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36 Pages
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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


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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 Poincar´e’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) isminimalif there exists no proper CR submanifoldNMthroughpof the same CR dimension asM. It isholomorphically nondegenerateif there exists no nontrivial holomorphic vector field tangent toMnearp(see [21]). An algebraic deformation of (M, p) is a real-algebraic family of germs atp of real-algebraic CR submanifolds (Ms, p)sRkinCn, defined forsRknear 0, such thatM0=M. We say that two such deformations (Ms, p)sRkand (Nt, p)tRkarebiholomorphically equivalentthere exists a germ of a real-if analytic diffeomorphismϕ: (Rk,0)(Rk,0) and a holomorphic submersion B: (Czn×Cku,(p,0))(Cn, p) such thatz→B(z, s) is a biholomorphism sending (Ms, p) to (Nϕ(s), p) for allsRkclose to 0. We shall say that such
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a pair (B, ϕ) is a biholomorphism between the two deformations1. We also say that they arealgebraically equivalentif one can chooseϕandBto be furthermore algebraic. Our main result is as follows.
Theorem 1.1.Two algebraic deformations of minimal holomorphically nondegenerate real-algebraic CR submanifolds ofCnare algebraically equiv-alent if and only if they are biholomorphically equivalent.
For a completely trivial deformation (i.e.,k= 0), Theorem 1.1 is actu-ally a consequence of the algebraicity theorem of Baouendiet al.[2], where they prove that every local biholomorphism sending holomorphically nonde-generate and minimal real-algebraic generic submanifolds ofCnmust neces-sarily be algebraic. This is not necessarily true for biholomorphisms between deformations, even for constant ones, as the following example shows. Example 1.1.Consider the Lewy hypersurfaceMinC(2z,w)defined by Imw=|z|2, and consider the trivial deformationMs=MforsRk. A biholomorphic map of (Ms,0)sRkto itself which is not algebraic is e.g., given byϕ(s) =s,B(z, w, s) = (esz, e2sw).
The main point of this example is that one cannot expect a biholomor-phism between two deformations to be algebraic. However, in Example 1.1, all the “fibers”Msof the deformations are the same. It is not difficult to show, by using the mentioned result of [2], that the conclusion of Theo-rem 1.1 holds when all the fibers of the deformation are algebraically equiv-alent. One approach which has been successful for more general deformations is to approximate a given biholomorphism by algebraic ones; Theorem 1.1 is a consequence of such an approximation statement. Theorem 1.2.Let(Ms, p)sRkand(Nt, p)tRkbe algebraic deformations of real-algebraic holomorphically nondegenerate minimal CR submanifolds ofCn, and assume that(B, ϕ)is a biholomorphism between(Ms, p)sRkand (Nt, p)tRk. Then for every integer >0there exists an algebraic biholo-morphism(B, ϕ)between(Ms, p)sRkand(Nt, p)tRkwhich agrees with (B, ϕ)up to orderat(p,0).
Under the stronger hypothesis that (M0, p) is “finitely nondegenerate”, Theorem 1.2 was proved by Baouendiet al.(see [7]). However, the weakening of the nondegeneracy assumption makes it impossible to use their methods.
1By slight abuse of language, we shall always identify the mapϕ: (Rk,0)(Rk,0) with its complexification from (Ck,0) to (Ck,0).
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Let us restate our results in a more geometric fashion. For this, we use the following notation. We say that two germs of real-algebraic CR subman-ifolds (M, p) and (M, p) ofCNare biholomorphically equivalent, and write N (M, p)h(M, p) if there exists a germ of a biholomorphismH: (C, p)(CN, p) and a neighborhoodUofpinCNsuch thatH(MU)M(we shall abbreviate this by writingH(M)M). We say that (M, p) and (M, p) are algebraically equivalent, and write (M, p)a(M, p), if there exists such a biholomorphism which is furthermore algebraic. Let us recall that ifMCNis a real-algebraic CR submanifold, for everyqMthere exists a unique germ of a real-algebraic submanifoldWq throughqwith the property that every (small) piecewise differentiable curve starting atq, whose tangent vectors are in the complex tangent space, has its image contained inWq(see [2]).Wqis referred to as thelocal CR orbit ofq. We shall say that (M, p) hasconstant orbit dimensionif dimWqis constant forqclose byp. The geometric counterpart of Theorem 1.1 can now be stated as follows.
Theorem 1.3.Let(M, p)be a germ of a holomorphically nondegenerate real-algebraic CR submanifold, which is in addition of constant orbit dimen-sion. Assume(M, p)is a germ of a real-algebraic submanifold ofCNfor which(M, p)h(M, p). Then(M, p)a(M, p). Also Theorem 1.3 is a consequence of an approximation theorem, which can be stated as follows.
Theorem 1.4.Let(M, p)CNa germ of a real-algebraic CR subman-be ifold which is holomorphically nondegenerate and of constant orbit dimen-sion. Then for every real-algebraic CR submanifoldMCNand every pos-itive integer, ifh: (CN, p)CNis the germ of a biholomorphic map sat-isfyingh(M)M, there exists an algebraic biholomorphismh: (CN, p)CNsatisfyingh(M)Mwhich agrees withhup to orderatp.
Let us briefly recall why a CR submanifold of constant orbit dimension is a deformation of its CR orbits: for a germ of a real-algebraic CR man-ifold (M, p) which is of constant orbit dimension, there exists an integer k∈ {0, . . . , N}and a real-algebraic submersionS: (M, p)(Rk,0) such thatS1(S(q)) =Wq=:MS(q)for allqMnearp. The level sets ofS therefore foliateMbyminimalreal-algebraic CR submanifolds, and thus Mis an algebraic deformation ofM0[7] or Lemma 2.1). In addition,(see a local biholomorphism sending two such real-algebraic CR submanifolds is also a biholomorphism of the associated deformations, since CR orbits of the