abstract: This is a joint work 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 $\alphaf(x)$ exists and coincides with the first dynamical degree $\deltaf$ 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, I would like to explain our new progress on KSC, especially for varieties with larger irregularity and irregular threefolds, together with Zariski dense orbit problem with explicit examples. Our approach is geometric while the original problems are of arithmetic nature.