Center for Complex Geometry

Sung Rak Choi, Subadditivity of Okounkov Bodies
2021-08-19 11:00 - 12:00 (Asia/Seoul) | on-line
Speaker

Sung Rak Choi
Yonsei University

We will investigate the subadditivity theorem of Okounkov bodies for algebraic fiber spaces. As an application, we obtain the subadditivity of the numerical Kodaira dimension and the restricted volume for algebraic fiber spaces. As a byproduct, we obtain a criterion of birational isotriviality in terms of Okounkov bodies when the general fiber is of general type. As a special case, we prove some variants of the Iitaka conjecture. We expect that our results will provide a new approach toward the Iitaka conjecture.
This is an ongoing research, in collaboration with Dr. Jinhyung Park.

Jinhyung Park, Comparing Numerical Iitaka Dimensions
2021-08-12 11:00 - 12:00 | on-line
Speaker

Jinhyung Park
Sogang University

There are several definitions of the “numerical” Iitaka dimensions of a pseudoeffective divisor, which are numerical analogues to the Iitaka dimension. Recently, Lesieutre proved that notions of numerical Iitaka dimensions do not coincide. In this talk, we prove that many of numerical Iitaka dimensions are equal to the notion introduced by Boucksom-Demailly-Paun-Peternell and that some other invariants introduced by Nakayama and Lehmann can be arbitrarily larger than this notion. We also show some properties of abundant divisors. This is joint work with Sung Rak Choi.

2021 Pacific Rim Complex and Symplectic Geometry Conference
2021-07-12 - 2021-07-16 | on-line
https://cgp.ibs.re.kr/activities/conferences/337

Yonghwa Cho, Cohomology of Divisors on Burniat Surfaces
2021-07-08 14:00 - 15:00 | B266, IBS, Korea, Republic of
Speaker

Yonghwa Cho
KIAS

A (primary) Burniat surface is a complex surface of general type that can be obtained as a bidouble cover of del Pezzo surface with K2 = 6. The Picard group is an abelian group of rank 4 with torsion part isomorphic to (Z/2)6. Alexeev studied the divisors on Burniat surfaces, and observed that the irreducible components of ramification divisors span the semigroup of effective divisors. Based on Alexeev’s result, I will describe the method for computing cohomology of arbitrary divisors on Burniat surfaces. If time permits, (non-)existence question about Ulrich bundles will be discussed as an application.

Sukmoon Huh, Logarithmic Sheaves on Projective Surfaces
2021-07-08 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Sukmoon Huh
Sungkyunkwan University

A logarithmic sheaf is a sheaf of differential one-forms on a variety with logarithmic poles along a given divisor. One of the main problems on this object is to see whether Torelli property holds, i.e. whether two different divisors define two non-isomorphic logarithmic sheaves. In this talk, after reviewing some basic material, we are going to introduce two new approaches to obtain Torelli property. This is a joint work in progress with S. Marchesi, J. Pons-Llopis and J. Valles.

Yong Hu, Noether-Severi Inequality and Equality for Irregular Threefolds of General Type
2021-07-01 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Yong Hu
KIAS

For complex smooth irregular 3-folds of general type, I will introduce the optimal Noether-Severi inequality. This answers an open question of Zhi Jiang in dimension three. Moreover, I will also completely describe the canonical models of irregular 3-folds attaining the Noether-Severi equality. This is a joint work with Tong Zhang.

Pak Tung Ho, Chern-Yamabe Problem
2021-06-23 16:00 - 17:00 | on-line
Speaker

Pak Tung Ho
Sogang University

I will explain what the Chern-Yamabe problem is, and talk about the Chern-Yamabe flow which is a geometric flow approach to solve the Chern-Yamabe problem.
I will also mention other results related to the Chern-Yamabe problem.

Baohua Fu, Normalized Tangent Bundle, Pseudoeffective Cone and Varieties with Small Codegree
2021-06-17 11:00 - 12:00 | on-line
Speaker

Baohua Fu
Chinese Academy of Science

We propose a conjectural list of Fano manifolds of Picard number one whose normalized tangent bundle is pseudoeffective and we prove it in various situations by relating it to the complete divisibility conjecture of Russo and Zak on varieties with small codegrees. The pseudoeffective cone of the projectivized tangent bundle of a rational homogeneous space of Picard number one is explicitly determined by studying the total dual VMRT and the geometry of stratified Mukai flops. This is a joint work with Jie LIU.

Dano Kim, Canonical Bundle Formula and Degenerating Families of Volume Forms
2021-06-16 16:00 - 17:00 | on-line
Speaker

Dano Kim
Department of Mathematical Sciences, Seoul National University

We will talk about a metric version of Kawamata’s canonical bundle formula for log Calabi-Yau fibrations: the L2 metric carries singularity described by the discriminant divisor and the moduli part line bundle has a singular hermitian metric with vanishing Lelong numbers. This answers a folklore conjecture arising from work of Kawamata and Tsuji and a question of Eriksson, Freixas i Montplet and Mourougane. It has immediate applications to L2 extension theorems which was our starting point.

Joonyeong Won, K-stability, Kähler-Einstein Metric on Fano Varieties and Sasaki-Einstein Metric on 5-dimensional Smale Manifolds
2021-06-03 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Joonyeong Won
KIAS

We discuss on recent progresses of the existence problem of Kähler-Einstein metric on Fano varieties by K-stability of them and also the existence problem of Sasaki-Einstein metric on 5-dimensional Smale manifolds via K-stability of weighted hypersurface log del Pezzo surfaces.

Hoseob Seo, On Singularities of Toric Plurisubharmonic Funcitons
2021-06-02 16:00 - 17:00 | B266, IBS, Korea, Republic of
Speaker

Hoseob Seo
Research Institute of Mathematics, Seoul National University

In this talk, we discuss recent progresses on singularities of toric plurisubharmonic functions. First, we review the notion of Newton convex bodies of toric plurisubharmonic functions on a polydisk D(0,r) ⊂ Cn. As an application, we show that the cluster points of jumping numbers of J(φ)0 are not accumulated and give a precise characterization of the set of those cluster points. These generalize a recent result of Kim and Seo from n = 2.
On the other hand, we extends a result by Guan, which proved when toric plurisubharmonic functions of the form log(∑ |z|ai) have a decreasing equisingular approximation with analytic singularities. We give a criterion for the existence of a decreasing equisingular toric approximation with analytic singularities for a given toric plurisubharmonic function. This is a joint work with Jongbong An.

Lucas Kaufmann, Commuting Pairs of Endomorphisms
2021-05-27 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Lucas Kaufmann
IBS, Center for Complex Geometry

The study of functional equations is at the origin of the early developments of the iteration theory of polynomials and rational functions, carried out by Fatou, Julia, Ritt and others. Among these equations, the commutation relation f g = g f is particularly interesting. In this talk I will discuss the following problem: can we classify commuting pairs of holomorphic endomorphisms of the complex projective space?
We will see that the relation f g = g f gives rise to special symmetries of several dynamical objects attached to these maps, such as their invariant currents and measures, their Julia sets and so on. This rigidity allows us to understand the structure of these maps and even give a full description in low dimensions.

Lucas Kaufmann, Introduction to Dynamics in Several Complex Variables
2021-05-20 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Lucas Kaufmann
IBS, Center for Complex Geometry

The field of complex dynamics deals with the study of the iteration of a map from a complex manifold to itself. The one dimensional theory is more than one-hundred years old and is now very well developed.
Due to the fundamental differences between complex analysis in one and higher dimensions, the study of higher dimensional dynamical systems is much more recent and is still an active research domain, with interesting connections to complex geometry, algebraic geometry, number theory, mathematical physics etc.
The aim of this talk is to survey standard results on the dynamics of self-maps in complex manifolds. I will focus on the case of endomorphisms of the complex projective space and use it to illustrate how the tools of pluripotential theory and the theory of currents are crucial in their study.
In particular, I aim to talk about Green currents and measures, equidistribution theorems and, if time permits, relations with the intersection theory of currents.

Jeong-Seop Kim, Stability of Symmetric Powers of Vector Bundles on a Curve
2021-05-13 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Jeong-Seop Kim
KAIST

For a stable vector bundle E on a smooth projective curve, it is known that the symmetric powers Sk E are semi-stable and are stable for all k > 0 in sufficiently general. Moreover, if E has rank 2, then Sk E is destabilized by a line subbundle if and only if the ruled surface PC(E) admits a k-section of zero self-intersection. In this talk, concentrating on the case of rank 2, we will find answers to the questions of which E has strictly semi-stable Sk E, and how many such E there are. Also, we will introduce relations between such E and the orthogonal bundles when k = 2, and Nori’s finite bundles when k ≥ 3.

Young-jun Choi, Existence of a Complete Holomorphic Vector Field via the Kähler-Einstein Metric
2021-05-12 16:00 - 17:00 | B266, IBS, Korea, Republic of
Speaker

Young-jun Choi
Pusan National University

A fundamental problem in Several Complex Variables is to classify bounded pseudoconvex domains in the complex Euclidean space with a noncompact automorphism group, especially with a compact quotient. In the results of Wong-Rosay and Frankel, they make use of the “Scaling method” for obtaining an 1-parameter family of automorphisms, which generates a holomorphic vector field.
In this talk, we discuss the existence of a nowhere vanishing complete holomorphic vector field on a strongly pseudoconvex manifold admitting a negatively curved Kähler-Einstein metric and discrete sequence of automorphisms by introducing the scaling method on potentials of the Kähler-Einstein metric.
This is a joint work with Kang-Hyurk Lee in Gyenongsang National University.

Hosung Kim, The Space of Rational Curves on a General Hypersurface of Projective Space
2021-04-22 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Hosung Kim
IBS, Center for Complex Geometry

In 1979, the work of Mori had brought out the importance of the study of rational curves in higher-dimensional geometry. In 1990s, applying Mori’s bend-and-break method, Campana and Kollar-Miyaoka-Mori proved that any Fano manifold is rationally connected. Since then the family of raional curves on Fano maniflolds has been considerably studied especially about the dimension and irreducibility. The expected dimension of the family of rational curves of degree e in a hypersurface of degree d in Pn is e(n-d+1)+n-4, and it has been conjectured that this family is irreducible and has dimension of the expected dimension if d ≤ n-1 and n > 3. This conjecture has been proven for d ≤ n-2 and e arbitrary by Riedl and Yang in 2019 based on bend-and-break. For d = n-1, some results for low e were obtained by Tseng in 2017. In this seminar I am going to introduce a technique based on degenerating the ambient projective space to a 2-component fan and simultaneously degenerate a general hypersurface of a projective space to a subscheme of the fan. Using this method, Ziv Ran reduced the original problem to that on the space of rational curves in Pn which are some secant to a certain (d,d-1) complete intersection, and proved the cases when e < d ≤ n-1 and n > 4.

Taeyong Ahn, Positive Closed Currents and Super-potentials
2021-04-21 16:00 - 17:00 | B266, IBS, Korea, Republic of
Speaker

Taeyong Ahn
Inha University, Department of Mathematics Education

In this talk, we briefly review the notion and properties of positive closed currents and super-potentials. As an application, we discuss the equidistribution of positive closed currents on the projective space. We also discuss the difficulty of the extension of the result to a general compact Kähler manifold.

Seungjae Lee, L2 Extension of Holomorphic Jets on Complex Hyperbolic Forms
2021-04-15 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Seungjae Lee
IBS, Center for Complex Geometry

As the continuation of the previous talk, I discuss an L2 extension problem of holomorphic jets on compact complex hyperbolic forms. Let Γ be a cocompact torsion-free lattice in the automorphism group Aut(Bn) and Ω be a quotient Bn × Bn given by diagonal action of Γ. In the setting, Ω becomes a ball-fiber bundle over Σ = Bn / Γ. Since we can identify symmetric differentials on Σ and jets of holomorphic function on D which is the maximal compact analytic variety on Ω, it is natural to expect that holomorphic function on Ω can be derived by symmetric differentials. In this context, M. Adachi (2017) extends holomorphic jets on D to weighted L2 holomorphic functions on Σ for the n=1 case. In 2020, A. Seo and S. Lee generalized his result by developing a Hodge type identity on SmTΣ*. In this talk, I will explain recent progress and if time is permitted, I sketch the proof of our result. This is joint work with A. Seo.

Seungjae Lee, Symmetric Differentials on Complex Hyperbolic Forms
2021-04-08 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Seungjae Lee
IBS, Center for Complex Geometry

Let Γ be a cocompact torsion-free lattice in the automorphism group of complex unit ball Bn, Aut(Bn). In this talk, we discuss the existence of symmetric differentials on the compact ball quotient Σ = Bn / Γ. Since Σ has a Kähler metric induced by the Bergman metric on the complex unit ball Bn, it has symmetric differentials on SmTΣ* if m is sufficiently large. Unfortunately, finding the smallest degree m which guarantees a symmetric differential on SmTΣ* is difficult in even compact ball quotient cases. Instead of this, I will prove that m ≥ n+2 is a sufficient condition to give a symmetric differential on SmTΣ*. To achieve this goal, I will explain how to induce symmetric differentials by using a recursive formula for ∂-operators and Poincaré series. This is joint work with A. Seo.

Jihun Yum, Characterization of Diederich-Fornaess and Steinness Indices in Complex Manifolds
2021-04-07 16:00 - 18:00 | B266, IBS, Korea, Republic of
Speaker

Jihun Yum
IBS, Center for Complex Geometry

Let Ω be a relatively compact pseudoconvex domain in a complex manifold X with smooth boundary ∂Ω. The Diederich-Fornaess index and the Steinness index of Ω are defined by
DF(Ω) := supρ { 0 < η < 1 : -(-ρ)η is strictly plurisubharmonic on Ω ∩ U for some neighborhood U of ∂Ω },
S(Ω) := infρ { η > 1 : ρη is strictly plurisubharmonic on Ωc ∩ U for some neighborhood U of ∂Ω },
where ρ is a defining function for Ω.
In the previous talk, we have seen that two indices are completely characterized by D’Angelo 1-form when the ambient space is X = Cn. In this talk, we generalize the formulas for a relatively compact pseudoconvex domains in a (general) complex manifold X. Since the formulas do not hold anymore in general, unfortunately, we introduce 4 kinds of each of the Diederich-Fornaess and Steinness indices. Then we give some non-degeneracy conditions for these indices agree. Also, we exam the geometric meaning of the D’Angelo 1-form when the boundary ∂Ω is Levi-flat.

Qifeng Li, Rigidity of Wonderful Group Compactifications under Fano Deformations
2021-04-01 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Qifeng Li
IBS, Center for Complex Geometry

For a complex connected semisimple linear algebraic group G of adjoint type and of rank n, De Concini and Procesi constructed its wonderful compactification X, which is a smooth Fano variety of Picard number n enjoying many interesting properties. In this talk, we will show that the wonderful compactification X is rigid under Fano deformations. Namely, for any family of smooth Fano varieties over a connected base, if one fiber is isomorphic to X, then so are all other fibers. This answers a question raised by Bien and Brion in their work on the local rigidity of wonderful varieties.

Sungmin Yoo, Bergman Metrics and Kähler-Einstein Metrics on Projective Manifolds
2021-03-25 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Sungmin Yoo
IBS, Center for Complex Geometry

After Yau suggested the problem of approximations of Kähler-Einstein metrics by Bergman type metrics, various types of Bergman metrics have been developed and studied by Tian, Donaldson, Tsuji, etc. They showed that if a polarized manifold admits a Kähler-Einstein metric, there exist a sequence of Bergman type metrics which converges to the Kähler-Einstein metric. In this talk, we will survey their results and discuss some remaining problems in this subject.

Jihun Yum, Characterization of Diederich-Fornaess and Steinness Indices in Cn
2021-03-24 16:00 - 18:00 | B266, IBS, Korea, Republic of
Speaker

Jihun Yum
IBS, Center for Complex Geometry

Let Ω be a bounded pseudoconvex domain in Cn with smooth boundary ∂Ω. The Diederich-Fornaess index and the Steinness index of Ω are defined by
DF(Ω) := supρ { 0 < η < 1 : -(-ρ)η is strictly plurisubharmonic on Ω ∩ U for some neighborhood U of ∂Ω },
S(Ω) := infρ { η > 1 : ρη is strictly plurisubharmonic on Ωc ∩ U for some neighborhood U of ∂Ω },
where ρ is a defining function for Ω.
First, we see basic properties and known results about the Diederich-Fornaess and Steinness indices. Also, we see the relation between two indices on a 1-parameter family of domains in C2, called worm domains, constructed by Diederich and Fornaess.

We characterize the Diederich-Fornaess and Steinness indices in terms of a special 1-form, which we call D’Angelo 1-form. These formulas are the most important in this talk. After giving a sketch of the proof, we show many applications and corollaries of the formulas. Especially, we prove that

two indices are invariant under CR-diffeomorphisms,
semi-continuity of two indices.

Sungmin Yoo, Fiberwise Kähler-Ricci Flows on Families of Strongly Pseudoconvex Domains
2021-03-18 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Sungmin Yoo
IBS, Center for Complex Geometry

A study on the positive variation of Kähler-Einstein metrics is first developed by Schumacher. More precisely, he has proved that the fiberwise Kähler-Einstein metrics on a family of canonically polarized compact Kähler manifolds is positive-definite on the total space. Later, Berman constructed the fiberwise Kähler-Ricci flow which converges to the fiberwise Kähler-Einstein metrics and showed that the positivity is preserved under this flow. In this talk, I will explain how to generalize these to a family of bounded strongly pseudoconvex domains, which is an important examples of non-compact complete Kähler manifolds. This is joint work with Young-Jun Choi.

Hyungryul Baik, Limits of Canonical Metrics in Low-dimensions
2021-01-28 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Hyungryul Baik
KAIST

For a tower of finite normal covers of graphs or surfaces, one can consider a sequence of metrics on the base given by pull-back of canonical metric of the covers. We show that such a sequence has a limit and it depends only on the cover approximated by the tower up to scaling. The case of compact Riemann surface where the tower approximates the universal cover is due to Kazhdan. In this talk, I will explain why one might be interested in such a problem, give a brief introduction to the main tool – L2-theory, and discuss some future directions. This talk is based on a joint work with Farbod Shokrieh and Chenxi Wu.

Seung-Jo Jung, Hodge Ideals and Spectra of Hypersurface Singularities
2021-01-21 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Seung-Jo Jung
Jeonbuk National University

Recently Mustata-Popa introduced a generalisation of multiplier ideals, which is called Hodge ideals. This talk introduces the Hodge ideals and explains the relations with other invariants, e.g. Bernstein-Sato polynomials, Hodge spectra, log canonical thresholds, minimal exponents etc. Mainly this talk concerns the relation between Hodge ideals and Hodge spectra. This talk is based on the joint paper with I-K. Kim, M. Saito, and Y. Yoon.

Sangbum Yoo, Spectral Data for Principal Higgs Bundles over a Singular Curve
2021-01-14 16:00 - 17:00 | B266, IBS, Korea, Republic of
Speaker

Sangbum Yoo
Gongju National University of Education

Spectral data for Higgs bundles over a smooth curve has been studied by several mathematicians. The studies in this direction are originated by N.J.Hitchin. Specially, it has contributed to the studies on the fibers of the Hitchin map. In this talk, I will introduce spectral data for Higgs bundles over a smooth curve and over a singular curve, and then present my ongoing project about spectral data for principal Higgs bundles over a singular curve.

Sung Yeon Kim, Nonsolvability of Lewy Operator and Non-realizable CR Structures
2020-12-16 16:00 - 17:00 | B266, IBS, Korea, Republic of
Speaker

Sung Yeon Kim
IBS, Center for Complex Geometry

Minseong Kwon, Integrability of G-structures III
2020-12-10 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Minseong Kwon
KAIST

This is a working seminar to introduce the notion of an integrable G-structure and its obstruction class. In the previous two talks, we discussed the definition of the k-th order structure tensor of a G-structure. In the third talk, we will discuss how the structure tensors can be characterized in terms of classical invariants, namely the curvature tensor and the torsion tensor of a given connection. The main reference is the paper ‘The Integrability Problem for G-structures’ (1965) written by Victor Guillemin.

Jihun Yum, Isometric Embedding of Kähler Manifolds and the Diastatic Function
2020-12-09 16:00 - 17:00 | B266, IBS, Korea, Republic of
Speaker

Jihun Yum
IBS, Center for Complex Geometry
