HOLOMORPHIC CORRESPONDENCES BETWEEN CR MANIFOLDS.

C. DENSON HILL AND RASUL SHAFIKOV

1. Introduction.

One of the interesting phenomena in several complex variables is the analytic continuation of a germ of a biholomorphic map f : M → M′, defined at a point p ∈ M, where M, M′ ⊂ ℂ^{n} are real-analytic hypersurfaces. Already Poincaré [Po] observed that a biholomorphic map sending an open piece of a unit sphere in ℂ^{2} to another such open piece must be an automorphism of the unit ball. This was proved for ℂ^{n} in [Ta] and [Al]. Clearly such an extension is possible only for n > 1, thus showing the very special nature of CR maps between real-analytic CR manifolds.

Pinchuk [Pi1],[Pi3] proved that the germ of a non-degenerate holomorphic map from a realanalytic strictly pseudoconvex hypersurface M in ℂ^{n} to a unit sphere in ℂ^{N}, N ≥ n, extends as a holomorphic map along any path on M. Webster [We] proved that the germ of a biholomorphic map between real-algebraic Levi non-degenerate hypersurfaces in ℂ^{n} extends as an algebraic map, and also gave sufficient conditions for the map to be rational. In that paper he studied the geometric properties of Segre varieties, which were originally introduced in [Se].

Much attention was devoted to the generalization of Webster’s theorem to the case of different dimensions and higher codimensions. In this situation both M and M′ are assumed to be real algebraic submanifolds or sets; that is, defined by the zero locus of a system of real polynomials. Under certain conditions, it then turns out that a locally defined holomorphic map between such objects must necessarily be algebraic. See Baouendi, Ebenfelt and Rothschild [BER1], Huang[Hu], Sharipov and Sukhov [SS], Baouendi, Huang and Rothschild [BHR], Coupet, Meylan and Sukhov [CMS], Zaitsev [Za], Merker [Me1] and many additional references contained therein. The special case of hyperquadrics was also considered in Tumanov [Tu], Forstnerič [Fo], Sukhov [Su] and other papers.

On the other hand, much less is known if at least one of the submanifolds is not assumed to be real-algebraic. In this case the map need not be algebraic; however, analytic continuation along a real-analytic hypersurface is also possible. Pinchuk [Pi2] proved that a germ of a biholomorphic map f from a strictly pseudoconvex, real-analytic, non-spherical hypersurface M to a compact, strictly pseudoconvex, real-analytic hypersurface M′ ⊂ ℂ^{n} extends holomorphically along any path on M. A similar result was shown in [Sh1] for the case when M is essentially finite, smooth, real-analytic and M′ ⊂ ℂ^{n} is compact, real-algebraic and strictly pseudoconvex. Levi non-degeneracy of the target hypersurface ensures that the extended map is single valued. If the target hypersurface is just assumed to be compact and smooth real-algebraic, the extension in general will be multiple-valued as was proved in [Sh2]. This naturally leads to consideration of holomorphic correspondences, a multiple-valued generalization of holomorphic mappings.

In this paper we study the analytic continuation of germs of holomorphic mappings from smooth real-analytic CR submanifolds of arbitrary codimension to compact smooth real-algebraic generic submanifolds in ℙ^{N} of general codimension. The continuation that we obtain is a holomorphic correspondence from a neighborhood of the submanifold in the pre-image to ℙ^{N}. This is analogous

Date: July 30, 2003.

2 C. DENSON HILL AND RASUL SHAFIKOV

to the algebraicity of the map asserted in the case when both submanifolds are real-algebraic. We also study some applications to maps between pseudoconcave CR submanifolds in ℙ^{n}. It is rather surprising that under certain conditions, a local CR map between such objects turns out to be the restriction of a rational, or even a linear map in ℙ^{n} without the assumption of algebraicity of the submanifold in the pre-image.

We remark that the proofs of the main results of this paper differ significantly from those utilized in [Sh2], where the main construction essentially uses the fact that the Segre varieties have codimension one.

In the next section we present the main results. In Section 3 we give some background on CR manifolds, Segre varieties and holomorphic correspondences. Section 4 contains the proof of the local extension of a holomorphic map as a holomorphic correspondence. In Section 5 we prove the global extension. The last section contains applications of the main theorem to pseudoconcave CR submanifolds in ℙ^{n}.

The second author would like to acknowledge the support of the Max-Planck-Institute for mathematics in Bonn during the preparation of this paper. We would also like to thank S. Popescu and S. Lu for a number of very helpful discussions.

2. Statement of Results.

2.1. Analytic Continuation. By a holomorphic correspondence we mean a complex analytic subset A ⊂ X × X′, where X and X′ are complex manifolds, such that (i) dim A ≡ dim X, and (ii) the projection π : A → X is proper. It is natural to define a multiple valued map F = π′ ∘ π^{−1} : X → X′ associated with A. A holomorphic correspondence F is called a finite correspondence, if F (p) and F^{−1}(p′) are finite sets for any p ∈ X and p′ ∈ X′. We say that F splits at a point q ∈ X if there exist a neighborhood U_{q} of q and an integer k such that F |_{U_{q}} is just the union of k holomorphic maps F^{j} : U_{q} → X′, j = 1, …, k.

Recall that if M is an abstract paracompact real analytic CR manifold of type (m, d), then by [AF] there exists a complex manifold X of dimension n = m + d such that M can be generically embedded into X as a CR submanifold.

Consider the following situation: M is a smooth real-analytic minimal CR manifold of type (m, d), m, d > 0; M′ ⊂ ℙ^{N} is a compact smooth real-algebraic essentially finite generic submanifold of type (m, d′), d′ > 0. The main results of the paper are the following.

