Center for Complex Geometry

Junyan Zhao, Moduli of Curves of Genus 6 and K-stability
2023-04-25 11:00 - 12:00 (Asia/Seoul) | 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.

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.

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.

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.

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.

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.

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.

Dongsoo Shin, Deformations of Sandwiched Surface Singularities and the Minimal Model Program
2023-02-28 10:00 - 10:50 | 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.

Hosung Kim, Lagrangian Fibration Structure on the Cotangent Bundle of a Del Pezzo Surface of Degree 4
2023-02-27 17:00 - 17:50 | B236-1, IBS, Korea, Republic of
Speaker

Hosung Kim
IBS CCG

(This is a part of Seminars on Algebraic Surfaces and Related Topics.)
The cotangent bundle of a complex projective manifold carries a natural holomorphic symplectic 2-form. The existence of a natural Lagrangian fibration structure of these non-compact complex manifolds has not been studied very much. In this talk, I will present a natural Lagrangian fibration structure on the map from the cotangent bundle of a del Pezzo surface of degree 4. This is a joint work with Prof. Yongnam Lee.

Yonghwa Cho, Nodal Surfaces and Cubic Discriminants
2023-02-27 15:50 - 16:40 | B236-1, IBS, Korea, Republic of
Speaker

Yonghwa Cho
IBS CCG

(This is a part of Seminars on Algebraic Surfaces and Related Topics.)
In this talk, I will explain how to associate a nodal surface in P3 with a cubic hypersurface, generalizing the method by Togliatti who constructed quintics with 31 nodes via a discriminant of a nodal cubic 4-folds. For low degrees(≤5), these constructions help to understand the classification problem of nodal surfaces, especially when the surface has the maximal number of nodes. For higher degrees the things get more complicated. I will explain our recent result on sextics proving that every nodal sextics with maximal number of nodes admit Togliatti type descriptions. This talk is based on joint works with Fabrizio Catanese, Stephen Coughlan, Davide Frapporti, Michael Kiermaier, and Sascha Kurz.

Guolei Zhong, Smooth Projective Surfaces with Pseudo-effective Tangent Bundles
2023-02-27 14:40 - 15:30 | B236-1, IBS, Korea, Republic of
Speaker

Guolei Zhong
IBS CCG

(This is a part of Seminars on Algebraic Surfaces and Related Topics.)
A vector bundle over a projective manifold is said to be pseudo-effective if the tautological line bundle of its Grothendieck projectivization is pseudo-effective. In this talk, I will show that a smooth non-uniruled projective surface S has pseudo-effective tangent bundle if and only if S is minimal and has vanishing second Chern class. Moreover, I will describe the non-rational ruled surface and its blow-up which has pseudo-effective tangent bundles. This is a joint work with Jia Jia and Yongnam Lee.

Giancarlo Urzua, N-resolutions
2023-02-27 13:30 - 14:20 | B236-1, IBS, Korea, Republic of
Speaker

Giancarlo Urzua
UC Chille

(This is a part of Seminars on Algebraic Surfaces and Related Topics.)
I will introduce N-resolutions, which are the negative analog of the Kollár–Shepherd-Barron (1988) P-resolutions of a 2-dimensional cyclic quotient singularity. (We instead work with the corresponding M-resolutions of Benkhe-Christophersen (1994).) I will start by describing an algorithm to find all of them based on the explicit algorithm for P-resolutions in Park-Park-Shin-Urzúa (2018) (that geometrically recovers Christophersen-Stevens’ zero continued fractions correspondence (1991)), which in turn is based on the explicit MMP described by Hacking-Tevelev-Urzúa (HTU 2017). I will also describe another way to find N-resolutions via antiflips (HTU 2017) starting with an M-resolution, showing an action of the braid group on all its associated Wahl resolutions. This will bring us to Hacking exceptional collections (2013-2016) on surfaces that are Q-Gorenstein smoothings of particular singular surfaces, where Karmazyn-Kuznetsov-Shinder (2022) have described their derived categories via derived categories of the Kalck-Karmazyn algebras (2017). This can be put together through Kawamata’s bundles (2018-2022), and I will describe our main theorem on semi-orthogonal decompositions defined by these M- and N-resolutions. I will end with applications to all simply-connected Dolgachev surfaces. I will mention open problems. This is on the joint recent work with Jenia Tevelev. This computer program finds all M- and N-resolutions.

