Center for Complex Geometry

Ming Xiao, On Some Mapping Problems between Bounded Symmetric Domains
2022-06-08 11:00 - 12:00 (Asia/Seoul) | on-line
Speaker

Ming Xiao
UCSD

Bounded symmetric domains are an important class of geometric objects in complex analysis and geometry, which possess a high degree of symmetry. They often serve as the model cases in the study of many rigidity phenomena. In this talk, we will discuss two mapping problems between bounded symmetric domains — proper maps and isomeric maps. We will present some recent results along these lines.

Christian Schnell, A New Approach to Degenerating Variations of Hodge Structure
2022-06-07 11:00 - 12:00 | on-line
Speaker

Christian Schnell
Stony Brook Univ.

The theory of variations of Hodge structure has many applications in algebraic geometry. Most of these are based on the results by Schmid, Cattani, Kaplan, Kashiwara, and Kawai from the 1970s and 1980s. I will describe a new approach that proves these results — such as the Hodge norm estimates or the nilpotent orbit theorem — more easily (and that no longer requires quasi-unipotent local monodromy).

Sung Rak Choi, On the Thresholds of Potential Pairs
2022-05-31 16:45 - 17:45 | B266, IBS, Korea, Republic of
Speaker

Sung Rak Choi
Yonsei Univ.

Choi-Park first introduced and develped the notion of potential pairs. The notion was designed to control the singularities of the outcome of the ‘anticanonical’ minimal model program. In this talk, after reviewing the properties of potnetial klt pairs, we examine the ACC property of the potential lc thresholds. This talk is based on the joint work in progress with Sungwook Jang.

Sheng Meng, Equivariant Kähler Model for Fujiki's Class
2022-05-31 15:30 - 16:30 | B266, IBS, Korea, Republic of
Speaker

Sheng Meng
KIAS

Let X be a compact complex manifold in Fujiki’s class C, i.e., admitting a big (1,1)-class [α]. Consider Aut(X) the group of biholomorphic automorphisms and Aut[α](X) the subgroup of automorphisms preserving the class [α] via pullback. We show that X admits an Aut[α](X)-equivariant Kähler model: there is a bimeromorphic holomorphic map σ : X~ → X from a Kähler manifold X~ such that Aut[α](X) lifts holomorphically via σ.
There are several applications. We show that Aut[α](X) is a Lie group with only finitely many components. This generalizes an early result of Fujiki and Lieberman on the Kähler case.We also show that every torsion subgroup of Aut(X) is almost abelian, and Aut(X) is finite if it is a torsion group.
This is a joint work with Jia Jia.

Xu Wang, An Explicit Estimate of the Bergman Kernel for Positive Line Bundles
2022-05-25 16:00 - 17:00 | on-line
Speaker

Xu Wang
NTNU – Norwegian University of Science and Technology

We shall give an explicit estimate of the lower bound of the Bergman kernel associated to a positive line bundle.
In the compact Riemann surface case, our result can be seen as an explicit version of Tian’s partial C0-estimate.

Lukasz Kosinski, Extension Property and Interpolation Problems
2022-05-11 16:00 - 17:00 | on-line
Speaker

Lukasz Kosinski
Jagiellonian University

A subset V of a domain Ω has the extension property if for every holomorphic function p on V there is a bounded holomorphic function φ on Ω that agrees with p on V and whose sup-norm on Ω equals the sup-norm of p on V. Within the talk, we shall study mutual relations between extension property and interpolations problems.

Brendan Hassett, Recent Progress and Questions on Stable Rationality
2022-05-10 11:00 - 12:00 | on-line
Speaker

Brendan Hassett
ICERM / Brown Univ.

A complex variety X is rational if its field of meromorphic functions is isomorphic to C(t1, …, td), the function field of projective space Pd. It is stably rational if X × Pm is rational for some m. Topological and complex invariants give criteria for whether varieties are stably rational but a simple characterization remains elusive. We present results for varieties admitting fibrations in conics, quadric hypersurfaces, and rational surfaces using techniques like decomposition of the diagonal. We close with some outstanding open problems. (joint with Kresch and Tschinkel)

Hoang-Chinh Lu, Monge-Ampère Volumes on Compact Hermitian Manifolds
2022-05-04 16:00 - 17:00 | on-line
Speaker

Hoang-Chinh Lu
Université Paris-Saclay, Orsay

We investigate in depth the behaviour of Monge-Ampère volumes of quasi-psh functions on a given compact hermitian manifold. We prove that the property for these Monge-Ampère volumes to stay bounded away from zero or infinity is a bimeromorphic invariant. We show in particular that a conjecture of Demailly-Paun holds true if and only if such Monge-Ampère volumes stay bounded away from infinity. This is a joint work with Vincent Guedj.

