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Workshop on Algebraic Geometry, Complex Dynamics and their Interaction
(4 - 7 January 2011)

Organizing Committee · Visitors and Participants · Overview · Activities · Venue

 

 Organizing Committee

 

 

 Visitors and Participants

 

 

 Overview

 

The purpose of the workshop is to report the recent progress in Algebraic Geometry, especially in the area of Birational Geometry in connection with the MMP (minimal model program), the equivariant MMP and the Dynamics of Symmetries on compact complex manifolds.


Background on the equivariant MMP and Dynamics of Symmetries on compact manifolds:
For the Algebro-geometric part, the main theme is Birational Geometry. One would like to classify algebraic varieties X according to their Kodaira dimension κ(X). This κ(X) measures certain positivity of the determinant line bundle KX = det(TX*) of the cotangent bundle TX* of X.


Minimal Model Conjecture (MMC). Every algebraic variety X of Kodaira dimension κ(X) ≥ 0 is birational to a so called minimal model Xm, in the sense that Xm has only mild singularities, and the canonical divisor KXm is numerically effective.


The MMC in the above general form is still open at this moment. However, some breakthroughs have been made recently. For algebraic varieties X of general type, the MMC has been proved (i.e., the existence of a minimal model Xm of X has been confirmed), independently by Yum-Tong Siu, and Birkar-Cascini-Hacon-McKernan. As a consequence, the minimal model Xm has a so called Einstein-Käher metric of negative curvature, which in turn, implies the inequalities (in some sense) c1(Xm)2 ≤ 2(n+1)/n c2(Xm) involving the first two Chern classes when Xm is smooth and of dimension n. This, together with Noether's inequality (when Xm is of dimension n = 2) c1(Xm)2 ≥ (1/5) c2(Xm) – 36/5, very effectively narrow down the region for algebraic varieties to exist:

 

As a further step after MMP, during the workshop, one may explore the way to have an equivariant MMP for a pair (X, G) of an algebraic variety X together with a group G of symmetries on X. To be interesting, G is assumed to be infinite and of positive topological entropy. One may speculate the following:


Question A. Can one say the following? All dynamically interesting symmetries on an algebraic variety (or more generally on a compact Kähler manifold) are built up from those on the following three types of varieties (called the basic building blocks): Calabi-Yau variety, Compact complex torus and Rationally connected variety.


The initial motivation to consider this equivariant MMP is due to a conjecture formulated by Shouwu Zhang (and commonly known as the Dynamic Manin-Mumford conjecture (DMM)): A subvariety Y of an algebraic variety X (or compact Kähler manifold) with a polarized self map f from X to itself, is f-preperiodic if and only if Y contains a Zariski-dense subset of f-preperiodic points (a slight modification of it is needed though, as noticed by Ghioca and Tucker). The DMM is only known for Compact complex tori and power maps of the projective n-space. If the Question A is affirmatively answered, then one could reduce, to some extent, the DMM to that on the three building blocks in Question A. Some discussion around the above problems is expected during the workshop.


 Activities


During the 4-day workshop, it is planned to have two lectures and three lectures in the morning and afternoon, respectively. The lectures are mostly to be delivered by overseas experts. Each day, there will be a mixture of lectures on Algebraic Geometry and Dynamics, so that there will be enough interactions between the two groups of experts.

 

Tuesday, 4 Jan 2011

09:00am - 09:25am

Registration

09:25am - 09:30am

Opening Remarks

Louis Chen, Institute for Mathematical Sciences
De-Qi Zhang, National University of Singapore

 

Chair: De-Qi Zhang, National University of Singapore

09:30am - 10:30am

Unramified endomorphisms of algebraic surfaces

Masayoshi Miyanishi, Osaka University, Japan

10:30am - 11:00am

--- Coffee Break ---

11:00am - 12:00nn

Some results about the fundamental groups of hyperelliptic fibrations and applications

Rajendra V. Gurjar, Tata Institute of Fundamental Research, India

12:00nn - 01:30pm

--- Lunch ---

 

