Center for Complex Geometry

Chuyu Zhou, Lecture 1: Constructible Properties of Various Domains for a Family of Couples
2024-08-06 16:00 - 17:30 (Asia/Seoul) | B236-1, IBS, Korea, Republic of
Speaker

Chuyu Zhou
Yonsei University

In this lecture, I will recall some basic knowledge on K-stability and some background on wall crossing in proportional setting. Then we plan to conduct a comparison between proportional case and non-proportional case. Under the comparison, we will define several domains associated to a family of couples and present their constructible properties.

Chuyu Zhou, Lecture 2: Non-linear Wall Crossing Theory
2024-08-07 16:00 - 17:30 | B236-1, IBS, Korea, Republic of
Speaker

Chuyu Zhou
Yonsei University

In this lecture, we will talk about two properties of K-semistable domains in non-proportional setting. One is the finiteness criterion, which states that the number of domains is finite for a family of couples. The other is about the shape of each domain, which states that they are semi-algebraic sets. These two properties together lead to a satisfactory wall crossing theory for K-stability in non-proportional case. Time permits, I will present a comparison between GIT-stability and K-stability in non-proportional setting.

Xiaojun Huang, Bounding a Levi-flat Hypersurface in a Stein Manifold
2024-08-22 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Xiaojun Huang
Rutgers Univ

Let M be a smooth real codimension two compact submanifold in a Stein manifold. We will prove the following theorem: Suppose that M has two elliptic complex tangents and suppose CR points are non-minimal. Assume further that M is contained in a bounded strongly pseudoconvex domain. Then M bounds a unique smooth up to M Levi flat hypersurface M̂ that is foliated by Stein subspace diffeomorphic to the ball. Moreover, M̂ is the hull of holomorphy of M. This subject has a long history of investigation from E. Bishop and Harvey-Lawson. I will discuss both the history and techniques used in the proof of the above mentioned theorem.

Changho Han (한창호, Korea University)
2024-10-13 - 2024-10-14
Changho Han (한창호)
Visitor (2024.10.13-2024.10.14) from Korea University
Office: –

Changho Han, Trigonal Curves and Associated K3 Surfaces
2024-10-14 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Changho Han
Korea university

K3 surfaces, as a generalization of elliptic curves, have a rich amount of geometric properties. Recalling that elliptic curves are double covers of rational curves branched over 4 distinct points, there are K3 surfaces that are cyclic triple covers of rational surfaces; Artebani and Sarti classified such generic K3 surfaces depending on lattice invariants. Such K3 surfaces admit Kulikov and KSBA degenerations, each leading to toroidal and KSBA compactifications of the moduli spaces of such K3 surfaces. As joint works in progress with Valery Alexeev, Anand Deopurkar, and Philip Engel, I will explain how to use trigonal curves (triple covers of rational curves) to obtain aforementioned degenerations, leading to more explicit understandings of boundaries of those compactifications: such as classifications of generic members and the dimensions.

Justin Lacini, On Log del Pezzo Surfaces in Positive Characteristic
2024-10-15 14:00 - 15:00 | B236-1, IBS, Korea, Republic of
Speaker

Justin Lacini
Princeton university

A log del Pezzo surface is a normal surface with only Kawamata log terminal singularities and anti-ample canonical class. Over the complex numbers, Keel and McKernan have classified all but a bounded family of log del Pezzo surfaces of Picard number one. In this talk we will extend their classification to positive characteristic. In particular, we will prove that for p>3 every log del Pezzo surface of Picard number one admits a log resolution that lifts to characteristic zero over a smooth base. As a consequence, we will see that Kawamata-Viehweg vanishing holds in this setting. Finally, we will conclude with some counterexamples in characteristic two, three and five.