Ngoc-Son Duong, Proper Holomorphic Maps from the Complex 2-ball into the 3-dimensional Classical Domain of Type IV
2022-04-27 16:00 - 17:00 | on-line
Speaker

Ngoc-Son Duong
University of Vienna

In this talk, we will discuss a complete classification of proper holomorphic maps from the unit ball in complex two dimensional space into the Cartan’s classical domain of type IV in complex three
dimensional space that extend smoothly to some boundary point. This classification (which is a consequence of a classification of CR maps from the 3-sphere into the tube over the future light cone) consists of 4 algebraic maps. Among them, two previously known maps are isometric with respect to the canonical Bergman metrics and two other maps give counterexamples to a recent conjecture of Xiao-Yuan for the case of maps from the complex 2-ball. This is a joint work with Michael Reiter.

Tsz On Mario Chan, Analytic Adjoint Ideal Sheaves via Residue Functions
2022-04-20 16:00 - 17:00 | B266, IBS, Korea, Republic of
Speaker

Tsz On Mario Chan
Pusan National University

In this talk, we introduce a modification of the analytic adjoint ideal sheaves. The original analytic adjoint ideal sheaves were studied by Guenancia and Dano Kim. The modified version makes use of the residue functions with respect to log-canonical (lc) measures, giving a sequence of adjoint ideal sheaves which provide a scheme-theoretic description of the lc centres of a given lc pair (X, S) (where X is a complex manifold) defined by multiplier ideal sheaves of quasi-psh functions with suitable regularity assumptions. We also discuss how the use of the residue functions helps to yield a local L2 extension result without using the Ohsawa-Takegoshi extension theorem, and to provide residue L2 norms on unions of lc centres which are invariant under any modifications of the ambient space X. Since some notions of singularities in birational geometry (namely, klt and lc) can be described via integrability conditions, the residue norms can be useful in the study of the inversion of adjunction.

Slawomir Dinew, Extension Through Small Sets in Complex Analysis
2022-04-13 16:00 - 17:00 | on-line
Speaker

Slawomir Dinew
Jagiellonian University, Krakow

Extension problems through a small singular set appear throughout complex
analysis. After a short reminder of some classical results we shall focus
on problems of extending (pluri)subharmonic functions. In particular we
shall focus on new techniques coming from PDEs that lead to resolutions of
several questions in the field. The talk is partially based on joint works
with Zywomir Dinew.

Keiji Oguiso, On Kawaguchi-Silverman Conjecture for Birational Automorphisms of Irregular Threefolds
2022-04-12 11:00 - 12:00 | on-line
Speaker

Keiji Oguiso
Univ. of Tokyo

This is a joint work in progress with Professors Jungkai-Alfred Chen and Hsueh-Yung Lin.
We study the main open parts of Kawaguchi-Silverman Conjecture (KSC), asserting that for a birational self-map f of a smooth projective variety X defined over K, the arithmetic degree αf(x) exists and coincides with the first dynamical degree δf for any K-point x of X with a well-defined Zariski dense f-orbit. Here K is an algebraic closure of the field of rational numbers. To make KSC meaningful, it is also important to study existence of K-point with Zariski dense f-orbit.
In this talk, after a brief introduction of KSC with known results and some difficulties for non-morphism case, I would like to explain our new progress on KSC and Zariski dense orbit problem especially for irregular threefolds. Our approach is geometric while problems are of arithmetic nature.

Radu Laza, Deformations of Singular Fano and Calabi-Yau Varieties
2022-04-05 11:00 - 12:00 | on-line
Speaker

Radu Laza
Stony Brook University

It is well known that Calabi-Yau manifolds have good deformation theory, which is controlled by Hodge theory. By work of Friedman, Namikawa, M. Gross, Kawamata, Steenbrink and others, some of these results have been extended to Calabi-Yau threefolds with canonical singularities. In this talk, I will report on further extensions in two directions: in dimension 3, we sharpen and clarify some of the existing results, and, secondly, we obtain some higher dimensional analogues. I will also briefly explain the related case of Fano varieties, where stronger results hold. One surprising aspect of our study is the role played by higher du Bois and higher rational singularities, notions that were recently introduced by Mustata, Popa, Saito and their collaborators. This is joint work with Robert Friedman.

