Institute for Mathematical Sciences Event Archive
Online registration form Enquiries |
|
Workshop on Algebraic Geometry, Complex Dynamics and their Interaction
(4 - 7 January 2011)
Organizing Committee · Visitors and Participants · Overview · Activities · Venue
- De-Qi Zhang (National University of Singapore)
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.
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 |
|
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 |
|
11:15am - 11:30am |
--- Coffee Break --- |
11:30am - 12:30pm |
|
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 |
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 |
|
11:15am - 11:30am |
--- Coffee Break --- |
11:30am - 12:30pm |
|
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 |
Atsushi Moriwaki, Kyoto University, Japan |
10:30am - 11:00am |
--- Coffee Break --- |
11:00am - 12:00nn |
Bifurcation currents and equidistribution in parameter space |
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.
Organizing Committee · Visitors and Participants · Overview · Activities · Venue
