## Activities

1. YMSC Courses Mini Courses Seminars Lectures Conferences

## Algebraic Geometry Seminar

Speaker：Jonathan Lai
Organizer：Caucher Birkar, Santai Qu, Bingyi Chen
Time：Every Thur. 15:30-16:30
Venue：Zoom Online

Schedule

Upcoming talks

Speaker: Jonathan Lai (Imperial College London)

Date: 26 Jan, 2023

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Compactified Mirror Families for Log Calabi-Yau Surfaces

Abstract: In joint work with Yan Zhou, we show that given a smooth affine log Calabi-Yau surface U with maximal boundary, the ring of regular functions has a canonical vector space basis, up to scaling. In order to obtain this, we investigate the mirror family of Looijenga pairs (Y,D), where U=Y\D, as constructed by Gross-Hacking-Keel. These mirror families come with a basis parameterized by the integral tropical points of U. The key point we make is identifying U as a fiber of the mirror family of (Y,D). This is achieved through canonically compactifying the mirror family and computing periods of the compactified family using techniques from Ruddat-Siebert.

Speaker: Sho Tanimoto (University of Wisconsin)

Date: 23 Feb, 2023

Time: 15:30-16:30

Zoom Meeting ID: TBA

Title: TBA

AbstractTBA

Speaker: Robert Lazarsfeld (Stony Brook University)

Date: March 2, 2023

Time: TBA

Zoom Meeting ID: TBA

Title: TBA

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Speaker: Stefan Kebekus (Albert-Ludwigs-Universität Freiburg)

Date: 9 March, 2023

Time: TBA

Zoom Meeting ID: TBA

Title: TBA

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Speaker: Bong Liang ()

Date: 16 March, 2023

Time: TBA

Zoom Meeting ID: TBA

Title: TBA

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Speaker: Mao Sheng ()

Date: 30 March, 2023

Time: TBA

Zoom Meeting ID: TBA

Title: TBA

AbstractTBA

Speaker: Kris Shaw (University of Oslo)

Date: April 13, 2023

Time: TBA

Zoom Meeting ID: TBA

Title: TBA

AbstractTBA

Past talks

Speaker: Jordan Ellenberg (University of Wisconsin)

Date: 19th January, 2023

Time: 9:00-10:00 am

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Sparsity of rational points on moduli spaces

Abstract: Varieties, if they are at all complicated, are expected to have very few rational points. This might mean “there are only finitely many rational points” or “the rational points are contained in a proper closed subvariety.” Statements like these are extremely difficult to prove in any degree of generality, with Faltings’ finiteness theorem for rational points on high-genus curves a notable exception. In this talk, I’ll explain how to prove theorems about “sparsity” of rational points, a weaker notion which asks that the number of such with height less than B grows more slowly than any power of B; it turns out that theorems of this kind can be proven for many varieties appearing as moduli spaces (though these still make up a very special subclass among the varieties we’d like to know about.) The two main ingredients are: 1) a classical trick allowing us to “trade” a single equation that’s hard to solve for many equations that are easier to solve; 2) theorems of Heath-Brown type which provide bounds on points of bounded height on varieties which are uniform in the sense that they barely depend on what variety you’re studying. These are very useful when proving theorems about varieties of which we know almost nothing. This is joint work with Brian Lawrence and Akshay Venkatesh.

Speaker: Zhan Li (Southern University of Science and Technology)

Date:  Jan 122023

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title Morrison-Kawamata cone conjecture for Calabi-Yau fiber spaces

Abstract: In this talk, I will explain the relationship between the Morrison-Kawamata cone conjecture for Calabi-Yau fiber spaces and the existence of Shokurov polytopes. For K3 fibrations, this enables us to establish the existence of (weak) fundamental domains for movable cones. This is joint work with Hang Zhao.

Speaker: Guillaume Tahar (BIMSA)

Date: 5 January, 2023

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

TITLE: Isoresidual fibration and resonance arrangements

ABSTRACT: Meromorphic 1-forms on the Riemann sphere with prescribed orders of singularities form strata endowed with period coordinates. Fixing residues at the poles defined a fibration of any stratum to the vector space of configurations of residues. For strata of 1-forms with only one zero, the isoresidual fibration is a cover of the space of configurations of residues ramified over an arrangement of complex hyperplanes. We give a formula to compute the degree of this cover and investigate its monodromy.The results are obtained using the dictionary between complex analysis and flat geometry of translation surfaces. The qualitative geometry of the latter translation surfaces is then classified by decorated trees, reducing the computation of the degree of the cover to a combinatorial problem. For strata with several zeroes, isoresidual fibers are complex manifolds endowed with a matrix-valued meromorphic differential. Singularities of the differential give insights on the topological invariants of the fibers. This is a joint work with Quentin Gendron.

