Explicit Birational Geometry seminar
Organizers: Meng Chen (Fudan), Zhi Jiang (SCMS), Chen Jiang (SCMS), Jingjun Han (SCMS)
The purpose of this seminar is to introduce recent results on explicit birational geometry to researchers and students at Fudan University.
This semester (2023 Fall) the standard time is Friday 16:00-17:00 (Beijing). If you are interested in giving a talk on your results, please contact the organizers.
Upcoming talks
2023 Fall
Date: 2023, Sep. 8 (FRI), 14:30-15:30 (Beijing)
Location: SCMS 102
Speaker: Long Wang (Fudan University)
Title: Arithmetic degrees of dominant rational self-maps
Abstract: We discuss a conjecture of Kawaguchi and Silverman about arithmetic degrees of dominant rational self-maps defined over number fields. Some new results and an application to the existence of Zariski dense orbits will be given. This is based on joint work with Yohsuke Matsuzawa.
Date: 2023, Sep. 8 (FRI), 16:00-17:00 (Beijing)
Location: SCMS 102
Speaker: Chuyu Zhou (École Polytechnique Fédérale de Lausanne)
Title: K-semistable domain and wall crossing for K-stability
Abstract: In this talk, we will define K-semistable domain of a pair consisting of a Fano variety and multiple boundaries. We will present many important properties of the K-semistable domain. Based on this, we will see a wall crossing theory for K-stability. Time permits, we will also talk about some interesting examples.
Archive
2023 Spring
Date: 2023, Jun. 8 (THU), 9:30-12:00 (Beijing)
Location: SCMS 106
Speaker: Jihao Liu (Northwestern University)
Title: Canonical bundle formula
Abstract: In this talk I will discuss the history and the results on the canonical bundle formula (cbf) and how they are usually
applied, including the following: 1. Cbf for elliptic fibrations. 2. Cbf for Iitaka fibrations. 3. Cbf for lc-trivial fibrations (contractions). 4. Cbf for generically finite morphisms and lc-trivial morphisms. 5. Cbf for generalized pairs. 6. Subadjunction formulas. 7. Cbf on Kähler varieties. 8. Cbf for foliations. 9. Abundance and log abundance of the moduli part. 10. New proof of the Cbf for lc-trivial fibrations via foliation. 11. Cbf in characteristic p.
Date: 2023, Jun. 7 (WED), 14:00-15:30 (Beijing)
Location: SCMS 106
Speaker: Jihao Liu (Northwestern University)
Title: Symphony of generalized pairs, foliations, and Kähler manifolds
Abstract: Generalized pairs, foliations, and Kähler manifolds are three objects in the study of birational geometry. In this talk, I will discuss the connection between these three seemingly unrelated topics from the viewpoint of the minimal model program. Through exploring these connections, I will discuss the potential implications of these topics
Date: 2023, May 31 (WED), 16:00-17:00 (Beijing)
Date: 2023, Jun. 2 (FRI), 16:00-17:00 (Beijing)
Date: 2023, Jun. 5 (MON), 16:00-17:00 (Beijing)
Date: 2023, Jun. 7 (WED), 16:00-17:00 (Beijing)
Location: SCMS 106
Speaker: Lingyao Xie (University of Utah)
Title: Introduction to toric varieties (I-IV)
Abstract: Toric varieties are geometric objects defined by combinatorial information. They provide a wonderful introduction to algebraic geometry and can be used to construct various interesting examples. In these series, I will give a brief introduction to this rich subject and show that there is an explicit way to compute the minimal log discrepancies of a toric variety (pair), which is really hard to do in general (especially in higher dimensions).
Date: 2023, Apr. 14 (FRI), 14:20-15:20 (Beijing)
Location: SCMS 102
Speaker: Lei Wu (Zhejiang University)
Title: Logarithmic cotangent bundles, Chern classes, and applications
Abstract: Using MacPherson's Euler obstruction function, one can identify the abelian group of constructible functions with the group of algebraic cycles on a smooth complex algebraic variety. Kashiwara's local index formula gives an alternative approach to this identification by using characteristic cycles for holonomic D-modules (they are Lagrangian cycles in the cotangent bundle). This identification then enables us to define Chern classes of algebraic cycles by using characteristic cycles. In this talk, I will first explain how to obtain Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement by using logarithmic cotangent bundles motivated by D-module theory. Then I will discuss applylications of such Chern classes in understanding Chern-Mather classes of very affine varieties and in proving the Involution Conjecture of Huh and Sturmfels in likelihood geometry. This work is joint with Maxim, Rodriguez, and Wang.