Nam-Hoon Lee, Mirror Pairs of Calabi-Yau Threefolds from Mirror Pairs of Quasi-Fano Threefolds
2022-03-29 16:45 - 17:45 | B266, IBS, Korea, Republic of
Speaker

Nam-Hoon Lee
Hongik Univ.

We present a new construction of mirror pairs of Calabi-Yau manifolds by smoothing normal crossing varieties, consisting of two quasi-Fano manifolds. We introduce a notion of mirror pairs of quasi-Fano manifolds with anticanonical Calabi-Yau fibrations using conjectures about Landau-Ginzburg models. Utilizing this notion, we give pairs of normal crossing varieties and show that the pairs of smoothed Calabi-Yau manifolds satisfy the Hodge number relations of mirror symmetry. We consider quasi-Fano threefolds that are some blow-ups of Gorenstein toric Fano threefolds and build 6518 mirror pairs of Calabi-Yau threefolds, including 79 self-mirrors.

Dongsoo Shin, Deformations of Sandwiched Surface Singularities and the Semistable Minimal Model Program
2022-03-29 15:30 - 16:30 | B266, IBS, Korea, Republic of
Speaker

Dongsoo Shin
Chungnam National Univ.

A sandwiched surface singularity is a rational surface singularity that admits a birational map to the complex projective plane. de Jong and van Straten [Duke Math J 1998] prove that deformations of sandwiched surface singularities are induced from special deformations of germs of plane curve singularities (called picture deformations). On the other hand, Kollár and Shepherd-Barron [Invent Math 1988] conjecture that deformations of any rational surface singularities are described by special partial modifications (called P-modifications). So there are two different descriptions of deformations of sandwiched surface singularities. We provide a way to find a correspondence between picture deformations and P-resolutions (roughly speaking, normal P-modifications with mild singularities) using the semistable minimal model program for complex 3-folds. As applications, 1. we provide correspondence between various deformation theories of cyclic quotient surface singularities 2. We give proofs of Kollár conjecture for certain weighted homogeneous surface singularities.

Hoseob Seo, On L2 Extension from Singular Hypersurfaces
2022-03-24 16:20 - 17:20 | B234
Speaker

Hoseob Seo
IBS CCG

In L2 extension theorems from an irreducible singular hypersurface in a complex manifold, important roles are played by certain measures such as the Ohsawa measure, which determines when a given function can be extended. In this talk, we show that the singularity of the Ohsawa measure can be identified in terms of algebraic geometry.
Using this, we give an analytic proof of the inversion of adjunction in this setting. These considerations enable us to compare various positive and negative results on L2 extension from singular hypersurfaces. In particular, we generalize a recent negative result of Guan and Li which places limitations on strengthening such L2 extension by employing a less singular measure in the place of the Ohsawa measure. This is joint work with Dano Kim.

Yonghwa Cho, Nodal Sextics and Even Sets of Nodes
2022-03-24 15:00 - 16:00 | B234
Speaker

Yonghwa Cho
IBS CCG

It is a classical question to ask how many nodes may a surface contain. For sextics, the maximum number of nodes is 65, and is attained by Barth’s example. We ask further: are all sextics with 65 nodes like Barth’s example? To find an answer, we study even sets of nodes on sextic surfaces, and prove that all sextics with 65 nodes share the same structure of even sets. This establishes a resemblance between Barth’s sextic and other sextics with 65 nodes. This talk is based on a joint work with Fabrizio Catanese, Michael Kiermaier, and Sascha Kurz.

Atsushi Ito, Projective Normality of General Polarized Abelian Varieties
2022-03-08 11:00 - 12:00 | on-line
Speaker

Atsushi Ito
Okayama Univ.

Projective normality is an important property of ample line bundles on algebraic varieties.
In this talk, I will explain that a general g-dimensional polarized abelian variety is projectively normal if χ(X, L) > 22g-1.
We note that this bound is sharp. A key tool is basepoint-freeness threshold, which is introduced by Jiang-Pareschi recently.

Guolei Zhong, Strictly Nef Divisors on Singular Varieties
2022-03-03 16:20 - 17:20 | TBA
Speaker

Guolei Zhong
IBS CCG

A Q-Cartier divisor on a normal projective variety is said to be strictly nef, if it has positive intersection with every integral curve. It has been a long history for people to measure how far a strictly nef divisor is from being ample. In this talk, I will survey some recent progress on this topic, especially the generalizations of Campana-Peternell’s conjecture and Serrano’s conjecture. My talk is based on some joint works with Jie Liu, Wenhao Ou, Juanyong Wang and Xiaokui Yang.