Theorem 2.1. Suppose that ω ⊂ M is a relatively compact connected open subset, and f : ω → M′ is a real-analytic CR map such that df|_{H_{p}} : H_{p}M → H_{f(p)}M′ is an isomorphism between the holomorphic tangent spaces for almost all p ∈ ω. Let q ∈ ∂ω. Then there exists a neighborhood U_{q} ⊂ X of q such that f extends to U_{q} as a holomorphic correspondence F : U_{q} → ℙ^{N} with F (U_{q} ∩ M) ⊂ M′.

We remark that in Theorem 2.1 we may assume that f : ω → M′ is just a smooth CR mapping, since by the result in [Da], f is in fact real-analytic. Also note that we do not claim in the above theorem that the extended correspondence near q is finite.

Theorem 2.2. Assume in addition that M is essentially finite and d′ ≥ d. Let p ∈ M, U_{p} ⊂ X be a neighborhood of p, and let f : U_{p} → ℙ^{N} be a holomorphic mapping of maximal rank such that f(U_{p} ∩ M) ⊂ M′. Suppose that M_{1} is a relatively compact simply-connected open subset of M containing p. Then

(1) There exists a neighborhood U ⊂ X of M_{1} such that f extends as a finite holomorphic correspondence F : U → ℙ^{N} with F (U ∩ M_{1}) ⊂ M′.

HOLOMORPHIC CORRESPONDENCES BETWEEN CR MANIFOLDS. 3

(2) F splits into holomorphic mappings at every point q ∈ M_{1} \ F^{−1}(Σ′), where Σ′ is the set of points on M′ near which the Segre map is not locally injective.

To illustrate the conclusions of Theorem 2.2 we consider a simple example. Let p and q be coprime positive integers, and M and M′ be the CR hypersurfaces in ℙ^{n} given in homogeneous coordinates by

(1)  M = {z ∈ ℙ^{n} : |z_{0}|^{2p} + ⋯ + |z_{k}|^{2p} − |z_{k+1}|^{2p} − ⋯ − |z_{n}|^{2p} = 0},
(2)  M′ = {z′ ∈ ℙ^{n} : |z′_{0}|^{2q} + ⋯ + |z′_{k}|^{2q} − |z′_{k+1}|^{2q} − ⋯ − |z′_{n}|^{2q} = 0},

with k < n. The finite holomorphic correspondence F (z) = (z_{0}^{p/q}, …, z_{n}^{p/q}) maps M to M′, is p^{n} : q^{n} valued, and F splits into q^{n} holomorphic mappings outside the branching locus of F. If we choose a point p ∈ M outside the branching locus, and if at p we choose the germ of one of the q^{n} branches of F, then Theorem 2.2 reproduces the entire correspondence F.

Furthermore, we obtain the following.

Theorem 2.3. Let M be a simply-connected compact smooth real-analytic Levi non-degenerate CR manifold of type (n − 1, 1), n > 1. Let M′ be the hyperquadric in ℙ^{n} given in homogeneous coordinates by

(3)  |z_{0}|^{2} + |z_{1}|^{2} + ⋯ + |z_{k}|^{2} − |z_{k+1}|^{2} − ⋯ − |z_{n}|^{2} = 0.

Suppose that ω is a connected open set in M, and f : ω → M′ is a CR map that is a local homeomorphism. Then M and M′ are globally CR equivalent; hence M has a CR embedding as a hypersurface in ℙ^{n}. In the special case where 0 < k < n − 1, and M is apriori a CR submanifold of ℙ^{n}, then f is the restriction to M of a linear automorphism of ℙ^{n}.

Note that Theorem 2.3 includes both the case when M′ is a sphere in ℂ^{n}, which was proved before in [Pi1], and the case when M′ is a compact pseudoconcave hyperquadric. If M is not assumed to be simply-connected, then f in general may not extend to a global map from M to M′ as examples in [BS] show for the case when M′ is a sphere.

Theorems 2.1, 2.2 and 2.3 can be considered as a generalization of the results on algebraicity of a local CR map between real-algebraic submanifolds. Non-algebraic holomorphic correspondences can be easily constructed from the examples considered in [BS].

Our proof of Theorems 2.1 and 2.2 is based on the technique of Segre varieties. As it was mentioned in the introduction, Webster was the the first to use Segre varieties in the context of holomorphic mappings. His ideas were further developed in [DW], [DF2], [DFY], [DP1] and other papers. Our main construction is based on the technique of these papers. A somewhat different approach was developed in [BJT], [BR] and [BER1]. We rely on the characterization of minimality in terms of Segre sets proved in [BER1].

Let us briefly explain the idea of the main construction. Let p = 0 ∈ M, and let U_{0} be a small neighborhood of the origin, where the map f is defined. The first step is to show that f extends as a holomorphic correspondence F_{1} to a neighborhood U_{1} of Q_{0}, the Segre variety of the origin. This can be understood as follows. If w ∈ Q_{0}, then 0 ∈ Q_{w}, and f(U_{0} ∩ Q_{w}) is a complex subvariety in the target space which passes through f(0). If f(U_{0} ∩ Q_{w}) ⊂ Q′_{w′} for some w′, then we set F_{1}(w) = w′. For w close to the origin, f(Q_{w} ∩ U_{0}) ⊂ Q′_{f(w)} by the invariance property of Segre varieties, and thus the extension defined in this way agrees with f on U_{0}. For essentially finite manifolds, if w′ is sufficiently close to M′, then there exists a finite number of points which have the same Segre variety as w′. Therefore in general F_{1} may not be single valued. The graph of F_{1} can be described as

(4)  A_{1} = {(w, w′) ∈ U_{1} × ℙ^{N} : f(Q_{w} ∩ U_{0}) ⊂ Q′_{w′}}.

