Center for Complex Geometry

Morse-Bott functions from complex structures
2026-05-27 11:00 - 12:00 (Asia/Seoul) | B236-1, IBS, Korea, Republic of
Speaker

Hyunmoon Kim
Ewha Womans University

We present a quadratic Morse-Bott function on the real Grassmannian of a symplectic vector space from a compatible linear complex structure. Its stable manifolds generalize the Lagrangian, symplectic, isotropic, and coisotropic Grassmannians to include the Grassmannians of linear subspaces that are neither isotropic, coisotropic, nor symplectic. The negative gradient flow deformation retracts these stable manifolds to compact homogeneous spaces for the unitary group.

Some geometric problems on G-varieties of complexity 1
2026-06-16 16:00 - 17:00, 2026-06-17 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Yan Li
Beijing Institute of Technology

Let $G$ be a connected, reductive, linear algebraic group that acts on a normal variety $X$, and $B$ be a Borel subgroup group of $G$. The complexity of the $G$-action on $X$ was defined by E. B. Vinberg in 1985 as the codimension of $B$-orbit at a sufficiently general position. $G$-varieties of complexity 0 are the well-known spherical varieties, which have been widely studied by many authors in the past decades. $G$-varieties of complexity 1 also provides rich examples, the basic theory on geometrical structure was founded by D. A. Timashev in 1997.
In the first lecture, I will give an overview on the classification theory, as well as some fundamental theorems on geometrical structure of $G$-varieties of complexity 1 which are mainly established by D. A. Timashev. In the second one, I will introduce some recent works on geometrico-analysis problems on $G$-varieties of complexity 1, such as K-stability and its weighted version. We will start from the former works of Suss, Ilten, Langlois, Terpereau, etc. until our recent ones.

Birational contractions of (overline{mathrm{M}}_{g,n}) and their dependence on the characteristic
2026-07-01 16:30 - 17:30 | B236-1, IBS, Korea, Republic of
Speaker

Daebeom Choi
University of Pennsylvania

In this talk, we discuss the existence and nonexistence of certain birational contractions of (overline{mathrm{M}}_{g,n}). Somewhat surprisingly, this depends on the characteristic of the base field: many such contractions exist only in positive characteristic. We present a precise form of this phenomenon and discuss two examples that highlight the difference between characteristic zero and positive characteristic. The first is a simple and explicit contraction that exists only in positive characteristic, and the second is a modular interpretation of the morphisms associated with psi classes on (overline{mathrm{M}}_{1,n}). We also offer some speculation on why such characteristic-dependent phenomena arise.

Mutation of Fano Simplices and Markov type equations
2026-07-03 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Yunhyung Cho
Sungkyunkwan University

It is well known that there is a bijective correspondence between the set of positive integer solutions to the Markov equation and the set of Fano triangles mutation equivalent to the Fano triangle of $mathbb{P}^2$.
In this talk, we establish a higher dimensional generalization of this correspondence for arbitrary Fano simplices of any dimension. On the polyhedral side, we introduce a distinguished class of facets, called admissible facets, and show that their number is preserved under facet mutation. As a consequence, facet mutation classes of Fano simplices carry natural exchange graph structures whose valency is equal to the number of admissible facets. On the arithmetic side, we associate to each Fano simplex a weighted Markov-type equation together with a distinguished positive integer solution, and show that the corresponding arithmetic mutations, given by Vieta involutions, are compatible with facet mutations. More precisely, the assignment from Fano simplices to Diophantine data intertwines combinatorial mutations with arithmetic mutations, thereby relating the mutation dynamics of Fano simplices to the arithmetic dynamics of positive integer solutions. Finally, we introduce a piecewise linear transformation on dual polytopes, called a sliding operator, which realizes combinatorial mutation in the dual picture. As applications, we obtain a volume formula for dual simplices in terms of the associated Diophantine data and recover the multiplicity change formula under mutation.

Indeterminacy of period map for hypersurfaces
2026-07-09 16:30 - 17:30 | B236-1, IBS, Korea, Republic of
Speaker

Sung Gi Park
Princeton University

I will discuss the indeterminacy locus of the period map from the moduli space of hypersurfaces to the period domain and its compactifications. In the case of quartic K3 surfaces, the result verifies the expectation of Laza and O’Grady that the period map from the GIT compactification is regular precisely on the semi-log canonical locus.

