Center for Complex Geometry

Luca Rizzi, Local Systems, Algebraic Foliations and Fibrations
2022-06-14 16:30 - 17:30 (Asia/Seoul) | TBA
Speaker

Luca Rizzi
IBS-CCG

Given a semistable fibration f : X → B I will show a correspondence between foliations on X and local systems on B. Building up on this correspondence we will find conditions that give maximal rationally connected fibrations in terms of data on the foliation. We will develop the ideas of the Castelnuovo-de Franchis theorem in the case of p-forms and show when, under some natural conditions, a line subbundle of the sheaf of p-forms induces the Iitaka fibration. Based on a joint work with Prof. Francesco Zucconi.

Jinhyun Park (박진현, KAIST)
2022-08-15 - 2023-08-14
Jinhyun Park (박진현)
Visitor (2022.8.15-2023.8.14) from KAIST
Office: B248

Guolei Zhong, Dynamical Characterization of Projective Toric Varieties
2022-09-01 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Guolei Zhong
IBS-CCG

As a fundamental building block of the equivariant minimal model program, the rationally connected variety plays a significant role in the classification of projective varieties admitting non-isomorphic endomorphisms. Twenty years ago, Nakayama confirmed Sato’s conjecture that, a smooth projective rational surface is toric if and only if it admits a non-isomorphic endomorphism. In this talk, I will survey some recent progress on a higher dimensional analogue of Nakayama’s result. This talk is based on some joint works with Jia Jia, Sheng Meng and De-Qi Zhang.

Ziquan Zhuang, Boundedness of Singularities and Minimal Log Discrepancies of Kollár Components
2022-09-13 10:00 - 11:00 | on-line
Speaker

Ziquan Zhuang
Johns Hopkins U

Several years ago, Chi Li introduced the local volume of a klt singularity in his work on K-stability. The local-global analogy between klt singularities and Fano varieties, together with recent study in K-stability lead to the conjecture that klt singularities whose local volumes are bounded away from zero are bounded up to special degeneration. In this talk, I will discuss some recent work on this conjecture through the minimal log discrepancies of Kollár components.

Benjamin McMillan, The Range of the Killing Operator
2022-09-15 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Benjamin McMillan
IBS-CCG

The Killing operator in (semi) Riemannian geometry has well understood kernel: the infinitesimal symmetries of a given metric. At the next level, the range of the Killing operator can be interpreted as those perturbations of the metric that result from a mere change of coordinates—in contexts like general relativity, the trivial perturbations. The range of the Killing operator, albeit important, has not been as well understood as its kernel. I will describe how one can use the projective nature of the Killing operator to reduce the question to that of the range of a connection on an associated bundle, and then how one can seek to understand the range of an arbitrary connection. This process yields a fairly complete answer on locally symmetric spaces.
Based on joint work with Federico Costanza, Mike Eastwood, and Thomas Leistner.

Mihai Paun (University of Bayreuth)
2022-09-21 - 2022-10-20
Mihai Paun
Visitor (2022.9.21-2022.10.20) from University of Bayreuth
Office: B253

Jakub Witaszek, Quasi-F-splittings
2022-09-27 10:00 - 11:00 | on-line
Speaker

Jakub Witaszek
Princeton U

What allowed for many developments in algebraic geometry and commutative algebra was a discovery of the notion of a Frobenius splitting, which, briefly speaking, detects how pathological positive characteristic Fano and Calabi-Yau varieties can be. Recently, Yobuko introduced a more general concept, a quasi-F-splitting, which captures much more refined arithmetic invariants. In my talk, I will discuss on-going projects in which we develop the theory of quasi-F-splittings in the context of birational geometry and derive applications, for example, to liftability of singularities. This is joint work with Tatsuro Kawakami, Hiromu Tanaka, Teppei Takamatsu, Fuetaro Yobuko, and Shou Yoshikawa.

