Center for Complex Geometry

Luca Rizzi, An Approach to the Fujita Decomposition via Massey Products and Castelnuovo-de Franchis Theorem
2022-03-03 15:00 - 16:00 (Asia/Seoul) | 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.

Sandor Kovacs, Hodge Sheaves for Singular Families
2022-02-08 11:00 - 12:00 | on-line
Speaker

Sandor Kovacs
Univ. of Washington

This is a report on joint work with Behrouz Taji. Given a flat projective morphism f : X → B of complex varieties, assuming that B is smooth, we construct a functorial system of reflexive Hodge sheaves on B . If in addition, X is also smooth then this system gives an extension of the Hodge bundle underlying the VHS of the smooth locus of f . This in turn provides a criterion that all VHSs of geometric origin must satisfy. As an independent application we prove a singular version of Viehweg’s conjecture about base spaces of families of maximal variation.

Jongbaek Song, Regular Hessenberg Varieties and Toric Varieties
2022-01-27 11:00 - 12:00 | TBA
Speaker

Jongbaek Song
KIAS

A Hessenberg variety is a subvariety of the flag variety (G/B) determined by two parameters: one is an element of the Lie algebra of G and the other is a B-submodule containing the Lie algebra of B, known as a Hessenberg space. In this talk, we focus on elements in the regular locus of the Lie algebra and the Hessenberg space determined by negative simple roots. Then, we aim to figure out cohomological relationship of these Hessenberg varieties with a certain class of toric varieties having orbifold singularities. The main result raises an interesting topic concerning toric varieties with symmetries by reflections. This is a joint work with M. Masuda, T. Horiguchi and J. Shareshian.

Rostislav Devyatov, Multiplicity-free Products of Schubert Divisors and an Application to Canonical Dimension
2022-01-25 16:20 - 17:20 | 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.
