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.