4 C. DENSON HILL AND RASUL SHAFIKOV

The next step of the proof is to define inductively the analytic continuation of f to bigger sets. This can be achieved using Segre sets (see the next section for definitions). Briefly, once the analytic continuation of f as a correspondence F_{1} is established, we can use a similar construction to define the extension of F_{1} along the Segre varieties of points on Q_{0}. Let

(5)  A_{2} = {(w, w′) ∈ U_{2} × ℙ^{N} : F_{1}(Q_{w} ∩ U_{1}) ⊂ Q′_{w′}},

where U_{2} is a neighborhood of Q_{0}^{2} = ∪_{w∈Q_{0}}Q_{w}. We note that for w′ close to M′ the condition for (w, w′) in (5) roughly speaking means that all branches of F_{1} map Q_{w} into Q′_{w′}. We can repeat this process for A_{3}, using the extension A_{2}, etc.

Finally, the crucial observation is that after a certain number of iterations, we obtain an extension of f to a neighborhood of the origin, which is independent of the choice of U_{0}, where f was originally defined. This follows from the minimality of M and the property of Segre sets proved in [BER1]. Furthermore, for compact minimal manifolds such neighborhoods can be chosen of uniform size for all points on M, which allows us to extend f to any simply-connected relatively compact subset of M.

2.2. Pseudoconcave CR manifolds. Our main application of Theorem 2.2 concerns pseudoconcave CR submanifolds embedded into projective spaces. We recall that a CR manifold is called pseudoconcave, if at each point its Levi form has at least one positive and at least one negative eigenvalue in every characteristic conormal direction.

Theorem 2.4. Consider a connected C^{∞}-smooth compact pseudoconcave CR submanifold M of ℙ^{n}, having type (m, d) with m, d > 0.

(a) Let f : M → ℙ^{N} be a continuous CR map. Then f is the restriction to M of a rational map F : ℙ^{n} → ℙ^{N}.

(b) Assume n = m + d, so that M is generic in ℙ^{n}. Let f : M → ℙ^{n} be a CR map which is a local diffeomorphism onto f(M). Then f is the restriction of a linear automorphism of ℙ^{n}.

For a locally biholomorphic map from a compact smooth pseudoconcave hypersurface in ℙ^{n} to ℙ^{n} part (b) was first shown in [Iv]. Our proof of Theorem 2.4 is based on the result of [HN5], where it is shown that any CR meromorphic function on M is necessarily rational.

Combining Theorem 2.4 with Theorem 2.2, we obtain the following results.

Theorem 2.5. Let M ⊂ ℙ^{n} (resp. M′ ⊂ ℙ^{N}) be a compact smooth real-analytic pseudoconcave essentially finite CR submanifold of type (m, d) (resp. (m, d′)), m, d, d′ > 0 and d′ ≥ d. Let M be simply connected and let M′ ⊂ ℙ^{N} be generic real-algebraic, and such that the Segre map associated with M′ is locally injective. Let p ∈ M, U_{p} be a neighborhood of p in ℙ^{n}, and let f : U_{p} ∩ M → M′ be a germ of a smooth CR map of maximal rank. Then

(a) f is the restriction of a rational map F : ℙ^{n} → ℙ^{N};

(b) If n = N = m + d and the Segre map associated with M is locally injective, then f is the restriction of a linear automorphism of ℙ^{n}.

If M, M′ ⊂ ℙ^{n} are both of hypersurface type (i.e. d = d′ = 1), then f may be assumed to be just a local CR homeomorphism, since in this case f extends smoothly to a neighborhood of p, and by [HN2], the Jacobian of f does not vanish at p.

Corollary 2.6. If a real-analytic submanifold M, satisfying the conditions of Theorem 2.5, is locally CR equivalent to a real-algebraic CR submanifold M′ satisfying Theorem 2.5, then M is necessarily real-algebraic.

HOLOMORPHIC CORRESPONDENCES BETWEEN CR MANIFOLDS. 5

Remark. The above results concerning pseudoconcave CR manifolds hold also under a weaker assumption on M and M′, namely, instead of pseudoconcavity it is enough to assume that M and M′ satisfy the so-called Property E. For a generic M this means that for each p ∈ M every local CR function defined near p extends to a holomorphic function in a full neighborhood of p in the ambient space. For details see [HN5].

3. CR manifolds, Segre varieties and holomorphic correspondences.

An abstract smooth CR manifold of type (m, d) consists of a connected smooth paracompact manifold M of dimension 2m + d, a smooth subbundle HM of T M of rank 2m, which is called the holomorphic tangent space of M, and a smooth complex structure J on the fibers of HM. Let T^{0,1}M be the complex subbundle of the complexification ℂHM of HM, which corresponds to the −i eigenspace of J:

(6)  T^{0,1}M = {Y + iJY |Y ∈ HM}.

We also require that the formal integrability condition

(7)  [C^{∞}(M, T^{0,1}M), C^{∞}(M, T^{0,1}M)] ⊂ C^{∞}(M, T^{0,1}M)

holds. We call m the CR dimension of M and d the CR codimension. M is called minimal at p ∈ M, if there exists no local CR manifold N ⊂ M passing through p having CR dimension m, but strictly smaller real dimension.

The characteristic bundle H^{0}M is defined to be the annihilator of HM in T^{∗}M. Its purpose is to parametrize the Levi form, which for every p ∈ M is defined for v ∈ H_{p}^{0}M and Y ∈ H_{p}M by

(8)  L(v; Y) = dṽ(Y, JY) = ⟨v, [JỸ , Ỹ]⟩,

where ṽ ∈ C^{∞}(M, H^{0}M) and Ỹ ∈ (M, HM) are smooth extensions of v and Y. For each fixed v it is a Hermitian quadratic form for the complex structure J_{p} on H_{p}M. A CR manifold M is said to be pseudoconcave if the Levi from L(v, ·) has at least one negative and one positive eigenvalue for every p ∈ M and every nonzero v ∈ H_{p}^{0}M.

Let M and M′ be two abstract smooth CR manifolds, with holomorphic tangent spaces HM and HM′. A smooth map f : M → M′ is CR if f_{∗}(HM) ⊂ HM′, and f_{∗}(Jw) = J′f_{∗}(w) for every w ∈ HM. A CR embedding of an abstract CR manifold M into a complex manifold X is a CR map which is an embedding. We say that the embedding is generic if the complex dimension of X is (m + d).

