OAI.Release061.main

The theorem states that, for a natural number n, take V a complex affine algebraic set in ℂⁿ (the common zero set of some family of complex polynomials) and U a subset of V that is open in V's subspace topology, connected, semialgebraic, and bounded. Semialgebraic here means U lies in the smallest class of subsets of ℂⁿ, viewed as ℝ²ⁿ via real and imaginary parts, that contains the zero sets and strict positivity sets of real polynomials in those coordinates and is closed under complement and binary union. Let Γ be a discrete topological group acting on U with a proper action such that the orbit space is compact, and suppose every element γ acts by a holomorphic map, meaning that near each point of U the map p ↦ γ·p agrees on U with a complex-analytic map defined on an open neighborhood in ℂⁿ. Then U is smooth, meaning each point has a neighborhood W in U, open in U, that is biholomorphic (a homeomorphism that is holomorphic in both directions in the same local-analytic sense) to an open subset of some ℂᵐ. Moreover, there exist m and a set D in ℂᵐ that is a bounded symmetric domain, meaning D is open, connected and bounded and each point p of D has a biholomorphic involution of D fixing p and having p as an isolated fixed point, such that U is biholomorphic to D.