Date: 2023, Apr. 14 (FRI), 15:50-16:50 (Beijing)
Location: SCMS 102
Speaker: Zhiyu Tian (Peking University)
Title: Algebraic equivalence and stable maps
Abstract: I will describe a technique to lift algebraic equivalence of one cycles on smooth projective varieties to deformation equivalence between stable maps, and some of its applications in geometry and arithmetic. This is joint work with János Kollár.
Date: 2023, Mar. 31 (FRI), 13:30-14:30 (Beijing)
Location: SCMS 102
Speaker: Zijun Zhou (Shanghai Jiao Tong University)
Title: Virtual Coulomb branch and quantum K-theory
Abstract: In this talk, I will discuss a virtual variant of the quantized Coulomb branch constructed by Braverman-Finkelberg-Nakajima, where the convolution product is modified by a virtual intersection. The resulting virtual Coulomb branch acts on the moduli space of quasimaps into the holomorphic symplectic quotient T^*N///G. When G is abelian, over the torus fixed points, this representation is a Verma module. The vertex function, a K-theoretic enumerative invariant introduced by A. Okounkov, can be expressed as a Whittaker function of the algebra. The construction also provides a description of the quantum q-difference module. As an application, this gives a proof of the invariance of the quantum q-difference module under the variation of GIT.
Date: 2023, Mar. 31 (FRI), 15:00-16:00 (Beijing)
Location: SCMS 102
Speaker: Yalong Cao (RIKEN)
Title: From curve counting on Calabi-Yau 4-folds to quasimaps for quivers with potentials
Abstract: I will start by reviewing an old joint work with Davesh Maulik and Yukinobu Toda on relating Gromov-Witten, Gopakumar-Vafa and stable pair invariants on compact Calabi-Yau 4-folds. For non-compact CY4 like local curves, similar invariants can be studied via the perspective of quasimaps to quivers with potentials. In a joint work in progress with Gufang Zhao, we define a virtual count for such quasimaps and prove a gluing formula. Computations of examples will also be discussed.
Date: 2023, Mar. 1 (WED), 15:30-16:30 (Beijing)
Location: SCMS 406
Speaker: Chenyu Bai (Université de Paris)
Title: Geometry of Calabi--Yau type Chow varieties of linear subspaces in cubic hypersurfaces
Abstract: For any given integer r\geq 0, there is a positive integer n depending on r, such that for any very general cubic n-fold, the Chow variety of r-dimensional linear subspaces is a Calabi--Yau type manifold. I will present some aspects of the geometry of these Calabi--Yau manifolds: their dimensions, moduli numbers, decompositions of CH_0, etc. A main tool is a self rational map of high degree on it. This is a work in progress.
Date: 2023, Mar. 1 (WED), 16:30-17:30 (Beijing)
Location: SCMS 106
Speaker: Junpeng Jiao (Tsinghua University)
Title: On singularities of Fano fibrations and degenerations of Fano varieties
Abstract: The Shokurov-Mckernan conjecture predicts that for a Fano fibration X\rightarrow Z, there is a relationship between the log discrepancies of singularities of X and Z. In this talk, we discuss the singularities of Fano fibrations, we show the connection between the Shokurov-Mckernan conjecture and the degenerations of Fano varieties, and we prove a birational boundedness result on the flat degenerations of Fano varieties.
2022 Fall
Date: 2022, Dec. 16 (FRI), 16:00-17:00 (Beijing)
Location: SCMS 102
Speaker: Guodu Chen (Westlake University)
Title: On effective log Iitaka fibrations
Abstract: We study the relationship between Iitaka fibrations and the conjecture on the boundedness of complements, assuming the good minimal model conjecture. The main result is that the conjecture on the boundedness of complements implies the effective log Iitaka fibration conjecture assuming the good minimal model conjecture. As a consequence, the effective log Iitaka fibration conjecture holds in dimension 3. This is an ongoing joint work with Jingjun Han and Jihao Liu.