If M is a real-analytic CR manifold of type (m, d), then by [AH], M is locally CR embeddable. Furthermore, by [AF] there exists a complex manifold X such that M can be globally generically embedded into X. Consider a connected open set ω on M. When M is real-analytic, and f is a real-analytic CR function in ω, then there is a connected open set Ω_{f} in X, with ω = M ∩ Ω_{f}, and a holomorphic extension f̃ of f to Ω_{f}. When M is a generic C^{∞}-smooth pseudoconcave CR submanifold of X, then there exists a connected open set Ω in X, with ω = M ∩ Ω, such that any CR distribution in ω has a unique holomorphic extension to Ω. See [BP], [NV], [HN3], [HN4].

Most of our considerations of real-analytic CR manifolds will be local, and therefore by the above mentioned results we can assume without loss of generality that M is a generically embedded CR submanifold of some open set in ℂ^{n}, where n = m+d. Note here that a compact pseudoconcave CR manifold cannot be embedded (even non-generically) into any Stein manifold (see e.g. [HN1]). Therefore, for application purposes the main theorem is formulated for the target submanifold embedded into ℙ^{N}.

6 C. DENSON HILL AND RASUL SHAFIKOV

Let M be a generic smooth real-analytic submanifold of ℂ^{n} and let p ∈ M. Then in a sufficiently small neighborhood U of p, M is given by

(9)  M = \{z ∈ U : ρ_{j}(z, \overline{z}) = 0, j = 1, … d\},

where each ρ_{j} is a real-valued real-analytic function and

(10)  \overline{∂}ρ_{1} ∧ ⋯ ∧ \overline{∂}ρ_{d} ≠ 0 on M ∩ U.

We set ρ = (ρ_{1}, ρ_{2}, ..., ρ_{d}). There exists a biholomorphic change of coordinates near p, z = (ξ, ζ) ∈ ℂ^{m} × ℂ^{d} = ℂ^{n} such that in the new coordinates, p = 0, and M is given by

(11)  Im ζ = ϕ(ξ, \overline{ξ}, Re ζ),

where ϕ is a vector-valued real-analytic function with ϕ(0) = 0 and dϕ(0) = 0.

If U is sufficiently small, to every point w ∈ U we can associate to M its so-called Segre variety in U defined as

(12)  Q_{w} = \{z ∈ U : ρ_{j}(z, \overline{w}) = 0, j = 1, …, d\} ,

where ρ_{j}(z, \overline{w}) is the complexification of the defining functions of M. Another important variety associated with the submanifold M and the neighborhood U is

(13)  I_{w} = {z ∈ U : Q_{w} = Q_{z}} .

From the reality condition on the defining functions the following simple but important properties of Segre varieties follow:

(14)  z ∈ Q_{w} ⇔ w ∈ Q_{z},
(15)  z ∈ Q_{z} ⇔ z ∈ M,
(16)  w ∈ M ⇔ I_{w} ⊂ M.

If 0 ∈ M, then from (10) and the implicit mapping theorem, there exist a local change of coordinates near the origin, and a pair of small neighborhoods U and U_{0} of the origin, U ⋐ U_{0}, where U_{0} is given in the product form

(17)  U_{0} = ′U_{0} × Ũ_{0}, ′U_{0} ⊂ ℂ^{m}, Ũ_{0} ⊂ ℂ^{d},

such that for every w ∈ U, the set Q_{w} ∩ U_{0} can be represented as the graph of a holomorphic mapping. That is

(18)  Q_{w} ∩ U_{0} = \{z = (ξ, ζ) ∈ ′U_{0} × Ũ_{0} : ζ = h(ξ, \overline{w})\},

where h(ξ, \overline{w}) is holomorphic in ξ and \overline{w}. Thus Q_{w} is a complex submanifold of U of complex codimension d.

The main use of Segre varieties comes from the fact that they are invariant under biholomorphic mappings. More precisely, given a holomorphic map f : U → U′, sending a generic smooth realanalytic submanifold M to another such submanifold M′, f(p) = p′ implies f(Q_{p} ∩ U) ⊂ Q′_{p′} for p sufficiently close to M. An analogous property holds also for holomorphic correspondences.

The proof of the basic properties of Segre varieties in higher codimensions is similar to the hypersurface case and can be found in [BER2] or [Me2].

A real analytic submanifold M is called essentially finite at p ∈ M, if I_{p} = {p} in a small neighborhood of p. The Segre map is defined by λ : w → Q_{w}. A manifold M being essentially finite now means that the Segre map is finite near M. It can be shown (see e.g. [Me2]) that any generic Levi non-degenerate CR submanifold of ℂ^{n} is essentially finite. Moreover, if M is a compact generic submanifold of ℂ^{n}, then it is automatically essentially finite, since by [DF1], any

HOLOMORPHIC CORRESPONDENCES BETWEEN CR MANIFOLDS. 7

compact real-analytic subset of ℂ^{n} does not contain any non-trivial germs of complex-analytic varieties. See also [BER2].

In [BER1] the authors introduced so-called Segre sets. We briefly recall this construction here. Let M be a generic smooth real-analytic submanifold of ℂ^{n}, 0 ∈ M and let Q_{0} = Q_{0}^{1} be the usual Segre variety of 0 as defined in (12). Define

(19)  Q_{0}^{j} = ⋃_{z∈Q_{0}^{j−1}} Q_{z}, j > 1.

Then

(20)  Q_{0}^{1} ⊂ Q_{0}^{2} ⊂ ⋯ ⊂ Q_{0}^{j} ⊂ · · · .