Speaker: Takumi Murayama (Purdue University)

Date:  December 22, 2022

Time: 8:30-9:30 am

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero

Abstract: In 2010, Birkar, Cascini, Hacon, and McKernan proved a relative version of the minimal model program for projective morphisms of complex quasi-projective varieties, called the relative minimal model program with scaling. Their result is now fundamental to our understanding of the birational classification of quasi-projective varieties and has numerous applications. In this talk, I will discuss recent joint work with Shiji Lyu that establishes the relative minimal model program with scaling for excellent schemes, excellent algebraic spaces, and analytic spaces simultaneously in equal characteristic zero. This not only recovers previous results for complex varieties, complex algebraic spaces, and complex analytic spaces, but also greatly extends the scope of the relative minimal model program with scaling to a broader class of geometric spaces, including formal schemes, rigid analytic spaces, and Berkovich spaces, all in equal characteristic zero. Our results for (non-algebraic) schemes and rigid analytic spaces were previously only known in dimensions 3 and 2, respectively, and our results for formal schemes and Berkovich spaces are completely new.

Speaker: Yoshinori Gongyo (The University of Tokyo)

Date: December 15, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Generalized complexity and Mukai’s type conjecture.

Abstract: We discuss some variants of Mukai’s conjecture for the characterization of projective spaces. We discuss the relation of the complexity of generalized pairs and such kind conjectures. I will talk based on the joint work with Joaquin Moraga.

Speaker: Masafumi Hattori (Kyoto University)

Date: 8 Dec, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: CM minimization and special K-stability

Abstract: Odaka proposed a conjecture predicting that the degrees of CM line bundles for families with fixed general fibers are strictly minimized if the special fibers are K-stable. This conjecture is called CM minimization and a quantitative strengthening of the conjecture of separatedness of moduli spaces of K-stable varieties (K-moduli). This conjecture was already shown for K-ample (Wang-Xu), Calabi-Yau (Odaka) and Fano varieties (Blum-Xu). In this talk, we introduce a new class, special K-stable varieties, and settle CM minimization for them, which is a generalization of the above results. In addition, we would like to explain an important application of this, construction of moduli spaces of uniformly adiabatically K-stable klt trivial fibrations over curves as a separated Deligne-Mumford stack in a joint work with Kenta Hashizume to appear. This is based on arXiv:2211.03108.

Speaker: Caucher Birkar (Tsinghua University)

Date: Dec 1, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Stable minimal models and their moduli.

Abstract: In this talk I will introduce stable minimal models and discuss recent results on existence of moduli spaces of such models. The moduli theory works for models of arbitrary non-negative Kodaira dimension.

Speaker: Ivan Fesenko (The University of Warwick)

Date: 24 Nov, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Refined Zariski cohomology numbers and applications

Abstract: Zariski cohomology groups of number fields (d=1) or arithmetic surfaces (d=2) miss the right duality with respect to i->d-i. We still do not know how to define the correct object H^1(D) for a fractional divisor D of a number field. However, following Iwasawa, Tate, Weil,..., A. Borisov it is possible to define cohomology numbers h^i when d=1, and get the right duality.

Towards d=2 our recent joint work with W. Cerniawska and P. Dolce defines and studies h^i for surfaces, using two-dimensional geometric adeles and selective integration over their subquotients. This leads to a beautiful adelic formula for the Euler characteristic of the surface and an extended divisor on it, and to a new higher adelic interpretation of the Arakelov intersection pairing. This study also provides a missing ingredient in the adelic study to the rank part of the Birch-Swinnerton-Dyer-Tate conjecture for elliptic curves over number fields.

Speaker: Andrea Fanelli (Université de Bordeaux)

Date: 17 Nov, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Fano varieties, rational curves and rational simple connectedness

Abstract: The goal of this talk is to survey « higher dimensional » notions of rational connectedness for Fano varieties, namely rational simple connectedness and higher Fano conditions. The former involves rational connectedness of some moduli spaces of rational curves and existence of « very twisting » surfaces, while the latter is purely numerical. I will explain how these conditions are related, at least conjecturally, to the existence of rational sections for Fano fibrations over surfaces.

