Center for Complex Geometry

Jungkai Chen, Threefold Divisorial Contraction to Curves
2024-05-17 16:00 - 17:00 (Asia/Seoul) | B236-1, IBS, Korea, Republic of
Speaker

Jungkai Chen
National Taiwan University

The minimal model program works pretty well in dimension three. However, the explicit classification of divisorial contractions to points was completed quite recently thanks to the work of Kawamata, Hayakawa, Kawakita and more. In this talk, we are going to describe threefold divisorial contractions to curves. We will demonstrate some more possible higher dimensional extensions. This is joint work in progress with Jheng-Jie Chen and Hsin-Ku Chen.

Ching-Jui Lai, Anticanonical Volume of Singular Fano Threefolds
2024-05-13 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Ching-Jui Lai
National Cheung Kung University

The set of canonical Fano threefolds form a bounded family by results of Kawamata, Mori-Miyaoka-Kollar-Tagaki, and in a much more general setting by Birkar. In particular, the anticaonical volume –KX3 is bounded. An optimal lower bound is 1/330 by the work of Chen-Chen. In this talk, we discuss the problem of identifying an optimal upper bound of –KX3.

Hsueh-Yung Lin, Motivic Invariants of Birational Automorphisms of Threefolds
2024-05-13 14:30 - 15:30 | B236-1, IBS, Korea, Republic of
Speaker

Hsueh-Yung Lin
National Taiwan University

The motivic invariant c(f) of a birational automorphism f : X – → X measures the difference between the birational types of the exceptional divisors of f and those of the inverse f-1. In general c(f) is nonzero: this is the case when f is some Cremona transformation, which leads to new explanation as to why Cremona groups are not simple. In the other direction, which will be the main focus of the talk, we show that the invariant c(f) always vanishes when X is a complex projective threefold. (joint work in progress with E. Shinder.)

Shigeyuki Kondo, A Review on Enriques Surfaces: Moduli, Automorphism Groups and Positive Characteristics, II
2024-05-08 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Shigeyuki Kondo
Nagoya University

The Enriques surface was discovered, in 1894 by Federigo Enriques, as a counter-example of a rationality problem.
First I would like to recall the moduli space and the automorphism groups of Enriques surfaces over the complex numbers.
In the later half, I shall mention a recent progress in positive characteristic, especially in characteristic two.

Shigeyuki Kondo, A Review on Enriques Surfaces: Moduli, Automorphism Groups and Positive Characteristics, I
2024-05-07 16:00 - 17:00 | B236-1, IBS, Korea, Republic of

Yong Hu, Noether Inequality for Irregular Threefolds of General Type
2024-03-26 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Yong Hu
Shanghai Jiao Tong University

Let X be a smooth irregular 3-fold of general type. In this talk, we will prove that the optimal Noether inequality vol(X) ≥ (4/3) pg(X) holds if pg(X) ≥ 16 or if X has a Gorenstein minimal model. Moreover, when X attains the equality and pg(X) ≥ 16, its canonical model will be explicitly described. This is a joint work with Tong Zhang.

Gebhard Martin, Automorphisms of del Pezzo Surfaces III
2024-03-14 15:30 - 16:30 | B236-1, IBS, Korea, Republic of
Speaker

Gebhard Martin
Universität Bonn

Motivated by the classification of finite subgroups of the Cremona group of the plane, I will survey old and new results on automorphism groups of del Pezzo surfaces. In particular, I will report on joint work with Igor Dolgachev on the classification of automorphism groups of smooth del Pezzo surfaces and on joint work with Claudia Stadlmayr on the classification of vector fields on mildly singular del Pezzo surfaces. Special emphasis will be placed on the theory of actions of infinitesimal group schemes and characteristic-free methods in classical algebraic geometry.

Gebhard Martin, Automorphisms of del Pezzo Surfaces II
2024-03-12 16:00 - 17:00 | B236-1, IBS, Korea, Republic of

Claudia Stadlmayr, Which Rational Double Points Occur on del Pezzo Surfaces?
2024-03-07 15:30 - 16:30 | B236-1, IBS, Korea, Republic of
Speaker

Claudia Stadlmayr
Technische Universität München

Canonical surface singularities, also called rational double points (RDPs), can be classified according to their dual resolution graphs, which are Dynkin diagrams of types A, D, and E. Whereas in characteristic different from 2, 3, and 5, rational double points are “taut”, that is, they are uniquely determined by their dual resolution graph, this is not necessarily the case in small characteristics. To such non-taut RDPs Artin assigned a coindex distinguishing the ones with the same resolution graph in terms of their deformation theory. In 1934, Du Val determined all configurations of rational double points that can appear on complex RDP del Pezzo surfaces. In order to extend Du Val’s work to positive characteristic, one has to determine the Artin coindices to distinguish the non-taut rational double points that occur. In this talk, I will explain how to answer the question “Which rational double points (and configurations of them) occur on del Pezzo surfaces?” for all RDP del Pezzo surfaces in all characteristics. This will be done by first reducing the problem to degree 1 and then exploiting the connection to (Weierstraß models of) rational (quasi-)elliptic surfaces.

