Center for Complex Geometry

Shigeru Mukai, Fano 3-folds, Lagrangian Fibration and Leech-K3 Geometry II
2023-05-09 15:00 - 16:00 (Asia/Seoul) | 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 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.

Kyeong-Dong Park, K-stability of Fano Spherical Varieties, I
2023-08-07 15:00 - 16:30 | B266, IBS, Korea, Republic of
Speaker

Kyeong-Dong Park
Gyeongsang National University

The aim of this seminar is to provide participants with a comprehensive understanding of the paper “K-stability of Fano spherical varieties” by Thibaut Delcroix. For a reductive algebraic group G, a normal G-variety is called spherical if it contains an open B-orbit, where B is a fixed Borel subgroup of G. The class of spherical varieties contains several important families which were studied independently, for example, toric varieties, rational homogeneous varieties, group embeddings, horospherical varieties, symmetric varieties, and wonderful varieties. They are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties. We discuss Delcroix’s combinatorial criterion for K-stability of a smooth Fano spherical variety in terms of the barycenter of its moment polytope with respect to the Duistermaat-Heckman measure and data associated to the corresponding spherical homogeneous space.
(1) Spherical varieties, colored fans, and algebraic moment polytopes
(2) Criterion for K-stability of Fano spherical varieties and its applications
(3) Equivariant test configurations with horospherical central fiber, and modified Futaki invariant

Kyeong-Dong Park, K-stability of Fano Spherical Varieties, II
2023-08-08 10:30 - 12:00 | B266, IBS, Korea, Republic of

Kyeong-Dong Park, K-stability of Fano Spherical Varieties, III
2023-08-08 15:00 - 16:30 | B266, IBS, Korea, Republic of

KSCV Workshop #26
2023-08-16 - 2023-08-18 | B109, IBS, Korea, Republic of
Speakers
Taeyong Ahn (Inha University)
Ye-Won Luke Cho (Pusan National University)
Young-Jun Choi (Pusan National University)
Pham Hoang Hiep (Vietnam Academy of Science and Technology)
Dinh Tuan Huynh (Hue University of Education-Hue University)
Łukasz Kosiński (Jagiellonian University)
Kang-Hyurk Lee (Gyeongsang National University)
Man-Chun Lee (The Chinese University of Hong Kong)
Hoseob Seo (Institute for Basic Science)
Guolei Zhong (Institute for Basic Science)
Abstracts
PDF File
Schedule
Day 1: August 16 (Wednesday)

~10:00
Registration

10:00~11:00
Kang-Hyurk Lee
The method of potential rescaling and a characterization of the unit ball

11:00~11:30
Coffee Break

11:30~12:30
Dinh Tuan Huynh (online)
Universal entrie curves in projective spaces with slow growth

12:30~14:30
Lunch

14:30~15:30
Taeyong Ahn
Equidistribution of positive closed currents on a compact Kähler manifold

15:30~16:00
Coffee Break

16:00-17:00
Guolei Zhong
Compact Kähler threefolds with the action of an abelian group of maximal dynamical rank

18:00~20:00
Speakers Dinner

Day 2: August 17 (Thursday)

10:00~11:00
Young-Jun Choi
An injectivity theorem on snc compact Kähler spaces

11:00~11:30
Coffee Break

11:30~12:30
Pham Hoang Hiep (online)
Singularity invariants of plurisubharmonic functions

12:30~14:30
Lunch

14:30~15:30
Man-Chun Lee
Kähler-Ricci flow on non-compact manifolds and application

15:30~16:00
Coffee Break

16:00-17:00
Ye-Won Luke Cho
Continuity of singular Kähler-Einstein potentials

17:30-19:00
Banquet

Day 3: August 19 (Friday)

10:00~11:00
Hoseob-Seo
Ohsawa measures on singular hypersurfaces and its applications

11:00~11:30
Coffee Break

11:30~12:30
Łukasz Kosiński
A Gehring-Hayman inequality

Organizers
Dano Kim (Seoul National University)
Sung-Yeon Kim (Institute for Basic Science)
N. Cuong Nguyen (Korea Advanced Institute of Science and Technology)
Aeryeong Seo (Kyungpook National University)
Venue
B109, IBS, Korea
Registration
Please submit Google form by July 28.
More Information
• How to get to IBS-CCG
• From Hotel to IBS
• IBS Cafeteria (Lunch)
Participants
Sui-Chung Ng (East China Normal University)
Bou Kang (KAIST)
Mario Chan (Pusan National University)
Jisoo Byun (Kyungnam University)
Hoseob Seo (Institute for Basic Science)
Sungmin Yoo (Incheon National University)
Seungjae Lee (Institute for Basic Science)
Han-Wool Lee (Korea University)
Hanjin Lee (Handong Global University)
Jiwon Jeong (Pusan National University)
Ye-Won Luke Cho (Pusan National University)
Boo Rim Choe (Korea University)
Hyunil Choi (Pusan National University)
Jihun Yum (Institute for Basic Science)
Kang-Hyurk Lee (Gyeongsang National University)
Do young Choi (KAIST)
Kang-Tae Kim (POSTECH)
Chiwon Yoon (KAIST)
Supported by Institute for Mathematical Convergence, KNU

Takayuki Koike, On Some Variants of Ueda's Lemma and its Application
2023-09-05 16:00 - 17:00 | on-line
Speaker

Takayuki Koike
(Osaka Metropolitan University)

In this talk, we explain our results on some variants of Ueda’s lemma on L∞-estimates for Cech coboundary operators that hold uniformly for all flat holomorphic line bundles on compact Kähler manifolds. This talk is partially based on joint work with Y. Hashimoto and T. Uehara.Title : Non-measure hyperbolicity of K3 and Enriques surfaces.