Speaker: Jian Xiao (Tsinghua University)

Date: Nov 3, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Geometric inequalities inspired by algebraic/analytic geometry

Abstract: Geometric inequalities reveal relation between different geometric quantities, 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 from algebraic and analytic geometry.

Speaker: Joe Waldron (Michigan State University)

Date: October 27, 2022

Time: 8:30-9:30 am

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Birational geometry in mixed characteristic

AbstractI will describe recent progress in the log minimal model program for threefolds in mixed characteristic. In particular I will discuss new techniques which enable us to circumvent the failure of Kodaira vanishing in mixed characteristics, which is one of the main obstacles to working in this situation.

Speaker: Santai Qu (Tsinghua University)

Date:  Oct 13, 2022

Time: 15:30 - 16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Bounding irrationality of degenerations of Fano fibrations

Abstract: In this talk, I will introduce a recent result about bounding degrees of irrationality of degenerations of klt Fano fibrations of arbitrary dimensions. This proves the generically bounded case of a conjecture proposed by C. Birkar and K. Loginov for log Fano fibrations of dimensions greater than three. Our approach depends on a method to modify the klt Fano fibration to a toroidal morphism of toroidal embeddings with bounded general fibres. Moreover, we show that every fibre of the toroidal morphism is bounded and has mild singularities if we replace the birational modifications by alterations. This is a joint work with Prof. C. Birkar.

Speaker: Tristan Collins (Massachusetts Institute of Technology)

Date:  Oct 6, 2022

Time: 8:00-9:00 am

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Complete Calabi-Yau Metrics on the Complement of Two Divisors

Abstract: In 1990 Tian-Yau proved that if Y is a Fano manifold and D is a smooth anti-canonical divisor, the complement X=Y\D admits a complete Calabi-Yau metric. A long standing problem has been to understand the existence of Calabi-Yau metrics when D is singular. I will discuss the resolution of this problem when D=D_1+D_2 has two components and simple normal crossings. I will also explain a general picture which suggests the case of general SNC divisors should be inductive on the number of components. This is joint work with Y. Li.

Speaker: Jean-Louis Colliot-Thelene (CNRS et Université Paris-Saclay)

Date: September 29, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title : Zero-cycles on geometrically rational surfaces

Abstract: Let k be a field of characteristic zero. In 1974, D. Coray

showed that on a smooth cubic surface over k, with a point over a field extension K of k of degree prime to 3, there exists such a point over a field extension K over k of degree 1, 4 or 10 (as of today unsurpassed result). We show how a combination of generisation, specialisation, Bertini theorems and "large" fields avoids lengthy considerations of degeneracy cases in Coray's proof, and leads to more results. For smooth cubic surfaces with a rational point, we then show that any zero-cycle of degree at least 10 is Chow-rationally equivalent to an effective cycle. Using the k-birational classification of geometrically rational k-surfaces (Enriques, Manin, Iskovskikh, Mori), by a case by case discussion, we show how these two results have analogues for arbitrary smooth, projective, geometrically rational surfaces.

Speaker: Mircea Mustata (University of Michigan)

Date: September 22, 2022

Time: 10:00–11:00 am

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Invariants of singularities from D-module theory and higher versions of rational and Du Bois singularities

Abstract: I will describe an invariant of hypersurface singularities, Saito's minimal exponent, that can be viewed as a refined

version of the log canonical threshold. I will also briefly discuss a recent generalization to locally complete intersections.

Finally, I will explain the relevance of these invariants to k-Du Bois and k-rational singularities, two "higher order" versions of

the classical notions of Du Bois and rational singularities.

Speaker: Costya Shramov (HSE University)

Date: September 15, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Conic bundles

AbstractConsider a conic bundle over a smooth incomplete curve C, i.e. a smooth surface S with a proper surjective morphism to C such that the push-forward of the structure sheaf of S coincides with the structure sheaf of C, and the anticanonical class of S is ample over C. If the base field is perfect, a conic bundle always extends to a conic bundle over a completion of C. I will tell about a necessary and sufficient condition for the existence of such an extension in the case of an arbitrary base field. The talk is based on a joint work in progress with V.Vologodsky.