David Sykes, CR Hypersurfaces, Studying 2-nondegenerate Structures via Absolute Parallelisms
2024-10-22 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

David Sykes
IBS CCG

The basic problem of finding (local) biholomorphisms mapping one real hypersurface in a complex space onto another is only well understood for a limited class of hypersurfaces, and has a fundamental relationship to their induced CR geometries. Following a light historical survey of major results in the area, this talk will present a solution to the local equivalence problem for 2-nondegenerate CR hypersurfaces, developed in joint work with Igor Zelenko, given by canonically encoding the geometries into absolute parallelisms.

Naoto Yotsutani (Kagawa University)
2024-10-26 - 2024-10-29
Naoto Yotsutani
Visitor (2024.10.26-2024.10.29) from Kagawa University
Office: –

Naoto Yotsutani, Bott Manifolds with the Strong Calabi Dream Structure
2024-10-29 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Naoto Yotsutani
Kagawa university

We prove that if the Futaki invariant of a polarized Bott manifold (X, L) for any ample line bundle L vanishes, then X is isomorphic to the products of the projective lines. This talk is based on a work joint with Kento Fujita (algebro-geometrical approach), and another independent work joint with Hajime Ono and Yuji Sano (convex geometrical approach).

Giancarlo Urzua (Pontificia Universidad Catolica de Chile)
2024-11-08 - 2024-11-13
Giancarlo Urzua
Visitor (2024.11.8-2024.11.13) from Pontificia Universidad Catolica de Chile
Office: –

Giancarlo Urzua, The Birational Geometry of Markov Numbers
2024-11-12 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Giancarlo Urzua
Pontificia Universidad Catolica de Chile

The projective plane is rigid. However, it may degenerate to surfaces with quotient singularities. After the work of Bădescu and Manetti, Hacking and Prokhorov 2010 classified these degenerations completely. They are Q-Gorenstein partial smoothings of P(a2, b2, c2), where a, b, c satisfy the Markov equation x2+y2+z2=3xyz. Let us call the corresponding degenerations Markovian planes. They are part of a bigger picture of degenerations with Wahl singularities, where there is an explicit MMP whose final results are either K nef, smooth deformations of ruled surfaces, or Markovian planes. Although it is a final result of MMP, we can nevertheless run MMP on small modifications of Markovian planes to obtain new numerical/combinatorial data for Markov numbers via birational geometry. New connections with Markov conjecture (i.e. Frobenius Uniqueness Conjecture) are byproducts. This is joint work with Juan Pablo Zúñiga (Ph.D. student at UC Chile), the pre-print can be found here https://arxiv.org/abs/2310.17957.

Izzet Coskun, The Geometry of Moduli Spaces of Sheaves on P2
2024-11-27 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Izzet Coskun
University of Illinois Chicago

In this talk, I will explain how to use Bridgeland stability conditions to compute the ample and effective cones of moduli spaces of sheaves on the projective plane. I will describe the birational geometry of these moduli spaces and give applications to the higher rank interpolation problem. This talk is based on joint work with Jack Huizenga, John Kopper and Matthew Woolf.

Izzet Coskun, The Higher Rank Brill-Noether Problem on Surfaces
2024-11-28 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Izzet Coskun
University of Illinois Chicago

In this talk, I will explain how to use Bridgeland stability conditions to compute the cohomology of a general stable sheaf on a K3 or abelian surface. This talk is based on joint work with Howard Nuer and Kota Yoshioka.

Luca Schaffler, An Explicit Wall Crossing for the Moduli Space of Hyperplane Arrangements
2024-12-06 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Luca Schaffler
Roma Tre University

The moduli space of hyperplanes in projective space has a family of geometric and modular compactifications that parametrize stable hyperplane arrangements with respect to a weight vector. Among these, there is a toric compactification that generalizes the Losev-Manin moduli space of points on the line. We study the first natural wall crossing that modifies this compactification into a non-toric one by varying the weights. As an application of our work, we show that any Q-factorialization of the blow up at the identity of the torus of the generalized Losev-Manin space is not a Mori dream space for a sufficiently high number of hyperplanes. Additionally, for lines in the plane, we provide a precise description of the wall crossing. This is joint work with Patricio Gallardo.