Date: 2022, Dec. 16 (FRI), 14:30-15:30 (Beijing)
Location: SCMS 102
Speaker: Chuanhao Wei (Westlake University)
Title: Kodaira-type vanishings via Nonabelian Hodge Theory
Abstract: In the past decade, Mochizuki has completed the spectacular theory of mixed Twistor D-modules. In this talk, I will first briefly introduce this result. Then, I will show that Kodaira-type vanishing still holds under the setting of mixed Twistor D-modules, which generalizes Saito vanishing under the setting of mixed Hodge Modules. I will also introduce a version of Kawamata-Viehweg vanishing with Q-divisors, and an effective global generalization theorem as an application.
Date: 2022, Dec. 14 (FRI), 10:00-11:30 (Beijing)
Location: Tencent Meeting ID: 127226670 Password: 200438
Speaker: Santai Qu (Tsinghua University)
Title: Bounding irrationality of degenerations of Fano fibrations
Abstract: In this talk, I will introduce a recent result about boundingdegreesofirrationality of degenerations of klt Fano fibrations of arbitrarydimensions.This proves the generically bounded case of a conjecture proposedbyC.Birkar and K. Loginov for log Fano fibrations of dimensions greaterthanthree. Our approach depends on a method to modify the klt Fanofibrationto a toroidal morphism of toroidal embeddings with boundedgeneral fibres.Moreover, we show that every fibre of the toroidal morphismisboundedand has mild singularities if we replace the birational modificationsbyalterations. This is a joint work with Prof. C. Birkar.
Date: 2022, Dec. 2 (FRI), 16:00-17:00 (Beijing)
Location: SCMS 102
Speaker: Zheng Zhang (ShanghaiTech University)
Title: Cubic threefolds with an involution and their intermediate Jacobians
Abstract: We study the moduli space of cubic threefolds admitting an involution via the period map sending such a cubic threefold to the invariant/anti-invariant part of the intermediate Jacobian. Our main result is global Torelli holds for the period map. Key ingredients of the proof include a description of the invariant/anti-invariant part of the intermediate Jacobian as a Prym variety and a detailed study of certain positive dimensional fibers of the corresponding Prym map. The proof also relies on the results of Donagi-Smith, Ikeda and Naranjo-Ortega on related Prym maps. This is joint work with S. Casalaina-Martin and L. Marquand.
Date: 2022, Dec. 2 (FRI), 14:30-15:30 (Beijing)
Location: SCMS 102
Speaker: Ziwen Zhu (Tongji University)
Title: Equivariant K-stability and alpha invariants
Abstract: Tian's criterion in terms of alpha invariant is one of the practical tools to determine K-stability of Fano varieties. However, the criterion fails to be applicable in many cases. For some varieties with large symmetry, an equivariant version of Tian's criterion works. In this talk, I will discuss some recent development on equivariant K-stability and mention some applications. I will also talk about a recent project related to higher codimensional alpha invariants which potentially generalizes Tian's criterion. Part of my talk is based on joint work with Yuchen Liu.
Date: 2022, Nov. 18 (FRI), 16:00-17:00 (Beijing)
Location: SCMS 102, Tencent Meeting 353 294 624, password: 872760
Speaker: Meng Sheng (East China Normal University)
Title: Log Calabi-Yau structure of projective threefolds admitting polarized endomorphisms
Abstract: Let X be a normal projective variety admitting a polarized endomorphism. It was conjectured by Broustet and Gongyo that X is log Calabi-Yau and they proved the surface case. In this talk, I'll explain my recent progress towards this conjecture.