Speaker: Osamu Fujino (Kyoto University)

Date: September 8, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Abstract: In this talk, I will explain adjunction and inversion of adjunction for log canonical centers of arbitrary codimension. Our result depends on the existence of log canonical modifications and the theory of basic slc-trivial fibrations. Hence it depends on the minimal model program for log canonical pairs and the theory of variations of mixed Hodge structure. This talk is based on a joint work with Kenta Hashizume.

Speaker: Yingying Zhang (Tsinghua University)

Date: September 1, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Deformation of Fano manifolds

Abstract: In this talk, we will discuss the existence of the Kahler-Einstein metrics on a Fano manifold under the deformation of complex structures. This leads to understanding of the Weil-Petersson metric on the moduli space of Fano Kahler-Einstein manifolds. We will also talk about a plurisubharmonic function on the Teichmuller space of Kahler-Einstein manifolds of general types. The talk is based on the joint work with H.-D. Cao, X. Sun and S.-T. Yau.

Speaker: Shin-ichi Matsumura (Tohoku University)

Date: June 16, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Abundance theorem for minimal compact Kaehler manifolds with vanishing second Chern class

Abstract: In this talk, I would like to discuss the abundance conjecture for minimal compact Kaehler manifolds with nef cotangent bundle. I explain that the second Chern class vanishes if and only if the canonical bundle has the numerical dimension 0 or 1. In this case, I show that the canonical bundle is semi-ample. If time permits, I give a relation between the variation of the fibers of the Iitaka fibration and a certain semi-positivity of the cotangent bundle.

Speaker: Yu Zou (Fudan University)

Date: June 9, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: On explicit birational geometry of weak Q-Fano 3-folds

Abstract: For a canonical weak Q-Fano 3-fold, we investigate the upper bound of the anti-canonical volume -K^3 and birationality of the m-th anti-canonical map induced by |-mK|. In this talk, I will report on my recent work concerning these problems. This is joint work with Chen Jiang.

Speaker: Professor Xiaotao Sun (Tianjin University)

Date: June 2, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Slope inequalities and applications

Abstract: We prove several slope inequalities for a relative minimal surface fibration in positive characteristic. As an application, we prove that $\chi(\mathcal O_S)>0$ holds for all smooth projective minimal surfaces $S$ of general type, which answers completely a question of Shepherd-Barron.

Speaker: Professor Yi Hu (The University of Arizona)

Date: May 26, 2022

Time: 8:30--9:30, 9:50--10:50, 11:20--12:20 (3 lectures).

Zoom Meeting ID: 276 366 7254

Passcode: YMSC

Lecture 1: Equations of canonical birational models of thin Schubert cells.

Time: 8:30--9:30

Abstract: By Mnev's universality, thin Schubert cells exhibit all possible singularities in algebraic geometry. In this first lecture, we introduce certain standard birational models of thin Schubert cells and describe their defining equations in certain smooth ambient schemes. These equations may be considered as the normalized equations of singularities. Based upon these normalized equations, we will then describe three sequential blowups of the smooth ambient schemes, collectively, called h-blowups.

Lecture 2: Birational transforms of thin Schubert cells after h-blowups

Time: 9:50--10:50

Abstract: We will continue to study the h-blowups, and more importantly, describe certain birational transforms of thin Schubert cells under the h-blowups, in terms of defining equations.

Lecture 3: Smoothness and applications to resolution of singularities

Time: 11:20--12:20

Abstract: Based upon the descriptions of their defining equations, we will prove that the birational transforms of thin Schubert cells after the final h-blowup are smooth. This immediately implies that every singularity type over a perfect field admits a resolution. By reviewing Lafforgue's version of Mnev's universality, we will apply the above to prove that every affine or projective singular variety over a perfect field admits a resolution.

Speaker: Mihai Paun (Universität Bayreuth)

Date: 19 May, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Hermite-Einstein metrics in singular setting and applications

Abstract: We will report on a ongoing joint project with J. Cao, P. Graf, P. Naumann, T. Peternell and X. Wu. Our main goal is to revisit and improve an important result due to S. Bando and Y.-T. Siu. I will discuss some of the aspects of the proof, as well as an application to Chern classes inequalities (which initially was the principal motivation for our work).