Yen-An Chen, Toric Fano Foliations
2024-12-11 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Yen-An Chen
National Taiwan University

In recent years, there are significant developments of the minimal model program for foliated varieties. It is intriguing to ask if Fano foliations form a bounded family. It is anticipated that Borisov-Alexeev-Borisov conjecture also holds in the context of foliations. In this talk, I will discuss the boundedness of the toric Fano adjoint foliated structure with mild singularities. This is a joint work in progress with Chih-Wei Chang.

Luca Schaffler, Unimodal Singularities and Boundary Divisors in the KSBA Moduli of a Class of Horikawa Surfaces
2024-12-11 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Luca Schaffler
Roma Tre University

Smooth minimal surfaces of general type with K2=1, pg=2, and q=0 constitute a fundamental example in the geography of algebraic surfaces, and the 28-dimensional moduli space M of their canonical models admits a modular compactification M via the minimal model program. We describe eight new irreducible boundary divisors in such compactification parametrizing reducible stable surfaces. Additionally, we study the relation with the GIT compactification of M and the Hodge theory of the degenerate surfaces that the eight divisors parametrize. Time permitting, we will discuss recent progress aimed at generalizing these techniques to study the boundary of compact moduli of other types of stable surfaces. This is joint work with Patricio Gallardo, Gregory Pearlstein, and Zheng Zhang.

Sungmin Yoo, Convergence of Sequences of the Bergman Type Volume Forms
2024-12-30 10:30 - 11:20 | B236-1, IBS, Korea, Republic of
Speaker

Sungmin Yoo
Incheon National University

Following the Yau-Tian-Donaldson conjecture, the construction of sequences of Bergman-type metrics converging to a canonical metric on a polarized manifold has been studied by many mathematicians including Tian, Donaldson, Tsuji, Berman, Berndtsson, and others. In this talk, I will introduce my recent findings on the uniform convergence of Tsuji’s Bergman kernel sequence to the volume form of the Kahler-Einstein metric on a uniformly squeezing domain.

Yonghwa Cho, Double Point Divisors from Projections
2024-12-30 11:30 - 12:20 | B236-1, IBS, Korea, Republic of
Speaker

Yonghwa Cho
Gyeongsang National University

Consider a smooth projective variety of codimension e. A general projection from a linear subspace of dimension (e-2) is birational, hence the non-isomorphic locus forms a proper closed subset of X. Mumford showed that this non-isomorphic locus is not merely a closed subset, but is naturally endowed with a divisor structure. We call it a double point divisor from outer projection. In this talk I will discuss the positivity property of double point divisors including our recent proof of very ampleness, except for some exceptional cases. This work is based on a joint work with Jinhyung Park.

Sung Wook Jang, Potential Log Discrepancy and Minimal Model Program I
2025-01-07 15:00 - 16:30 | B236-1, IBS, Korea, Republic of
Speaker

Sung Wook Jang
IBS CCG

Minimal model program (abbreviated as MMP) is a central problem in birational geometry. The MMP is a sequence of divisorial contractions or flips, which makes the canonical divisor closer to a nef divisor. If the MMP successfully terminates, then we have either a minimal model or a Mori fiber space. We will briefly show how to run an MMP and introduce recent progress concerned with the MMP.

Sung Wook Jang, Potential Log Discrepancy and Minimal Model Program II
2025-01-08 15:00 - 16:30 | B236-1, IBS, Korea, Republic of
Speaker

Sung Wook Jang
IBS CCG

We are interested in an anticanonical divisor and hope to establish the MMP for an anticanonical divisor. We believe that the beginning point is the potential log discrepancy that controls singularities of a possible resulting model of MMP for an anticanonical divisor. In this talk, we will give the definition and properties of the potential log discrepancy. We can also define a plc threshold in terms of the potential log discrepancy and show that plc thresholds satisfy ACC.

