Center for Complex Geometry

Izzet Coskun, The Geometry of Moduli Spaces of Sheaves on P2
2024-11-27 16:00 - 17:00 (Asia/Seoul) | 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.

Lorenzo Barban, Geometric realization of birational maps among Mori dream spaces
2025-03-20 15:00 - 16:30 | B236-1, IBS, Korea, Republic of
Speaker

Lorenzo Barban
IBS CCG

Given a birational map ϕ among normal projective varieties, a geometric realization of ϕ is a normal projective C*-variety such that the birational map among geometric quotients parametrizing general orbits coincides with ϕ. Geometric realizations can be thought of as a projective algebraic version of the notion of cobordism coming from Morse theory. After recalling basic facts about Mori dream spaces, we show that any birational map among Mori dream spaces admits a geometric realization. In the context of toric varieties, we present a SageMath function to explicitly compute the polytope of the geometric realization. We conclude providing explicit examples of this construction.

Mihai Paun (Universität Bayreuth)
2025-03-26 - 2025-05-02
Mihai Paun
Visitor (2025.3.26-2025.5.2) from Universität Bayreuth
Office: B249

Mihai Paun, Positivity of quotients of holomorphic tensor fields (Lecture I)
2025-03-31 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Mihai Paun
U. Bayreuth

I will present the main results and techniques in the preprint arxiv:2502.02183 (joint with J. Cao). This was motivated by the recent preprint by W. Ou, cf. arXiv:2501.1808 (especially by the algebraicity criteria for foliations on compact Kähler manifolds in this article).
The first lecture is dedicated to the proof of the aforementioned algebraicity criteria. Two main techniques are coming into the picture here. One is the usual substitute for the absence of ample line bundles on compact Kähler manifolds, i.e. the Monge-Ampère equation (combined with the mass concentration method). The second one concerns basic properties of Lelong numbers of closed positive currents.
In the second lecture I will survey a few results concerning the positivity of direct images, and the finally I will explain the proof of the main results (i.e. the complete generalisation to the Kähler case of the results obtained in collaboration with F. Campana).

Mihai Paun, Positivity of quotients of holomorphic tensor fields (Lecture II)
2025-04-01 11:00 - 12:00 | B236-1, IBS, Korea, Republic of

Thibaut Delcroix, Weighted Kähler geometry and semisimple principal fibrations (Lecture I)
2025-04-02 14:00 - 15:00 | B236-1, IBS, Korea, Republic of
Speaker

Thibaut Delcroix
Université de Montpellier

In Kähler geometry, especially in the questions of existence of canonical Kähler metrics, the volume form ωn associated with a Kähler form ω plays a central role. In weighted Kähler geometry, we consider a Kähler manifold X equipped with a Hamiltonian action of a torus T, a moment map μ and associated moment polytope Δ=μ(X). We fix a weight function v:Δ → (0,+∞), then replace the volume form ωn by v ◦ μ ωn. One can then define new canonical Kähler metrics: weighted solitons and weighted cscK metrics (introduced by Lahdili), which include most classical canonical Kähler metrics.
I will first introduce this weighted setting, then the (analytic) weighted delta invariant, a number that encodes the existence of weighted solitons. I will then present a sufficient condition of existence of weighted cscK metrics, in line with the J-flow approach of Song-Weinkove. Then I will focus on the semisimple principal fibration cosntruction, a construction of varieties from a principal torus bundle and a fiber. The link with weighted Kähler geometry is that, under assumptions on the principal bundle, the Kähler geometry of the total space reduces to the weighted Kähler geometry of the fiber.

Mihai Paun, Hermitian-Einstein metrics for stable coherent sheaves and Bogomolov-Gieseker inequality(Lecture III)
2025-04-02 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Mihai Paun
U. Bayreuth

I will survey the main results obtained in the articles arXiv:2501.18088 and arXiv:2405.10003. There will be no “serious” proofs, I will nevertheless try to highlight the main ingredients in our arguments.

Jinhyung Park, Effective gonality theorem on weight-one syzygies of algebraic curves
2025-04-03 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Jinhyung Park
KAIST

In 1986, Green-Lazarsfeld raised the gonality conjecture asserting that the gonality gon(C) of a smooth projective curve C of genus g can be read off from weight-one syzygies of a sufficiently positive line bundle L, and also proposed possible least degree of L, that is 2g+gon(C)-1. In 2015, Ein-Lazarsfeld proved the conjecture when deg(L) is sufficiently large, but the effective part of the conjecture remained widely open and was reformulated explicitly by Farkas-Kemeny a few years ago. We show an effective vanishing theorem for weight-one syzygies, which implies that the gonality conjecture holds if deg(L) is at least 2g+gon(C) or equal to 2g+gon(C)-1 and C is not a plane curve. As Castryck observed that the gonality conjecture may not hold for a plane curve when deg(L)=2g+gon(C)-1, this result is the best possible and thus gives a complete answer to the gonality conjecture. This is joint work with Wenbo Niu.

Thibaut Delcroix, Weighted Kähler geometry and semisimple principal fibrations (Lecture II)
2025-04-04 14:00 - 15:00 | B236-1, IBS, Korea, Republic of

Thibaut Delcroix, Weighted Kähler geometry and semisimple principal fibrations (Lecture III)
2025-04-07 14:00 - 15:00 | B236-1, IBS, Korea, Republic of

Thibaut Delcroix, Weighted Kähler geometry and semisimple principal fibrations (Lecture IV)
2025-04-09 14:00 - 15:00 | B236-1, IBS, Korea, Republic of

Long Li, Plurisubharmonic functions and Sasaki geometry
2025-04-16 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Long Li
ShanghaiTech University

In this talk, we will discuss the recent progress on the zero mass conjecture for plurisubharmoinc functions, raised by Guedj and Rashkovskii. For a local plurisubharmonic function with an isolated singularity at the origin, the conjecture states that the zero Lelong number (at the singularity) implies the zero residual Monge-Ampere mass at the same point. We confirm this conjecture under the circular symmetry condition, with the aid from Sasaki geometry. Furthermore, we will also apply our technique to more general plurisubharmonic functions in certain cases.