Note:  https://cloud.tsinghua.edu.cn/f/2126e5b2d547475db4df/

Speaker: Richard P W Thomas (Imperial College London)

Date: 12 May, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Rank r DT theory from rank 1

Abstract: Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants. By the MNOP conjecture these rank 1 “abelian” invariants are determined by the GW invariants of X. Along the way we also express rank r DT invariants in terms of invariants counting “D4-D2-D0 branes”: rank 0 sheaves supported on surfaces in X. These invariants are predicted by physicists’ S-duality to be governed by (vector-valued, mock) modular forms.

Speaker: Bernd Sturmfels (MPI Leipzig and UC Berkeley)

Date: 5 May, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Beyond Linear Algebra

Abstract: Our title challenge the audience to venture beyond linear algebra when designing models and numerical algorithms for solving them. Algebraic geometry is the key to this. We discuss recent advances in the study of critical point equations from optimization and statistics, and we explore the role of algebra in the study of linear PDE with constant coefficients.

Reference: arxiv.org/abs/2108.09494

Speaker: Junpeng Jiao (The University of Utah)

Date: 28 April, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Boundedness of polarised Calabi-Yau fibrations.

Abstract: In this talk, we investigate the boundedness of good minimal models with intermediate Kodaira dimensions. We prove that good minimal models are bounded modulo crepant birational when the base (canonical models) are bounded and the general fibers of the Iitaka fibration are in a bounded family of polarized Calabi-Yau pairs. As a corollary, we prove that smooth Calabi-Yau varieties with a polarized fibration structure are bounded modulo flop.

Speaker: Radu Laza (Stony Brook University)

Date: 21 April, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Deformations of singular Fano and Calabi-Yau varieties

Abstract: It is well known that Calabi-Yau manifolds have good deformation theory, which is controlled by Hodge theory. By work of Friedman, Namikawa, M. Gross, Kawamata, Steenbrink and others, some of these results have been extended to Calabi-Yau threefolds with canonical singularities. In this talk, I will report on further extensions in two directions: in dimension 3, we sharpen and clarify some of the existing results, and, secondly, we obtain some higher dimensional analogues. I will also briefly explain the related case of Fano varieties, where stronger results hold. One surprising aspect of our study is the role played by higher du Bois and higher rational singularities, notions that were recently introduced by Mustata, Popa, Saito and their collaborators. This is joint work with Robert Friedman.

Speaker: Caucher Birkar (Tsinghua University)

Date: 14 April, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Volume of canonical Fano 4-folds

Abstract: In this talk I will describe a result on effective bound for the anti-canonical volume of Fano 4-folds with canonical singularities.

Speaker: Jintai Ding (YMSC, Tsinghua University Ding Lab, BIMSA)

Date: 7 April, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Multivariate public key cryptosystems – a Post-quantum Candidate

Abstract: Multivariate public key cryptosystems (MPKC) are one of the four main families of post-quantum public key cryptosystems. In a MPKC, the public key is given by a set of quadratic polynomials and its security is based on the hardness of solving a set of multivariate polynomials. This lecture gives a general introduction to the multivariate public key cryptosystems including the main designs, the main attack tools and the mathematical theory behind. We will present state of the art research in the area.

Speaker: Yuri Manin (Max Planck Institute for Mathematics)

Date: 31 March, 2022

Time: 16:30-17:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Introduction to algebraic supergeometry and supersymmetric (susy) modular operads

Abstract: Main goal of this talk is an expostion of the necessary background for presentation of the recent research article [KeMaWu22]. The basic notions of algebraic supergeometry can be arranged into a system of definitions and constructions parallel to the one of Grothendieck schemes. Drastically new phenomena appear only when we begin studying the superversions of tangent vector fields and, dually, difierential 1-forms.

Reference: [KeMaWu22] E. Keßler, Yu. Manin, Y. Wu. Moduli spaces of SUSY curves and their operads. arXiv:2202.10321. 21 pp.

Speaker: Hamid Abban (Loughborough University)

Date: 24 March, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: On K-stability of Fano varieties

Abstract: K-stability is an algebraic notion that detects the existence of Kahler-Einstein metrics on Fano varieties. I will give an overview of the theory of K-stability from a birational geometer’s perspective. Then I go through the existing methods of verifying K-stability for a given Fano variety before introducing the new method (joint work with Ziquan Zhuang) which is based on linear algebra and induction. Several results will be illustrated.