Luca Rizzi, An Approach to the Fujita Decomposition via Massey Products and Castelnuovo-de Franchis Theorem
2022-03-03 15:00 - 16:00 | TBA
Speaker

Luca Rizzi
IBS CCG

Let f be a semistable fibration between a smooth complex variety of dimension n and a smooth complex curve. By a famous result of Fujita the direct image of the relative dualizing sheaf is the direct sum of a unitary flat vector bundle and an ample vector bundle. In this talk we focus our attention on the former: unitary flat bundles are in one-to-one correspondence with local systems and monodromy representations and I will show how these local systems are related to the de Rham closed one forms and top forms on the fibers of f. I will also give an introduction on the construction of Massey products and show its close relation with both a famous result by Castelnuovo and de Franchis and these local systems. Thanks to the presented techniques it is possible to give some results on the finiteness of the monodromy associated to the unitary flat bundle; equivalently on its semi-ampleness.

Duc-Viet Vu, Moser-Trudinger Inequalities and Complex Monge-Ampere Equations
2022-03-02 16:00 - 17:00 | on-line
Speaker

Duc-Viet Vu
Cologne

I present a version of the Moser-Trudinger inequality in the setting of complex geometry. As a very particular case, the result already gives a new Moser-Trudinger inequality for functions in the Sobolev space W1,2 of a domain in R2. As an application, we deduce a new necessary condition for the complex Monge-Ampere equation for a given measure on a compact Kahler manifold to admit a Holder continuous solution. This is a joint work with Tien-Cuong Dinh and George Marinescu.

Jeong-Seop Kim, Positivity of Tangent Bundles of Fano Threefolds
2022-02-24 11:00 - 12:00 | TBA
Speaker

Jeong-Seop Kim
KAIST

As well as the Hartshorne-Frankel conjecture on the ampleness of tangent bundle, it has been asked to characterize a smooth projective variety X whose tangent bundle TX attains certain positivity, e.g., nefness, k-ampleness, or bigness. But for the ampleness, the complete answers are not known even within the class of smooth Fano varieties, only partial answers are known in the case of lower dimension or lower Picard number, some of which rely on classification theorems. On the bigness of TX, the characterization has been done recently in the case of dimension 2 (Höring-Liu-Shao) and dimension 3 with Picard number 1 (Höring-Liu) using a special divisor on P(TX), called the total dual VMRT. In this talk, I will briefly review the classification of Fano threefolds and the theory of total dual VMRT. Then I will introduce some criteria to determine the bigness of TX, and announce a result on the bigness of TX in the case of dimension 3 with higher Picard number. This is joint work with Hosung Kim and Yongnam Lee.

Kang-Hyurk Lee, Smoothly Bounded Domain with a Compact Quotient
2022-02-23 11:00 - 12:00 | on-line
Speaker

Kang-Hyurk Lee
GNU

The Wong-Rosay theorem says that a smoothly bounded domain covering a compact complex manifold is biholomorphically equivalent to the unit ball. The general methodology of this theorem is the affine rescaling method. In this talk, I will introduce the potential rescaling method, an alternative of the affine rescaling. This is a method to construct a specific potential function of the Kahler-Einstein which possesses a complete holomorphic vector field. I will also give a generalization of the Wong-Rosay theorem. This is a collaborated work with Y.-J. Choi and A. Seo.

Junho Peter Whang, Decidable Diophantine Problems on Character Varieties
2022-02-22 16:30 - 17:30 | B266, IBS, Korea, Republic of
Speaker

Junho Peter Whang
Seoul National Univ.

(This is a part of Arithemetic Geometry Day in IBS-CCG.)
Character varieties of manifolds are basic objects in geometry and low-dimensional topology. We motivate the Diophantine study of their integral points. After discussing an effective finite generation theorem for integral points on SL2-character varieties of surfaces, we present new decidability results for certain Diophantine problems on character varieties with other coefficient groups and manifolds.

WonTae Hwang, Jordan Constants of Simple Abelian Varieties over Fields of Positive Characteristic
2022-02-22 15:15 - 16:15 | B266, IBS, Korea, Republic of
Speaker

WonTae Hwang
Jeonbuk National Univ.

(This is a part of Arithemetic Geometry Day in IBS-CCG.)
We compute the Jordan constants of simple abelian surfaces over fields of positive characteristic, with the aid of a similar computation on the Jordan constants of some arithmetic objects. As an update, we also briefly record a similar recent result on the case of simple abelian fourfolds over finite fields.

