Topological entropy, Salem numbers and automorphisms of complex surfaces
Speaker: Keiji Oguiso (The University of Tokyo)

Time: 14h, Monday, October 29, 2018
Location: Room 612, Building A6, Institute of Mathematics
Abstract: In this talk, I would like to explain some unexpected beautiful relations among algebraic geometry of compact connected complex K"ahler surface automorphisms, complex dynamics and number theory, after works of Cantat, Dinh-Sibony, McMullen and my joint works with H'el`ene Esnault and Xun Yu.

Complexity of behaviours of orbits of points under iterations of selfmap is the central object in complex dymanics. The topological entropy, which is a non-negative real number, is the most fundamental quantity to measure the complexity.

Though the topological entropy is highly transcendental in nature, the topological entropies of comlex K"ahler surface automorphisms, if strictly positive, turn out to be very algebraic; the natural logarithm of very interesting special algebriac integers, called Salem numbers. Also, the classes of compact K"ahler surfaces with automorphisms of
strictly positive entropy are very restrictive ones, namely, rational surfaces (i.e., a surface bimeromorphic to the complex projective plane) and surfaces bimeromorphic to 2 dimensional complex tori, K3 surfaces or Enriques surfaces.
After briefly recalling these facts, I would like to explain the following (somewhat unexpected) applications:

  1. Counter-example of Kodaira problem on algebraic approximation of compact K"ahler manifolds by smooth projective varieties;
  2. Existence of non-liftable automorphisms of supersingular K3 surfaces (in positive characteristic) to characteristic zero;
  3. Determination of minimal strictly positive entropy of automorphisms for each possible classes of surfaces, rational surfaces, 2 dimensional complex tori, K3 surfaces and Enriques surfaces. Especially the determination for Enriques surface automorphisms was difficult remaining problems, and settled quite recently by my joint work with Xun Yu.