Speaker: Fujita Kento (Osaka University)

Date: 17 March, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: The Calabi problem for Fano threefolds

Abstract: There are 105 irreducible families of smooth Fano threefolds, which have been classified by Iskovskikh, Mori and Mukai. For each family, we determine whether its general member admits a Kaehler-Einstein metric or not. This is a joint work with Carolina Araujo, Ana-Maria Castravet, Ivan Cheltsov, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Suess and Nivedita Viswanathan.

SpeakerPaolo Cascini (Imperial College London)

Date: 10 March, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: On the Minimal Model for algebraically integrable foliations.

Abstract: Every fibration, or more in general, every dominant rational map between normal varieties, defines a natural foliation, which is called algebraically integrable. The canonical sheaf of such a foliation behaves, in many aspects, as the canonical sheaf of a normal variety. I will describe some recent results in this direction, such as a cone theorem, and some applications on the canonical bundle formula. In particular, this provides a proof of a conjecture by Shokurov. This is joint work with Ambro, Shokurov and Spicer.

Speaker: Ivan Cheltsov (The University of Edinburgh)

Date: 3 March, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: Equivariant birational geometry of three-dimensional projective space.

Abstract: We will describe G-equivariant birational geometry of the three-dimensional projective space in the case when G is a finite group that G does not fix a point and does not leave a pair of skew lines invariant. In particular, we describe all possibilities for G such that the projective space is G-rigid, i.e. it is not G-birational to a conic bundles and it is not G-birational to a del Pezzo fibration. For these groups, we will explicitly describe all G-Mori fibre spaces that are G-birational to the projective space. This is a joint project with Arman Sarikyan.

Speaker: Meng Chen (Fudan University)

Date: Feb 24, 2022

Time: 15:30-16:30

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Title: On explicit birational geometry of higher dimensional varieties

Abstract: In this seminar, I will talk about some new progress on estimating the lower bound of canonical volumes and the upper bound of canonical stability indices for higher dimensional varieties of general type. I will mainly introduce the construction of examples with small volumes.

Speaker: Professor Shigeru Mukai (Research Institute for Mathematical Sciences)

Title: Prime Fano threefolds and Leech-like lattices

Date: Dec 23, 2021

Time: 15:30-16:30

Zoom Meeting ID: 849 963 1368

Passcode: YMSC

Abstract: I computed the automorphism groups of certain two models of supersingular K3 surfaces with Artin invariant one in Beijing 2019. One is over the binary and the other over ternary field. Recently I found two more models over the binary and quaternary fields. I will report their relation with prime Fano threefolds classified into 10 deformation types of degree (-K)^3=2g-2 with g=2,…,10 and 12, by Fano-Iskovskih. Our four models are related with g=7, 8, 9, 10 via three Leech-like lattices; Leech, Barnes-Wall and Coxeter-Todd. Another Leech-like lattice is related with g=12 via the regular 4-polytope {3,3,5}.

Speaker: Professor De-Qi Zhang (National University of Singapore)

Title: Jordan Property for Automorphism Groups of Compact Complex Varieties

Date: Dec 16, 2021

Time: 15:30 - 16:30

Zoom Meeting ID: 849 963 1368

Passcode: YMSC

Abstract: We report some recent progress on the Jordan property of the automorphism group Aut(X) of an algebraic variety X or a compact complex manifold (close to be Kahler), i.e., whether every finite subgroup of it is almost commutative.

Speaker: Professor Jungkai Chen (National Taiwan University)

Title: On recent development of geography of varieties of general type

Date: Dec 9, 2021

Time: 15:30-16:30

Zoom Meeting ID: 849 963 1368

Passcode: YMSC

Abstract: Among varieties of general of fixed dimension, there are some known (or expected) constraints on their birational invariant. It is particularly interesting to study those varieties whose invariants achieve optimal values. In this talk, I am going to introduce some recent developments along this direction. One type of result is about asymptotic behavior of invariants, as dimension approaches infinity.

The other type of result is about explicit description of varieties with extremal invariants. 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. Various geometric properties of Horikawa varieties can be described very explicitly in the joint work with Bangere and Gallego. Various geometric structure of threefolds of general type with extremal invariants will be introduced as well.

