Keiji Oguiso, On Kawaguchi-Silverman Conjecture for Birational Automorphisms of Irregular Threefolds

This is a joint work in progress with Professors Jungkai-Alfred Chen and Hsueh-Yung Lin.

We study the main open parts of Kawaguchi-Silverman Conjecture (KSC), asserting that for a birational self-map f of a smooth projective variety X defined over K, the arithmetic degree α_f(x) exists and coincides with the first dynamical degree δ_f for any K-point x of X with a well-defined Zariski dense f-orbit. Here K is an algebraic closure of the field of rational numbers. To make KSC meaningful, it is also important to study existence of K-point with Zariski dense f-orbit.

In this talk, after a brief introduction of KSC with known results and some difficulties for non-morphism case, I would like to explain our new progress on KSC and Zariski dense orbit problem especially for irregular threefolds. Our approach is geometric while problems are of arithmetic nature.