc Ho (Institute of Science and Tech., Austria)
2022/02/25  GMT 8:15-9:15   Jinhyun Park (KAIST)

National Center for Theoretical Sciences

2021/12/31 GMT 7:00-8:00
Speaker: Kien Huu Nguyen (KU Leuven, Belgium)
Place: Online (Zoom ID: 896 0321 0669 Password: 340252)
Title: Exponential sums modulo $p^m$ for Deligne polynomials
Abstract: TBA

2021/12/31 GMT 8:15-9:15
Speaker: Xuan Viet Nhan Nguyen (BCAM, Spain)
Place: Online (Zoom ID: 896 0321 0669 Password: 340252)
Title: Smooth approximation in polynomially bounded o-minimal structures
Abstract: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a $C^p$ semialgebraic function , $p \in \mathbb{N}$. Shiota (1986) proved that given a positive continuous semi-algebraic function $\varepsilon: \mathbb{R}^n \to \mathbb{R}$, there is a $C^\infty$ semialgebraic function $g: \mathbb{R}^n \to \mathbb{R}$ such that $|D^\alpha (f - g)| < \varepsilon$ for every $|\alpha|\leq p$. In this talk, we show that the theorem still holds if we replace semialgebraic functions by definable functions in polynomially bounded o-minimal structures that allow smooth cell decompositions. Some applications are also given. This is a joint work with Anna Valette.

2021/12/31 GMT 7:00-8:00
Speaker: Lei Wu (KU Leuven, Belgium)
Place: Online (Zoom ID: 896 0321 0669 Password: 340252)
Title: D-modles, motivic integral and hypersurface singularities
Abstract: This talk is an invitation to the study of monodromy conjecture for hypersurfaces in complex affine spaces. I will recall two different ways to understand singularities of hypersurfaces in complex affine spaces. The first one is to use D-modules to define the b-function (also known as the Bernstein-Sato polynomial) of a polynomial (defining the hypersurface). The other one uses motivic integrals and resolution of singularities to obtain the motivic/topological zeta function of the hypersurface. The monodromy conjecture predicts that these two ways of understanding hypersurface singularities are related. Then I will discuss some known cases of the conjecture for hyperplane arrangements.

2021/12/31 GMT 8:15-9:15
Speaker: Wenhao Ou (AMSS, CAS)
Place: Online (Zoom ID: 896 0321 0669 Password: 340252)
Title: Projective varieties with strictly nef anticanonical divisor
Abstract: A conjecture of Campana-Peternell presumes that, if the anticanonical divisor of a projective variety X has strictly positive intersection with all curves, then the manifold is Fano. We show that if X is klt, then it is rationally connected. This provides an evidence to the conjecture. Furthermore, if the dimension is at most three, then we prove that X is Fano. This is joint with Jie Liu, Juanyong Wang, Xiaokui Yang and Guolei Zhong.

2021/12/17 GMT 7:00-8:00
Speaker: Shuai Guo (Peking University)
Place: Online (Zoom ID: 896 0321 0669 Password: 340252)
Title: Structure of higher genus Gromov-Witten invariants of the quintic threefolds
Abstract: The computation of the Gromov-Witten (GW) invariants of the compact Calabi Yau 3-folds is a central and yet difficult problem in geometry and physics. In a seminal work in 1993, Bershadsky, Cecotti, Ooguri and Vafa (BCOV) introduced the higher genus B-model in physics. During the subsequent years, a series of conjectural formulae was proposed by physicists based on the BCOV B-model,  which effectively calculates the higher genus GW potential from lower genus GW potentials and a finite ambiguity. In this talk, we will introduce some recent mathematical progresses in this direction. This talk is based on the joint works with Chang-Li-Li and the joint works with Janda-Ruan.

2021/12/17 GMT 8:15-9:15
Speaker: Yang Zhou (Fudan University)
Place: Online (Zoom ID: 896 0321 0669 Password: 340252)
Title: Wall-crossing for K-theoretic quasimap invariants
Abstract: For a large class of GIT quotients, the moduli of epsilon-stable quasimaps is a proper Delinge-Mumford stack with a perfect obstruction theory. Thus K-theoretic epsilon-stable quasimap invariants are defined. As epsilon tends to infinity, it recovers the K-theoretic invariants; and as epsilon decreases, fewer and fewer rational tails are allowed in the domain curves. There is a wall and chamber structure on the space of stability conditions. In this talk, we will decribe a master space construction involoving the moduli spaces on the two sides of a wall, leading to the proof of a wall-crossing formula. A key ingredient is keeping track of the S_n-equivariant structure on the K-theoretic invariants.

