The Kawaguchi-Silverman conjecture for endomorphisms on affine surfaces.

Abstract: This talk is based on the preprint
https://arxiv.org/pdf/2104.05339.pdf by J. Jia, T. Shibata, J. Xie, and
D.-Q. Zhang.

For a quasi-projective variety $X$ and a finite surjective endomorphism
$f:X \longrightarrow X$ defined over $\overline{\mathbb{Q}}$, the
Kawaguchi-Silverman conjecture (KSC) is a conjecture predicting the
coincidence of the first dynamical degree $d_1(f)$ of $f$ and arithmetic
degree $\alpha_f(P)$ at a point $P \in X$ having Zariski dense $f$-orbit.
This conjecture is verified for certain algebraic varieties, but the case
of an open algebraic variety is hardly verified. Assuming $X$ is a smooth
affine surface such that the logarithmic Kodaira dimension of $X$ is
non-negative, the authors confirm KSC (when $\deg(f) \geq 2$) in this
preprint, which I will present in this talk.