Sung Wook Jang, Potential Log Discrepancy and Minimal Model Program III
2025-01-09 15:00 - 16:30 | B236-1, IBS, Korea, Republic of
Speaker

Sung Wook Jang
IBS CCG

We can run an MMP for an lc pair. However, in general, we do not know whether the MMP terminates or not. Nevertheless, we can show that special MMP terminates. Immediately, we can prove the existence of minimal models for certain pairs. Analogously, for an anticanonical divisor, if we impose appropriate conditions, then we can say that there exists an anticanonical minimal model. Furthermore, we will talk about ongoing project on anticanonical minimal model program.

Shuang Su, Volumes of Components of Lelong Upper Level Sets
2025-01-22 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Shuang Su
University of Cologne

In this talk, I will talk about the joint work with Duc-Viet Vu. We establish an optimal upper bound for the volumes of components of Lelong upper level sets of closed positive (1,1)-currents, in terms of non-pluripolar products of currents.

Combinatorics on flag varieties and related topics 2025
2025-02-10 - 2025-02-14 | Ajou University, Suwon, Korea, Republic of
Speakers
Tatsuyuki Hikita (RIMS)
Byung-Hak Hwang (KIAS)
Takeshi Ikeda (Waseda University)
Minyoung Jeon (University of Georgia)
Donggun Lee (IBS-CCG)
Eunjeong Lee (Chungbuk National University)
Seung Jin Lee (Seoul National University)
Mikiya Masuda (OCAMI)
Seonjeong Park (Jeonju University)
Martha Precup (Washington University in St. Louis)
Mark Skandera (Lehigh University)
John Shareshian (Washington University in St. Louis)
Foster Tom (Massachusetts Institute of Technology)
Organizers
Soojin Cho (Ajou University)
Yunhyung Cho (Sungkyunkwan University)
Jaehyun Hong (IBS-CCG)
JiSun Huh (Ajou University)
Eunjeong Lee (Chungbuk National University)
Seonjeong Park (Jeonju University)
Venue
Ajou University, Suwon, Republic of Korea
Registration
Please submit Google form by January 10th, 2025.
For details, see https://sites.google.com/view/flagvarieties2025/home.

Sung-Yeon Kim, Real orbits in flag manifolds
2025-02-26 15:00 - 16:30 | B236-1, IBS, Korea, Republic of
Speaker

Sung-Yeon Kim
IBS CCG

Let G​ be a complex semisimple Lie group, P​​ be a parabolic subgroup and G0​​ be a real form of G.​​ Then the flag manifold G/P​​ decomposes into finitely many G0-orbits. The complex structure of G/P​​ yields a natural homogeneous CR manifold structure on the real orbits such that all elements in G0 are CR automorphisms. Among them there is exactly one orbit of minimal dimension, which is compact. Shilov boundary of bounded symmetric domains are the well-known examples of such minimal orbits. In this talk, we study real orbits from the point of view of CR geometry. In particular we study the CR structures on the minimal orbits in Grassmannians.

Sung-Yeon Kim, Proper holomorphic maps between bounded symmetric domains
2025-02-27 15:00 - 16:30 | B236-1, IBS, Korea, Republic of
Speaker

Sung-Yeon Kim
IBS CCG

In this talk, we study the rigidity of proper holomorphic maps f: Ω→Ω‘​​ between irreducible bounded symmetric domains Ω​​ and Ω‘​​. First, we will define the moduli maps induced by f​​. This moduli maps are CR maps between real orbits in flag maniflods. If the rank difference is small, i.e., 2 ≤ rank(Ω‘)< 2 rank(Ω)-1​​, then there exists a moduli map which preserves the VMRT structures on the real orbits. Using this property, we will show that f: Ω → Ω‘​​ is a holomorphic totally geodesic isometric embedding with respect to Kobayashi metrics for certain pairs (Ω, Ω‘)​​. This is a joint work with N. Mok and A. Seo.

