Center for Complex Geometry

Eric Sommers, Some Slodowy Slices Associated to Special Nilpotent Orbits
2024-06-11 16:00 - 17:00 (Asia/Seoul) | B236-1, IBS, Korea, Republic of
Speaker

Eric Sommers
University of Massachusetts

Among the nilpotent orbits in a simple Lie algebra are the special nilpotent orbits, which play an important role in representation theory. Some of the geometry of the closure of a nilpotent orbit can be understood by taking a transverse slice to a smaller orbit in the closure. This talk concerns a classification of two types of such transverse slices: (1) those between adjacent special nilpotent orbits; and (2) those between a special nilpotent orbit and a certain non-special nilpotent orbit in its closure. The slices in part (1) exhibit a duality, which extends an observation of Kraft and Procesi for type A. The slices in part (2) are related to a conjecture of Lusztig on special pieces. This talk is based on two preprints with Baohua Fu, Daniel Juteau, and Paul Levy.

Cheol Hyun Cho, Floer Theory for the Variation Operator of an Isolated Singularity
2024-06-13 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Cheol Hyun Cho
Seoul National University

The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analogue for an isolated singularity. We define a new Floer cohomology, called monodromy Lagrangian Floer cohomology, which provides categorifications of the standard theorems on the variation operator and the Seifert form. We discuss examples of plane curve singularities and explain the relation between exceptional collections of non-compact curves and distinguished collections of vanishing cycles.

Myeongjae Lee, Connected Components of the Strata of Residueless Meromorphic Differentials
2024-06-18 11:00 - 12:00 | on-line
Zoom ID: 880 6763 5837
PW: [REDACTED]

Speaker

Myeongjae Lee
Stony Brook University

Strata of differentials are interesting objects studied in various fields such as Teichmuller dynamics, topology and algebraic geometry. Generalized strata are subsets of the strata of meromorphic differentials, where certain sets of residues summing up to zero. We present the classification of the connected components of the strata of residueless meromorphic differentials, using a degeneration technique to the boundary of a nice compactification of strata.

Kisun Lee, Introduction to Numerical Algebraic Geometry
2024-06-24 14:00 - 15:00 | B236-1, IBS, Korea, Republic of
Speaker

Kisun Lee
Clemson University

Numerical algebraic geometry employs numerical techniques for problems in algebraic geometry. This talk begins with a question reminding the meaning of solving a (polynomial) equation. It overviews the homotopy continuation as a method for finding solutions to a system of polynomial equations. After problems from algorithmic and application points of view in numerical nonlinear algebra are introduced, we discuss how the results from numerical techniques can be certified. No pre-knowledge from graduate-level algebraic geometry is assumed.

Kisun Lee, Numerical Certification and Certified Homotopy Tracking
2024-06-24 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Kisun Lee
Clemson University

A certified algorithm produces a solution and a certificate of correctness to a problem. Numerical certification studies certified algorithms for results obtained from numerical methods in algebraic geometry. In this talk, we discuss why numerical certification is needed in numerical algebraic geometry and introduce the Krawczyk homotopy as one of its methods. The Krawczyk homotopy exploits interval arithmetic to track the homotopy solution path without path jumping so that the correctness of its output is guaranteed. This is joint work with Timothy Duff.

Euisung Park, On Rank 3 Quadratic Equations of Projective Varieties
2024-06-25 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Euisung Park
Korea University

Many projective varieties are ideal-theoretically cut out by quadratic polynomials of rank less than or equal to 4. Classical constructions in projective geometry like rational normal scrolls and Segre-Veronese varieties are examples. Regarding this phenomenon, I would like to talk about the following two results in this talk. First, there are many projective varieties cut out ideal-theoretically by quadratic polynomials of rank 3. Second, there is a nice structure of the locus of rank 3 quadratic equations of a projective variety.

Jiewon Park, Hessian Estimates, Monotonicity Formulae, and Applications
2024-06-26 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Jiewon Park
KAIST

Various monotonicity formulae have profound applications in many different problems in geometric analysis. Quite often these formulae can be derived from pointwise Hessian estimates, also known as Li-Yau-Hamilton estimates or matrix Harnack inequalities. In this talk we will focus on this connection building upon Hessian estimates for the Green function, leading to novel monotonicity formulae on manifolds with certain curvature lower bounds and Einstein manifolds. Some of the results have counterparts for Kähler manifolds.