Complex Analytic Geometry
2022-10-05 - 2022-10-07 | B236-1, IBS, Korea, Republic of
Speakers

Young-Jun Choi (Pusan National U.)
Yoshinori Hashimoto (Osaka Metropolitan U.)
Dano Kim (Seoul National U.)
Takayuki Koike (Osaka Metropolitan U.)
Seungjae Lee (IBS-CCG)
Nguyen Ngoc Cuong (KAIST)
Mihai Paun (Bayreuth U.)
Martin Sera (Kyoto U. Advanced Science)
Jihun Yum (IBS-CCG)

Schedule

Oct. 5

Infinitesimal extension of twisted canonical forms and applications (part 1)
Mihai Paun
10:30-11:15

Weighted L2 holomorphic functions on ball fiber bundles over compact Kähler manifolds
Seungjae Lee
13:30-14:20

Weak solutions to Monge-Ampère type equations on compact Hermitian manifold with boundary
Nguyen Ngoc Cuong
14:40-15:30

Limit of Bergman kernels on a tower of coverings of compact Kähler manifolds
Jihun Yum
15:50-16:40

Oct. 6

Infinitesimal extension of twisted canonical forms and applications (part 2)
Mihai Paun
10:30-11:15

Curvature of higher direct images
Young-Jun Choi
13:30-14:20

Some recent results on constant scalar curvature Kähler metrics with cone singularities
Yoshinori Hashimoto
14:40-15:30

Projective K3 surfaces which contain Levi-flat hypersurfaces
Takayuki Koike
15:50-16:40

Oct. 7

Hermite-Einstein metrics on stable reflexive sheaves on Kaehler manifolds
Mihai Paun
10:30-11:15

Lelong numbers of direct images of generalized Monge-Ampère products
Martin Sera
13:30-14:20

Canonical bundle formula and degenerating families of volume forms
Dano Kim
14:40-15:30

Pak Tung Ho, The Weighted Yamabe Problem
2022-10-12 15:00 - 16:00 | B236-1, IBS, Korea, Republic of
Speaker

Pak Tung Ho
Sogang University

In this talk, I will explain what the weighted Yamabe problem is, and mention some related results that Jinwoo Shin (KIAS) and I obtained.

Aeryeong Seo, TBA
2022-10-12 16:15 - 17:15 | B266 and on-line
Speaker

Aeryeong Seo
Kyungpook National University

TBA

Jinhyun Park, A Reciprocity Theorem Arising from a Family of Algebraic Curves
2022-10-13 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Jinhyun Park
KAIST

The classical reciprocity theorem, also called the residue theorem, states that the sum of the residues of a rational (meromorphic) differential form on a compact Riemann surface is zero. Its generalization to smooth projective curves over a field is often called the Tate reciprocity theorem.
There is a different “multiplicative version” too. Here, instead of a rational form, one uses a pair of rational functions on a smooth projective curve, and instead of residues, one uses “Tame symbols”. The corresponding global result is called the Weil reciprocity. This result is elegantly reformulated in terms of the Milnor K-theory, and it is generalized to sequences of rational functions by A. Suslin. This Suslin reciprocity was recently strengthened by D. Rudenko, resolving a conjecture of A. Goncharov.
In this talk, let me sketch my recent work in-progress, that studies a different kind of reciprocity results coming from a proper family of algebraic curves over an algebraically closed field of characteristic 0.

Jaewoo Jeong, Hankel Index of Smooth Non-ACM Curves of Almost Minimal Degree
2022-10-27 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Jaewoo Jeong IBS CCG

The Hankel index of a real variety is a semi-algebraic invariant that quantifies the (structural) difference between nonnegative quadrics and sums of squares on the variety. Note that the Hankel index of a variety is difficult to compute and was computed for just few cases. In 2017, Blekherman, Sinn, and Velasco provided an captivating (lower) bound of the Hankel index of a variety by an algebraic invariant, Green-Lazarsfeld index, of the variety. In particular, if the variety X is an arithmetrically Cohen-Macaulay (ACM) variety of almost minimal degree, then the Hankel index of X equals to the Green-Lazarsfeld index of X plus one (which is the equality case of the bound). We study the Hankel index of smooth non-ACM curves of almost minimal degree. Note that the curve is the image of the projection of rational normal curves away from an outer point. It is known that the Green-Lazarsfeld index of the curve is determined by the rank of the center of the projection with respect to the rational normal curve. We found a new rank of the center that detects the Hankel index of the rational curves. In addition, it turns out that the rational curves are the first class of examples that the lower bound of the Hankel index is strict.