Date: 2022, Nov. 18 (FRI), 14:30-15:30 (Beijing)
Location: Tencent Meeting 353 294 624, password: 872760
Speaker: Jihao Liu (Northwestern University)
Title: 1-gap of R-complementary thresholds on surfaces and its applications
Abstract: In this talk, I will show that the optimal 1-gap of R-complementary thresholds of surfaces is equal to 1/13. I will provide several applications of this result, including 1) finding the optimal 1-gap of global log canonical threshold for surfaces, 2) finding the optimal lower bound of volumes of ample log surfaces with reduced boundary, and 3) finding the smallest minimal log discrepancy of kit Calabi-Yau surfaces. These results answer a question of V. Alexeev and W. Liu and a question of J. Kollár, and reprove a recent result of L. Esser, B. Totaro, and C. Wang. If time admits, I will discuss a conjecture on the 1-gap of R-complementary thresholds and the 1-gap of minimal log discrepancies in high dimensions. This is an ongoing joint work with V. V. Shokurov.
Date: 2022, Nov. 9 (WED), 10:30-11:30 (Beijing)
Location: Zoom Meeting 99487623924, password: 3264
Speaker: Lu Qi (Princeton University)
Title: Convexity of multiplicities of filtrations on local rings
Abstract: In this talk, I will discuss some convexity properties of multiplicities of filtrations on a local ring. In particular, the multiplicity function is convex along geodesics. As a major application, we get a new proof of a theorem due to Xu and Zhuang on the uniqueness of normalized volume minimizers. In order to characterize strict convexity, we introduce the notion of saturation of a filtration, which turns out to be useful in other settings. For example, it allows us to generalize a theorem of Rees on multiplicities of ideals and characterize when the Minkowski inequality for filtrations is an equality. This talk is based on joint work with Harold Blum and Yuchen Liu.
Date: 2022, Nov. 4 (FRI), 14:30-15:30 (Beijing)
Location: Tencent Meeting 900 651 060, password: 247436
Speaker: Ya Deng (Université de Lorraine)
Title: Hyperbolicity and fundamental groups of complex quasi-projective varieties
Abstract: Non-abelian Hodge theories in both Archimedean and non-Archimedean settings are robust tools in studying fundamental groups of algebraic varieties. In this talk I will explain the recent progress on these subjects. I will then give a sharp condition for the hyperbolicity of quasi-projective varieties whose fundamental groups admitting linear representations.
Date: 2022, Oct. 21 (FRI), 16:00-17:00 (Beijing)
Location: SCMS 102
Speaker: Xin Lv (East China Normal University)
Title: The Coleman-Oort conjecture on the finiteness of CM curves
Abstract: The Coleman-Oort conjecture predicts that for each fixed genus g>>0, there are at most finitely many complex algebraic curves of genus g whose Jacobians are abelian varieties with complex multiplication. In this talk, I would like to give an introduction to this conjecture and report our progress on it. It is based on joint works with Ke Chen and Kang Zuo.
Date: 2022, Oct. 21 (FRI), 14:30-15:30 (Beijing)
Location: SCMS 102
Speaker: Zili Zhang (Tongji Univeristy)
Title: Non-abelian Hodge correspondence and the P=W conjecture
Abstract: Fix a complex projective curve C and a reductive group G. There are three moduli spaces with the pair (C,G): the character variety M_B, the moduli of flat connections M_dR, and the moduli of Higgs bundles M_Dol. The non-abelian Hodge correspondence says there are natural homeomorphisms among the three moduli spaces, and hence identify the cohomology groups of them. The geometric structures of the moduli spaces induce various filtrations in the cohomology groups. De Cataldo-Hausel-Migliorini conjectured in 2012 that the Perverse filtration (P) of M_Dol is identical to the Hodge-theoretic weight filtation (W) of M_B; the P=W conjecture. We will introduce the background and recent progress of the nonabelian Hodge correspondence and the P=W conjecture. The talk is not aimed at specialists.
Date: 2022, Oct. 12 (WED), 16:00-17:00 (Beijing)
Location: Tencent Meeting 142 549 093, password: 312368
Speaker: Fei Hu (Univeristy of Oslo)
Title: An upper bound for polynomial log-volume growth of automorphisms of zero entropy
Abstract: Let $X$ be a normal projective variety of dimension $d\ge 2$ over an algebraically closed field and $f$ an automorphism of $X$.
Suppose that the pullback $f^*|_{\mathsf{N}^1(X)_\mathbf{R}}$ of $f$ on the space $\mathsf{N}^1(X)_\bR$ of numerical $\mathbf{R}$-divisor classes is unipotent and denote the index of the eigenvalue $1$ by $k+1$.
