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

Abstract

Let G ⊂ 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 ∂G is of finite type and the type r satisfies r ≤ 2d. The inequality is optimal as equality holds for the egg domains {|z| 2 + |w| 2s < 1}, s ∈ Z+, by D'Angelo's explicit formula for their Bergman kernels. Our results imply, in particular, that a smoothly bounded pseudoconvex domain G ⊂ 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 p q , reduced to lowest degrees, then its rational degree max{deg p, deg q} ≥ 6. Equality is achieved if and only if G is biholomorphic to the unit ball by a complex affine transformation of C 2 .