Indeed, let k be the smallest integer such that Q_{0}^{k} ⊄ Q_{0}^{k+1}. Clearly, k ≥ 2. If z ∈ Q_{0}^{k} \ Q_{0}^{k+1}, then there exists w ∈ Q_{0}^{k−1} such that z ∈ Q_{w}. By assumption, Q_{0}^{k−1} ⊂ Q_{0}^{k}. Therefore w ∈ Q_{0}^{k}, and Q_{w} ⊂ Q_{0}^{k+1}; in particular z ∈ Q_{0}^{k+1}, which is a contradiction.

According to [BER1] (see also [BER2]), there exists an integer j_{0}, 0 < j_{0} < ∞, such that ∪_{j≤j_{0}}Q_{0}^{j} contains a neighborhood of the origin in ℂ^{n}, provided that M is minimal at 0. We define

(21)  Ω_{0} = {z : z ∈ Q_{0}^{j}, j ≤ j_{0}}.

Moreover, if M is compact, or is a relatively compact open set of a bigger minimal submanifold, then there exists ϵ > 0 such that for any point p ∈ M, the neighborhood Ω_{p}, defined as in (21), contains a ball of radius ϵ centered at p.

Suppose now that the manifold M ⊂ ℙ^{n} is connected and defined by real polynomials. Then the Segre varieties associated with M can be defined globally as projective algebraic varieties in ℙ^{n}. Indeed, let M ∩ ℂ^{n} be given as a connected component of the set defined by

(22)  \{z ∈ ℂ^{n} : ρ_{j}(z, \overline{z}) = 0, j = 1, …, d\},

where ρ_{j} are real polynomials. We can projectivize each ρ_{j} to define M in ℙ^{n} in homogeneous coordinates

(23)  ẑ = [ẑ_{0} : ẑ_{1} : ⋯: ẑ_{n}], z_{k} = \frac{ẑ_{k}}{ẑ_{0}} , k = 1, …, n,

as a connected component of the set defined by

(24)  \{ẑ ∈ ℙ^{n} : ρ̂_{j}(ẑ, \overline{ẑ}) = 0\}.

We may define now the polar of M as

(25)  M̂^{c} = {(ẑ, ζ̂) ∈ ℙ^{n} × ℙ^{n} : ρ̂_{j}(ẑ, ζ̂) = 0, j = 1, …, d}.

Then M̂^{c} is a complex algebraic variety in ℙ^{n} × ℙ^{n}. Given τ ∈ ℙ^{n}, we set

(26)  Q̂_{τ} = M̂^{c} ∩ \{(ẑ, ζ̂) ∈ ℙ^{n} × ℙ^{n} : ζ̂ = \overline{τ}\}.

We define the projection of Q̂_{τ} to the first coordinate to be the Segre variety of τ. Alternatively, given (22) one can define the polar as

(27)  M^{c} = {(z, ζ) ∈ ℂ^{n} × ℂ^{n} : ρ_{j}(z, ζ) = 0, j = 1, …, d}.

The submanifold M ∩ ℂ^{n} can be recovered by intersecting M^{c} with the totally real subspace T = \{(z, ζ) ∈ ℂ^{2n} : ζ = \overline{z}\} and taking an appropriate connected component. Given w ∈ ℂ^{n}, we define

(28)  Q_{w}^{c} = M^{c} ∩ \{(z, ζ) ∈ ℂ^{2n} : ζ = \overline{w}\}.

8 C. DENSON HILL AND RASUL SHAFIKOV

The standard Segre variety Q_{w} can now be recovered by projecting Q_{w}^{c} to ℂ_{z}^{n}. The algebraic varieties M^{c} and Q_{w}^{c} can be projectivized which gives us objects geometrically equivalent to (25) and (26). Note that the closure \overline{T} ⊂ ℙ^{2n} of the set T is a smooth submanifold of ℙ^{2n}, and thus M can be identified with a connected component of \overline{M^{c}} ∩ \overline{T} .

We note that condition (10) implies that for w close to M, the Segre variety Q_{w} contains a connected component Q̃_{w} of dimension m. However in general, Q_{w} may have other components, which apriori may even have different dimension (higher than m). For w ∈ M, w ∈ Q̃_{w}, and for w close to M, Q̃_{w} is the component which near w is given by (18).

We will understand essential finiteness of a real algebraic submanifold M ⊂ ℙ^{n} in the sense that for every w ∈ M, the set

(29)  Ĩ_{w} := {z ∈ M : Q̃_{z} = Q̃_{w}}

is finite. In this case we can further show that generically the various Ĩ_{w} have the same number of points. Note that the set Ĩ_{w} is a globally defined object in ℙ^{n}.

Lemma 3.1. Let M be a compact smooth real-algebraic essentially finite generic CR submanifold in ℙ^{n}. Then there exist a neighborhood U ⊂ ℙ^{n} of M and an integer R ≥ 1 such that

(30)  #Ĩ_{w} = #{z ∈ U : Q̃_{z} = Q̃_{w}} = R

for almost all w ∈ U.

Proof. For each p ∈ M choose a product neighborhood U_{p} as in (17). Let U = ⋃_{p∈M} U_{p}. We define the map λ from U to the set of all algebraic varieties in ℙ^{n} of dimension m by letting λ(w) := Q̃_{w}. Then Q̃_{w} depends anti-holomorphically on w. Locally, near almost every w ∈ U, the varieties Q̃_{w} have the same algebraic degree. Since algebraic varieties of positive codimension do not divide U, the Q̃_{w} have fixed degree for almost all w in U. Algebraic varieties of fixed dimension and degree are known to be parametrized by the so-called Chow variety (see e.g. [Ha]), and the parametrization λ is an algebraic map between algebraic varieties.

