algebraic surfaces and related...

Algebraic Surfaces and Related Topics

School of Mathematical Sciences, Xiamen University

September 24-28，2018

Organizers: Valery Alexeev (Athens), Wenfei Liu (Xiamen), Bo Yang (Xiamen)

Schedule

Meeting Room：Shiyan Building 105, Haiyun Campus

Banquet and Dinner：Dafengyuan Restaurant

Sep 24 Sep 25 Sep 26 Sep 27 Sep 28

9:10-9:30 Opening

9:30-10:30 Keum Fujino 9:00-10:00

Bauer

Oguiso Jiang

Coffee

11:00-12:00 Lee Prokhorov 10:15-11:15

Catanese

Roulleau Cheltsov

Lunch 11:30-12:30

Hwang

14:00-15:00 Urzúa Deopurkar Free

Afternoon

Odaka Free

Afternoon 15:15-16:15 Yin Liu Gonzaléz-Alonso

Coffee

16:45-17:45 Gongyo Ascher Lü

18:00-19:00 Reception Banquet Dinner

Titles and Abstracts

Compactifying the moduli space of degree one del Pezzo surfaces via

elliptic fibrations

Kenneth Ascher

Abstract In this talk I will discuss a construction of various modular

compactifications of spaces of elliptic surfaces, analogous to Hassett's

moduli spaces of weighted stable curves. One application of this

construction is a compactification of the moduli space of degree one del

Pezzo surfaces. This is joint work with Dori Bejler

Rigid compact complex manifolds and the answer to a question of

Morrow-Kodaira

Ingrid Bauer

Abstract: TBD

Canonical maps of surfaces and of hypersurfaces in Abelian

varieties, and the Double Point Formula Fabrizio Catanese

Abstract I will first give an introduction, recalling the results of several recent

papers, dedicated to the search (and to the construction) of canonical

surfaces of high degree and low geometric genus pg = 4, 5, 6.

Of particular interest is the question of finding canonically embedded

surfaces in P5 (i.e., an embedding of the canonical model Z of S), which

relates to several questions of homological algebra (Walter's bundle

Pfaffians).

In fact, this question is open (and interesting) only for pg 6, since

for pg = 4, Z must be a 5-ic, and, for pg =5, joint work with Oguiso

allowed to show, also in the case where Z is singular, that the canonical

model Z must be a complete intersection of type (2, 4) or (3, 3).

This result for pg = 5 uses some generalizations of Severi’s double point

formula for varieties with isolated singularities, gotten together with

Oguiso, that I shall explain.

For pg = 6, degree d = 24 canonically embedded surfaces were obtained

by myself with some regular surfaces (q = 0), and by Cesarano in his

Ph.D. Thesis with a family of surfaces having q = 3, polarizations of

type (1, 2, 2) in an Abelian 3-fold.

Time permitting, I shall describe work in progress, and some older and

new results and conjectures, related to the question whether a general

hypersurface D in a polarized Abelian variety (not a ppav) has

a birational canonical map.

I shall then explain the relation of the double point formula with the

question: when is this canonical map (generically) an embedding ?

How to compute delta-invariant of Fano varieties?

Ivan Cheltsov

Abstract

Recently, Kento Fujita and Yuji Odaka introduced delta-invariant of Fano

varieties. It plays an important role in K-stability similar to alpha-invariant

introduced by Tian many years ago. Delta-invariant is more powerful than

alpha-invariant. But it is also much harder to compute it. In my talk I will

show how to estimate delta-invariants for some smooth and singular del

Pezzo surfaces. This is a joint work with Yanir Rubinshtein, Kewei Zhang,

Jihun Park and Costya Shramov.

Log surfaces of almost K3 type and curves of genus 4

Anand Deopurkar

Abstract

A log surface of "almost K3 type" is one where the log canonical divisor is

positive but very close to 0. We study compactified moduli spaces of such

log surfaces, constructed using the techniques of Kollár, Shepherd-Barron,

Alexeev, and Hacking. We completely describe such a compactification of

the moduli space of (X, D) where X is a quadric surface and D is a

canonical genus 4 curve, obtaining a new birational model of M4. This is

joint work with Changho Han. Minimal model theorem for log surfaces

Osamu Fujino

Abstract

I will explain recent developments of the minimal model theory for log

surfaces. The minimal model program, the finite generation of log

canonical rings, the abundance theorem, and so on, hold true under the

weaker assumption that we usually thought.

