Algebraic Bergman kernels and finite type domains in ${\mathbb{C}}^2$

Let $G\subset {\mathbb{C}}^2$ be a smoothly bounded pseudoconvex domain and assume that the Bergman kernel of $G$ is algebraic of degree $d$. We show that the boundary $\mathrm{\partial }G$ is of finite type and the type $r$ satisfies $r\leq 2d$\mathrm{.}Theinequalityisoptimalasequalityholdsfortheeggdomains${|z|^2+|w|^{2s}<1},$ $s\in {\mathbb{Z}}_{+}$, by D’Angelo’s explicit formula for their Bergman kernels. Our results imply, in particular, that a smoothly bounded pseudoconvex domain $G\subset {\mathbb{C}}^2$ cannot have rational Bergman kernel unless it is strongly pseudoconvex and biholomorphic to the unit ball by a rational map. Furthermore, we show that if the Bergman kernel of $G$ is rational of the form $\frac{p}{q}$, reduced to lowest degrees, then its rational degree $\max \{\text{deg }p,\text{deg }q\}\ge 6$. Equality is achieved if and only if $G$ is biholomorphic to the unit ball by a complex affine transformation of ${\mathbb{C}}^2$.