We prove an upper bound for polynomial log-volume growth $\mathrm{plov}(f)$ of $f$, or equivalently, for the Gelfand--Kirillov dimension of the twisted homogeneous coordinate ring associated with $(X,f)$, as follows: \[\mathrm{plov}(f) \le (k/2 + 1)d.\]
In characteristic zero, combining with the inequality $k\le 2(d-1)$ due to Dinh--Lin--Oguiso--Zhang, we obtain an optimal inequality that \[\mathrm{plov}(f) \le d^2,\]
which affirmatively answers questions of Cantat--Paris-Romaskevich and Lin--Oguiso--Zhang. This is joint work with Chen Jiang.
Date: 2022, Sep. 16 (FRI), 15:00-16:30 (Beijing)
Location: SCMS 102
Speaker: Jingjun Han (Fudan University)
Title: On ACC for minimal log discrepancies for exceptionally non-canonical pairs
Abstract: We reduce the termination of flips to the termination of terminal flips and the ACC conjecture for minimal log discrepancies for exceptionally non-canonical pairs. In particular, the ACC conjecture implies the termination of flips in dimension four. We also show the ACC conjecture holds in dimension three. This is an ongoing joint work with Jihao Liu.
2022 Spring
Date: 2022, Jul. 13 (WED), 10:00-11:00 (Beijing)
Location: Tencent Meeting 416 471 516, password: 247436
Speaker: Yunfeng Jiang (University of Kansas)
Title: The virtual fundamental class for the moduli space of general type surfaces
Abstract: Sir Simon Donaldson conjectured that there should exist a virtual fundamental class on the moduli space of surfaces of general type inspired by the geometry of complex structures on the general type surfaces. In this talk I will present a method to construct the virtual fundamental class on the moduli stack of lci (locally complete intersection) covers over the moduli stack of general type surfaces with only semi-log-canonical singularities. A tautological invariant is defined by taking the integration of the power of the first Chern class of the CM line bundle over the virtual fundamental class. This can be taken as a generalization of the tautological invariants on the moduli space of stable curves to the moduli space of stable surfaces.
Date: 2022, May 19 (THU), 9:30-10:30 (Beijing)
Location: Zoom Meeting 971 4102 0213, no password
Speaker: Jihao Liu (Northwestern University)
Title: ACC for mlds for terminal threefolds
Abstract: Recently, the speaker, J. Han, and Y. Luo proved Shokurov’s ascending chain condition (ACC) conjecture for minimal log discrepancies (mlds) for terminal threefolds. In this talk, I will discuss this result, investigate its related corollaries and applications. If I have enough time, then I will sketch a proof of this result.
Date: 2022, May 18 (WED), 15:00-16:00 (Beijing)
Location: Zoom Meeting 868 9869 7641, password: SCMS
Speaker: Yalong Cao (RIKEN iTHEMS)
Title: Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds
Abstract: Gromov-Witten invariants of holomorphic symplectic 4-folds vanish and one can consider the corresponding reduced theory. In this talk, we will explain a definition of Gopakumar-Vafa type invariants for such a reduced theory. These invariants are conjectured to be integers and have alternative interpretations using sheaf theoretic moduli spaces. Our conjecture is proved for the product of two K3 surfaces, which naturally leads to a closed formula of Fujiki constants of Chern classes of tangent bundles of Hilbert schemes of points on K3 surfaces. On a very general holomorphic symplectic 4-folds of K3^[2] type, our conjecture provides a Yau-Zaslow type formula for the number of isolated genus 2 curves of minimal degree. Based on joint works with Georg Oberdieck and Yukinobu Toda.
Date: 2022, April 27 (WED), 15:00-16:00 (Beijing)
Location: Zoom Meeting 840 2712 4061, password: SCMS
Speaker: Vladimir Lazić (Universität des Saarlandes)
Title: The Nonvanishing problem for varieties with nef first Chern class
Abstract: Let X be a threefold with mild singularities such that c1(X) is nef. In this talk I will present a very recent proof that then the numerical class of c1(X) is effective. This result (joint work with Thomas Peternell, Nikolaos Tsakanikas and Zhixin Xie) is the positive curvature counterpart of the famous result of Miyaoka from the 1980s, which showed that –c1(X) is effective as soon as it is nef.