On log CY structure of surfaces admitting non-trivial polarized

endomorphisms

Yoshinori Gongyo

Abstract

We treat mainly normal projective varieties admitting non-trivial polarized

endomorphism. We discuss the conjecture "normal projective varieties

admitting non-trivial polarized endomorphism have log Calabi--Yau

structure". We confirm this conjecture for surfaces. This is a joint work

with Amael Broustet.

Families of curves with prescribed rank of Higgs field.

Víctor Gonzaléz-Alonso

Abstract

The Hodge bundle of a (non-isotrivial, semistable) family of curves has

two nested subbundles: the flat unitary subbundle (spanned by flat sections

with respect to the Gauss-Manin connection), and the kernel of the Higgs

field, which has seminegative definite curvature and contains the flat

subbundle. Though there is no reason to expect equality, we also don't

know of examples showing a strict inclusion. In this talk I will present a

construction of such examples, with kernels of any possible rank, some of

them even over a projective base curve. Time permitting, I will show some

consequences of this construction for the geometry of the Torelli locus and

the classification of irregular fibred surfaces. This is work in progress with

Sara Torelli.

Cascades of toric log del Pezzo surfaces

DongSeon Hwang

Abstract

There have been numerous attempts to classify log del Pezzo surfaces.

To achieve this goal, we proposed an approach by using the concept of

a 'cascade', which can be viewed as a generalization of the

classification of the smooth del Pezzo surfaces. So far this approach

has been reasonably successful in the case when the log del Pezzo surface

is toric or is of Picard number one. In this talk, I will focus on the toric case

and explain the concept of cascades and how it leads to the classification.

Alpha invariants of minimal surfaces of general type and applications

to birational geometry of 3-folds.

Chen Jiang

Abstract

It is well-known that alpha-invariants play important roles in the study of

Fano varieties. In this talk I will introduce the analogue of alpha-invariants

for minimal varieties of general type. I will explain how the alpha

invariants of minimal surfaces appear naturally in the study of birational

geometry of 3-folds and give some interesting applications based on a joint

work with Jungkai Chen and Meng Chen.

Algebraic surfaces with minimal Betti numbers

Jonghae Keum

Abstract

The surfaces in the title are algebraic surfaces with the Betti numbers of

the complex projective plane, and are called Q-homology projective

planes. If such a surface has only quotient singularities, then its minimal

resolution is a smooth surface with pg =q=0. Fake projective planes and

the complex projective plane are smooth examples of a Q-homology

projective plane. I will begin with basic definitions and examples and

then describe recent progress in the study of such surfaces, singular ones

and fake projective planes. I will also discuss open questions.

Compact moduli space of plane curves via K-stability

Yuchen Liu

Abstract

We study the K-stability compactification of moduli of log Fano pairs (P2,

C) where C is a smooth plane curve of degree at least 4. We show that

when is small, the K-moduli compactification is isomorphic to the GIT

moduli space of plane curves. We establish a framework of wall-crossing

behaviors of these K-moduli spaces as increases. Specifically, we show

that the first wall-crossing of these K-moduli spaces are weighted

blow-ups. We describe all wall-crossings for degree 4, 5 and 6. We also

relate the final K-moduli spaces to Hacking's moduli spaces. This is joint

work in progress with K. Ascher, K. DeVleming, and P. Gallardo.

Deformation of a generically finite map to a hypersurface embedding

Yongnam Lee

Abstract

In this talk, we give a structure theorem for projective manifolds W0 with

the property of admitting a 1-parameter deformation where Wt is a

hypersurface in a projective manifold Zt. There structure is the one of

special iterated univariate coverings which we call of normal type, which

essentially means that the line bundles where the univariate coverings live

are tensor powers of the normal bundle to the image. We give applications

to the case where Zt is a projective space, respectively an Abelian variety.