Bo-Hae Im, A Hyperelliptic Curve Mapping to Specified Elliptic Curves
2022-02-22 14:00 - 15:00 | B266, IBS, Korea, Republic of
Speaker

Bo-Hae Im
KAIST

(This is a part of Arithemetic Geometry Day in IBS-CCG.)
We are interested in the existence and non-existence of rational curves on certain Kummer varieties which can be applied to the rank problem of quadratic twists of elliptic curves. In this talk, we prove that if the j-invariants of a 21-tuple of elliptic curves E1, …, E21 over C are algebraically independent over Q, there is no hyperelliptic curve which admits a non-trivial morphism to each of the Ei, which shows the non-existence of rational curves of the Kummer variety of the products of more than 20 elliptic curves.
This is a joint work with Michael Larsen and Sailun Zhan.

Arithemetic Geometry Day in IBS-CCG
2022-02-22 | B266, IBS, Korea, Republic of
List of Seminars

A Hyperelliptic Curve Mapping to Specified Elliptic Curves
Bo-Hae Im (KAIST)
14:00-15:00, IBS B266

Jordan Constants of Simple Abelian Varieties over Fields of Positive Characteristic
WonTae Hwang (Jeonbuk National Univ.)
15:15-16:15, IBS B266

Decidable Diophantine Problems on Character Varieties
Junho Peter Whang (Seoul National Univ.)
16:30-17:30, IBS B266

Gunhee Cho, The Lower Bound of the Integrated Carathéodory-Reiffen Metric and Invariant Metrics on Complete Noncompact Kähler Manifolds
2022-02-16 15:00 - 17:00 | on-line
Speaker

Gunhee Cho
UCSB

We seek to gain progress on the following long-standing conjectures in hyperbolic complex geometry: prove that a simply connected complete Kähler manifold with negatively pinched sectional curvature is biholomorphic to a bounded domain and the Carathéodory-Reiffen metric does not vanish everywhere. As the next development of the important recent results of D. Wu and S.T. Yau in obtaining uniformly equivalence of the base Kähler metric with the Bergman metric, the Kobayashi-Royden metric, and the complete Kähler-Einstein metric in the conjecture class but missing of the Carathéodory-Reiffen metric, we provide an integrated gradient estimate of the bounded holomorphic function which becomes a quantitative lower bound of the integrated Carathéodory-Reiffen metric. Also, without requiring the negatively pinched holomorphic sectional curvature condition of the Bergman metric, we establish the equivalence of the Bergman metric, the Kobayashi-Royden metric, and the complete Kähler-Einstein metric of negative scalar curvature under a bounded curvature condition of the Bergman metric on an n-dimensional complete noncompact Kähler manifold with some reasonable conditions which also imply non-vanishing Carathédoroy-Reiffen metric. This is a joint work with Kyu-Hwan Lee.

Chenyang Xu, K-stability of Fano Varieties
2022-02-16 11:00 - 12:00 | on-line
Speaker

Chenyang Xu
Princeton Univ.

K-stability of Fano varieties was initiated as a central topic in complex geometry, for its relation with the Kähler-Einstein metric. It turns out that the machinery of higher dimensional geometry, developed around the minimal model program, provides a fundamental tool to study it, and therefore makes it an active algebraic subject. This meeting of two well-studied fields has made a number of major conjectures solved. In this talk, I will survey the recent development.

Young-Hoon Kiem, A New Construction of the Moduli Space of Pointed Stable Curves of Genus 0
2022-02-10 11:00 - 12:00 | TBA
Speaker

Young-Hoon Kiem
Seoul National University

The moduli space of n points on a projective line up to projective equivalence has been a topic of research since the 19th century. A natural moduli theoretic compactification was constructed by Deligne and Mumford as an algebraic stack. Later, Knudsen, Keel, Kapranov and others provided explicit constructions by sequences of blowups. The known inductive constructions of Knudsen and Keel however are rather inconvenient when one wants to compute the cohomology of the compactified moduli space as a representation space of its automorphism group because the blowup sequences are not equivariant. I will talk about a new inductive construction of the much studied moduli space from the perspective of sheaf theory. In fact, we consider the moduli space of rank 1 stable pairs over the moduli space of n pointed stable curves of genus 0. By studing the wall crossing, we obtain an equivariant sequence of blowups which ends up with the moduli space of n+1 pointed stable curves of genus 0. As an application, we provide a closed formula of the character of the cohomology of the moduli space. We also provide a partial answer to a question of Manin and Orlov which asks whether the cohomology is a permutation representation or not. Based on a joint work with Jinwon Choi and Donggun Lee.
