Center for Complex Geometry

Sangbum Yoo, Spectral Data for Principal Higgs Bundles over a Singular Curve
2021-01-14 16:00 - 17:00 (Asia/Seoul) | 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