2021/12/03 GMT 7:00-8:00
Speaker: Jinhyun Park (KAIST)
Place: Online (Zoom ID: 896 0321 0669 Password: 340252)
Title: On motivic cohomology of singular algebraic schemes
Abstract: Motivic cohomology is a hypothetical cohomology theory for algebraic schemes, including algebraic varieties, over a given field, that can be seen as the counterpart in algebraic geometry to the singular cohomology theory in topology. Its construction was completed for smooth varieties, but for singular ones the situation was not clear.
In this talk, I will sketch some recent attempts of mine to provide an algebraic-cycle-based functorial model for the motivic cohomology of singular algebraic schemes, via formal schemes and some ideas from derived algebraic geometry. As this is very complicated, as an illustration I will give an example on the concrete case of the fat points, where the situation is simpler, but not still trivial.

2021/11/19 GMT 7:00-8:00
Speaker: Keiji Oguiso (University of Tokyo)
Place: Online (Zoom ID: 896 0321 0669 Password: 340252)
Title: Smooth complex projective rational varieties with infinitely many real forms
Abstract: This is a joint work with Professors Tien-Cuong Dinh and Xun Yu. The real form problem asks how many different ways one can describe a given complex variety by polynomial equations with real coefficients up to isomorphisms over the real number field. For instance, the complex projective line has exactly two real forms up to isomorphisms. This problem is in the limelight again after a breakthrough work due to Lesieutre in 2018. In this talk, among other relevant things, we would like to show that in each dimension greater than or equal to two, there is a smooth complex projective rational variety with infinitely many real forms. This answers a question of Kharlamov in 1999.

2021/11/19 GMT 8:15-9:15
Speaker: Yuki Hirano (Kyoto University)
Place: Online (Zoom ID: 896 0321 0669 Password: 340252)
Title: Equivariant tilting modules, Pfaffian varieties and noncommutative matrix factorizations
Abstract: It is known that a tilting bundle T on a smooth variety X induces a derived equivalence of coherent sheaves on X and finitely generated modules over the endomorphism algebra End(T). We prove that, in a suitable setting, a tilting bundle also induces an equivalence of derived matrix factorization categories. As an application, we show that the derived category of a noncommutative resolution of a linear section of a Pfaffian variety is equivalent to the derived matrix factorization category of a noncommutative gauged Landau-Ginzburg model.

2021/10/08 GMT 7:00-8:00
Speaker: Christian Schnell (Stony Brook University)
Place: Online (Zoom ID: 827 7229 2046 Password: 830392
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)
Place: Online (Zoom ID: 827 7229 2046 Password: 830392
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.

2021/09/24 GMT 7:00-8:00
Speaker: Kenta Hashizume (University of Tokyo)
Place: Online (Zoom ID: 827 7229 2046 Password: 830392
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.

2021/09/24 GMT 8:15-9:15
Place: Online (Zoom ID: 827 7229 2046 Password: 830392
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.

2021/9/10 GMT 1:00-2:00
Speaker: Kyoung-Seog Lee (Miami University)
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.

2021/9/10 GMT 2:15-3:15
Speaker: Insong Choe (Konkuk University)
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.

2021/9/3 GMT 1:00-2:00
Speaker: Jingjun Han (Johns Hopkins University)
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.

2021/9/3 GMT 2:15-3:15
Speaker: Jia Jia (National University of Singapore)
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.

2021/8/13 GMT 7:00-8:00
Speaker: Jihao Liu (University of Utah)
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.

2021/8/13 GMT 8:15-9:15
Speaker: Thomas Krämer (Humboldt-Universität zu Berlin)
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).

2021/7/30 GMT 1:00-2:00
Speaker: Hsian-Hua Tseng (Ohio State University)
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.

2021/7/30 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.

2021/7/16 GMT 7:00-8:00
Speaker: Qingyuan Jiang (University of Edinburgh)
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.

2021/7/16 GMT 8:15-9:15
Speaker: Le Quy Thuong (Vietnam National University)
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.

2021/7/2 GMT 7:00-8:00
Speaker: Han-Bom Moon (Fordham University, New York)
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.

2021/7/2 GMT 8:15-9:15
Speaker: Yifei Chen (Chinese Academy of Sciences)
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.

2021/6/18 GMT 7:00-8:00
Speaker: Mingshuo Zhou (Tianjin University)
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.

2021/6/18 GMT 8:15-9:15
Speaker: Zhi Jiang (Shanghai Center for Mathematical Sciences)
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
Speaker: Zhiyu Tian (BICMR-Beijing University)
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
Speaker: Joao Pedro dos Santos (Universite de Paris)
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
Speaker: Yuji Odaka (Kyoto University)
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
Speaker: YongJoo Shin (Chungnam National University)
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
Speaker: Yi Gu (Suzhou University)
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
Speaker: Takehiko Yasuda (Osaka University)
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
Speaker: Kuan-Wen Lai (University of Massachusetts Amherst)
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
Speaker: Yu-Shen Lin (Boston University)
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
Speaker: Weizhe Zheng (Morningside Center of Mathematics)
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
Speaker: Kestutis Cesnavicius (U. Paris Sud)
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
Speaker: Zhiyuan Li (Shanghai Center for Mathematical Sciences)
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
Speaker: Michael Kemeny (University of Wisconsin-Madison)
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
Speaker: 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
Speaker: 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.