Date: 2022, April 20 (WED), 16:00-17:00 (Beijing)
Location: Zoom Meeting 891 2163 6889, password: cremona
Speaker: Hsueh-Yung Lin (Taiwan University)
Title: Motivic invariants of birational maps and Cremona groups
Abstract: (Joint with E. Shinder and partly with S. Zimmerman) In characteristic zero, birational maps of projective varieties factorize through a sequence of blow-ups and blow-downs along smooth centers. We study to which extent these factorization centers are unique, and construct motivic invariants of birational maps which account for the non-uniqueness of centers. For surfaces over a perfect field, we prove the uniqueness of centers in the strongest possible sense. In higher dimension, the centers fail to be unique. Relying on the non-uniqueness, we provide new explanations of the non-simplicity of Cremona groups.
Date: 2022, April 13 (WED), 15:00-16:00 (Beijing)
Location: Tencent Meeting 839 603 441, password: 535575
Speaker: Yu Zou (Fudan University)
Title: On the anti-canonical geometry of weak Q-Fano 3-folds, III
Abstract: By a terminal weak Q-Fano 3-fold (resp. terminal Q-Fano 3-fold) we mean a normal projective one with at worst terminal singularities on which the anti-canonical divisor is nef and big (resp. ample). For a terminal weak Q-Fano 3-fold X, we show that the m-th anti-canonical map defined by |-mK_X| is birational for all m≥59. Furthermore, if X is a terminal Q-Fano 3-fold, then the m-th anti-canonical map defined by |-mK_X| is birational for all m≥58. (This is a joint work with Chen Jiang)
Date: 2022, March 24 (THU), 8:30-9:30 (Beijing)
Location: Zoom Meeting 952 6613 6871, password: SCMS
Speaker: Fanjun Meng (Northwestern University)
Title: Kodaira dimension of fibrations over abelian varieties
Abstract: The Kodaira dimension of smooth projective varieties is an important birational invariant. In this talk, we will discuss some conjectures on the behavior of Kodaira dimension proposed by Popa. We prove an additivity result for the log Kodaira dimension of algebraic fiber spaces over abelian varieties, a superadditivity result for algebraic fiber spaces over varieties of maximal Albanese dimension, as well as a subadditivity result for log pairs over abelian varieties. This is joint work with Mihnea Popa.
Date: 2022, March 10 (THU), 8:30-9:30 (Beijing)
Location: Zoom Meeting 960 4727 5610, password: SCMS
Speaker: Lingyao Xie (University of Utah)
Title: On the fixed part of pluricanonical systems for surfaces
Abstract: Assuming K_X is nef and big or ample, we study the behavior of |mK_X| when the Cartier index of K_X is not bounded. Especially we are interested in when |mK_X| is free in codim 1 (does not have fixed part). we show that in general there is no uniform m to ensure |mK_X| free in codim 1 (for klt variety) unless we have some extra assumption on the singularities. More precisely, we show that |mK_X| defines a birational map and has no fixed part for some bounded positive integer m for any 1/2-lc surface X such that K_X is big and nef. For every positive integer n>2, we construct a sequence of projective surfaces X_{n,i}, such that K_{X_{n,i}} is ample, mld(X_{n,i})>1/n for every i, the limit of mld(X_{n,i}) is 1/n, and for any positive integer m, there exists i such that |mK_{X_{n,i}}| has non-zero fixed part. This is a joint work with Jihao Liu.
Date: 2022, March 3 (THU), 16:00-17:00 (Beijing)
Location: SCMS 102
Speaker: Long Wang (University of Tokyo)
Title: Dynamical and arithmetic degrees of dominant rational self-maps
Abstract: We survey some recent results about a conjecture of Kawaguchi and Silverman concerning dynamical and arithmetic degrees of dominant rational self-maps defined over number fields.