Livia Campo, Fano 3-folds and Equivariant Unprojections
2022-11-01 15:00 - 16:00 | B266, IBS, Korea, Republic of
Speaker

Livia Campo
KIAS

The classification of terminal Fano 3-folds has been tackled from different directions: for instance, using the Minimal Model Program, via explicit Birational Geometry, and via Graded Rings methods. In this talk I would like to introduce the Graded Ring Database – an upper bound to the numerics of Fano 3-folds – and discuss the role it plays in the classification and construction of codimension 4 Fano 3-folds having Fano index 2.

Junho Choe, Constructions of Counterexamples to the Regularity Conjecture
2022-11-01 16:30 - 17:30 | B266, IBS, Korea, Republic of
Speaker

Junho Choe
KIAS

Castelnuovo-Mumford regularity, simply regularity, is one of the most interesting invariants in projective algebraic geometry, and the regularity conjecture due to Eisenbud and Goto says that the regularity can be controlled by the degree for any projective variety. But counterexamples to the conjecture have been constructed by some methods. In this talk we review the counterexample constructions including the Rees-like algebra method by McCullough and Peeva and the unprojection method.

Joaquín Moraga, Coregularity of Fano Varieties
2022-11-15 11:00 - 12:00 | on-line
Speaker

Joaquín Moraga
UCLA

In this talk, we will introduce the absolute coregularity of Fano varieties. The coregularity measures the singularities of the anti-pluricanonical sections. Philosophically, most Fano varieties have coregularity 0. In the talk, we will explain some theorems that support this philosophy. We will show that a Fano variety of coregularity 0 admits a non-trivial section in |-2KX|, independently of the dimension of X. This is joint work with Fernando Figueroa, Stefano Filipazzo, and Junyao Peng.

Andrea Petracci, A 1-dimensional Component of K-moduli of Del Pezzo Surfaces
2022-11-29 17:00 - 18:00 | on-line
Speaker

Andrea Petracci
Università di Bologna

Fano varieties are algebraic varieties with positive curvature; they are basic building blocks of algebraic varieties. Great progress has been recently made by Xu et al. to construct moduli spaces of Fano varieties by using K-stability (which is related to the existence of Kähler-Einstein metrics). These moduli spaces are called K-moduli.
In this talk I will explain how to easily deduce some geometric properties of K-moduli by using toric geometry and deformation theory. In particular, I will show how to construct a 1-dimensional component of K-moduli which parametrises certain K-polystable del Pezzo surfaces.

Yunhyung Cho (조윤형, Sungkyunkwan University)
2023-01-04 - 2023-01-06
Yunhyung Cho (조윤형)
Visitor (2023.1.4-2023.1.6) from Sungkyunkwan University
Office: –

JongHae Keum, Fake Projective Planes I
2023-01-12 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

JongHae Keum
KIAS

Fake projective planes (abbreviated as FPPs) are 2-dimensional complex manifolds with the same Betti numbers as the projective plane, but not isomorphic to it.
FPPs can be uniformized by a complex 2-ball. In other words, they are ball quotients having the minimum possible Betti numbers.
The existence of such a surface was first proved by Mumford in 1979, via 2-adic uniformization.
Not always algebraic varieties are described via polynomial equations: sometimes they are constructed via uniformization: this means, as quotients of certain domains in a complex vector space, called bounded symmetric domains, via the action of discontinuous groups. Then general theorems (as Kodaira’s) imply the algebraicity of these quotient complex manifolds. The problem concerning the algebro-geometrical properties of such varieties constructed via uniformization and especially the description of their projective embeddings (and the corresponding polynomial equations) lies at the crossroads of several allied fields: the theory of arithmetic groups and division algebras, complex algebraic and differential geometry, linear systems, use of group symmetries, and topological and homological tools in the study of quotient spaces. Of particular importance are the so-called ball quotients, especially in dimension 2, since they yield the surfaces with the maximal possible canonical volume K2 for a fixed value of the geometric genus pg.
In the first lecture I will introduce basic properties of FPPs and their position in the classification theory of algebraic surfaces.
In the second I will discuss recent progress on them, such as their derived categories, bicanonical maps and their equations.

JongHae Keum, Fake Projective Plane II
2023-01-13 11:00 - 12:00 | B236-1, IBS, Korea, Republic of

Kangjin Han, Secant variety and its singularity I
2023-01-26 16:00 - 17:00 | B266, IBS, Korea, Republic of
Speaker

Kangjin Han
DGIST

Secant variety (or more generally Join) construction is one of the main methods to construct a new geometric object from the original one in classical algebraic geometry. In this series of talks, we first consider some general facts on secant varieties and then focus on a specific topic, i.e. singularity of secants. For this purpose, we review some basic facts and known results in the literature and present some ideas to show (non-)singularity of points in the given secant. We also report a recent work with K. Furukawa on the singular loci of higher secant of Veronese varieties and others. The talks cover such topics as Terracini lemma, identifiability, tangential k-contact locus from geometric side and apolar ideal and defining equations via a Young flattening and so on from algebraic side.

