Algebraic and symplectic aspects in degenerations of complex surfaces
《복소곡면의 퇴화의 대수기하학적 그리고 사교위상수학적 측면》 기초연구실
2021.06.01--2024.02.29
한국연구재단(National Research Foundation of Korea) grant no. 2021R1A4A3033098
Embedded Files
본 기초연구실은 복소곡면이 퇴화되면서 나타나는 여러 현상을 대수기하학과 사교위상수학의 융합을 통해 연구하는 것을 목표로 한다. 이를 바탕으로 각 분야의 중요한 미해결 문제에 도전한다.
본 기초연구실은 복소곡면이 퇴화되면서 나타나는 여러 현상을 대수기하학과 사교위상수학의 융합을 통해 연구하는 것을 목표로 한다. 이를 바탕으로 각 분야의 중요한 미해결 문제에 도전한다.
학술활동 Activities
학술활동 Activities
ASADCS 세미나 목록 Seminar List
ASADCS 세미나 목록 Seminar List
2022-11-18 / 10:30--12:30 / ASADCS Seminar: Yonghwa Cho _ IBS-CCG
Cyclic quotient surface singularities and vector bundles
We continue the discussion of singularities of class T and exceptional vector bundles on their smoothing - the talk I gave at BRL-ASADCS in this March. We will quickly cover some basic notions including derived categories, exceptional objects, and singularities of class T and their Q-Gorenstein deformation. Then, we are going to discuss the results of Hacking, myself, Kawamata, Tevelev-Urzua which associate certain vector bundles with degenerations to cyclic quotient singularities.
Online seminar: zoom id 313 252 8553
Host: YongJoo Shin
2022-11-04 / 10:30--12:30 / ASADCS Seminar: Jeong-Seop Kim _ KIAS
Positivity of tangent bundles of smooth projective varieties
After Mori's solution to the Hartshorne's conjecture on the ampleness of tangent bundles of smooth projective varieties, there has been similar questions to characterize smooth projective varieties under certain positivity of their tangent bundles. For example, Campana and Peternell propose a conjecture on the nefness, and also, Solá Conde and Wiśniewski give a complete answer to the question of big and 1-ample tangent bundles. Recently, there are some progress on the question of which smooth projective varieties have big tangent bundles. For example, smooth toric varieties, and smooth Fano twofolds or prime Fano threefolds of higher degree are known to have big tangent bundles. In this talk, I will review the progress, and explain a method to find a total dual VMRT, which plays an important role in determining the bigness of tangent bundles. Then I will introduce results on the bigness in the cases of Fano threefolds of Picard number 2, and projective bundles over a smooth projective curve. This talk is based on a joint work with Hosung Kim and Yongnam Lee.
Online seminar: zoom id 313 252 8553
Host: YongJoo Shin
2022-10-28 / 11:00--12:00 / ASADCS Seminar: Hosung Kim _ IBS-CCG
Lagrangian fibration structure on the cotangent bundle of a del Pezzo surface of degree 4
In this talk, we will talk about a natural Lagrangian fibration structure on the map Φ from the cotangent bundle of a del Pezzo surface X of degree 4 to C^2. This is a joint work with Yongnam Lee.
Online: ZOOM ID 313 252 8553
Host: YongJoo Shin
2022-10-07 / 11:00--12:00 / ASADCS Seminar: Hakho Choi _ SNU QSMS
Rational homology disk smoothings and Lefschetz fibrations
In this talk, we construct genus-1 positive allowable Lefschetz fibrations representing rational homology disk smoothings of weighted homogeneous surface singularities whose resolution graphs are 3-legged with a central bad vertex.
Online: ZOOM ID 313 252 8553
Host: Dongsoo Shin
2022-07-08 / 11:00--12:00 / ASADCS Seminar: Myeonggi Kwon _ Sunchon National University
Contact topology of singularities and symplectic fillings
For an isolated hypersurface singularity, the intersection with a small sphere forms a smooth manifold, called the link of a singularity. It admits an additional geometric structure called a contact structure, and this turns out to be a fine invariant of singularities and provides an interesting playground to explore relationships between contact topology and singularity theory. In this talk, we briefly introduce results on contact topology of singularities in terms of symplectic fillings, exotic contact spheres, and Floer theory.
Online seminar: zoom id 313 252 8553
Host: Dongsoo Shin
2022-05-06 / 10:30--12:30 / ASADCS Seminar: In-Kyun Kim _ Yonsei University
An introduction to K-stability of Fano varieties II
The correspondence between the existence of K\"ahler-Einstein metrics and the K-stability of Fano varieties is one of the important problems. Using the results of Chen, Donaldson, Sun and Tian of this problem the concept of K-stability can give algebraic proofs for differential geometric problems. In this talk I will give an overview of the theory of K-stability and K-moduli spaces.
