Center for Complex Geometry

Algebraic Geometry Day at CCG in IBS
2021-12-09 | B266, IBS, Korea, Republic of
List of Seminars

On the Singular Loci of Higher Secants of Veronese Varieties
Kangjin Han (DGIST)
14:00-14:50, online

Manin’s Conjecture for a Log Del Pezzo Surface of Index 2
DongSeon Hwang (Ajou Univ.)
15:20-16:10, IBS B266

Ulrich Bundles on Cubic Fourfolds
Yeongrak Kim (Pusan National Univ.)
16:30-17:20, IBS B266

Sai-Kee Yeung, Almost Complex Structures and Complex Structures on Manifolds of Even Dimension at Least 6
2021-12-02 11:00 - 12:00 (Asia/Seoul) | on-line
Speaker

Sai-Kee Yeung
Purdue University

We would like to explain for each real even dimension 2n ≥ 6 some examples of compact differentiable manifolds supporting an almost complex structure which cannot be deformed to an integrable complex structure.

Kyoung-Seog Lee, Cox Rings and Geometry of Some Surfaces of General Type with pg=q=0
2021-11-23 11:00 - 12:00 | on-line
Speaker

Kyoung-Seog Lee
University of Miami

Cox ring is an important tool in modern algebraic geometry and several other branches of mathematics. In the first part of this talk, I will briefly review basic theory of Cox ring and explain how it connects birational geometry and geometric invariant theory. Then I will discuss how we can use Cox rings to study geometry of certain algebraic surfaces of general type with pg=q=0. The second part of this talk is based on several joint works (some in progress) with JongHae Keum and Davide Frapporti.

Giancarlo Urzúa, Wormholes: MMP, Topology, Continued Fractions
2021-11-16 09:30 - 10:30 | on-line
Speaker

Giancarlo Urzúa
Pontificia Universidad Catolica de Chile

We defined wormholes in https://arxiv.org/abs/2102.02177 (joint with Nicolás Vilches). Conjecturally it is a way to non-continuous travel in the KSBA compactification of the moduli space of surfaces of general type. It depends on a particular MMP. In that paper, we verified the conjecture in several cases, but many remain open. Beyond the certainty of the conjecture, it would be interesting to know about changes in the topology or differential structure after traveling through a wormhole. In this talk, I will exemplify what we know, and I will state open questions, which also include a mysterious combinatorial invariant delta that remains constant in this journey and seems to be part of some particular sequence of integers.

Jihun Yum, Limits of Bergman kernels on a Tower of Coverings of Compact Kähler Manifolds
2021-11-11 16:00 - 17:00 | B266, IBS, Korea, Republic of
Speaker

Jihun Yum
IBS, Center for Complex Geometry

The Bergman kernel BX, which is by the definition the reproducing kernel of the space of L2 holomorphic n-forms on a n-dimensional complex manifold X, is one of the important objects in complex geometry. In this talk, we observe the asymptotics of the Bergman kernels, as well as the Bergman metric, on a tower of coverings. More precisely, we show that, for a tower of finite Galois coverings {ϕj : Xj → X} of compact Kähler manifold X converging to an infinite Galois covering ϕ : X~ → X, the sequence of push-forward Bergman kernels ϕj*BXj locally uniformly converges to ϕ*BX~. Also, we show that if the canonical line bundle KX~ of X~ is very ample, then the canonical line bundle KXj of Xj is also very ample for sufficiently large j. This is a joint work with S. Yoo in IBS-CCG.

Jie Liu, Bigness of Tangent Bundles of Fano Manifolds with Zero Dimensional VMRT
2021-11-04 11:00 - 12:00 | TBA
Speaker

Jie Liu
Institute of Mathematics, AMSS, CAS

It is expected that the bigness of tangent bundle is a quite restrictive property for Fano manifolds, especially for those of Picard number one. In this talk, I will present our recent first attempt to tackle this problem. More precise, we will consider Fano manifolds of Picard number one and having zero-dimensional VMRT, and it turns out that in this case only the quintic del Pezzo threefold has big tangent bundle. This is based on my recent joint work with Andreas Höring.

