communications in analysis and geometry Volume 18, Number 5, 891–925, 2010 Holomorphic versus algebraic equivalence for deformations of real-algebraic Cauchy–Riemann manifolds Bernhard Lamel and Nordine Mir We consider (small) algebraic deformations of germs of real- algebraic Cauchy–Riemann submanifolds in complex space and study the biholomorphic equivalence problem for such deforma- tions. We show that two algebraic deformations of minimal holomorphically nondegenerate real-algebraic CR submanifolds are holomorphically equivalent if and only if they are algebraically equivalent. 1. Introduction Since Poincare's celebrated paper [19] published in 1907, there has been a growing literature concerned with the equivalence problem for real submani- folds 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 subman- ifolds 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 real- algebraic 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 .
- real
- every integer
- holomorphic versus algebraic equivalence
- holomorphically nondegenerate
- algebraic cr
- general deformations
- minimal real
- nontrivial holomorphic