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Higher Dimensional Algebraic Geometry, Holomorphic Dynamics and Their Interactions
(3 - 28 January 2017)

 

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

 

 Organizing Committee

 

Co-chairs

 Visitors and Participants

 

 

 Overview


In recent years we have seen great breakthroughs in the classification theory of higher dimensional compact algebraic varieties and complex manifolds. The seminal results are the proofs of finite generations of canonical rings of algebraic varieties by Caucher Birkar – Paolo Cascini – Christopher D. Hacon – James McKernan [BCHM06] using the algebraic method and Yum-Tong Siu [Siu08] using the analytic method. These results have profound influence on many areas of mathematics – including the study of higher dimensional dynamics and number theoretical dynamics. The interactions of algebraic geometry and the study of these dynamics is exactly the main theme of this program.

Here we recall the main conjectures in birational geometry which are still open.


Existence of minimal model conjecture: Every non-uniruled variety X is birational to a variety X’ which is a minimal variety, i.e., the cotangent line bundle of X’ is numerically effective.

 

Abundance conjecture: For every minimal variety X, some pluri-canonical system of X is base point free. Hence it defines a holomorphic fibration X -> Y where a very general fibre of the fibration has Kodaira dimension zero.


The consideration of three typical fibrations: Iitaka fibration, Albanese map and Maximal rationally connected fibrations reduces the classification of algebraic varieties to the following three building blocks:

 

1) (RC) Rationally connected varieties

2) (GT) Varieties of general type

3) (K0) Varieties of vanishing Kodaira dimension and irregularity


Now let us turn our attention to the study of holomorphic dynamics. We are interested in the symmetries f on a variety X. It is natural that one tries to apply the above geometric machineries to the study of these automorphisms or endomorphisms f of X, or even birational maps or rational self-maps f of X. For the Iitaka fibration I: X -> I(X), the albanese map albX: X -> Alb(X), and the maximal rationally connected (MRC) fibration r : X -> r(X), we can respectively reduce to the study of fibres by classical results of Pierre Deligne – Iku Nakamura - Kenji Ueno, using the universal properties of the Albanese map, and taking a special Chow reduction MRC in the sense of N. Nakayama. As in the geometric case, for the study of dynamics of space symmetries, the building blocks of symmetries are those on the three types of varieties (RC), (GT) and (K0) above.


However, here we encounter the obstacle from the dynamical side. On the one hand, from geometric side, for type (RC) rationally connected varieties X or type (K0) varieties X of vanishing Kodaira dimension and irregularity, we may apply the minimal model program by birational divisorial or flip contractions X -> X’ to arrive at some variety X’ which either has a Fano fibraiton X’ -> Y with general fibres F being Fano varieties, or has trivial first Chern class and irregularity. Except for the unsolved two conjectures above, the divisorial or flip contractions may not be equivariant w.r.t. a given symmetry f. Namely, we have:


Problem. How to make the minimal model program equivariant w.r.t. an automorphism?

From the above discussion, we see the close interaction of dynamics study and algebraic geometry. This is exactly the main theme of the proposal. Just to add: such interaction is made possible also because of the fundamental work of Mikhail Gromov [Gr77] and Yosef Yomdin [Yo87] saying that the topological entropy of a holomorphic self-map f on a compact Kähler variety X is just the logarithm of the spectral radius of the action of f on the full cohomology group of X. This latter result has been generalized to the case of meromorphic self-map f in the work of Tien-Cuong Dinh and Nessim Sibony [DS05].

Apart from the interaction of dynamics study and algebraic geometry, there is also the dynamics with number theoretic flavors. There are versions of Dynamical Manin-Mumford conjecture asserting some relation between subvarieties Y of a variety X with dense set of periodic points under the action of a symmetry f on X and the f-periodicity of Y itself. This is to generalize the classical result of Michel Raynaud [Ray83] on complex tori to a general variety.

In summary, the program is to discuss and update on the progress towards the two conjectures and one problem highlighted above.

 Activities


 

Please note the following public holiday.
- 28 Jan 2017, Chinese New Year

 

 Venue

 


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

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