Hodge structure on the singular cohomology of singular cubic fourfolds
2026-07-10 10:00 - 11:00 | B236-1, IBS, Korea, Republic of
Speaker

Hyunsuk Kim
University of Michigan

Considering the singular cohomology of a cubic fourfold yields a morphism from the moduli space of (smooth) cubic fourfolds to the period domain. Both spaces have natural compactifications, the GIT moduli space of cubic fourfolds parametrizing all GIT polystable objects, and the Baily-Borel compactification of the period domain, purely coming from group theory. This yields a birational map between two projective varieties. Laza and Looijenga independently gave a precise description of the birational geometry between these two spaces. Recently, Sung Gi Park gave a systematic understanding on this picture using Hodge-Du Bois theory and higher singularities, which have been developed by Friedman-Laza, Mustata-Popa, and many others.
In order to attack this question, it is vital to understand the Hodge structure on the singular cohomology of a singular member in this moduli problem, and also the Hodge theoretic properties of a degeneration to it. In a joint work with Kenny Ascher, Jennifer Li, Lisa Marquand, Sung Gi Park, and Sasha Viktorova, we systematically carry out the calculation of the Hodge-Du Bois diamond of all GIT polystable cubic fourfolds using techniques coming from Saito’s Hodge modules.

Intensive Lectures on the Algebraic Montgomery-Yang Problem
2026-08-07 - 2026-08-08 | 8101, KIAS
Speakers
Woohyeok Jo (Seoul National U.)
Jongil Park (Seoul National U.)
Kyungbae Park (Kangwon National U.)
Abstracts
PDF file
Schedule
August 7 (Fri)
10:30-12:00 Jongil Park
14:00-15:30 Woohyeok Jo
August 8 (Sat)
10:30-12:00 Kyungbae Park
14:00-16:00 Jongil Park & Kyungbae Park
Organizers
JongHae Keum (KIAS)
DongSeon Hwang (IBS-CCG)
Venue
8101, KIAS, Seoul, Korea

Compact Kähler varieties with pseudo-effective tangent sheaf and a dynamical application
2026-08-26 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Guolei Zhong
East China Normal University

Positivity properties of tangent sheaves are known to impose strong restrictions on the global geometry of projective varieties and, more generally, compact Kähler varieties. In this talk, after briefly reviewing the background, we will present several structural results for compact Kähler varieties with pseudo-effective tangent sheaf. As an application to complex dynamics, we will also establish a structure theorem for compact Kähler manifolds admitting a surjective holomorphic self-map with dominant topological degree. The talk is based on joint works with Shin-ichi Matsumura.

Flag variety bundles encoding pseudo-product structures
2026-09-02 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

David Sykes
IBS-CCG

This talk will introduce a natural construction for encoding codimension 1 pseudo-product structures into filtered manifolds amenable to analysis by Tanaka prolongation techniques, and its application to local geometry of CR hypersurfaces. Classically, Tanaka prolongation has only been applicable to maximally nondegenerate pseudo-product structures, while our new techniques require much less. The construction recursively produces levels in a tower of fiber bundles inheriting filtered manifold structures from the underlying geometry. In successful cases, it produces an induced filtered structure with finite-dimensional prolongation, bringing the underlying pseudo-product structure equivalence problems into the scope of classical Tanaka prolongation techniques. As a practical demonstration, we will cover the general construction applied to $(2k+3)$-dimensional $k$-nondegenerate CR hypersurfaces, a complete description of the resulting filtered manifolds’ Tanaka symbols, and their algebraic prolongations. The talk is based on joint work with Boris Doubrov and Igor Zelenko.