Gebhard Martin, Automorphisms of del Pezzo Surfaces I
2024-03-05 11:00 - 12:00 | B236-1, IBS, Korea, Republic of

Caucher Birkar, Stable Minimal Models
2024-02-20 17:00 - 18:00 | on-line
Zoom ID: 880 6763 5837
PW: [REDACTED]

Speaker

Caucher Birkar
Tsinghua University

In this talk I will introduce stable minimal models and discuss some related results and if times allows, problems.
(This seminar is a part of School and Workshop on Moduli, K-trivial Varieties, and Related Topics)

Grzegorz Kapustka, Projective Models of Nikulin Orbifolds
2023-12-11 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Grzegorz Kapustka
Jagiellonian University

We describe a locally complete family of projective irreducible holomorphic symplectic orbifolds as double covers of special complete intersections (3, 4) in P6. This is a joint work with C. Camere, A. Garbagnati and M. Kapustka.

Andreas Höring, Projective Manifolds with Pseudoeffective (Co)tangent Bundle, III, IV
2023-11-22 15:00 - 17:30 | B236-1, IBS, Korea, Republic of
Speaker

Andreas Höring
University of Nice

Symmetric holomorphic forms on projective manifolds have been studied from various angles in the last ten years : Brotbek-Darondeau and independently Song-Yan Xie have shown that complete intersections of sufficiently high degree and codimension have an ample cotangent bundle, so many symmetric forms. Vice versa Campana and Paun show that if the cotangent bundle is big, then X is of general type. In this series of lectures I will work under the assumption that the cotangent bundle is pseudoeffective. In general this positivity assumption is too weak to have any useful implications, but we will see that for the (co)tangent bundle it leads to surprisingly many restrictions on the geometry of the manifold. In particular we expect a nonvanishing property that is analogous to the famous nonvanishing conjecture for the canonical bundle. These talks are based on joint papers with Thomas Peternell and Junyan Cao.

Andreas Höring, Projective Manifolds with Pseudoeffective (Co)tangent Bundle, I, II
2023-11-20 15:00 - 17:30 | B236-1, IBS, Korea, Republic of

Yotsutani Naoto, Chow Stability and Non-symmetric Kähler-Einstein Toric Fano Manifolds
2023-10-25 11:00 - 12:00 | B266, IBS, Korea, Republic of
Speaker

Yotsutani Naoto
Kagawa University

In 2001, Donaldson proved that if a polarized manifold (X, L) admits constant scalar curvature Kähler metrics in c1(L), then (X, L) is asymptotically Chow stable whenever its automorphism group Aut(X, L) is discrete.
In this talk, we show that this result does not hold in the case where Aut(X, L) is not discrete.
This talk is based on a joint work with H. Ono and Y. Sano.

Guolei Zhong, Positivity of Tangent Sheaf of Projective Varieties
2023-10-10 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Guolei Zhong
IBS-CCG

In this talk, we discuss the structure of projective varieties with certain positive tangent sheaves. More precisely, when the tangent sheaf is almost nef or positively curved, we construct a well-defined MRC fibration and study the Fujita decomposition of reflexive sheaves. Besides, we show that the almost nefness of the tangent sheaf will impose rather restrictive condition on the singularities, and some typical examples will be given. This is based on my recent joint work with Masataka Iwai and Shin-ichi Matsumura.

Seung uk Jang, Vieta Involutions on Tropical Markov Cubics
2023-09-19 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Seung uk Jang
Université de Rennes 1

We discuss the algebraic dynamics on Markov cubics generated by Vieta involutions, in the tropicalized setting, including some introduction to the tropicalization. It turns out that there is an invariant subset of the tropicalized Markov cubic where the action by Vieta involutions can be modeled by that of the Farey reflection group on the hyperbolic plane.
Some applications in non-archimedean dynamics, number theory, or complex dynamical questions will also be introduced.

Chang-Yeon Chough, Introduction to algebraic stacks, VII, VIII
2023-07-26 10:00 - 12:15 | B236-1, IBS, Korea, Republic of
Speaker

Chang-Yeon Chough
Sogang Univ.

This is an 8 hours long lecture series on algebraic stacks, which have become an important part of algebraic geometry (for example, in the study of moduli spaces) since Deligne and Mumford established the foundation of the theory of stacks. This crash course will be following roughly “Algebraic Spaces and Stacks” by Martin Olsson. Our main goal is to set up a foundation for the theory of algebraic stacks, so that the attendees would be able to use it in their own research in the future.

Chang-Yeon Chough, Introduction to algebraic stacks, V, VI
2023-07-25 15:00 - 17:30 | B236-1, IBS, Korea, Republic of

Chang-Yeon Chough, Introduction to algebraic stacks, III, IV
2023-07-19 10:00 - 12:15 | B236-1, IBS, Korea, Republic of

Chang-Yeon Chough, Introduction to algebraic stacks, I, II
2023-07-18 15:00 - 17:30 | B236-1, IBS, Korea, Republic of

Shinnosuke Okawa, Semiorthogonal Decompositions and Relative Canonical Base Locus
2023-07-06 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Shinnosuke Okawa
Osaka University

Motivated by the DK hypothesis, some years ago I proved that SODs of the derived category of a smooth projective variety are strongly constrained by the base locus of the canonical linear system. In particular, this leads to the indecomposability of the derived category of varieties whose canonical bundles are globally generated (hence minimal). In this talk I will briefly recall this work and discuss its generalization to the relative settings. The latter implies new indecomposability results, including the case of minimal surfaces of positive irregularity. This talk is based on arXiv:2304.14048.