This is a joint work with Fabrizio Catanese.

Canonically fibered surfaces of general type

Xin Lü

Abstract

The canonical map of a surface of general type is called fibered, if its

canonical map induces a fibration. Beauville showed that the genus of

such a fibration is bounded from above by 5 when the geometric genus is

large. Examples with the genus equal to 2 or 3 have been constructed.

Xiao asked whether there exists a surface of general type with large

geometric genus, whose canonical map is fibered of genus greater than 3?

In this talk, I will report the first such examples with the genus equal to 4.

Non-variety geometrically compactified moduli of K3 surfaces

Yuji Odaka

Abstract

We give some explanation on our recent compactifications of classical

moduli varieties of (mainly) polarized K3 surfaces, among others

(Riemann surfaces, abelian varieties, etc), whose boundaries parametrize

some degeneration data rather than degenerate projective varieties. The

compactifications are not varieties which may disappoint some audience,

but we try to explain how they are still interesting even from

algebra-geometric perspective and are connected to many other fields e.g.

tropical geometry. Also, our compactifications give a convenient new

framework to describe the collapsing of Kähler-Einstein metrics, such as

K3 surfaces to spheres or the interval. Some of the details are in

arXiv:1406.7772 (Mg), 1705.05545 (mostly Ag), 1805.01724 (mainly K3,

joint with Y. Oshima).

A surface with discrete and non-finitely generated automorphism

group and infinitely many real forms

Keiji Oguiso

Abstract

This is a joint work with Professor Dinh at Singapore. We show that

there is a smooth complex projective variety, of any dimension greater

than or equal to two, whose automorphism group is discrete and not

finitely generated. Moreover, this variety admits infinitely many real

forms which are mutually non-isomorphic over R. Our result is

inspired by the work of Lesieutre and answers questions by Dolgachev,

Esnault and Lesieutre.

Degenerations of del Pezzo surfaces in Q-Gorenstein families Yuri Prokhorov

Abstract

Consider a family of del Pezzo surfaces over a disc whose total space is

Q-Gorenstein. We study the central fiber of such families under additional

assumption that it is normal and has at worst log canonical singularities.

This problem is motivated by the minimal model program and rationality

questions.

Construction of Nikulin configurations on some Kummer surfaces

Xavier Roulleau

Abstract

A Nikulin configuration on a K3 surface is a set C of 16 smooth disjoint

rational curves. By the results of Nikulin, any K3 surface X containing a

Nikulin configuration is a Kummer surface, which means that there exists

an abelian surface A such that X is the minimal resolution of the quotient

A/[-1] and the exceptional curves of the resolution X->A[-1] are the 16

curves of the Nikulin configuration C (this is denoted X=Km(A)).

In this talk, starting with a Kummer configuration C on some polarised

Kummer surface X, we will construct another Kummer configuration C’ on

X such that if A and A’ denotes the associated Abelian surfaces, although

one has: Km(A)=X=Km(A’), the Abelian surfaces A and A’ are not

isomorphic (unless X is a Jacobian Kummer surface). If we have enough

time we will derive some applications on the construction surfaces of

general type, like the Schoen surfaces. This is a joint work with Alessandra

Sarti.

Hyperbolic complete intersection surfaces

Giancarlo Urzúa

Abstract

A complex space X is said to be hyperbolic if every holomorphic map from

the complex plane to X is constant. Often this is referred as Brody

hyperbolic, and it coincides with Kobayashi hyperbolic (i.e. Kobayashi

pseudo-distance is a distance) when X is a compact complex manifold. Our

interest is in relation to a generalization of the famous Kobayashi

conjecture but for complete intersection surfaces (cis): A very general cis

of general type must be hyperbolic. In particular for each multidegree

(m1, ..., mn) with m1+...+mn > n+3, there should exist hyperbolic cis in Pn+2

.

