Center for Complex Geometry

Seungjae Lee, Cohomological Isomorphism of Symmetric Power of Cotangent Bundle of Ball Quotient and its Toroidal Compactification
2023-11-01 15:00 - 16:00 (Asia/Seoul) | B236-1, IBS, Korea, Republic of
Speaker

Seungjae Lee
IBS CCG

In this talk, we investigate the L2-Dolbeault cohomology of the symmetric power of cotangent bundles of ball quotients with finite volume, as well as their toroidal compactification. As a result, we establish a version of L2-Hodge decompostion for complex hyperbolic space forms with finite volume, under some mild condition for lattice.

Yotsutani Naoto, Chow Stability and Non-symmetric Kähler-Einstein Toric Fano Manifolds
2023-10-25 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Yotsutani Naoto
Kagawa University

In 2001, Donaldson proved that if a polarized manifold (X, L) admits constant scalar curvature Kähler metrics in c1(L), then (X, L) is asymptotically Chow stable whenever its automorphism group Aut(X, L) is discrete.
In this talk, we show that this result does not hold in the case where Aut(X, L) is not discrete.
This talk is based on a joint work with H. Ono and Y. Sano.

Workshop on Parabolic Geometry and Related Topics
2023-10-17 - 2023-10-20 | B109, IBS, Korea, Republic of
Speakers
Gil Bor (CIMAT)
Andreas Cap (U. Vienna)
Sean Curry (Oklahoma State U.)
Mike Eastwood (U. Adelaide)
Jaehyun Hong (IBS-CCG)
Jun-Muk Hwang (IBS-CCG)
Sung Yeon Kim (IBS-CCG)
Ilya Kossovskiy (Masaryk U.)
Boris Kruglikov (U. Tromso)
Omid Makhmali (PAS-CTP)
Benjamin McMillan (IBS-CCG)
Tohru Morimoto (Nara Women’s U.)
Igor Zelenko (Texas A&M)
Abstracts
PDF file
Schedule
Oct. 17
10:00-11:00 Igor Zelenko
11:30-12:30 Ilya Kossovskiy
15:00-16:00 Gil Bor
16:30-17:30 Sung Yeon Kim
Oct. 18
10:00-11:00 Andreas Cap
11:30-12:30 Jun-Muk Hwang
14:30-15:30 Benjamin McMillan
Oct. 19
10:00-11:00 Sean Curry
11:30-12:30 Omid Makhmali
15:00-16:00 Tohru Morimoto
16:30-17:30 Jaehyun Hong
Oct. 20
10:00-11:00 Boris Kruglikov
11:30-12:30 Mike Eastwood
Organizer
Jun-Muk Hwang (IBS-CCG)
Venue
B109, IBS, Daejeon, Korea
More Information
• How to get to IBS-CCG

Gil Bor, Bicycle Tracks, their Monodromy Invariants and Geodesics
2023-10-16 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Gil Bor
CIMAT

At first sight, the pair of front and back wheel tracks left by a passing bike on a sandy or muddy terrain seems like a random pair of curves. This is not the case. For example, one can usually distinguish between the front and back wheel tracks, and even the direction at which these were traversed, based solely on their shapes. You can try it for the following pair of paths.

Another example: If the front wheel traverses a small enough closed path (compared to the bike size), then, typically, the back track does not close up, by an amount approximated by the area enclosed by the front track and the bicycle length; this fact was utilized to build a simple area measuring mechanical device, now obsolete, called the Hatchet planimeter.
In recent years the subject has attracted attention due to newly discovered relations with the theory of completely integrable systems (the filament flow), sub-Riemannian geometry and elasticity theory. I will try to describe some of these developments and open questions.

Gil Bor, Cusps of Caustics by Reflection in a Convex Billiard Table
2023-10-16 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Gil Bor
CIMAT

Place a point light source inside a smooth convex billiard table (or mirror). The n-th caustic by reflection is the envelope of light rays after n reflections. Theorem: each of these caustics, for a generic point light source, has at least 4 cusps. Conjecture: there are exactly 4 cusps iff the table is an ellipse. Here are the 2nd, 5th and 8th caustics by reflection in an ellipse, each with 4 cusps (marked by gray disks; the light source is the white disk)

This is a billiard version of “Jacobi’s Last Geometric Statement”, concerning the number of cusps of the conjugate locus of a point on a convex surface, proved so far only in the n=1 case. (Joint work with Serge Tabachnikov, from Penn State, USA).

Guolei Zhong, Positivity of Tangent Sheaf of Projective Varieties
2023-10-10 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Guolei Zhong
IBS-CCG

In this talk, we discuss the structure of projective varieties with certain positive tangent sheaves. More precisely, when the tangent sheaf is almost nef or positively curved, we construct a well-defined MRC fibration and study the Fujita decomposition of reflexive sheaves. Besides, we show that the almost nefness of the tangent sheaf will impose rather restrictive condition on the singularities, and some typical examples will be given. This is based on my recent joint work with Masataka Iwai and Shin-ichi Matsumura.

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.

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.

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.

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.

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).

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).

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.

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

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

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.

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.

Chang-Yeon Chough, Introduction to algebraic stacks, VII, VIII
2023-07-26 10:00 - 12:15 | 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, V, VI
2023-07-25 15:00 - 17:30 | B236-1, IBS, Korea, Republic of

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
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, I
2023-07-20 11:00 - 12:00 | B236-1, IBS, Korea, Republic of

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

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

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).

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.

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.

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.

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.
