Finite Type Conditions on Reinhardt Domains

In this paper we prove that, if p is a boundary point of a smoothly bounded pseudoconvex Reinhardt domain in Cn, then the variety type at p is identical to the regular type. In this paper we study the finite type conditions on pseudoconvex Reinhardt domain. We prove that, if p is a boundary point of a smoothly bounded pseudoconvex Reinhardt domain in C, then the variety type at p is identical to the regular type. In a forthcoming paper, we will study the biholomorphically invariant objects (e.g., the Bergman kernel and metric, the Kobayashi and Carathéodory metrics) on a pseudoconvex Reinhardt domain of finite type. We first recall some definitions. A domain Ω ⊂ C is Reinhardt if (ez1, . . . , e zn) ∈ Ω whenever (z1, . . . , zn) ∈ Ω and 0 ≤ θj ≤ 2π, 1 ≤ j ≤ n. Denote Zj = {(z1, . . . , zn) ∈ C ; zj = 0}, for j = 1, . . . , n. Let Z = ⋃n j=1 Zj . Define L : C n \ Z → R by L(z1, . . . , zn) = (log |z1|, . . . , log |zn|) and L(z1, . . . , zn) = (log z1, . . . , log zn). where, in the second case, the logarithm takes the principle branch, and L is defined locally near every point of C \ Z. 1991 Mathematics Subject Classification. Primary 32F25, Secondary 52A50.