Date: 2022, February 16 (WED), 16:00-17:00 (Beijing)
Location: Zoom Meeting 921 8900 9193, password: SCMS
Speaker: Roberto Svaldi (École polytechnique fédérale de Lausanne)
Title: On the boundedness of elliptically fibered varieties
Abstract: In this talk, I will survey recent progress on the question of boundedness for elliptic varieties, in particular for the case of projective Calabi—Yau varieties.I will explain different aspects of the question and discuss some of the techniques involved in proving this type of results, with particular focus to the ideas coming from the MMP and their interaction with other results/conjectures on Calabi—Yau varieties, e.g., the Kawamata-Morrison Cone Conjecture. The talk will discuss results from joint collaborations with Di Cerbo, Di Cerbo and Birkar, and particularly the recent work in collaboration with Filipazzi and Hacon.
Date: 2022, January 5 (WED), 16:00-17:00 (Beijing)
Location: Tencent Meeting 317 724 263, password: 685922
Speaker: Guolei Zhong (IBS Center for Complex Geometry)
Title: Characterization of projective toric varieties from dynamical viewpoints
Abstract: As a fundamental building block of the equivariant minimal model program, the rationally connected variety plays a significant role in the classification of projective varieties admitting non-isomorphic endomorphisms. Twenty years ago, Nakayama confirmed Sato’s conjecture that, a smooth projective rational surface is toric if and only if it admits a non-isomorphic endomorphism. In this talk, I will survey some recent progress on a higher dimensional analogue of Nakayama’s result. This talk is based on some joint works with Jia Jia, Sheng Meng and De-Qi Zhang.
2021 Fall
Date: 2021, December 29 (WED), 16:00-17:00 (Beijing)
Location: Zoom Meeting 964 0959 8310, password: SCMS
Speaker: Taro Sano (Kobe University)
Title: On K-stability of Fano weighted hypersurfaces
Abstract: Weighted complete intersections are a rich source of examples of varieties. K-stability (or existence of Kähler-Einstein metrics) of explicit Fano varieties has been studied for a long time. In this talk, I will explain our results on the K-stability of some Fano weighted hypersurfaces. This is based on joint work with Luca Tasin.
Date: 2021, December 22 (WED), 16:00-17:00 (Beijing)
Location: Tencent Meeting 995 792 574, password: 382322
Speaker: Baohua Fu (Chinese Academy of Sciences)
Title: Normalized tangent bundle, pseudoeffective cone and varieties with small codegree
Abstract: We propose a conjectural list of Fano manifolds of Picard num- ber one whose normalized tangent bundle is pseudoeffective and we prove it in various situations by relating it to the complete divisibility conjecture of Russo and Zak on varieties with small codegrees. The pseudoeffective cone of the projectivized tangent bundle of a rational homogeneous space of Picard number one is explicitly determined by studying the total dual VMRT and the geometry of stratified Mukai flops. This is a joint work with Jie LIU
Date: 2021, December 15 (WED), 16:00-17:00 (Beijing)
Location: Tencent Meeting 256 496 451, password: 54321
Speaker: Junyan Cao (Université Côte d’Azur)
Title: On extension of pluricanonical forms defined on the central fiber of a Kahler family
Abstract: In this talk, we will discuss several results related to Y.-T. Siu's conjecture of deformational invariance of plurigenera for Kahler families. The main tools is the L^2 Hodge decomposition for singular metrics. This is a joint work with Mihai Paun.
Date: 2021, December 1 (WED), 16:00-17:00 (Beijing)
Location: Tencent Meeting 790 778 401, password: 428474
Speaker: Haidong Liu (Sun Yat-sen University)
Title: A snapshot of Serrano's conjecture
Abstract: Strictly nef divisors sit between nef divisors and ample divisors. A natural question is how far are strictly nef divisors from being ample. In this area, Serrano's conjecture predicts that after twisting by the canonical divisor a little bit, the adjoint strictly nef divisor should be ample. In this talk, I will survey some recent progress on this conjecture. (Part of my works is jointed with Roberto Svaldi, part is jointed with Shin-ichi Matsumura.)
