Holomorphic versus algebraic equivalence for deformations of real-algebraic Cauchy–Riemann manifolds

Abstract

Since Poincare’s celebrated paper [19] published in 1907, there has been a growing literature concerned with the equivalence problem for real submanifolds in complex space (see e.g., [4, 6, 7, 11, 13, 14, 22] for some recent works as well as the references therein). One interesting phenomenon, observed by Webster for biholomorphisms of Levi nondegenerate hypersurfaces [23], is that the biholomorphic equivalence of some types of real-algebraic submanifolds of a complex space implies their algebraic equivalence. In this paper, we show that this very phenomenon holds for algebraic deformations of germs of minimal holomorphically nondegenerate realalgebraic CR submanifolds in complex space. Let us recall that a germ of a real-algebraic CR submanifold (M,p) ⊂ (Cn, p) is minimal if there exists no proper CR submanifold N ⊂ M through p of the same CR dimension as M . It is holomorphically nondegenerate if there exists no nontrivial holomorphic vector field tangent to M near p (see [21]). An algebraic deformation of (M,p) is a real-algebraic family of germs at p of real-algebraic CR submanifolds (Ms, p)s∈Rk in Cn, defined for s ∈ Rk near 0, such that M0 = M . We say that two such deformations (Ms, p)s∈Rk and (Nt, p′)t∈Rk are biholomorphically equivalent if there exists a germ of a realanalytic diffeomorphism φ : (Rk, 0) → (Rk, 0) and a holomorphic submersion B : (Cz × Cu, (p, 0)) → (Cn, p′) such that z → B(z, s) is a biholomorphism sending (Ms, p) to (Nφ(s), p′) for all s ∈ Rk close to 0. We shall say that such