Pacific Rim Complex and Symplectic Geometry Conference
2024-07-29 - 2024-08-02 | IBS Science Culture Center, Daejeon, Korea, Republic of
Invited Speakers
Dongwook Choa (KIAS, Seoul)
Young-Jun Choi (Pusan National Univ.)
Siarhei Finski (École Polytechnique)
Hervé Gaussier (Univ. Grenoble-Alpes)
Masafumi Hattori (Kyoto Univ.)
Siqi He (AMSS, Beijing)
Ludmil Katzarkov (Univ. Miami)
Yusuke Kawamoto (ETH, Zurich)
Takayuki Koike (Osaka Metropolitan Univ.)
Yu-Shen Lin (Boston Univ.)
George Marinescu (Univ. Köln)
Yuichi Nohara (Meiji Univ.)
Semon Rezchikov (Princeton Univ.)
Harish Seshadri (Indian Inst. Science)
Rasul Shafikov (Univ. Western Ontario)
Li-Sheng Tseng (U.C. Irvine)
Seungook Yu (Postech)
Jihun Yum (Gyeongsang National Univ.)
Ruobing Zhang (Princeton Univ.)
Andrew Zimmer (Univ. Wisconsin)
Abstracts
PDF file
Schedule
July 29 (Monday)
(Morning session chaired by Jun-Muk Hwang)
10:00-11:00 Marinescu
11:20-12:20 Choi
12:30-14:00 Lunch
(Afternoon session chaired by Dano Kim)
14:00-15:00 Nohara
15:15-16:15 Gaussier
16:30-17:30 Yum
July 30 (Tuesday)
(Morning session chaired by Kaoru Ono)
10:00-11:00 Seshadri
11:20-12:20 Kawamoto
12:30-14:00 Lunch
(Afternoon session chaired by Yong-Geun Oh)
14:00-15:00 Rezchikov
15:15-16:15 Yu
16:30-17:30 Choa
18:00-20:00 Banquet
July 31 (Wednesday)
(Morning session chaired by Kengo Hirachi)
10:00-11:00 Koike
11:20-12:20 Zhang
12:30-14:00 Lunch
Free Afternoon
August 1 (Thursday)
(Morning session chaired by Jae-cheon Joo)
10:00-11:00 Zimmer
11:20-12:20 Lin
12:30-14:00 Lunch
(Afternoon session chaired by Kyoung-Seog Lee)
14:00-15:00 Katzarkov
15:15-16:15 Finski
16:30-17:30 Tseng
August 2 (Friday)
(Morning session chaired by Sungyeon Kim)
9:30-10:30 He
10:40-11:40 Hattori
11:50-12:50 Shafikov
12:50-14:00 Lunch
Scientific Committee
Xiuxiong Chen (Stony Brook Univ.)
Kengo Hirachi (Univ. Tokyo)
Jun-Muk Hwang (IBS-CCG)
Yong-Geun Oh (IBS-CGP & POSTECH)
Kaoru Ono (RIMS, Kyoto)
Yongbin Ruan (Zhejiang Univ.)
Organizing Committee
Jun-Muk Hwang (IBS-CCG)
Sungyeon Kim (IBS-CCG)
Yong-Geun Oh (IBS-CGP & POSTECH)
Venue
IBS Science Culture Center, Daejeon, Korea
Registration
Please submit Google form.
More Information
• How to get to IBS-CCG

Sheng Meng, On Surjective Endomorphisms of Projective Varieties
2024-08-06 14:00 - 15:00 | B236-1, IBS, Korea, Republic of
Speaker

Sheng Meng
East China Normal University

Let X be a normal projective variety over C. Let f be a surjective endomorphism of X. In this talk, I will try to explain our current program on the classification and the building blocks of (f, X), involving two main tools: equivariant minimal model program and dynamical Iitaka fibration. Two applications on the Kawaguchi-Silverman conjecture and the Zariski dense orbit conjecture will be given. This is based on several joint works with Junyi Xie, De-Qi Zhang and Guolei Zhong.

Chuyu Zhou, Lecture 1: Constructible Properties of Various Domains for a Family of Couples
2024-08-06 16:00 - 17:30 | 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.