Kangjin Han, Secant variety and its singularity II
2023-01-27 10:30 - 11:30 | B266, IBS, Korea, Republic of

Dennis The (University of Tromso, Norway)
2023-01-30 - 2023-02-08
Dennis The
Visitor (2023.1.30-2023.2.8) from University of Tromso, Norway
Office: –

Dennis The, A Cartan-theoretic Perspective on (2,3,5)-distributions
2023-02-02 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Dennis The
UiT The Arctic University of Norway

Generic rank 2 distributions on 5-manifolds, i.e. “(2,3,5)-distributions”, are interesting geometric structures arising in the study of non-holonomic systems, underdetermined ODE of Monge type, conformal 5-manifolds with special holonomy, etc. The origins of their study date to Élie Cartan’s “5-variables” paper of 1910, where he gave a tour-de-force application of his method of equivalence. In my talk, I’ll revisit the classification of homogeneous (2,3,5)-distributions from a modern “Cartan-theoretic” perspective. In particular, I’ll discuss from this viewpoint the exceptionality of the 3:1 ratio for two spheres rolling on each other without twisting or slipping.

Antonio Nigro (Universidade Federal Fluminense)
2023-02-03 - 2023-02-09
Antonio Nigro
Visitor (2023.2.3-2023.2.9) from Universidade Federal Fluminense
Office: –

Laurent Stolovitch (Universite Cote d'Azur, France)
2023-02-04 - 2023-02-10
Laurent Stolovitch
Visitor (2023.2.4-2023.2.10) from Universite Cote d’Azur, France
Office: –

Eunjeong Lee (이은정, Chungbuk National University)
2023-02-06 - 2023-02-09
Eunjeong Lee
Visitor (2023.2.6-2023.2.9) from Chungbuk National University
Office: –

Daniele Agostini, The Martens-Mumford Theorem and the Green-Lazarsfeld Secant Conjecture
2023-02-07 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Daniele Agostini
Eberhard Karls Universität Tübingen

The syzygies of a curve are the algebraic relation amongst the equation defining it. They are an algebraic concept but they have surprising applications to geometry. For example, the Green-Lazarsfeld secant conjecture predicts that the syzygies of a curve of sufficiently high degree are controlled by its special secants. We prove this conjecture for all curves of Clifford index at least two and not bielliptic and for all line bundles of a certain degree. Our proof is based on a classic result of Martens and Mumford on Brill-Noether varieties and on a simple vanishing criterion that comes from the interpretation of syzygies through symmetric products of curves.

Laurent Stolovitch, Introduction to Normal Form Theory of Holomorphic Vector Fields 1
2023-02-07 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Laurent Stolovitch
Universite Cote d’Azur

In this short lecture, I will introduce the notion of normal form and resonances. I will also explain the phenomenon of “small divisors” and give some fundamental results of holomorphic conjugacy to a normal form.

Laurent Stolovitch, Introduction to Normal Form Theory of Holomorphic Vector Fields 2
2023-02-09 11:00 - 12:00 | B236-1, IBS, Korea, Republic of

Yoon-Joo Kim, Isotrivial Fibrations of Compact Hyper-Kähler Manifolds
2023-02-10 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Yoon-Joo Kim
MPI-Bonn

A compact hyper-Kähler (HK) manifold and its Lagrangian fibration are higher-dimensional generalizations of a K3 surface and its elliptic fibration. A Lagrangian fibration f : X → B of a HK manifold is called isotrivial if its smooth fibers are all isomorphic to each other; this is the most special type of Lagrangian fibrations, generalizing the notion of an isotrivial elliptic fibration. In this talk, I will propose a possible classification scheme of HK manifolds admitting isotrivial Lagrangian fibrations. This method in particular rediscovers the two well-known constructions of HK manifolds, the K3[n] and generalized Kummer constructions. I will then specialize the discussion to HK fourfolds. This is joint work with Radu Laza and Olivier Martin.