Ngoc Cuong Nguyen, Equidistribution of Fekete points on projective manifolds
2025-02-27 17:00 - 18:00 | B236-1, IBS, Korea, Republic of
Speaker

Ngoc Cuong Nguyen
KAIST

We survey recent developments on the speed of convergence of Fekete points on projective manifolds where the equidistribution was proved by Berman and Boucksom (2011). In particular, the convergence speed can be obtained for a large class of polynomially cuspidal compact sets introduced by Pawłucki and Pleśniak (1988). These sets are (Cα, Cα’)-regular in the sense of Dinh, Ma and Nguyen (2017).

Ngoc Cuong Nguyen (KAIST)
2025-03-01 - 2026-02-15
Ngoc Cuong Nguyen
Visitor (2025.3.1-2026.2.15) from KAIST
Office: B248

Workshop on Hyperkähler Varieties and Related Topics
2025-03-10 - 2025-03-13 | B109, IBS, Korea, Republic of
Invited Speakers (three one-hour lectures)
Ekaterina Amerik (HSE U.)
Yoonjoo Kim (Columbia U.)
Zhiyuan Li (SCMS, Fudan U.)
Keiji Oguiso (U. Tokyo)
Abstracts
PDF file
Schedule
March 10 (Monday)
14:00-15:00 Registration
15:00-16:00 Amerik, Lecture I
16:00-16:30 Coffee Break
16:30-17:30 Li, Lecture I
18:00-20:00 Speakers dinner
March 11 (Tuesday)
11:00-12:00 Amerik, Lecture II
12:00-14:00 Lunch
14:00-15:00 Li, Lecture II
15:00-15:20 Coffee Break
15:20-16:20 Kim, Lecture I
16:20-16:40 Coffee Break
16:40-17:40 Oguiso, Lecture I
18:00-20:00 Banquet
March 12 (Wednesday)
11:00-12:00 Kim, Lecture II
12:00-14:00 Lunch
14:00-15:00 Oguiso, Lecture II
15:00-15:20 Coffee Break
15:20-16:20 Amerik, Lecture III
16:20-16:40 Coffee Break
16:40-17:40 Li, Lecture III
18:00-20:00 Speakers dinner
March 13 (Thursday)
11:00-12:00 Kim, Lecture III
12:00-14:00 Lunch
14:00-15:00 Oguiso, Lecture III
Organizers
Jun-Muk Hwang (IBS-CCG)
Yongnam Lee (IBS-CCG / KAIST)
Venue
B109, IBS, Daejeon, Korea
Registration
Please submit Google form from February 3 to 23.
More Information
• How to get to IBS-CCG

Lorenzo Barban, General results on C*-actions on projective varieties
2025-03-18 15:00 - 16:30 | B236-1, IBS, Korea, Republic of
Speaker

Lorenzo Barban
IBS CCG

In this lecture series we aim to describe the rich relation between C*-actions on complex normal projetive varieties and the birational maps among the associated geometric quotients. We will begin this first seminar by explaining a motivating example, called the Atiyah flop. We will then discuss general results on normal projective varieties admitting a C*-action, such as Sumihiro’s theorem and Białynicki-Birula theorem. We will then conclude by describing how much a C*-action affects the birational geometry of a C*-variety.

Lorenzo Barban, Geometric Invariant Theory for C*-actions
2025-03-19 15:00 - 16:30 | B236-1, IBS, Korea, Republic of
Speaker

Lorenzo Barban
IBS CCG

In this second talk, which is the technical core of the lecture series, we describe several tools to study C*-actions on projective varieties, such as the bandwidth, the AMvsFM Lemma, and the pruning of a variety. With this, we will be able to describe the -birational geometry of the- geometric quotients of a C* action on a pair (X, L), with X a normal projective variety and L an ample line bundle on X. We will conclude by characterizing C*-varieties with small bandwidth.