I will present some results (joint with Natalia Garcia-Fritz) on the

existence of hyperbolic cis. In particular we show explicit families of them

for each possible multidegree in Pn+2

when n>7. For P9, P

8, P

7 we only miss

the multidegree (2, ..., 2). For lower dimensional projective spaces (and by

adding what is known via other methods), we do not know examples for

infinitely many multidegrees, except for P3 where it is only unknown the

existence of a hyperbolic quintic.

Generalizations of the Bogomolov-Mumford theorem

Qizheng Yin

Abstract

The Bogomolov-Mumford theorem says that every K3 surface admits a

rational curve. I will discuss some generalizations of this theorem to higher

dimensions, e.g. the (conjectural) existence of uniruled divisors/rational

surfaces on holomorphic symplectic 4-folds. Many questions and

examples will be presented. Based on joint work with Georg Oberdieck

and Junliang Shen.

List of Participants 1 Valery Alexeev University of Georgia

2 Kenneth Ascher Princeton University

3 Ingrid Bauer Universität Bayreuth

4 Fabrizio Catanese Universität Bayreuth

5 Ivan Cheltsov University of Edinburgh

6 Anand Deopurkar Australian National University

7 Osamu Fujino Osaka University

8 Yoshinori Gongyo University of Tokyo

9 Victor Gonzaléz-Alonso Leibniz Universität Hannover

10 Dongseon Hwang Ajou University

11 Chen JIANG (江辰) KIPMU, University of Tokyo

12 JongHae Keum Korea Institute for Advanced

Study

13 Yongnam Lee Korea Advanced Institute of

Science and Technology

14 Yuchen LIU (刘雨晨) Yale University

15 Xin LÜ (吕鑫) East China Normal University

16 Yuji Odaka Kyoto University

17 Keiji Oguiso University of Tokyo

18 Yuri Prokhorov Steklov Mathematical Institute

19 Xavier Roulleau Université d’Aix-Marseille

20 Giancarlo Urzúa Pontificia Universidad Católica

de Chile

21 Qizheng YIN (訚琪峥) BICMR, Peking University

22 Jingshan Chen（陈敬珊） Tsinghua University

23 Cheng Gong（龚成） Soochow University

24 Yi Gu（顾怡） Soochow University

25 Songbo Ling（凌松波） BICMR, Peking University

26 Xiaolei Liu（刘小雷） Dalian University of Technology

27 Jun Lu（陆俊） East China Normal University

28 Fan Peng（彭帆） Guangxi Normal University

29 Jingsong Xu（许劲松） Xi’an Jiaotong-Liverpool

University

30 Wanyuan Xu(徐万元) Shanghai Normal University

31 Xun Yu(余讯) Tianjin University

32 Lei Zhang（张磊） University of Science and

Technology of China

33 Tong Zhang（张通） East China Normal University

36 Hang Zhao（赵航） Peking University

34 Haidong Liu（刘海东） Kyoto University

35 Bloss Patrick Leibniz Universität Hannover

37 WU Xian University of Georgia

38 Wenfei Liu（刘文飞） Xiamen University

39 Bo Yang（杨波） Xiamen University

40 Jie Sun（孙杰） Jianghan University

Accommodation and Arrival Information

Accommodation

Speakers: Lujiang-Mega Hotel

Address: No.382 Longhushan Road, Siming District, Xiamen

Tel: +86 (0) 592 2199099

Other participants not from Xiamen: Tianhai Huayuan Hotel

Address: No.9 Longhushan Road, Siming District, Xiamen

Tel: +86 (0) 592 2519888

Arrival information

From Xiamen (Gaoqi) International Airport (code: XMN) to Lujiang

Mega Hotel or Tianhai Huayuan Hotel

Taxi: It takes about 25 mins and the cost is about 50 RMB (an

additional 20% charge after 23:00).

From Xiamen North Railway Station to Lujiang Mega Hotel or

Tianhai Huayuan Hotel.



From Xiamen Railway Station to Lujiang Mega Hotel or Tianhai

Huayuan Hotel.



It you prefer to take the subway, a shuttle bus, or other public

transportation, please use baidu map or other mobile app to get the

route information.

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