Online seminar: zoom id 3132528553
Host: YongJoo Shin
2022-04-15 / 10:30--12:30 / ASADCS Seminar: In-Kyun Kim _ Yonsei University
An introduction to K-stability of Fano varieties I
The correspondence between the existence of K\"ahler-Einstein metrics and the K-stability of Fano varieties is one of the important problems. Using the results of Chen, Donaldson, Sun and Tian of this problem the concept of K-stability can give algebraic proofs for differential geometric problems. In this talk I will give an overview of the theory of K-stability and K-moduli spaces.
Online seminar: zoom id 3132528553
Host: YongJoo Shin
2022-03-31 / 13:00--15:00 / ASADCS Seminar: Kwangwoo Lee _ Yonsei University
Automorphisms of K3 surfaces with Picard number 2
For K3 surfaces with Picard number 2, it is known that if automorphism group is infinite then it is either infinite cyclic group or infinite dihedral group. We investigate some conditions on an automorphism of infinite order and on an involution. For this, we use Salem numbers of automorphisms of infinite order. Using these conditions, we also study which K3 surface with Picard number 2 can allow dihedral automorphisms groups. In this talk, We will review some results about lattices, Salem numbers and solvability of some Pell equations.
Online seminar: zoom id 3132528553
Host: YongJoo Shin
2022-03-25 / 11:00--12:30 / ASADCS Seminar: Yonghwa Cho _ IBS-CCG
Singularities of class T and exceptional vector bundles
Recent studies tell that $\mathbb Q$-Gorenstein degeneration of surfaces give rise to a certain collection of exceptional vector bundles. Such phenomena first observed in projective planes, where $\mathbb Q$-Gorenstein degenerations and semiorthogonal decompositions are both classified by Markov equation. It turns out that it is a glimpse of more general construction done by Hacking, which considers degenerations to Wahl singularities and exceptional vector bundles. We review his construction and explain our generalization to cyclic quotient singularities of class T. Also, we introduce a recent work of Kawamata which interprets the construction as a multi-pointed non-commutative deformation of sheaves.
Online seminar: zoom id 3132528553
Host: Dongsoo Shin
2022-03-11 / 11:00--12:30 / ASADCS Seminar: Hakho Choi _ KIAS
Symplectic fillings weighted homogeneous surface singularities and rational blowdowns
One way to understand a topology of symplectic $4$-manifolds is to study techniques for constructing symplectic $4$-manifolds. Rational blowdown, replacing certain plumbings of symplecitc $2$-spheres with rational homology ball, is one of the most efficient technique for cut-and-paste modification of symplectic $4$-manifolds. For example, it has been known that every minimal symplectic filling of quotient surface singularities and certain small Seifert fibered spaces with natural contact structure is obtained by sequence of rational blowdowns from their minimal resolution. On the other hand, there is minimal symplectic fillings of weighted homogeneous surface singularities which cannot be obtained by sequence of rational blowdowns. In this talk, we discuss necessary and sufficient condition for a minimal symplectic filling to be obtained by sequence of rational blowdowns in terms of curve configuration of the filling.
Online seminar: zoom id 3132528553
Host: Dongsoo Shin
2021-12-10 / 10:30--12:30 / ASADCS seminar: Yonghwa Cho _ KIAS
Surfaces with many nodes
In this talk, we discuss complex surfaces in a three-dimensional projective space having only nodes(ordinary double points) as singularities. One of the classical questions asks how many nodes such surfaces may contain. There is a general bound found by Miyaoka, but the optimal bound is only known for small degrees(not larger than 6). To find the optimal bound for quintics, Beauville introduced even sets of nodes, which turned out to be a powerful tool for the study of nodal surfaces. We briefly review Beauville's approach on quintics, and explain our attempts to study the even sets of nodes on sextics. This talk is based on joint works in progress with Fabrizio Catanese, Stephen Coughlan, Davide Frapporti, Michael Kiermaier, and Sascha Kurz.
Online seminar: zoom id 3132528553
Host: YongJoo Shin
2021-12-03 / 10:30--12:30 / ASADCS seminar: In-Kyun Kim _ Yonsei University
An introduction to log canonical thresholds
The log canonical threshold is an important invariant. Let (X, D) be a pair. Then the log canonical threshold provides a way of measuring how singular X and D are. In this talk, we study the definition, some properties of the log canonical threshold and examples. And we introduce some applications of log canonical thresholds. Finally we study the Hodge ideal which is a generalization of the multiplier ideal.