George Hitching, Brill-Noether Loci on Moduli Space of Symplectic Bundles over a Curve
2023-09-06 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

George Hitching
Oslo Metropolitan University

Let C be a smooth projective curve of genus g. The symplectic Brill-Noether locus S(k, 2n, K) parametrises stable bundles of rank 2n over C with at least k independent sections, and which admit a nondegenerate skewsymmetric bilinear form with values in the canonical bundle K. This is a symmetric determinantal variety whose tangent spaces are described by a symmetrised Petri map. We show for many values of g, n and k that S(k, 2n, K) is nonempty and has a component which is generically smooth and of the expected dimension. As an application, we show that for certain n and k, for any curve of large genus the usual Brill-Noether locus B(k, 2n, 2n(g-1)) has a component of excess dimension. This is joint work with Ali Bajravani (Tabriz/Berlin).

Thomas Peternell, Semipositive Tangent Bundles and Canonical Extensions
2023-09-07 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Thomas Peternell
University of Bayreuth

Given a projective complex manifold M with an ample polarization there is canonically associated an affine bundle Z over M. The question I will discuss is under which circumstances Z is an affine variety, or at least Stein. This is related to the global structure of M, specifically to the semipositivity of the tangent bundle. I will explain the main conjectures and recent results (joint work with A. Höring).

Yum-Tong Siu, [IBS-KAIST Seminar] Differential Relations for Multiplier Ideal Sheaves in ∂ Estimates
2023-09-07 15:00 - 16:00 | B236-1, IBS, Korea, Republic of
Speaker

Yum-Tong Siu
Harvard University

For sums of squares of real vector fields, Hörmander linked subelliptic estimates to the spanning property of iterated Lie brackets of vector fields. Kohn studied the more complicated analogue of subelliptic ​∂​​​​ estimates for weakly pseudoconvex domains, with vector-valued unknowns.
In the weak-solution approach to solving the ∂​​​ equation, multipliers for the test function are introduced so that estimates hold after multiplication by a multiplier. Kohn used the dual formulation of differential forms instead of vector fields so that (i) the Lie brackets of vector fields are replaced by differential relations to generate new multipliers and (ii) the spanning property of iterated Lie brackets is replaced by the constant function 1 being a multiplier.
We will focus on recent problems and results concerning Kohn’s theory of subelliptic estimates in terms of D’Angelo’s condition of finite type. The distant eventual goal is to explore the theory for differential relations to generate multipliers for subelliptic estimates for a general system of equations with compatibility conditions.

Fabrizio Catanese, [IBS-KAIST Seminar] Geometry and Dynamics of Geometric Endomorphisms of the Hesse Moduli Space of Elliptic Curves
2023-09-08 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Fabrizio Catanese
University of Bayreuth

We consider the geometric map C, called Cayleyan, associating to a plane cubic E the adjoint of its dual curve. We show that C and the classical Hessian map H generate a free semigroup. We begin the investigation of the geometry and dynamics of these maps, and of the geometrically special elliptic curves: these are the elliptic curves isomorphic to cubics in the Hesse pencil which are fixed by some endomorphism belonging to the semigroup generated by H, C. We point out how the dynamic behaviours of H and C differ drastically. Firstly, concerning the number of real periodic points: for H these are infinitely many, for C they are just 4. Secondly, the Julia set of H is the whole projective line, unlike what happens for all elements of the semigroup which are not iterates of H.

Seung uk Jang, Vieta Involutions on Tropical Markov Cubics
2023-09-19 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Seung uk Jang
Université de Rennes 1

We discuss the algebraic dynamics on Markov cubics generated by Vieta involutions, in the tropicalized setting, including some introduction to the tropicalization. It turns out that there is an invariant subset of the tropicalized Markov cubic where the action by Vieta involutions can be modeled by that of the Farey reflection group on the hyperbolic plane.
Some applications in non-archimedean dynamics, number theory, or complex dynamical questions will also be introduced.

Yaoxiong Wen, Generalized Quiver Mutation and PAX/PAXY Models
2023-09-21 15:00 - 16:00 | B236-1, IBS, Korea, Republic of
Speaker

Yaoxiong Wen
KIAS

During this presentation, I will discuss the generalized quiver mutation conjecture proposed by Prof. Ruan. Our focus will be on proving this conjecture for the Grassmannian bundle. Additionally, we will explore the determinantal variety and its two different desingularizations, known as PAX/PAXY models. We will demonstrate the relationship between these models through generalized quiver mutation and prove the PAX/PAXY correspondence in the level of quasimap I-functions. This talk is based on the ongoing research collaboration with Mark Shoemaker and Nathan Priddis.