Feng Shao, The Bigness of Tangent Bundles of Projective Manifolds
2021-10-28 11:00 - 12:00 | TBA
Speaker

Feng Shao
IBS, Center for Complex Geometry

Let X be a Fano manifold. While the properties of the anticanonical divisor –KX and its multiples have been studied by many authors, the positivity of the tangent bundle TX is much more elusive. In this talk, we give a complete characterization of the pseudoeffectivity and the bigness of TX for del Pezzo surfaces, hypersurfaces in the projective space and del Pezzo threefolds.

Zhi Jiang, On Syzygies of Abelian Varieties
2021-10-26 11:00 - 12:00 | on-line
Speaker

Zhi Jiang
SCMS, Fudan University

Syzygies of ample line bundles on abelian varieties have attracted lots of attentions in recent years. People tried to study these geometric objects by different methods, including Okounkov bodies, X-methods from MMP, and generic vanishing theory. We will report some progress on this subject based on the work of Jiang-Pareschi, Caucci, and Ito based on cohomological rank functions.

Yewon Jeong, Several Types of Dual Defective Cubic Hypersurfaces
2021-10-14 11:00 - 12:00 | TBA
Speaker

Yewon Jeong
IBS, Center for Complex Geometry

Given a hypersurface X = V(f) in a complex projective space, we say X is dual defective if the Gauss map of X, the restriction of the gradient map of f on X, has positive dimensional fibers. Especially for cubics, there is an interesting classification of them. We will study several types of dual defective cubic hypersurfaces and the relation between them.

Yuchen Liu, K-stability and Moduli of Quartic K3 Surfaces
2021-10-05 11:00 - 12:00 | on-line
Speaker

Yuchen Liu
Northwestern University

We show that K-moduli spaces of (P3, cS) where S is a quartic surface interpolates between the GIT moduli space and the Baily-Borel compactification as c varies in (0,1). We completely describe the wall crossings of these K-moduli spaces. As a consequence, we verify Laza-O’Grady’s prediction on the Hassett-Keel-Looijenga program for quartic K3 surfaces. This is based on joint work with K. Ascher and K. DeVleming.

Yoon-Joo Kim, The Dual Lagrangian Fibration of Compact Hyper-Kähler Manifolds
2021-09-30 11:00 - 12:00 | on-line
Speaker

Yoon-Joo Kim
Stony Brook University

A compact hyper-Kähler manifold is a higher dimensional generalization of a K3 surface. An elliptic fibration of a K3 surface correspondingly generalizes to the so-called Lagrangian fibration of a compact hyper-Kähler manifold. It is known that an elliptic fibration of a K3 surface is always “self-dual” in a certain sense. This turns out to be not the case for higher-dimensional Lagrangian fibrations. In this talk, we will explicitly construct the dual of Lagrangian fibrations of all currently known examples of compact hyper-Kähler manifolds.

Changho Han, Compact Moduli of Lattice Polarized K3 Surfaces with Nonsymplectic Cyclic Action of Order 3
2021-09-16 11:00 - 12:00 | on-line
Speaker

Changho Han
University of Georgia

Observe that any construction of “meaningful” compactification of moduli spaces of objects involve enlarging the class of objects in consideration. For example, Deligne and Mumford introduced the notion of stable curves in order to compactify the moduli of smooth curves of genus g, and Satake used the periods from Hodge theory to compactify the same moduli space. After a brief review of the elliptic curve case (how those notions are the same), I will generalize into looking at various compactifications of Kondo’s moduli space of lattice polarized K3 surfaces (which are of degree 6) with nonsymplectic Z/3Z group action; this involves periods and genus 4 curves by Kondo’s birational period map in 2002. Then, I will extend Kondo’s birational map to describe birational relations between different compactifications by using the slc compactifications (also known as KSBA compactifications) of moduli of surface pairs. The main advantage of this approach is that we obtain an explicit classification of degenerate K3 surfaces, which is used to find geometric meaning of points parametrized by Hodge-theoretic compactifications. This comes from joint works (in progress) with Valery Alexeev, Anand Deopurkar, and Philip Engel.

Sung Rak Choi, Subadditivity of Okounkov Bodies
2021-08-19 11:00 - 12:00 | 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.
