Dongsoo Shin, Deformations of Sandwiched Surface Singularities and the Minimal Model Program
2023-02-28 10:00 - 10:50 (Asia/Seoul) | B236-1, IBS, Korea, Republic of
Speaker

Dongsoo Shin
Chungnam National U.

(This is a part of Seminars on Algebraic Surfaces and Related Topics.)
We investigate the correspondence between three theories of deformations of rational surface singularities: de Jong and van Straten’s picture deformations, Kollár’s P-resolutions, and Pinkham’s smoothings of negative weights. We provide an explicit method for obtaining, from a given deformation in one theory, deformations in other theories that parameterize the same irreducible components of the deformation space of the singularity. We employ the semi-stable minimal model program significantly for this purpose. We prove Kollár conjecture for various sandwiched surface singularities as an application. This is a joint work with Heesang Park.

JongHae Keum, Mori Dream Surfaces of General Type with pg=0
2023-02-28 11:10 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

JongHae Keum
KIAS

(This is a part of Seminars on Algebraic Surfaces and Related Topics.)
The Cox ring of a variety is the total coordinate ring, i.e., the direct sum of all spaces of global sections of all divisors. When this ring is finitely generated, the variety is called Mori dream (MD). A necessary condition for being MD is the finite generatedness of Pic(X), i.e., the vanishing of the irregularity. Smooth rational surfaces with big anticanonical divisor are MD. So are all del Pezzo surfaces of any degree. A K3 surface or an Enriques surface with Picard number at least 3 is MD iff its automorphism group is finite.
In this talk I will consider the case of surfaces of general type with pg=0, and provide several examples that are MD. I will also provide non-minimal examples that are not MD. This is a joint work with Kyoung-Seog Lee.

Insong Choe (최인송, Konkuk University)
2023-03-01 - 2024-02-28
Insong Choe (최인송)
Visitor (2023.3.1-2024.2.28) from Konkuk University
Office: B255

Yunhyung Cho, Monotone Lagrangian Tori in Fano Varieties
2023-03-02 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Yunhyung Cho
Sungkyunkwan University

This is a survey talk of current progress of mirror symmetry of Fano varieties. For a given smooth Fano variety X, it has been conjectured that there exists a Laurent polynomial called a (weak) Landau-Ginzburg mirror (or weak LG mirror shortly) which encodes a quantum cohomology ring structure of X. Tonkonog proved that one can find a weak LG mirror using a monotone Lagrangian torus in X. In this talk I will explain how to find a monotone Lagrangian torus using a Fano toric degeneration of X. If time permits, I will also describe a monotone Lagrangian torus in a given flag variety.

Donghoon Jang, Circle Actions on Almost Complex Manifolds with Isolated Fixed Points
2023-03-09 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Donghoon Jang
Pusan National University

We briefly review group actions on manifolds and equivariant cohomology, which is cohomology of a manifold with a group action. We review classification results for circle actions on various types of manifolds in low dimensions. An almost complex manifold is a manifold with a complex structure on its tangent bundle; every complex or symplectic manifold is almost complex. For an action of the circle group on a compact almost complex manifold that has isolated fixed points, we study its properties and discuss the classification when the number of fixed points is small.

Insong Choe, Subsheaves of Maximal Rank in a Symplectic and Orthogonal Bundle over a Curve
2023-03-30 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Insong Choe
Kunkuk University

We first review the known results on the Quot schemes on a smooth algebraic curve. Next we explain how they can be generalized to the Lagrangian Quot scheme, which parametrizes Lagrangian subsheaves on a symplectic vector bundle. Also we discuss the parallel results for orthogonal bundles. This will be an overview of the findings over the past 15 years in collaboration with George H. Hitching and D. Cheong.