Online seminar: zoom id 3132528553
Host: YongJoo Shin
2021-11-26 / 10:30--12:30 / ASADCS seminar: Kyeong-Dong Park _ KIAS
Singularities of spherical varieties and their combinatorial criteria
For a reductive algebraic group G, a normal G-variety is called spherical if it contains an open orbit under the action of a Borel subgroup of G. The class of spherical varieties contains several important geometric objects which were studied independently, for example, toric varieties, flag varieties, horospherical varieties, group embeddings, and symmetric varieties. By the Luna-Vust theory, the normal equivariant embeddings of a given spherical homogeneous space are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties. The algebraic group action imposes restrictions on singularities and the Minimal Model Program works perfectly for spherical varieties. In this talk, we consider some types of singularities of spherical varieties and give combinatorial criteria of these singularities.
Online seminar: zoom id 3132528553
Host: YongJoo Shin
2021-11-05 / 10:30--12:30 / ASADCS seminar: Hyeonho Lho _ Chungnam National University
Introduction to mirror symmetry of singularity
Continuing the series of Choa's previous lectures, I will explain the mirror symmetry on singularities based on the Cohomological field theory.
Online seminar: zoom id 3132528553
Host: Dongsoo Shin
2021-10-29 / 10:30--12:30 / ASADCS seminar: Dongwook Choa _ KIAS
Introduction to algebro/symplectic geometry of hypersurface singularities and their homological mirror symmetries
I. Algebraic geometry of hypersurface singularities
II. Symplectic geometry of hypersurface singularities
III. Introduction to homological mirror symmetry between them
This lecture deals with hypersurface singularities, or germs of holomorphic function W. I will introduce two distinguished approach and their mirror symmetry. First one developed by a group of algebraists in 70's, and culminated to the invention of matrix factorization by D. Eisenbud. The second one is symplectic, and it starts from the Picard-Lefschetz theory of fibration. Although these two directions seem completely different, they are miraculously related each other under the name of homological mirror symmetry.
Online seminar: zoom id 3132528553
Host: Dongsoo Shin
2021-10-22 / 10:30--12:30 / ASADCS seminar: Dongwook Choa _ KIAS
Introduction to algebro/symplectic geometry of hypersurface singularities and their homological mirror symmetries
I. Algebraic geometry of hypersurface singularities
II. Symplectic geometry of hypersurface singularities
III. Introduction to homological mirror symmetry between them
This lecture deals with hypersurface singularities, or germs of holomorphic function W. I will introduce two distinguished approach and their mirror symmetry. First one developed by a group of algebraists in 70's, and culminated to the invention of matrix factorization by D. Eisenbud. The second one is symplectic, and it starts from the Picard-Lefschetz theory of fibration. Although these two directions seem completely different, they are miraculously related each other under the name of homological mirror symmetry.
Online seminar: zoom id 3132528553
Host: Dongsoo Shin
2021-10-08 / 10:30--12:30 / ASADCS Learning seminar: YongJoo Shin _ Chungnam National University
Keum-Naie surfaces with $K^2=4$ via bidouble covers
Let $S$ be a complex minimal surface of general type with $p_g(S)=0$ and $K_S^2=4$ whose bicanonical map is of degree $4$. Keum firstly constructed a four-dimensional family of the surfaces $S$ as double covers of Enriques surfaces branched along a reducible curve and eight nodes. Naie later constructed a six-dimensional family of the surfaces $S$ containing the surfaces constructed by Keum. Mendes Lopes and Pardini considered a six-dimensional family of the surfaces $S$ containing the surfaces constructed by Keum and Naie. Bauer and Catanese defined the Keum-Naie surfaces as the minimal resolution of the quotient of a double cover, of two elliptic curves branched in a $\mathbb{Z}_2^2$-invariant divisor of type $(4,4)$, by free $\mathbb{Z}_2^2$-action. Their surfaces form a six-dimensional family of the surfaces $S$ containing ones treated by Keum, Naie, and Mendes Lopes and Pardini.<br><br>In this talk we define a {\it Keum-Naie surface} with $K^2=4$ by the equivalence among ones of Naie, Mendes Lopes and Pardini, and Bauer and Catanese. Then we show that general Keum-Naie surfaces are constructed by bidouble covers of four-nodal del Pezzo surfaces of degree $4$. It implies that general Keum-Naie surfaces satisfy Bloch's conjecture by showing that the intermediate double covers of the bidouble covers have the Kodaira dimension less than $2$. Moreover our construction of general Keum-Naie surfaces with $K^2=4$ yields that there are two fibrations of hyperelliptic curves of genus three exactly with ten singular fibers on the each fibration, and leads to degenerations of the Keum-Naie surfaces to the surfaces constructed by Keum and to the surfaces with $K^2=3$ studied by Chen.
