Center for Complex Geometry

Rostislav Devyatov, Multiplicity-free Products of Schubert Divisors and an Application to Canonical Dimension
2022-01-25 16:20 - 17:20 (Asia/Seoul) | B266, IBS, Korea, Republic of
Speaker

Rostislav Devyatov
KAIST

In the first part of my talk I am going to speak about Schubert calculus. Let G/B be a flag variety, where G is a linear simple algebraic group, and B is a Borel subgroup. Schubert calculus studies (in classical terms) multiplication in the cohomology ring of a flag variety over the complex numbers, or (in more algebraic terms) the Chow ring of the flag variety. This ring is generated as a group by the classes of so-called Schubert varieties (or their Poincare duals, if we speak about the classical cohomology ring), i. e. of the varieties of the form BwB/B, where w is an element of the Weyl group. As a ring, it is almost generated by the classes of Schubert varieties of codimension 1, called Schubert divisors. More precisely, the subring generated by Schubert divisors is a subgroup of finite index. These two facts lead to the following general question: how to decompose a product of Schubert divisors into a linear combination of Schubert varieties. In my talk, I am going to address (and answer if I have time) two more particular versions of this question: If G is of type A, D, or E, when does a coefficient in such a linear combination equal 0? When does it equal 1?
In the second part of my talk, I will define canonical dimension of varieties (which, roughly speaking, measures how hard it is to get a rational point in a given variety) and canonical dimension of algebraic groups (which, roughly speaking, measures how complicated the torsors of an algebraic group can be). Then I will state a theorem about an upper estimate on the canonical dimension of the group and its torsors following from the fact that a certain coefficient we obtained in the first part of my talk (i. e. the coefficient in the decomposition of a product of Schubert divisors into a linear combination of Schubert varieties) equals 1. As a result, we will get some explicit numerical estimates on canonical dimension of simply connected simple split algebraic groups of type A, D, and E.

Sanghoon Baek, Relationship between the Chow and Grothendieck Rings for Generic Flag Varieties
2022-01-25 15:00 - 16:00 | B266, IBS, Korea, Republic of
Speaker

Sanghoon Baek
KAIST

Consider the canonical morphism from the Chow ring of a smooth variety X to the associated graded ring of the coniveau filtration on the Grothendieck ring of X. In general, this morphism is not injective. However, Nikita Karpenko conjectured that these two rings are isomorphic for a generic flag variety X of a semisimple group G, where he confirmed the conjecture for a simple group G of type A or C. Recently, this conjecture was disproved by Nobuaki Yagita for some spin groups G. We will discuss further counter-examples using the K-theoretical Pieri formula for highest orthogonal grassmannians. This is joint work with Nikita Karpenko.

Han-Bom Moon, Derived Category of Moduli of Vector Bundles II
2022-01-20 11:00 - 12:00 | TBA
Speaker

Han-Bom Moon
Fordham University

The derived category of a smooth projective variety is an object expected to encode much birational geometric information. Recently, there have been many results on decomposing derived categories into simpler building blocks. In the first lecture, I will provide an elementary introduction to two independent topics — 1. the definition and basic properties of the derived category and 2. moduli spaces of vector bundles on a curve. In the second lecture, I will present recent progress on the structure of the derived category of the moduli space. Most of the lectures will be accessible to graduate students with basic knowledge of algebraic geometry. The second lecture is based on ongoing joint work with Kyoung-Seog Lee.

Han-Bom Moon, Derived Category of Moduli of Vector Bundles I
2022-01-19 10:00 - 12:00 | TBA

Kento Fujita, The Calabi Problem for Fano Threefolds
2022-01-18 11:00 - 12:00 | on-line
Speaker

Kento Fujita
Osaka Univ.

There are 105 irreducible families of smooth Fano threefolds, which have been classified by Iskovskikh, Mori and Mukai. For each family, we determine whether its general member admits a Kähler-Einstein metric or not. This is a joint work with Carolina Araujo, Ana-Maria Castravet, Ivan Cheltsov, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Suess and Nivedita Viswanathan.