Seminars on Algebraic Surfaces and Related Topics
2023-02-27 - 2023-02-28 | B236-1, IBS, Korea, Republic of
Schedule

Feb. 27

N-resolutions
Giancarlo Urzua (UC Chille)
13:30-14:20

Smooth Projective Surfaces with Pseudo-effective Tangent Bundles
Guolei Zhong (IBS-CCG)
14:40-15:30

Nodal Surfaces and Cubic Discriminants
Yonghwa Cho (IBS-CCG)
15:50-16:40

Lagrangian Fibration Structure on the Cotangent Bundle of a Del Pezzo Surface of Degree 4
Hosung Kim (IBS-CCG)
17:00-17:50

Dinner
18:20-20:00

Feb. 28

Deformations of Sandwiched Surface Singularities and the Minimal Model Program
Dongsoo Shin (Chungnam National U.)
10:00-10:50

Mori Dream Surfaces of General Type with pg=0
JongHae Keum (KIAS)
11:10-12:00

Lunch
12:00-13:00

Korea-Japan Conference in Algebraic Geometry
2023-02-13 - 2023-02-17 | IBS Science Culture Center, Daejeon, Korea, Republic of
Speakers
Yonghwa Cho (IBS-CCG)
Junho Choe (KIAS)
Yoshinori Gongyo (Tokyo U.)
Kenta Hashizume (Kyoto U.)
Sukmoon Huh (Sungkyunkwan U.)
WonTae Hwang (Jeonbuk National U.)
Seung-Jo Jung (Jeonbuk National U.)
Yeongrak Kim (Pusan National U.)
Tasuki Kinjo (IPMU, Tokyo)
Tatsuki Kuwagaki (Kyoto U.)
Shin-ichi Matsumura (Tohoku U.)
Yosuke Matsuzawa (Osaka Metropolitan U.)
Jinhyung Park (KAIST)
Kenta Sato (Kyushu U.)
Joonyeong Won (Ehwa Womans U.)
Shou Yoshikawa (RIKEN iTHEMS)
Abstracts
PDF File
Schedule
Day 1: February 13 (Monday)

~10:00
Registration

10:00~11:00
Jinhyung Park
Syzygies of tangent developable surfaces and K3 carpets via secant varieties
Jun-Muk Hwang
(Chair)

11:00~11:30
Coffee Break

11:30~12:30
Shin-ichi Matsumura
The nonvanishing problem for varieties with nef anticanonical bundle

12:30~14:30
Lunch

15:00~16:00
Seung-Jo Jung
On Milnor numbers and Tjurina numbers of hypersurface singularities
Yujiro Kawamata
(Chair)

16:00~16:20
Coffee Break

16:20-17:20
Yosuke Matsuzawa
Zariski dense orbit conjecture and arithmetic degrees of cohomologically hyperbolic maps

18:00~20:00
Speakers Dinner

Day 2: February 14 (Tuesday)

10:00~11:00
Yoshinori Gongyo
Generalized complexities and the Mukai type conjecture
Yongnam Lee
(Chair)

11:00~11:30
Coffee Break

11:30~12:30
Shou Yoshikawa
Quasi-F-splitting

12:30~14:30
Lunch

15:00~16:00
Sukmoon Huh
Torelli problem on logarithmic sheaves
Sijong Kwak
(Chair)

16:00~16:20
Coffee Break

16:20-17:20
Kenta Sato
General hyperplane section of log canonical threefolds in positive characteristic

Day 3: February 15 (Wednesday)

10:00~11:00
WonTae Hwang
Jordan constants of groups in connection with abelian varieties in positive characteristic
Dongseon Hwang
(Chair)

11:00~11:30
Coffee Break

11:30~12:30
Tasuki Kinjo
Euler characteristic for stacks

12:30~18:00
Excursion

Day 4: February 16 (Thursday)

10:00~11:00
Junho Choe
Various parallels between projective varieties and secant varieties
Jaehyun Hong
(Chair)

11:00~11:30
Coffee Break

11:30~12:30
Tatsuki Kuwagaki
Some examples of Hodge-Fukaya theory

12:30~14:30
Lunch

15:00~16:00
Joonyeong Won
Twisted Kaehler-Einstein metric on del Pezzo surfaces
JongHae Keum
(Chair)