Let Y = λ(U). Then since M is essentially finite, dim Y = n. It follows (see e.g. [Mu]) that there exists an algebraic variety Z ⊂ Y such that for any q ∈ Y \ Z,

(31)  #λ^{−1}(q) = deg(λ) := R,

where R is a positive integer. From (31) the assertion follows. □

Let X and X′ be complex manifolds and D ⊂ X, D′ ⊂ X′ be open sets. Recall that if A ⊂ D × D′ is a holomorphic correspondence, then π : A → D is proper. A is called a proper holomorphic correspondence, if π′ : A → D′ is also proper. Throughout the paper we will identify the multiple valued map F := π′ ∘ π^{−1} with its graph A, so given a point p ∈ D,

(32)  F (p) = {p′ ∈ D : π′ ∘ π^{−1}(p)}

is a compact subset of D′. Given a complex analytic subset G′ ⊂ D′, F^{−1}(G′) is a complex analytic subset of D. Indeed π′^{−1}(G′) is clearly analytic, and since π is proper, it follows from Remmert’s theorem that π(π′^{−1}(G)) is analytic in D. F is called a finite-valued holomorphic correspondence if F (p) is a finite set for any p ∈ D, and a finite correspondence if in addition F^{−1}(p′) is finite for any p′ ∈ X′. If X′ is a Stein manifold, then any proper holomorphic correspondence F is automatically finite-valued. To see this observe that by the proper embedding theorem, X′ can be viewed as a submanifold of ℂ^{n′} for some n′ > 1, and a compact complex analytic set F (p) must be discrete. Similarly, if X and X′ are both Stein, then any proper holomorphic correspondence is finite.

HOLOMORPHIC CORRESPONDENCES BETWEEN CR MANIFOLDS. 9

Given a finite-valued holomorphic correspondence A ⊂ D×D′, there exists a complex subvariety S ⊂ D (possibly empty) such that for any point p ∈ D\S, there exists a neighborhood U_{p} ⊂ D\S, such that F splits into k holomorphic maps F^{j} : U_{p} → D′, j = 1, …, k, that represent F. The integer k is independent of p, and the F^{j} are called the branches of F.

Given a locally complex analytic set A in X of pure dimension p, we say that A extends analytically to an open set U ⊂ X, A ∩ U ≠ ∅, if there exists a (closed) complex-analytic set A^{∗} in U such that (i) dim A^{∗} ≡ p, (ii) A ∩ U ⊂ A^{∗} and (iii) every irreducible component of A^{∗} has a nonempty intersection with A of dimension p. By the uniqueness theorem for analytic sets such analytic continuation of A is uniquely defined. From this we define the analytic continuation of holomorphic correspondences:

Definition 3.2. Let D ⊂ X and D′ ⊂ X′ be open sets and let A ⊂ D × D′ be a holomorphic correspondence. We say that A extends as a holomorphic correspondence to an open set U ⊂ X, D ∩U ≠ ∅, if there exists an open set U′ ⊂ X′ such that A∩(U ×U′) ≠ ∅, A extends analytically to a set A^{∗} ⊂ U × U′, and π : A^{∗} → U is proper.

If A and A^{∗} are both finite-valued, then A^{∗} may have more branches in D ∩ U than A. The following lemma gives a simple criterion for the extension to have the same number of branches.

Lemma 3.3. Let A^{∗} ⊂ U ×X′ be a finite-valued holomorphic correspondence which is an analytic extension of a finite correspondence A ⊂ D × D′. Suppose that for any z ∈ (D ∩ U),

(33)  #{π^{−1}(z)} = #{π^{∗−1}(z)},

where π : A → D and π^{∗} : A^{∗} → U are the projections. Then A ∪ A^{∗} is a holomorphic correspondence in (D ∪ U) × X′.

The proof is the same as in [Sh2], Lemma 2.

4. Local continuation as a correspondence.

In this section we prove Theorem 2.1. We may assume the following situation: Ω ⊂ ℂ^{n} is a connected open set, Ω ∩ M = ω, f : Ω → ℙ^{N} is a holomorphic map, and f(Ω ∩ M) ⊂ M′. Let q ∈ ∂Ω ∩ M and let U be a neighborhood of q such that for any z ∈ U, the Segre variety Q_{z} is defined in a strictly larger set, and can be represented as in (18). We show that f extends to some neighborhood of q as a holomorphic correspondence. We choose a point a ∈ ω such that df|_{H_{a}M} is an isomorphism, and so close to q that q ∈ Ω_{a}, where Ω_{a} is defined as in (21). Fix some neighborhood U_{a} ⊂ Ω of a. We will show that f|_{U_{a}} extends as a holomorphic correspondence to Ω_{a}, in particular to a neighborhood of q.

Assume for simplicity that a = 0. Choose a small neighborhood U_{0} of the origin and shrink U_{a} in such a way, that for any w ∈ U_{0}, the set Q_{w} ∩ U_{a} is non-empty and connected. Let U′ ⊂ ℙ^{N} be the neighborhood of M′ as in Lemma 3.1. Define

(34)  A_{0} = {(w, w′) ∈ U_{0} × U′ : f(Q_{w} ∩ U_{a}) ⊂ Q′_{w′}}.

Then A_{0} is a complex-analytic subset of U_{0} × U′. In the case when M and M′ are hypersurfaces, this was shown in [Sh2], the proof in the general case is analogous (see also similar constructions later in this section). Note that in (34),

(35)  dim Q_{w} = dim Q′_{w′} = dim f(Q_{w} ∩ U_{a}) = m.

Indeed, since df|_{H_{0}M} is an isomorphism, dim Q_{0} = dim f(Q_{0} ∩ U_{0}). Without loss of generality we may assume that M near 0 and M′ near f(0) are chosen as in (11). Let J_{f}(z) be the Jacobian matrix associated with the map f. Then in the chosen coordinate systems, df|_{H_{0}M} being an

10 C. DENSON HILL AND RASUL SHAFIKOV

isomorphism means that the principal minor of J_{f}(0) of size m × m has a non-zero determinant. The same property also holds for points sufficiently close to the origin. Therefore, after shrinking U_{0}, (35) holds for all w in U_{0}. After further shrinking U_{0} if necessary, we may assume that if (w, w′) ∈ A_{0}, then w′ ∈ Ĩ′_{f(w)}, which is a finite set by the assumption on M′. It follows that dim A_{0} ≡ n. Finally, (16) implies that if U_{0} is sufficiently small, then A_{0} has no limit points on U_{0} × ∂U′, and therefore the natural projection from A_{0} to the first coordinate is proper. This shows that A_{0} defines a holomorphic correspondence. Denote the corresponding multiple valued map by F_{0}.