Speaker: Professor Victor Batyrev (University of Tuebingen)

Title: On combinatorial and algorithmic aspects of the Minimal Model Program

Date: Dec 2, 2021

Time: 15:30-16:30

Zoom Meeting ID: 849 963 1368

Passcode: YMSC

Abstract: The talk is devoted to a simple combinatorial algorithm for explicit constructing minimal models of complex algebraic varieties defined by Laurent polynomials in algebraic torus of arbitrary dimension d>1. The considered algorithm proposes explicit combinatorial formulas for the Hodge-theoretic topological invariants of obtained minimal models. In the talk I will explain the main ideas of the algorithm and the corresponding combinatorial formulas in a way comprehesible for undergraduate students.

Speaker: Professor Yi Hu, (The University of Arizona)

Title: Local Resolution of Singularities

Date: Nov 25, 2021

Time: 11-12 am

Zoom Meeting ID: 981 9384 1924

Passcode: YMSC

Abstract: Mnev's universality theorem asserts that every singularity type over the ring of integers appears in some thin Schubert cell of the Grassmannian Gr(3,E) for some vector space E. We construct sequential blowups of Gr(3,E) such that certain induced birational transforms of all thin Schubert cells become smooth over prime fields. This implies that every singular variety X defined over a prime field admits local resolutions. For a singular variety X over a general perfect field k, we spread it out and deduce that X/k admits local resolution as well.

Speaker: Professor Burt Totaro (UCLA)

Title: Varieties of general type with doubly exponential asymptotics

Date: 18 Nov, 2021

Time: 9-10 am

Zoom Meeting ID: 843 5849 9857

Passcode: YMSC

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.)

Speaker: Professor Yuri Prokhorov (Steklov Mathematical Institute & HSE University)

TitleTowards a classification of Q-Fano threefolds

DateNov 11, 2021

Time1530-1630

Zoom Meeting ID: 849 963 1368

Passcode: YMSC

AbstractWe survey recent progress in  classification of singular Fano threefolds with special emphasis on birational transformations between them.

Speaker: Sean Keel (University of Texas)

Title: Mirror symmetry, analytic disks, and (if time permits) moduli of log CY pairs.

Date: Nov 4, 2021

Time: 2200-2300

Zoom Meeting ID850 5273 8208
Passcode
YMSC

Abstract: I will explain my recent construction, joint with Tony Yu, of the mirror to an affine log CY variety of an algebra with a canonical Mori theoretic basis and structure constants given by naive counts of Berkovich analytic disks. My main goal will be to convince you of the simplicity of the construction, both conceptually, and in technical detail. I will assume a basic understanding of algebraic geometry, but no background in Berkovich geometry. If time permits, I will explain the connection to our conjecture, joint with Paul Hacking, that very natural moduli spaces of log CY pairs are (up to finite cover) toric varieties.

Speaker: Professor Valery Alexeev (University of Georgia)

Title:  Compact moduli spaces of K3 surfaces
Date: October 28, 2021
Time: 9:00-10:00 am
Zoom Meeting ID
228 011 0844
Passcode
YMSC

Abatract: I will explain recent results on modular, geometrically meaningful compactifications of moduli spaces of K3 surfaces, most of which are joint with Philip Engel. A key notion is that of a recognizable divisor: a canonical choice of a divisor in a multiple of the polarization that can be canonically extended to any Kulikov degeneration. For a moduli of lattice-polarized K3s with a recognizable divisor we construct a canonical stable slc pair (KSBA) compactification and prove that it is semi toroidal. We prove that the rational curve divisor is recognizable, and give many other examples.

Speaker: Prof. Yujiro Kawamata, University of Tokyo

Title: Semi-orthogonal decomposition and smoothing

Date: October 21, 2021
Time: 3:30-4:30
Abstract: A sheaf F on a normal projective variety X is called pre-tilting if all higher self-extensions vanish.  F generates a subcategory in a bounded derived category of coherent sheaves D(X) which is equivalent to a bounded derived category D(R) of finitely generated modules over a finite dimensional associative algebra R = End(F), and D(X) has a corresponding semi-orthogonal decomposition.  I will investigate what happens to F, R and D(X) if X has a deformation to a smooth projective variety Y.  I take examples where X is a weighted projective surface which are deformed to a projective plane or a quadric Y by Q-Gorenstein smoothing.  Then I will prove in general that a pre-tilting object F arising from a singularity which allows Q-Gorenstein smoothing is deformed to a direct sum of exceptional vector bundles found by Hacking.  In particular, R is deformed to a direct product of matrix algebras.

