Guolei Zhong, Strictly Nef Divisors on Singular Varieties
2022-03-03 16:20 - 17:20 (Asia/Seoul) | 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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)

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.

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.

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.

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.

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).

Ming Xiao, On Some Mapping Problems between Bounded Symmetric Domains
2022-06-08 11:00 - 12:00 | 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.

Luca Rizzi, Local Systems, Algebraic Foliations and Fibrations
2022-06-14 16:30 - 17:30 | 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.