Date: 2021, November 24 (WED), 16:00-17:00 (Beijing)
Location: Zoom 839 1555 5712, password: 962388
Speaker: Jungkai Chen (Taiwan University)
Title: On extremal varieties of general type
Abstract: Among varieties of general of fixed dimension, there are some known (or expected) constraints on their birational invariant. It is particular interesting to study those varieties whose invariant achieve optimal values. For example, let d_X be the dimension of image of canonical map. If d_X=n, then one has that the canonical volume is greater than or equal to 2(p_g-n). If equality holds, then we call X to be a Horikawa variety. In this talk, we are going to describe various geometric properties of Horikawa varieties. If time permits, we will also address some relation with threefold of general type on the Noether line and irregular threefold on the Noether-Severi line.
Date: 2021, November 17 (WED), 16:00-17:00 (Beijing)
Location: Tencent Meeting 662 884 493, password: 369116
Speaker: Qifeng Li (IBS Center for Complex Geometry)
Title: Deformation rigidity of wonderful group compactifications
Abstract: For a complex connected semisimple linear algebraic group G of adjoint type, De Concini and Procesi constructed its wonderful compactification, which is a smooth Fano equivariant embedding of G enjoying many interesting properties. In this talk, we will discuss on the properties of wonderful group compactifications, especially the deformation rigidity of them. This is a joint work with Baohua Fu.
Date: 2021, November 11 (THU), 8:30-9:30 (Beijing) Note: unusual date and time
Location: Zoom Meeting 971 4102 0213, password: 200438
Speaker: Burt Totaro (University of California, Los Angeles)
Title: Varieties of general type with doubly exponential asymptotics
Abstract: We construct smooth projective varieties of general type with the smallest known volume and others with the most known vanishing plurigenera in high dimensions. The optimal volume bound is expected to decay doubly exponentially with dimension, and our examples achieve this decay rate. We also consider the analogous questions for other types of varieties. For example, in every dimension we conjecture the terminal Fano variety of minimal volume, and the canonical Calabi-Yau variety of minimal volume. In each case, our examples exhibit doubly exponential behavior. (Joint work with Louis Esser and Chengxi Wang.)
Date: 2021, November 3 (WED), 16:00-17:00 (Beijing)
Location: Tencent Meeting 184 722 527, password: 825882
Speaker: Jian Xiao (Tsinghua University)
Title: Geometric inequalities inspired by algebraic geometry
Abstract: Geometric inequalities reveal relation between different geometric invariants, such as volume, surface area, width, diameter, etc. By the correspondences between convexity and positivity, such as mixed volumes of convex bodies and intersection numbers of divisors, we present a series of new geometric inequalities inspired by positivity results in algebraic and analytic geometry.
Date: 2021, October 20 (WED), 16:00-17:00 (Beijing)
Location: Tencent Meeting 200 981 650, password: 505955
Speaker: Jie Liu (Chinese Academy of Sciences)
Title: Bigness of tangent bundles of Fano manifolds with zero dimensional VMRT
Abstract: It is expected that the bigness of tangent bundle is a quite restrictive property for Fano manifolds, especially for those of Picard number one. In this talk, I will present our recent first attempt to tackle this problem. More precise, we will consider Fano manifolds of Picard number one and having zero-dimensional VMRT, and it turns out that in this case only the quintic del Pezzo threefold has big tangent bundle. This is based on my recent joint work with Andreas Höring.
Date: October 13 (WED), 16:00-17:00 (Beijing)
Location: SCMS 102
Speaker: Yong Hu (Shanghai Jiao Tong University)
Title: Noether-Severi inequality and equality for irregular threefolds of general type
Abstract: We prove the optimal Noether-Severi inequality that $\vol(X) \ge \frac{4}{3} \chi(\omega_{X})$ for all smooth and irregular $3$-folds $X$ of general type over $\CC$. This answers an open question of Z. Jiang in dimension three. For those $3$-folds $X$ attaining the equality, we completely describe their canonical models and show that the topological fundamental group $\pi_1(X) \simeq \ZZ^2$. As a corollary, we obtain for the same $X$ another optimal inequality that $\vol(X) \ge \frac{4}{3}h^0_a(X, K_X)$ where $h^0_a(X, K_X)$ stands for the continuous rank of $K_X$, and we show that $X$ attains this equality if and only if $\vol(X) = \frac{4}{3}\chi(\omega_{X})$. This is a joint work with Tong Zhang.