Paul-Andi Nagy, Einstein Deformations of Hyperkaehler Cones
2021-12-28 16:00 - 17:00 | B236, IBS, Korea, Republic of
Speaker

Paul-Andi Nagy
IBS CCG

For a hyperkaehler cone with compact link (M, g) we describe the Einstein deformation theory of g and relate it to the algebraic geometry of the twistor space Z of M. This is joint work with Uwe Semmelmann.

Olivier Martin, Measures of Association for Algebraic Varieties
2021-12-28 11:00 - 12:00 | on-line
Speaker

Olivier Martin
Stony Brook Univ.

I will discuss recent work in collaboration with R. Lazarsfeld which explores the following question: Given varieties X and Y of the same dimension how far are they from being birational? I will define various “measures of association” which quantify the failure of X and Y to be birational and present partial results, heuristics, and several open problems. For instance, given an n-fold Z dominating very general hypersurfaces X and Y in Pn+1 of degrees d,e > 2n+1, we show that the degrees of the projections Z—>X and Z—>Y are at least d and e. Moreover, given very general hyperelliptic curves X and Y, any hyperelliptic curve in XxY is contracted by the projection to X or the projection to Y.

Yeongrak Kim, Ulrich Bundles on Cubic Fourfolds
2021-12-09 16:30 - 17:20 | B266, IBS, Korea, Republic of
Speaker

Yeongrak Kim
Pusan National Univ.

(This is a part of Algebraic Geometry Day at CCG in IBS.)
Ulrich bundles are geometric objects corresponding to maximally generated maximal Cohen-Macaulay modules, whose existence has several interesting applications in commutative algebra, homological algebra, and linear algebra. After a pioneering work of Beauville and Eisenbud-Schreyer, existence and classification of Ulrich bundles become important questions also in projective geometry. For instance, they could help to understand the cone of cohomology tables of coherent sheaves on the underlying projective variety, determinantal representations of hypersurfaces, and determinantal representations of Cayley-Chow forms. In this talk, I will discuss construction of Ulrich bundles on smooth cubic fourfolds. Unlike smooth cubic surfaces or threefolds, the smallest possible rank of Ulrich bundles on a smooth cubic fourfold may vary if it is special, i.e., X contains certain surfaces which are not homologous to complete intersections. On the other hand, a (very) general cubic fourfold does not have an Ulrich bundle of rank <6. I will explain how to construct a rank 6 Ulrich bundle on an arbitrary smooth cubic fourfold. This is a joint work with Daniele Faenzi.

DongSeon Hwang, Manin’s Conjecture for a Log Del Pezzo Surface of Index 2
2021-12-09 15:20 - 16:10 | B266, IBS, Korea, Republic of
Speaker

DongSeon Hwang
Ajou Univ.

(This is a part of Algebraic Geometry Day at CCG in IBS.)
Manin’s conjecture predicts the asymptotic behavior on the number of rational points of bounded anticanonical height on Fano varieties. In this talk, I will explain how the geometry governs the arithmetic in the case of a log del Pezzo surface with A4 and K5 singularities using the torsor method. This talk is based on joint work in progress with Ulrich Derenthal.

Kangjin Han, On the Singular Loci of Higher Secants of Veronese Varieties
2021-12-09 14:00 - 14:50 | on-line
Speaker

Kangjin Han
DGIST

(This is a part of Algebraic Geometry Day at CCG in IBS.)
For a projective variety X in PN, the k-secant variety σk(X) is defined to be the closure of the union of k-planes in PN spanned by k-points of X. In this talk, we consider singular loci of higher secant varieties of the image of the d-uple Veronese embedding of projective n-space, νd(Pn). For the singular loci of k-secant of νd(Pn), it has been known only for k≤3. First, I will review some basic notions and results and then explain projective techniques with respect to an explicit calculation of the Gauss map of X and computation for conormal space via a certain type of Young flattening. As investigating geometry of moving tangents along subvarieties, we determine the (non-)singularity of so-called ‘subsecant loci’ of k-secant of νd(Pn) for arbitrary k. This is a joint work with Katsuhisa Furukawa.

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 | 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.