Online seminar: zoom id 3132528553
Host: Dongsoo Shin
2021-09-17 / 11:40--12:40 / ASADCS Learning seminar: Dongsoo Shin _ Chungnam National University
T-singularities, P-resolutions, and the Minimal model program
We briefly review basics on deformations of quotient surface singularities, which would be an algebro-geometric version of Hakho Choi's recent talks about symplectic fillings of such singularities. The talk is based on the paper by H. Park, J. Park, D. Shin, Urzua [Adv Math 2018].
Online seminar: zoom id 3132528553
Host: Dongsoo Shin
2021-09-17 / 10:30--11:30 / ASADCS Learning seminar: Jaekwan Jeon _ Chungnam National University
Describing Milnor fibers of smoothings of cyclic quotient surface singularities
Milnor fibers of smoothings of cyclic quotient surface singularities can be described as \underline{k}, triangulations, incidence matrices and P-resolutions. I will explain their relations by explicit examples and reinterpret this relation in the view of MMP. This learning seminar is based on Chapter 6 and 7 of Némethi, A., Popescu-Pampu, P.: On the Milnor fibres of cyclic quotient surface singularities. Proc. Lond. Math. Soc. (3) 101(2), 554–588 (2010)
Online seminar: zoom id 3132528553
Host: Dongsoo Shin
2021-08-23 / 10:30--12:30 / ASADCS seminar: Hakho Choi _ KIAS
Minimal symplectic fillings of weighted homogeneous surface singularities II
One of the main research topics in symplectic 4-manifold topology is to classify symplectic fillings of certain 3-manifolds equipped with a natural contact structure. For example, there is a natural contact structure on the link of a weighted homogeneous surface singularities, so called the ‘Milnor fillable’ contact structure, which is given by complex tangencies $JTL\cap TL$. In this talk, we will discuss classification and surgery description of the minimal symplectic fillings of the link of weighted homogeneous surface singularities with the Milnor fillable contact structure.
Online seminar: zoom id 3132528553
Host: Kyungbae Park
2021-08-20 / 13:00--15:00 / ASADCS seminar: Honglu Fan _ University of Geneva
Structures in Gromov-Witten theory II
In the series of talks I would like to go through the basics of Gromov-Witten theory. I will cover WDVV equations and quantum cohomology. Depending on how far we can go, I might also talk about more advanced topics and recent progress.
Online seminar: zoom id 81514859393
Host: Hyeonho Lho
2021-08-19 / 13:00--15:00 / ASADCS seminar: Honglu Fan _ University of Geneva
Structures in Gromov-Witten theory I
In the series of talks I would like to go through the basics of Gromov-Witten theory. I will cover WDVV equations and quantum cohomology. Depending on how far we can go, I might also talk about more advanced topics and recent progress.
Online seminar: zoom id 81514859393
Host: Hyeonho Lho
2021-08-09 / 10:30--12:30 / ASADCS seminar: Hakho Choi _ KIAS
Minimal symplectic fillings of weighted homogeneous surface singularities I
One of the main research topics in symplectic 4-manifold topology is to classify symplectic fillings of certain 3-manifolds equipped with a natural contact structure. For example, there is a natural contact structure on the link of a weighted homogeneous surface singularities, so called the ‘Milnor fillable’ contact structure, which is given by complex tangencies $JTL\cap TL$. In this talk, we will discuss classification and surgery description of the minimal symplectic fillings of the link of weighted homogeneous surface singularities with the Milnor fillable contact structure.
Online seminar: zoom id 3132528553
Host: Kyungbae Park
ASADCS 학술대회 목록 Workshop List
ASADCS 학술대회 목록 Workshop List
2024.01.22--01.27: Topology of 4-manifolds and related topics
2024.01.18--01.20: Intensive lectures on low-dimensional topology
2023.02.21: One-day workshop on derived category
2023.02.01--02.03: 복소곡면 특이점의 변형의 기하와 위상 (Geometry and Topology of Deformations of Complex Surface Singularities)
2022.02.13--02.15: 복소곡면의 퇴화에 관한 워크샵 (2022 Workshop on degeneration of complex surfaces)
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