16:00~16:20
Coffee Break

16:20-17:20
Kenta Hashizume
On effective base point freeness for klt pairs

17:30~19:00
Banquet

Day 5: February 17 (Friday)

10:00~11:00
Yeongrak Kim
Ulrich bundles on cubic fourfolds
Keiji Oguiso
(Chair)

11:00~11:30
Coffee Break

11:30~12:30
Yonghwa Cho
Nodal (hyper-)surfaces

Organizers
Osamu Fujino (Kyoto U.)
DongSeon Hwang (IBS-CCG)
Jun-Muk Hwang (IBS-CCG)
Yongnam Lee (IBS-CCG)
Keiji Oguiso (U. Tokyo)
Shunsuke Takagi (U. Tokyo)
Registration
Please submit Google form by January 31.
More Information
• Food Zone Map
• How to get to IBS-CCG

Yoon-Joo Kim, Isotrivial Fibrations of Compact Hyper-Kähler Manifolds
2023-02-10 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Yoon-Joo Kim
MPI-Bonn

A compact hyper-Kähler (HK) manifold and its Lagrangian fibration are higher-dimensional generalizations of a K3 surface and its elliptic fibration. A Lagrangian fibration f : X → B of a HK manifold is called isotrivial if its smooth fibers are all isomorphic to each other; this is the most special type of Lagrangian fibrations, generalizing the notion of an isotrivial elliptic fibration. In this talk, I will propose a possible classification scheme of HK manifolds admitting isotrivial Lagrangian fibrations. This method in particular rediscovers the two well-known constructions of HK manifolds, the K3[n] and generalized Kummer constructions. I will then specialize the discussion to HK fourfolds. This is joint work with Radu Laza and Olivier Martin.

Laurent Stolovitch, Introduction to Normal Form Theory of Holomorphic Vector Fields 2
2023-02-09 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Laurent Stolovitch
Universite Cote d’Azur

In this short lecture, I will introduce the notion of normal form and resonances. I will also explain the phenomenon of “small divisors” and give some fundamental results of holomorphic conjugacy to a normal form.

Laurent Stolovitch, Introduction to Normal Form Theory of Holomorphic Vector Fields 1
2023-02-07 16:00 - 17:00 | B236-1, IBS, Korea, Republic of

Daniele Agostini, The Martens-Mumford Theorem and the Green-Lazarsfeld Secant Conjecture
2023-02-07 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Daniele Agostini
Eberhard Karls Universität Tübingen

The syzygies of a curve are the algebraic relation amongst the equation defining it. They are an algebraic concept but they have surprising applications to geometry. For example, the Green-Lazarsfeld secant conjecture predicts that the syzygies of a curve of sufficiently high degree are controlled by its special secants. We prove this conjecture for all curves of Clifford index at least two and not bielliptic and for all line bundles of a certain degree. Our proof is based on a classic result of Martens and Mumford on Brill-Noether varieties and on a simple vanishing criterion that comes from the interpretation of syzygies through symmetric products of curves.

Eunjeong Lee (이은정, Chungbuk National University)
2023-02-06 - 2023-02-09
Eunjeong Lee
Visitor (2023.2.6-2023.2.9) from Chungbuk National University
Office: –

Laurent Stolovitch (Universite Cote d'Azur, France)
2023-02-04 - 2023-02-10
Laurent Stolovitch
Visitor (2023.2.4-2023.2.10) from Universite Cote d’Azur, France
Office: –

Antonio Nigro (Universidade Federal Fluminense)
2023-02-03 - 2023-02-09
Antonio Nigro
Visitor (2023.2.3-2023.2.9) from Universidade Federal Fluminense
Office: –

Dennis The, A Cartan-theoretic Perspective on (2,3,5)-distributions
2023-02-02 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Dennis The
UiT The Arctic University of Norway

Generic rank 2 distributions on 5-manifolds, i.e. “(2,3,5)-distributions”, are interesting geometric structures arising in the study of non-holonomic systems, underdetermined ODE of Monge type, conformal 5-manifolds with special holonomy, etc. The origins of their study date to Élie Cartan’s “5-variables” paper of 1910, where he gave a tour-de-force application of his method of equivalence. In my talk, I’ll revisit the classification of homogeneous (2,3,5)-distributions from a modern “Cartan-theoretic” perspective. In particular, I’ll discuss from this viewpoint the exceptionality of the 3:1 ratio for two spheres rolling on each other without twisting or slipping.

