[Paper Review] Holomorphic versus algebraic equivalence for deformations of real-algebraic CR manifolds

Bernhard Lamel, Nordine Mir|arXiv (Cornell University)|Jun 16, 2011

Holomorphic and Operator Theory19 references3 citations

TL;DR

This paper establishes that for minimal holomorphically nondegenerate real-algebraic CR submanifolds in complex space, two algebraic deformations are biholomorphically equivalent if and only if they are algebraically equivalent. The key result relies on approximating biholomorphic maps between deformations by algebraic ones up to arbitrary order, leveraging the holomorphic nondegeneracy and minimality of the CR structures to ensure algebraicity of the equivalence.

ABSTRACT

We consider (small) algebraic deformations of germs of real-algebraic CR submanifolds in complex space and study the biholomorphic equivalence problem for such deformations. 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.

Motivation & Objective

• To resolve the equivalence problem for algebraic deformations of real-algebraic CR submanifolds in complex space.
• To determine whether biholomorphic equivalence between such deformations implies algebraic equivalence.
• To extend known results on algebraicity of biholomorphisms to the setting of deformations.
• To establish a bridge between holomorphic and algebraic equivalence in CR geometry under minimal and nondegeneracy conditions.

Proposed method

• Use the concept of algebraic deformations: real-algebraic families of germs of CR submanifolds parameterized by a real parameter space.
• Define biholomorphic equivalence via a holomorphic submersion and a real-analytic diffeomorphism of the parameter space.
• Define algebraic equivalence by requiring both the diffeomorphism and the submersion to be algebraic.
• Prove that any biholomorphic equivalence between such deformations can be approximated by algebraic equivalences up to any finite order.
• Leverage the holomorphic nondegeneracy and minimality of the CR submanifolds to ensure the existence of such approximations.
• Apply the algebraicity theorem of Baouendi-Ebenfelt-Rothschild in the non-degenerate case and extend it to deformations via jet approximation.

Experimental results

Research questions

• RQ1 Under what conditions does biholomorphic equivalence between algebraic deformations of CR submanifolds imply algebraic equivalence?
• RQ2 Can a biholomorphic map between two algebraic deformations of minimal holomorphically nondegenerate CR manifolds be approximated by algebraic maps?
• RQ3 Does the holomorphic nondegeneracy and minimality of the CR structure force the equivalence to be algebraic?
• RQ4 Is the algebraicity of the equivalence preserved under deformation, even when the fibers are not fixed?
• RQ5 What is the role of CR orbits and orbit dimension in determining the algebraicity of the equivalence?

Key findings

• Two algebraic deformations of minimal holomorphically nondegenerate real-algebraic CR submanifolds are biholomorphically equivalent if and only if they are algebraically equivalent.
• For any integer ℓ > 0, a biholomorphic equivalence between such deformations can be approximated by an algebraic biholomorphism up to order ℓ at the base point.
• The approximation result holds even when the fibers of the deformation are not algebraically equivalent, overcoming limitations of prior methods.
• The proof relies on the fact that holomorphic vector fields tangent to the submanifold must vanish under nondegeneracy, forcing the map to be independent of certain variables.
• The result extends the classical algebraicity theorem of Baouendi-Ebenfelt-Rothschild to the deformation setting.
• The conclusion holds even when the CR manifold is not generic, by reducing to the generic case via algebraic decomposition.