DMM seminar

Speaker: Keiji Oguiso (U Tokyo)
Date (JST): Mon, Dec 21, 2015, 14:00 - 17:00
Place: Seminar Room A
Abstract: We work over C. The main result of this talk is:
Theorem 1.1. There are 4-dimensional smooth projective abelian varieties, hyperkahler
manifolds, Calabi-Yau manifolds, rational manifolds with primitive biregular automorphisms
of positive entropy.
A birational automorphism f ∈ Bir (X) of a projective variety X is called primitive if
there is no dominant rational map ψ : X --→ B with 0 < dim B < dim X preserved by f.
A biregular automorphism of f ∈ Aut (X) of a smooth projective variety X is of positive
entropy, via Gromov-Yomdin theorem, iff the spectral radius of f*| H*(X, Z) is strictly
greater than 1.
Proof of Theorem 1.1 ovbiously consists of the two parts:
(1) How to find candidates (X, f), especially, those which f ∈ Aut (X) of positive entropy.
(2) How to check the primitivity of candidates (X, f).
For (2), we use the notion of (relative) dynamical degrees and their basic properties
due to Dinh-Sibony, Dinh-Nguyen-Troung, which are useful renements of the notion and
properties of entropy of the automorphism due to Gromov-Yomdin.
In the rst half of my talk, After giving a few examples and counterexamples, I would
like to explain back ground materials:
(i) reason why I consider particular classes of manifolds, smooth projective abelian vari-
eties, hyperkahler manifolds, Calabi-Yau manifolds, rational manifolds;
(ii) the notion of (relative) dynamical degrees and their basic properties.
In the second half of my talk, after brief review of relevant results known in lower di-
mension, I would like to give a proof of Theorem 1.1.
Remarks: Break; 15:00 - 15:30