Speaker: Yitwah Cheung, Tsinghua University

Date: September 30, 2021
Time: 3:30-4:30
Abstract: The Hodge bundle of a Riemann surface of genus g is a complex vector bundle of rank g over the moduli space of closed Riemann surfaces of genus g.  On this space, there is an action by SL(2,R) that is poorly understood in algebraic geometric terms.  The recent work of Eskin and Mirzakhani completed the (initial phase of the) classification of all orbit closures and invariant probability measures of this action using methods of ergodic theory.  It would be desirable to extend the Teichmuller action to one on an appropriate bundle over the moduli space of a higher dimensional complex manifold, which would then allow the methods of ergodic theory to bear fruit.  In this talk, my goal is to give an idea of the new area of research that would open up, should this extension be possible.

Title:  Kohn-Rossi cohomology, complex Plateau problem and Rigidity of CR morphisms

Speaker: Stephen Yau (Tsinghua University)

Date: 2021-9-23

Abstract: In this talk, we investigate the relationship between compact strongly pseudoconvex CR manifolds and the singularities of their Stein fillings. We first compute the dimensions of Kohn-Rossi cohomology groups with values in holomorphic vector bundles in terms of local cohomology groups. As an application, we solve the classical complex Plateau problem for compact strongly pseudoconvex CR manifolds when its Stein fillings has only isolated complete intersection singularities. Finally, using Kohn-Rossi cohomology, we obtain some sufficient conditions for the non-existence of CR morphisms between the links of isolated complete intersection singularities with different embedding codimensions. This is a joint work with Xiankui Meng.

Title: Hodge ideals and roots of the Bernstein-Sato polynomial

Speaker: Bingyi Chen (Tsinghua University)

Date: 2021-9-16

Abstract: Hodge ideals, which are developed by Mustata and Popa, are important invariants of hypersurface singularities that arise naturally from Saito's theory of mixed Hodge modules. There are two diffierent approaches to study Hodge ideals: birational geomerty method by means of log resolution and D-module method by means of V-filtration. The theory of Hodge ideals connects these two diffierent fields and leads to a number of striking applications. In this talk, I will introduce the theory of Hodge ideals and talk about its application on the bound of roots of reduced Bernstein-Sato polynomial.

Title:  Measure the positivity of tangent bundles of Fano varieties

Speaker: Jie Liu  (Chinese Academy of Sciences)

Date: 2021-9-09

Abstract: While the properties of the anticanonical divisor $-K_X$ of a Fano variety $X$ and its multiples have been studied by many authors, the positivity of the tangent bundle $T_X$ is much more elusive. In this talk, I will introduce some invariants to measure the positivity of the tangent bundles of Fano varieties via their anti-canonical bundles and then I will discuss various interesting examples to illustrate the general situations. This is based on joint works with Baohua Fu, Andreas Höring and Feng Shao.

Title: Explicit boundedness of canonical Fano 3-folds: known results and open problems

Speaker: Chen Jiang (Fudan University)

Date2021-9-02
Abstract:  Motivated by the classification of canonical Fano 3-folds, we are interested in boundedness results on diffrent kinds of canonical Fano 3-folds, such as anticanonical systems, indices, degrees, and so on. I will summarize known results with some progress (based on joint works with Meng Chen and Yu Zou) and open problems in this area.

Title: Recent progress towards Shokurov’s ACC conjecture for mld

Speaker: Dr. Jihao Liu, Northwest Univ

Date2021-8-19

Abstract: Shokurov's ascending chain condition (ACC) conjecture for minimal log discrepancies (mlds) is a core conjecture in birational geometry and has a deep relationship with the minimal model program. In particular, in 2004, Shokurov shows that the ACC conjecture and the lower-semicontinuity conjecture for mlds will imply the termination of flips and will thus complete the minimal model program. In this talk, I will discuss some recent progress towards Shokurov's ACC conjecture and some related questions. Some parts of this talk are joint works with Jingjun Han, V.V. Shokurov, Liudan Xiao, and Lingyao Xie.

Title: Sarkisov program and automorphism of affine spaces

Speaker: Yifei Chen (Chinese Academy of Sciences)

Date2021-8-12

Abstract: Sarkisov program roughly says that birational maps of Mori fiber spaces can be factored through four basic links. In the talk, I will briefly introduce Sarkisov program, as well as its application for Cremona groups of rank 2 and automophisms of affine plane.