Dennis The (University of Tromso, Norway)
2023-01-30 - 2023-02-08
Dennis The
Visitor (2023.1.30-2023.2.8) from University of Tromso, Norway
Office: –

Kangjin Han, Secant variety and its singularity II
2023-01-27 10:30 - 11:30 | B266, IBS, Korea, Republic of
Speaker

Kangjin Han
DGIST

Secant variety (or more generally Join) construction is one of the main methods to construct a new geometric object from the original one in classical algebraic geometry. In this series of talks, we first consider some general facts on secant varieties and then focus on a specific topic, i.e. singularity of secants. For this purpose, we review some basic facts and known results in the literature and present some ideas to show (non-)singularity of points in the given secant. We also report a recent work with K. Furukawa on the singular loci of higher secant of Veronese varieties and others. The talks cover such topics as Terracini lemma, identifiability, tangential k-contact locus from geometric side and apolar ideal and defining equations via a Young flattening and so on from algebraic side.

Kangjin Han, Secant variety and its singularity I
2023-01-26 16:00 - 17:00 | B266, IBS, Korea, Republic of

JongHae Keum, Fake Projective Plane II
2023-01-13 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

JongHae Keum
KIAS

Fake projective planes (abbreviated as FPPs) are 2-dimensional complex manifolds with the same Betti numbers as the projective plane, but not isomorphic to it.
FPPs can be uniformized by a complex 2-ball. In other words, they are ball quotients having the minimum possible Betti numbers.
The existence of such a surface was first proved by Mumford in 1979, via 2-adic uniformization.
Not always algebraic varieties are described via polynomial equations: sometimes they are constructed via uniformization: this means, as quotients of certain domains in a complex vector space, called bounded symmetric domains, via the action of discontinuous groups. Then general theorems (as Kodaira’s) imply the algebraicity of these quotient complex manifolds. The problem concerning the algebro-geometrical properties of such varieties constructed via uniformization and especially the description of their projective embeddings (and the corresponding polynomial equations) lies at the crossroads of several allied fields: the theory of arithmetic groups and division algebras, complex algebraic and differential geometry, linear systems, use of group symmetries, and topological and homological tools in the study of quotient spaces. Of particular importance are the so-called ball quotients, especially in dimension 2, since they yield the surfaces with the maximal possible canonical volume K2 for a fixed value of the geometric genus pg.
In the first lecture I will introduce basic properties of FPPs and their position in the classification theory of algebraic surfaces.
In the second I will discuss recent progress on them, such as their derived categories, bicanonical maps and their equations.

JongHae Keum, Fake Projective Planes I
2023-01-12 16:00 - 17:00 | B236-1, IBS, Korea, Republic of

Yunhyung Cho (조윤형, Sungkyunkwan University)
2023-01-04 - 2023-01-06
Yunhyung Cho (조윤형)
Visitor (2023.1.4-2023.1.6) from Sungkyunkwan University
Office: –

Andrea Petracci, A 1-dimensional Component of K-moduli of Del Pezzo Surfaces
2022-11-29 17:00 - 18:00 | on-line
Speaker

Andrea Petracci
Università di Bologna

Fano varieties are algebraic varieties with positive curvature; they are basic building blocks of algebraic varieties. Great progress has been recently made by Xu et al. to construct moduli spaces of Fano varieties by using K-stability (which is related to the existence of Kähler-Einstein metrics). These moduli spaces are called K-moduli.
In this talk I will explain how to easily deduce some geometric properties of K-moduli by using toric geometry and deformation theory. In particular, I will show how to construct a 1-dimensional component of K-moduli which parametrises certain K-polystable del Pezzo surfaces.

Joaquín Moraga, Coregularity of Fano Varieties
2022-11-15 11:00 - 12:00 | on-line
Speaker

Joaquín Moraga
UCLA

In this talk, we will introduce the absolute coregularity of Fano varieties. The coregularity measures the singularities of the anti-pluricanonical sections. Philosophically, most Fano varieties have coregularity 0. In the talk, we will explain some theorems that support this philosophy. We will show that a Fano variety of coregularity 0 admits a non-trivial section in |-2KX|, independently of the dimension of X. This is joint work with Fernando Figueroa, Stefano Filipazzo, and Junyao Peng.