Models for elliptic surfaces with a bisection
2026-09-04 16:30 - 17:30 | B236-1, IBS, Korea, Republic of
Speaker

Sönke Rollenske
Marburg University

In the study of (relatively minimal) elliptic surfaces $fcolon S to C$ with a section, the Weierstrass model plays an important role. We discuss two different models for elliptic surfaces with a bisection (or more generally an $f$-nef divisor which has degree two on the fibres): a birational model induced by the involution on the general fibre and a relative determinantal model coming from the relative section ring. The presence of multiple fibres gives rise to new phenomena, which will be illustrated for Enriques surfaces or rational elliptic surfaces. (joint work with Anna Ulivi and Aline Zanardini)

Compact complex parallelisable nilmanifolds with unobstructed deformations
2026-09-08 16:30 - 17:30 | B236-1, IBS, Korea, Republic of
Speaker

Sönke Rollenske
Marburg University

Compact quotients of non-abelian complex nilpotent Lie groups are non-Kähler manifolds with trivial canonical bundle, the most prominent example being the Iwasawa manifold. We give a precise characterisation of the (rare) cases where deformations of such manifolds are unobstructed in terms of so-called verbal ideals in free nilpotent Lie-algebras. On our way, we will have to review some basic ideas of deformation theory. (joint work with M. Paulsen and K. Wehler)

TBA
2026-09-09 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Yewon Luke Cho
Gyeongsang National U

TBA

KSBA moduli compactifications and weighted stable hyperplane arrangements
2026-09-10 16:30 - 17:30 | B236-1, IBS, Korea, Republic of
Speaker

Xian Wu
Yonsei University

We begin with an overview of the framework of KSBA (Kollár—Shepherd-Barron—Alexeev) moduli spaces and stable degenerations, emphasizing some phenomena that distinguish the higher-dimensional cases from the Deligne–Mumford moduli of stable curves. We then survey several existing explicit constructions of KSBA compactifications. As a detailed example, we discuss the construction of KSBA weighted stable hyperplane arrangements via geometric invariant theory (GIT) and matroid tilings due to Valery Alexeev. In particular, here GIT is applied on the construction of the family/stable objects rather than on the construction of the compactified moduli space. Finally, we discuss our ongoing joint work with Hanlong Fang, Patricio Gallardo, and Luca Schaffler on an explicit KSBA wall-crossing morphism to the Losev–Manin space.

Cylinders in Fano threefolds
2026-09-17 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

In-Kyun Kim
HCMC KIAS

A variety $X$ is called cylindrical if it contains a Zariski-open subset isomorphic to $Z times mathbb A^1$ for some affine variety $Z$. If such an open subset is the complement of the support of an effective $mathbb{Q}$-divisor $D$ satisfying $D sim_{mathbb Q} -K_X$, then it is called an anticanonical polar cylinder. The study of cylinders in Fano varieties lies at the intersection of birational geometry and affine geometry. In particular, over an algebraically closed field of characteristic zero, a cylindrical Fano threefold is rational. In recent years, many results have been obtained on which Fano varieties admit cylinders and which do not. In this talk, I will discuss several results on the existence and non-existence of cylinders in Fano threefolds, together with some of the main geometric methods used to prove them.

Conference on Complex Geometry
2026-11-09 - 2026-11-13 | IBS Science Culture Center, Daejeon, Korea, Republic of
Speakers
Rodolf Aguilar Aguilar (CIMAT, Guanajuato)
Cong Ding (Shenzhen U.)
Cecil Gachet (U. Bochum)
Jaehyun Hong (IBS-CCG)
Xiaojun Huang (Rutgers U.)
Lucas Kaufmann (U. Orleans)
Yuta Kusakabe (Kyushu U.)
Minseong Kwon (AMSS, Beijing)
Shin-ichi Matsumura (Tohoku U.)
Suichung Ng (ECNU, Shanghai)
Dan Popovici (U. Toulouse)
Vasily Rogov (MPI, Leipzig)
Min Ru (U. Houston)
Aeryeong Seo (Kyungpook National U.)
Laurent Stolovitch (U. Nice)
Sheng-Li Tan (ECNU, Shanghai)
Wing-Keung To (National U. Singapore)
I-Hsun Tsai (National Taiwan U.)
Julie Wang (Academia Sinica, Taipei)
Kwok Kin Wong (Shenzhen U.)
Abstracts
TBA
Schedule
TBA
Organizers
Philippe Eyssidieux (U. Grenoble)
Jun-Muk Hwang (IBS-CCG)
Sung Yeon Kim (IBS-CCG)
Ngaiming Mok (U. Hong Kong)
Venue
Science Culture Center, IBS, Daejeon, Korea
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