Chair: Akira Fujiki, Osaka University, Japan

01:30pm - 02:30pm

Foliations by Riemann surfaces

John Fornaess, University of Michigan, USA

02:30pm - 03:00pm

--- Coffee Break ---

03:00pm - 04:00pm

Stability conditions and Bogomolov-Gieseker type inequalities

Yukinobu Toda, IPMU, Japan

04:00pm - 04:30pm

--- Coffee Break ---

04:30pm - 05:30pm

Extremal Gromov-Witten invariants of the Hilbert scheme of points on surfaces

Wei-Ping Li, The Hong Kong University of Science & Technology, Hong Kong

Wednesday, 5 Jan 2011

08:45am - 09:00am

Registration

 

Chair: Masayoshi Miyanishi, Osaka University, Japan

09:00am - 10:00am

A remark on the abundance conjecture

Yujiro Kawamata, University of Tokyo, Japan

10:00am - 10:15am

--- Coffee Break ---

10:15am - 11:15am

Dynamics of Kaehler-Einstein metrics

Vincent Guedj, Université Aix-Marseille, France

11:15am - 11:30am

--- Coffee Break ---

11:30am - 12:30pm

The Noether function arising from the geography

Meng Chen, Fudan University, China

12:30pm - 02:00pm

--- Lunch ---

 

Chair: Rajendra V. Gurjar, Tata Institute of Fundamental Research, India

02:00pm - 03:00pm

Algebraic Montgomery-Yang problem on homology projective planes

JongHae Keum, Korea Institute for Advanced Study, Korea

03:00pm - 03:15pm

--- Coffee Break ---

03:15pm - 04:15pm

Slodowy slices and universal Poisson deformations

Yoshinori Namikawa, Kyoto University, Japan

04:15pm - 04:25pm

--- Coffee Break ---

04:25pm - 05:25pm

Entire holomorphic curves and the quasi-Albanese map

Steven Lu, Université du Québec à Montréal, Canada

05:30pm - 09:30pm

Conference Dinner
Venue: Jumbo Seafood Restaurant
East Coast Seafood Centre
Blk 1206 East Coast Parkway #01-07/08
Singapore 449883
* Shuttle bus leaves IMS at 5.30pm for dinner, and leaves dinner place at 9.30pm for IMS.

Thursday, 6 Jan 2011

08:45am - 09:00am

Registration

 

Chair: John Fornaess, University of Michigan, USA

09:00am - 10:00am

Pluripotential theory in a non-archimedean setting

Mattias Jonsson, University of Michigan, USA

10:00am - 10:15am

--- Coffee Break ---

10:15am - 11:15am

On Enriques manifolds

Keiji Oguiso, Osaka University, Japan

11:15am - 11:30am

--- Coffee Break ---

11:30am - 12:30pm

Extensions of multiply twisted pluri-canonical forms

Chin-Lung Wang, National Taiwan University, Taiwan

12:30pm - 02:00pm

--- Lunch ---

Friday, 7 Jan 2011

09:15am - 09:30am

Registration

 

Chair: Mattias Jonsson, University of Michigan, USA

09:30am - 10:30am

Birational Arakelov geometry

Atsushi Moriwaki, Kyoto University, Japan

10:30am - 11:00am

--- Coffee Break ---

11:00am - 12:00nn

Bifurcation currents and equidistribution in parameter space

Romain Dujardin, École Polytechnique, France

12:00nn - 01:30pm

--- Lunch ---

 

Chair: Yujiro Kawamata, University of Tokyo, Japan

01:30pm - 02:30pm

Injectivity radius and gonality of a compact Riemann surface

Jun-Muk Hwang, Korea Institute for Advanced Study, Korea

02:30pm - 03:00pm

--- Coffee Break ---

03:00pm - 04:00pm

Kodaira dimension of irregular varieties

Jungkai Alfred Chen, National Taiwan University, Taiwan

04:00pm - 04:30pm

--- Coffee Break ---

04:30pm - 05:30pm

Kähler cones on non-Kähler surfaces

Akira Fujiki, Osaka University, Japan


Students and researchers who are interested in attending these activities and who do not require financial aid are requested to complete the online registration form.

The following do not need to register:

  • Those invited to participate.
  • Those applying for financial support.


 Venue

 

 

Organizing Committee · Visitors and Participants · Overview · Activities · Venue

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