Donggun Lee, Birational Geometry of Generalized Hessenberg Varieties and the Generalized Shareshian-Wachs Conjecture
2023-04-06 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Donggun Lee
IBS-CCG

Hessenberg varieties are subvarieties of flag varieties with interesting properties in both algebro-geometric and combinatorial perspectives. The Shareshian-Wachs conjecture connects their cohomology with the chromatic quasi-symmetric functions of the associated graphs, which are refinements of the chromatic polynomials. In this talk, we introduce generalized Hessenberg varieties and study their birational geometry via blowups. As a result, natural maps from Hessenberg varieties to projective spaces or the permutohedral varieties are decomposed into explicit blowups and projective bundle maps. As a byproduct, we also provide an elementary proof of the Shareshian-Wachs conjecture and its natural generalization. This is joint work with Prof. Young-Hoon Kiem.

Jihun Yum, Stochastic Bergman Geometry
2023-04-19 16:00 - 18:00 | B266, IBS, Korea, Republic of
Speaker

Jihun Yum
IBS-CCG

For a bounded domain Ω in Cn, let P(Ω) be the set of all (real) probability distributions on Ω. Then, in general, P(Ω) becomes an infinite-dimensional smooth manifold and it always admit a natural Riemannian pseudo-metric, called the Fisher information metric, on P(Ω). Information geometry studies a finite-dimensional submanifold M, which is called a statistical model, in P(Ω) using geometric concepts such as Riemannian metric, distance, connection, and curvature, to better understand the properties of statistical models M and provide insights into the behavior of learning algorithms and optimization methods.
In this talk, we first introduce a map Φ : Ω → P(Ω) and prove that the pull-back of the Fisher information metric on P(Ω) is exactly same as the Bergman metric of Ω. This map provides a completely new perspective that allows us to view Bergman geometry from a stochastical viewpoint. We will discuss the following 4 things.
1. The relation between Φ and the Kobayashi map ι : Ω → CP∞.
2. A Stochastic formula for the holomorphic sectional curvature of the Bergman metric.
3. A Stochastic condition for injectivity of a proper holomorphic surjective map between two bounded domains.
4. The central limit theorem on Ω.
This is a joint work with Gunhee Cho at UC Santa Barbara University.

Junyan Zhao, Moduli of Curves of Genus 6 and K-stability
2023-04-25 11:00 - 12:00 | on-line
Speaker

Junyan Zhao
University of Illinois Chicago

A general curve C of genus 6 can be embedded into the unique quintic del Pezzo surface X5 as a divisor of class -2KX5. This embedding is unique up to the action of the symmetric group S5. Taking a double cover of X5 branched along C yields a K3 surface Y. Thus the K-moduli spaces of the pair (X5, C) can be studied via wall-crossing and by relating them to the Hassett-Keel program for C and the HKL program for Y. On the other hand, X5 can be embedded in P1 × P2 as a divisor of class O(1,2), under which -2KX is linearly equivalent to OX(2,2). One can study the VGIT-moduli spaces in this setting. In this talk, I will compare these four types of compactified moduli spaces and their different birational models given by wall-crossing.

Hoseob Seo, On L2 Extension from Singular Hypersurfaces
2023-04-26 16:00 - 18:00 | B266, IBS, Korea, Republic of
Speaker

Hoseob Seo
IBS CCG

In L2 extension theorems from a singular hypersurface in a complex manifold, important roles are played by certain measures such as the Ohsawa measure which determine when a given function can be extended. We show that the singularity of the Ohsawa measure can be identiﬁed in terms of singularity of pairs from algebraic geometry. Using this, we give an analytic proof of the inversion of adjunction in this setting. Then these considerations enable us to compare various positive and negative results on L2 extension from singular hypersurfaces. In particular, we generalize a recent negative result of Guan and Li which places limitations on strengthening such L2 extension by employing a less singular measure in the place of the Ohsawa measure. This is joint work with Dano Kim.

Shigeru Mukai, Fano 3-folds, Lagrangian Fibration and Leech-K3 Geometry I
2023-05-09 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Shigeru Mukai
RIMS, Kyoto University

PDF file

Shigeru Mukai, Fano 3-folds, Lagrangian Fibration and Leech-K3 Geometry II
2023-05-09 15:00 - 16:00 | B236-1, IBS, Korea, Republic of

Shigeru Mukai, Fano 3-folds, Lagrangian Fibration and Leech-K3 Geometry III
2023-05-11 11:00 - 12:00 | B236-1, IBS, Korea, Republic of

Workshop on Moduli, K-stability, Fano varieties, and related topics
2023-05-15 - 2023-05-19 | IBS Science Culture Center, Daejeon, Korea, Republic of
Speakers
Arnaud Beauville (University of Nice)
Fabrizio Catanese (University of Bayreuth)
Thibaut Delcroix (University of Montpellier)
Kento Fujita (Osaka University)
Young-Hoon Kiem (KIAS)
Shigeru Mukai (RIMS, Kyoto University)
Yuri Prokhorov (Steklov Mathematical Institute)
Constantin Shramov (Steklov Mathematical Institute)
Abstracts
PDF File
Schedule
Day 1: May 15 (Monday)

~10:00
Registration

10:00~11:00
Shigeru Mukai
Moduli of curves of genus 11, and prime Fano 3-folds of adjacent poristic genera, I

11:00~11:20
Coffee Break

11:20~12:20
Kento Fujita
The Calabi problem for Fano threefolds, I

12:20~15:00
Lunch

15:00~16:00
Shigeru Mukai
Moduli of curves of genus 11, and prime Fano 3-folds of adjacent poristic genera, II

16:00~16:30
Coffee Break

16:30-17:30
Kento Fujita
The Calabi problem for Fano threefolds, II

18:00~20:00
Dinner for Speakers

Day 2: May 16 (Tuesday)

10:00~11:00
Yuri Prokhorov
On the classification of singular Fano threefolds, I

11:00~11:20
Coffee Break

11:20~12:20
Constantin Shramov
Conic bundles, I

12:20~15:00
Lunch

15:00~16:00
Yuri Prokhorov
On the classification of singular Fano threefolds, II

16:00~16:30
Coffee Break

16:30-17:30
Constantin Shramov
Conic bundles, II

Day 3: May 17 (Wednesday)

10:00~11:00
Arnaud Beauville
Symmetric tensors on the intersection of two quadrics and Lagrangian fibration, I

11:00~11:20
Coffee Break

11:20~12:20
Thibaut Delcroix
Effective K-stability of spherical varieties, I

13:20~20:00
Excursion

Day 4: May 18 (Thursday)

10:00~11:00
Arnaud Beauville
Symmetric tensors on the intersection of two quadrics and Lagrangian fibration, II

11:00~11:20
Coffee Break

11:20~12:20
Thibaut Delcroix
Effective K-stability of spherical varieties, II

12:20~15:00
Lunch

15:00~16:00
Young-Hoon Kiem
Counting surfaces in projective varieties, I

16:00~16:30
Coffee Break

16:30-17:30
Fabrizio Catanese
Status of the classification and old and new constructions for surfaces of general type with pg=q=2, I

18:00~20:00
Banquet

Day 5: May 19 (Friday)

10:00~11:00
Young-Hoon Kiem
Counting surfaces in projective varieties, II

11:00~11:20
Coffee Break

11:20~12:20
Fabrizio Catanese
Status of the classification and old and new constructions for surfaces of general type with pg=q=2, II

Organizers
Yongnam Lee (IBS-CCG/KAIST)
Jihun Park (IBS-CGP/POSTECH)
Main Hotel
Lotte City Hotel Daejeon (4-30 Doryong-dong, Yuseong-gu, Daejeon)
Registration
Please submit Google form by April 21.
More Information
• How to get to IBS-CCG
• From Hotel to IBS
• IBS Cafeteria (Lunch)

Changho Han, Compact Moduli of K3 Surfaces with a Given Nonsymplectic Cyclic Action
2023-05-23 16:30 - 17:30 | B236-1, IBS, Korea, Republic of
Speaker

Changho Han
University of Waterloo

To construct a moduli space which is itself a compactification of a given moduli space, one needs to enlarge the class of objects in consideration (e.g. adding certain singular curves to the class of smooth curves). After a brief review of the compactifications of the moduli of elliptic curves, I will generalize into looking at various compactifications of the moduli of K3 surfaces with nonsymplectic cyclic actions, and then discuss how those compactifications are birationally related to each other. As an application, I will apply this framework into Kondo’s moduli space of sextic K3 surfaces with Z/3Z action. Results come from joint works (in progress) with Valery Alexeev, Anand Deopurkar, and Philip Engel.