We shrink the neighborhood U_{0} of the origin, where F_{0} is defined, and choose a ’thin’ neighborhood U_{1} of Q_{0} ∩ U such that for any w ∈ U_{1}, the set Q_{w} ∩ U_{0} is non-empty and connected. Note that from (14), 0 ∈ Q_{w} for any w ∈ Q_{0}. Denote now by A_{0} the graph of F_{0} in U_{0} × ℙ^{N}. We define the set A_{1} as follows:

(36)  A_{1} = {(w, w′) ∈ U_{1} × ℙ^{N} : F_{0}(Q_{w} ∩ U_{0}) ⊂ Q′_{w′}} .

Then A_{1} is a complex-analytic subset of U_{1} × ℙ^{N}. To verify this assertion we prove the following:

1. A_{1} ≠ ∅. Indeed, by the invariance property of Segre varieties, f(Q_{w} ∩ U_{0}) ⊂ Q′_{f(w)} for w sufficiently close to the origin. Then

(37)  F_{0}(Q_{w} ∩ U_{0}) ⊂ Q′_{w′},

where w′ ∈ F_{0}(w). To see this, suppose that z ∈ Q_{w} ∩U_{0} and z′ ∈ F_{0}(z). By construction, F_{0}(Q_{z} ∩ U_{0}) ⊂ Q′_{z′}. If w is sufficiently close to the origin, w ∈ Q_{z} ∩ U_{0}, and therefore w′ ∈ F_{0}(w) ⊂ Q′_{z′}. This implies z′ ∈ Q′_{w′}. And since z ∈ Q_{w} ∩ U_{0} was arbitrary, (37) holds. Moreover, F_{0}(w) = Ĩ′_{w′}, and thus

(38)  A_{0}|_{(U_{1}∩U_{0})×ℙ^{N}} ⊂ A_{1},

in particular A_{1} is non-empty.

2. A_{1} is a complex-analytic set in a neighborhood of any of its points. Indeed, let (w_{0}, w′_{0}) ∈ A_{1}, z_{0} ∈ Q_{w_{0}} ∩ U_{0} be an arbitrary point, and let U_{z_{0}} ⊂ U_{0} be a small neighborhood of z_{0}. Since Q_{w} ∩U_{0} is connected for all w ∈ U_{1}, F_{0}(Q_{w} ∩U_{z_{0}}) ⊂ Q′_{w′} implies F_{0}(Q_{w} ∩U_{0}) ⊂ Q′_{w′}. Therefore in (36) U_{0} can be replaced with U_{z_{0}}. Choose U_{w_{0}} so small that Q_{w} ∩U_{z_{0}} ≠ ∅ for all w ∈ U_{w_{0}}. Since the pre-image of an analytic set under a holomorphic correspondence is an analytic set, S_{w′} := F_{0}^{−1}(Q′_{w′}) is an analytic subset of U_{0}. Let U′_{w′_{0}} be so small that S_{w′} ∩ U_{z_{0}} ≠ ∅ for all w′ ∈ U′_{w′_{0}}. Let S_{w′} near z_{0} be given by

(39)  S_{w′} = \{z ∈ U_{z_{0}} : ϕ_{j}(z, \overline{w}′) = 0, j = 1, …, j̃\},

where the ϕ_{j} depend holomorphically on \overline{w}′. We may assume that U_{z_{0}} is chosen as in (17), and therefore z ∈ Q_{w} simply means z = (ξ, ζ) and ζ = h(ξ, \overline{w}). Then the condition F_{0}(Q_{w} ∩ U_{z_{0}}) ⊂ Q′_{w′} is equivalent to

(40)  ϕ_{j} ((ξ, h(ξ, \overline{w})), \overline{w}′) = 0, ξ ∈ ′U_{z_{0}}, j = 1, …, j̃.

This is an infinite system of holomorphic equations (after conjugation) which defines A_{1} as an analytic set in U_{w_{0}} × U′_{w′_{0}}.

3. A_{1} is closed in U_{1} × ℙ^{N}. Indeed, suppose that (w^{j}, w′^{j}) → (w^{0}, w′^{0}), as j → ∞, where (w^{j}, w′^{j}) ∈ A_{1} and (w^{0}, w′^{0}) ∈ U_{1} × ℙ^{N}. Since Q_{w^{j}} → Q_{w^{0}}, and Q′_{w′^{j}} → Q′_{w′^{0}} as j → ∞, by analyticity F_{0}(Q_{w^{0}} ∩ U_{0}) ⊂ Q′_{w′^{0}}, which implies that (w^{0}, w′^{0}) ∈ A_{1} and thus A_{1} is a closed set.

HOLOMORPHIC CORRESPONDENCES BETWEEN CR MANIFOLDS. 11

It follows from 1–3 that A_{1} is a complex-analytic subset of U_{1} × ℙ^{N}. Let π_{1} : A_{1} → U_{1} and π′_{1} : A_{1} → ℙ^{N} be the coordinate projections. Since ℙ^{N} is compact, π_{1} is proper. We consider only the irreducible components of A_{1} of dimension n which contain A_{0}. The union of all such components we denote again by A_{1}. Thus A_{1} is an analytic continuation of A_{0} as a holomorphic correspondence. Denoting U_{0} ∩ U_{1} again by U_{0}, we may assume from the uniqueness theorem for analytic sets that

(41)  A_{1}|_{U_{0}×ℙ^{N}} = A_{0}.