Shinnosuke Okawa, Moduli Space of Semiorthogonal Decompositions
2023-07-04 16:00 - 17:00 | B236-1, IBS, Korea, Republic of
Speaker

Shinnosuke Okawa
Osaka University

Semiorthogonal decomposition (SOD) is a central notion in the study of triangulated categories. In particular, SODs of the bounded derived category of coherent sheaves of a variety (SODs of the variety, for short) have profound relations to its geometry. In this talk I discuss the moduli functor which classifies SODs of the fibers of smooth projective morphisms. The main result is that it is an algebraic space which is locally etale over the target of the morphism. I will explain the main points of the proof, various applications and open problems. This talk is based on the joint work arXiv:2002.03303 with Andrea Ricolfi and Pieter Belmans.

Sung Gi Park, Kodaira Dimension and Hyperbolicity for Smooth Families of Varieties
2023-06-20 11:00 - 12:00 | on-line
Speaker

Sung Gi Park
Harvard University

In this talk, I will discuss the behavior of positivity, hyperbolicity, and Kodaira dimension under smooth morphisms of complex quasi-projective manifolds. This includes a vast generalization of a classical result: a fibration from a projective surface of non-negative Kodaira dimension to a projective line has at least three singular fibers. Furthermore, I will explain a proof of Popa’s conjecture on the superadditivity of the log Kodaira dimension over bases of dimension at most three. These theorems are applications of the main technical result, namely the logarithmic base change theorem.

Changho Han, Compact Moduli of K3 Surfaces with a Given Nonsymplectic Cyclic Action
2023-05-23 16:30 - 17:30 | B236-1, IBS, Korea, Republic of
Speaker

Changho Han
University of Waterloo

To construct a moduli space which is itself a compactification of a given moduli space, one needs to enlarge the class of objects in consideration (e.g. adding certain singular curves to the class of smooth curves). After a brief review of the compactifications of the moduli of elliptic curves, I will generalize into looking at various compactifications of the moduli of K3 surfaces with nonsymplectic cyclic actions, and then discuss how those compactifications are birationally related to each other. As an application, I will apply this framework into Kondo’s moduli space of sextic K3 surfaces with Z/3Z action. Results come from joint works (in progress) with Valery Alexeev, Anand Deopurkar, and Philip Engel.

Shigeru Mukai, Fano 3-folds, Lagrangian Fibration and Leech-K3 Geometry III
2023-05-11 11:00 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

Shigeru Mukai
RIMS, Kyoto University

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Shigeru Mukai, Fano 3-folds, Lagrangian Fibration and Leech-K3 Geometry II
2023-05-09 15:00 - 16:00 | B236-1, IBS, Korea, Republic of

Shigeru Mukai, Fano 3-folds, Lagrangian Fibration and Leech-K3 Geometry I
2023-05-09 11:00 - 12:00 | B236-1, IBS, Korea, Republic of

Junyan Zhao, Moduli of Curves of Genus 6 and K-stability
2023-04-25 11:00 - 12:00 | on-line
Speaker

Junyan Zhao
University of Illinois Chicago

A general curve C of genus 6 can be embedded into the unique quintic del Pezzo surface X5 as a divisor of class -2KX5. This embedding is unique up to the action of the symmetric group S5. Taking a double cover of X5 branched along C yields a K3 surface Y. Thus the K-moduli spaces of the pair (X5, C) can be studied via wall-crossing and by relating them to the Hassett-Keel program for C and the HKL program for Y. On the other hand, X5 can be embedded in P1 × P2 as a divisor of class O(1,2), under which -2KX is linearly equivalent to OX(2,2). One can study the VGIT-moduli spaces in this setting. In this talk, I will compare these four types of compactified moduli spaces and their different birational models given by wall-crossing.

JongHae Keum, Mori Dream Surfaces of General Type with pg=0
2023-02-28 11:10 - 12:00 | B236-1, IBS, Korea, Republic of
Speaker

JongHae Keum
KIAS

(This is a part of Seminars on Algebraic Surfaces and Related Topics.)
The Cox ring of a variety is the total coordinate ring, i.e., the direct sum of all spaces of global sections of all divisors. When this ring is finitely generated, the variety is called Mori dream (MD). A necessary condition for being MD is the finite generatedness of Pic(X), i.e., the vanishing of the irregularity. Smooth rational surfaces with big anticanonical divisor are MD. So are all del Pezzo surfaces of any degree. A K3 surface or an Enriques surface with Picard number at least 3 is MD iff its automorphism group is finite.
In this talk I will consider the case of surfaces of general type with pg=0, and provide several examples that are MD. I will also provide non-minimal examples that are not MD. This is a joint work with Kyoung-Seog Lee.