Paul-Andi Nagy, Introduction to Feix-Kaledin Construction
2023-06-14 15:30 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Paul-Andi Nagy
IBS-CCG

TBA

Minseong Kwon, Introduction to Grauert Tubes
2023-06-16 15:30 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Minseong Kwon
KAIST

TBA

Sung Gi Park, Kodaira Dimension and Hyperbolicity for Smooth Families of Varieties
2023-06-20 11:00 - 12:00 | on-line
Speaker

Sung Gi Park
Harvard University

In this talk, I will discuss the behavior of positivity, hyperbolicity, and Kodaira dimension under smooth morphisms of complex quasi-projective manifolds. This includes a vast generalization of a classical result: a fibration from a projective surface of non-negative Kodaira dimension to a projective line has at least three singular fibers. Furthermore, I will explain a proof of Popa’s conjecture on the superadditivity of the log Kodaira dimension over bases of dimension at most three. These theorems are applications of the main technical result, namely the logarithmic base change theorem.

Shin-Young Kim, Minimal Rational Curves on Complete Symmetric Varieties
2023-06-22 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Shin-Young Kim
IBS-CGP

We describe the families of minimal rational curves on any complete symmetric variety, and the corresponding varieties of minimal rational tangents. In particular, we prove that these varieties are homogeneous and that for non-exceptional irreducible wonderful varieties, there is a unique family of minimal rational curves. We relate these results to the restricted root system of the associated symmetric space. In particular, for certain Fano wonderful symmetric varieties, the VMRT has two connected components. This is a joint work with M.Brion and N. Perrin.

Shinnosuke Okawa, Moduli Space of Semiorthogonal Decompositions
2023-07-04 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Shinnosuke Okawa
Osaka University

Semiorthogonal decomposition (SOD) is a central notion in the study of triangulated categories. In particular, SODs of the bounded derived category of coherent sheaves of a variety (SODs of the variety, for short) have profound relations to its geometry. In this talk I discuss the moduli functor which classifies SODs of the fibers of smooth projective morphisms. The main result is that it is an algebraic space which is locally etale over the target of the morphism. I will explain the main points of the proof, various applications and open problems. This talk is based on the joint work arXiv:2002.03303 with Andrea Ricolfi and Pieter Belmans.

Shinnosuke Okawa, Semiorthogonal Decompositions and Relative Canonical Base Locus
2023-07-06 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Shinnosuke Okawa
Osaka University

Motivated by the DK hypothesis, some years ago I proved that SODs of the derived category of a smooth projective variety are strongly constrained by the base locus of the canonical linear system. In particular, this leads to the indecomposability of the derived category of varieties whose canonical bundles are globally generated (hence minimal). In this talk I will briefly recall this work and discuss its generalization to the relative settings. The latter implies new indecomposability results, including the case of minimal surfaces of positive irregularity. This talk is based on arXiv:2304.14048.

Qifeng Li, Rigidity of Projective Symmetric Manifolds of Picard Number 1 Associated to Composition Algebras
2023-07-11 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Qifeng Li
Shandong University

To each complex composition algebra A, there associates a projective symmetric manifold X(A) of Picard number 1. The vareity X(A) is closed related with Freudenthal’s Magic Square, which is a square starting from the adjiont varieties of F4, E6, E7 and E8. In a recent joint work with Yifei Chen and Baohua Fu, we obtain the deformation rigidity of X(A). In this talk, we will introduce the construction of X(A) from Freudenthal’s Magic Square, the geometric properties of them, and finally the deformation rigidity of X(A).

Chang-Yeon Chough, Introduction to algebraic stacks, I, II
2023-07-18 15:00 - 17:30 | B236-1, IBS, Korea, Republic of
Speaker

Chang-Yeon Chough
Sogang Univ.

This is an 8 hours long lecture series on algebraic stacks, which have become an important part of algebraic geometry (for example, in the study of moduli spaces) since Deligne and Mumford established the foundation of the theory of stacks. This crash course will be following roughly “Algebraic Spaces and Stacks” by Martin Olsson. Our main goal is to set up a foundation for the theory of algebraic stacks, so that the attendees would be able to use it in their own research in the future.

Chang-Yeon Chough, Introduction to algebraic stacks, III, IV
2023-07-19 10:00 - 12:15 | B236-1, IBS, Korea, Republic of

Patrick Brosnan, How Markman Saves the Hodge Conjecture (for Weil Type Abelian Fourfolds) from Kontsevich, I
2023-07-20 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Patrick Brosnan
University of Maryland

I’ll explain what I know about two very interesting pieces of work:
(1) Markman’s proof of the Hodge conjecture for Weil type abelian fourfolds of discriminant 1.
(2) Kontsevich’s tropical approach to looking for counterexamples to the Hodge conjecture for Weil type abelian varieties.
Then I’ll explain a simple observation of mine, which implies that Kontsevich’s approach cannot work for abelian fourfolds of any discriminant.
I’ll start out by explaining the statement of (1) precisely along with the geometric input that is required in (2). Then I’ll formulate my observation, which is really about why the discriminant determines what types of degenerations an abelian fourfold of Weil type can have. Along the way, I’ll take the opportunity to say a little bit about Markman’s proof of (1).

Patrick Brosnan, How Markman Saves the Hodge Conjecture (for Weil Type Abelian Fourfolds) from Kontsevich, II
2023-07-25 11:00 - 12:00 | B236-1, IBS, Korea, Republic of

Chang-Yeon Chough, Introduction to algebraic stacks, V, VI
2023-07-25 15:00 - 17:30 | B236-1, IBS, Korea, Republic of

Chang-Yeon Chough, Introduction to algebraic stacks, VII, VIII
2023-07-26 10:00 - 12:15 | B236-1, IBS, Korea, Republic of

Gunhee Cho, Non-measure Hyperbolicity of K3 and Enriques Surfaces
2023-08-03 16:00 - 17:00 | B266, IBS, Korea, Republic of
Speaker

Gunhee Cho
UCSB

By exploiting the upper semicontinuity of the Kobayashi-Eisenman pseudo volume (and pseudometric) under deformations of complex structures, we establish the non-measure hyperbolicity of K3 surfaces—which M. Green and P. Griffiths verified for certain cases in 1980—holds for all K3 surfaces. Our result provides a stronger condition than the Kobayashi non-hyperbolicity for any complex K3 surface, which had been demonstrated by L. Kamenova, S. Lu, and M. Verbitsky. An interesting consequence is the non-measure hyperbolicity of Enriques surfaces. This is the joint work with David R. Morrison.

Minseong Kwon, Spherical Geometry of Hilbert Schemes of Conics in Adjoint Varieties
2023-08-07 14:00 - 14:50 | B266, IBS, Korea, Republic of
Speaker

Minseong Kwon
KAIST

For each rational homogeneous space, the space of lines is now well-understood and can be described in terms of the induced group action. It is natural to consider rational curves of higher degree, and in this talk, we discuss geometry of conics in adjoint varieties, which are rational homogeneous spaces associated to simple Lie algebras. Each adjoint variety is equipped with a hyperplane distribution called the contact distribution, and we show that smooth conics transverse to the contact distribution form a homogeneous symmetric variety if the adjoint variety is of Picard number 1. This enables us to view the Hilbert scheme of conics as a spherical variety, and we compute its colored fan by using the description of the space of lines and the Hilbert-Chow morphism.