We set F_{1} := π′_{1} ∘ π_{1}^{−1} : U_{1} → ℙ^{N}. Note that in general, F_{1} may not be finite-valued. However, by the Cartan-Remmert theorem (see e.g. [Lo]) combined with Remmert’s proper mapping theorem, the set of points

(42)  Σ_{1} = {z ∈ U_{1} : dim π_{1}^{−1}(z) > 0}

is a complex subvariety of U_{1}. Since dim A_{1} ≡ n, we have dim Σ_{1} < n, in particular, Σ_{1} does not divide U_{1}, and A_{1}|_{(U_{1}\Σ_{1})×ℙ^{N}} is a finite-valued holomorphic correspondence.

By construction, if (w, w′) ∈ A_{1}, then for any (z, z′) ∈ A_{1} such that z ∈ Q_{w} ∩U_{0}, we necessarily have z′ ∈ Q′_{w′}. We wish to conclude the same for all z in Q_{w} ∩ U_{1}. The difficulty is that in general, Q_{w} ∩ U_{1} may have more than one connected component. To prove the assertion we argue by contradiction, and assume that there exists a point (w^{0}, w′^{0}) ∈ A_{1} and (z, z′) ∈ A_{1}, such that z ∈ Q_{w^{0}} ∩ U_{1}, but z′ ∉ Q′_{w′^{0}}. Connect w^{0} and the origin with a smooth path γ contained in U_{1}. We may assume that w^{0} is the first point on γ for which the desired property does not hold. Without loss of generality we may also assume that for all p ∈ γ between the origin and w^{0} (excluding w^{0}) there exists a small neighborhood U_{p} such that whenever z ∈ U_{p}, all components of Q_{z} ∩ U_{1} are mapped by F_{1} into the same Segre variety. For each point p ∈ γ between the origin and w^{0}, we construct the following set:

(43)  A_{p} = {(w, w′) ∈ U(Q_{p}) × ℙ^{N} : F_{1}(Q_{w} ∩ U_{p}) ⊂ Q′_{w′}} ,

where U_{p} ⊂ U_{1} is a neighborhood of p and U(Q_{p}) is a neighborhood of Q_{p} ∩ U, which are chosen in such a way that U_{p} satisfies the property described above and that for any w ∈ U(Q_{p}), the set Q_{w} ∩ U_{p} is connected. Repeating the argument that was used for A_{1}, one can prove that each A_{p} is a complex analytic set, which defines a holomorphic correspondence. For p = 0, this is just the set A_{1}. Moreover, for any p between the origin and w^{0} we have

(44)  A_{1}|_{(U(Q_{p})∩U_{1})×ℙ^{N}} ⊂ A_{p}|_{(U(Q_{p})∩U_{1})×ℙ^{N}}.

Indeed, suppose that w ∈ U(Q_{p})∩U_{1} and (w, w′) ∈ A_{1}. Let z ∈ Q_{w}∩U_{p} be an arbitrary point, and z′ ∈ F_{1}(z). Then F_{0}(Q_{z} ∩ U_{0}) ⊂ Q′_{z′}. From (41) we have F_{1}(Q_{z} ∩ U_{0}) ⊂ Q′_{z′}. By the assumption on U_{p}, F_{1}(Q_{z} ∩ U_{1}) ⊂ Q′_{z′}, in particular F_{1}(w) ⊂ Q′_{z′}. Therefore, w′ ∈ Q′_{z′}, and z′ ∈ Q′_{w′}. Since z ∈ Q_{w} ∩ U_{p} was arbitrary, it follows that F_{1}(Q_{w} ∩ U_{p}) ⊂ Q′_{w′}. But this means that (w, w′) ∈ A_{p}, and thus (44) holds.

For any p, Q_{p}∩U is a connected set in U(Q_{p}) and therefore is mapped by A_{p} into the same Segre variety. By continuity and from (44) we conclude that F_{1}(Q_{w_{0}} ∩ U_{1}) ⊂ Q′_{w′_{0}}, which contradicts the assumption. Thus for any (w, w′) ∈ A_{1}, if (z, z′) ∈ A_{1} and z ∈ Q_{w} ∩ U_{1}, then z′ ∈ Q′_{w′}.

We now define recursively for j > 1 the following sets:

(45)  A_{j} = {(w, w′) ∈ U_{j} × ℙ^{N} : F_{j−1}(Q_{w} ∩ U_{j−1}) ⊂ Q′_{w′}} .

Here the open set U_{j} is defined as follows. Suppose that the set A_{j−1} ⊂ U_{j−1} × ℙ^{N} is already defined and U_{j−1} ⊂ U is some connected open set. We let U_{j} be the set of points w in U such that Q_{w} ∩U_{j−1} ≠ ∅. Furthermore, after shrinking at each step, if necessary, the sets U_{k} for k < j,

12 C. DENSON HILL AND RASUL SHAFIKOV

we may assume that U_{k−1} ⊂ U_{k} for 1 ≤ k ≤ j. Note that it follows from the construction that Q_{0}^{k} ⊂ U_{k} for 1 ≤ k ≤ j.

We claim that for all j > 0, A_{j} is a complex-analytic subset of U_{j} × ℙ^{N}, which satisfies the following properties:

(i) A_{j}|_{U_{j−1}×ℙ^{N}} = A_{j−1}; (ii) A_{j} defines a holomorphic correspondence F_{j} : U_{j} → ℙ^{N}; (iii) for any (w, w′) ∈ A_{j}, if (z, z′) ∈ A_{j} and z ∈ Q_{w} ∩ U_{j}, then z′ ∈ Q′_{w′}.

Condition (iii) can be understood in the sense that the map F_{j}, associated with A_{j}, sends all connected components of Q_{w} ∩ U_{j} into Q′_{w′} provided that (w, w′) ∈ A_{j}.

The proof is by induction. The case j = 1 is already proved. Suppose that A_{j−1} is as claimed. We show that the set defined by (45) is also a holomorphic correspondence satisfying properties (i)–(iii).

(i) Let w ∈ U_{j−1}, and (w, w′) ∈ A_{j−1}. