Center for Complex Geometry

Hosung Kim, The Space of Rational Curves on a General Hypersurface of Projective Space
2021-04-22 11:00 - 12:00 (Asia/Seoul) | 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

Nguyen Ngoc Cuong, Hölder Continuous Solutions to Complex Monge-Ampère Equations and its Applications II
2020-12-02 16:00 - 17:00 | B266, IBS, Korea, Republic of
Speaker

Nguyen Ngoc Cuong
KAIST

The Monge-Ampère equations provide Kähler-Einstein metrics on projective manifolds with negative or zero first Chern classes thanks to the AubinYau and Yau theorems. However, most projective manifolds do not have a negative definite or trivial first Chern class. The study of the canonical metric on these manifolds leads to study degenerate Monge-Ampère equations both on the right hand side and on the background form. It turns out that Hölder continuity is the best regularity we can hope for the solution (quasi-plurisubharmonic potential) to the equation. We discuss situations and criterions such that we will have this property and some applications.

More precisely, let X be a compact Kähler manifold of dimension n and ω a Kähler form on X. We consider the complex Monge-Ampère equation (ddcu+ω)n = µ, where µ is a given positive measure on X of suitable mass and u is an ω-plurisubharmonic function. We show that the equation admits a Hölder continuous solution if and only if the measure µ, seen as a functional on a complex Sobolev space W∗(X), is Hölder continuous. Here, denote by W1,2(X) the Sobolev space of real valued functions f on X such that both f and df are of class L2(X). Then, the complex Sobolev space W∗(X), introduced by Dinh-Sibony, is the space of all functions f ∈ W1,2(X) such that
df ∧ dcf ≤ T
for some closed positive (1, 1)-current T on X.

A similar result is also obtained for the complex Monge-Ampère equations on domains of Cn.

In the first talk we give motivations and background to study weak solutions of Monge-Ampère equations.

In the second talk we focus on the several criterions such that the solution is Hölder continuous, or alternatively we give the estimate of Hölder norm that depends very weak on the datum such as Lp-norm, p > 1, of the right hand sides.

This is based on joint work with Tien-Cuong Dinh and Slawomir Ko lodziej.

Minseong Kwon, Integrability of G-structures II
2020-11-26 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 first talk, I discussed the definition of an integrable G-structure and introduced the existence theorem for the structure tensor. In this talk, I will construct the structure tensor of a G-structure, and then prove that the structure tensor is indeed an obstruction class of the integrability which is lying in the Spencer cohomology group. The main reference is the paper ‘The Integrability Problem for G-structures’ (1965) written by Victor Guillemin.

Kyeong-Dong Park, Kähler-Einstein Metrics on Smooth Fano Symmetric Varieties with Picard Number One
2020-11-19 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Kyeong-Dong Park
IBS-CGP

Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces and they are interesting examples of spherical varieties. I will prove that all smooth Fano symmetric varieties with Picard number one admit Kähler-Einstein metrics using a combinatorial criterion for K-stability of Fano spherical varieties obtained by Delcroix. To this end, I will present their algebraic moment polytopes and the barycenter of each moment polytope with respect to the Duistermaat-Heckman measure. This talk is based on joint work with Jae-Hyouk Lee and Sungmin Yoo.

Jihun Yum, Classification of Domains that Admit the Bergman-Einstein Metric II
2020-11-18 16:00 - 17:00 | B266, IBS, Korea, Republic of

Jihun Yum, Classification of Domains that Admit the Bergman-Einstein Metric I
2020-11-11 16:00 - 17:00 | B266, IBS, Korea, Republic of

Eunjeong Lee, Geometry of Flag Varieties and Related Combinatorics
2020-11-10 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Enjeong Lee
IBS-CGP

For a semisimple algebraic group G and a Borel subgroup B, the homogeneous space G/B, called the flag variety, is a smooth projective variety which has a fruitful connection with G-representations. Indeed, the set of global sections H0(G/B, L) is an irreducible G-representation for a very ample line bundle L on G/B. On the other hand, string polytopes are combinatorial objects which encode the characters of irreducible G-representations. One of the most famous examples of string polytopes is the Gelfand–Cetlin polytope, and there might exist combinatorially different string polytopes. The string polytopes are related to the flag varieties via the theory of Newton–Okounkov bodies. In this talk, we will study Gelfand–Cetlin type string polytopes, their enumerations, and we will present small toric resolutions of certain string polytopes. This talk is based on several collaborations with Yunhyung Cho, Jang Soo Kim, Yoosik Kim, and Kyeong-Dong Park.

Insong Choe, Minimal Rational Curves on the Moduli Spaces of Symplectic and Orthogonal Bundles over a Curve
2020-11-05 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Insong Choe
Konkuk University

Let M be the moduli of vector bundles over a curve of fixed determinant. It is known that the Hecke curves are rational curves of minimal degree on M passing through a general point of M. We prove a similar result for the moduli of symplectic and orthogonal bundles. In particular, I explain how to construct an analogue of Hecke curves on the symplectic and orthogonal settings, and how to parameterize them. This is based on a joint work with Kiryong Chung and Sanghyeon Lee.

Nguyen Ngoc Cuong, Hölder Continuous Solutions to Complex Monge-Ampère Equations and its Applications
2020-11-04 11:00 - 12:00 | B266, IBS, Korea, Republic of

Minseong Kwon, Integrability of G-structures
2020-10-28 11:00 - 12:00 | B266, IBS, Korea, Republic of

Jun-Muk Hwang, VMRT of Unbendable Rational Curves Subordinate to Contact Structures
2020-10-20 11:00 - 12:00 | B236-1, IBS, Korea, Republic of

Sungyeon Kim, Proper Holomorphic Maps between Bounded Symmetric Domains
2020-10-14 10:30 - 11:30 | B236, IBS, Korea, Republic of

Hosung Kim, The Abel-Jacobi Mapping for Fano Threefolds
2020-10-06 11:00 - 12:00 | B236, IBS, Korea, Republic of

Jaehyun Hong, Stability of Tangent Bundles of Fano Manifolds
2020-09-22 11:00 - 12:00 | B236, IBS, Korea, Republic of

Qifeng Li, Unbendable Rational Curves of Goursat Type
2020-09-15 11:00 - 12:00 | B236, IBS, Korea, Republic of
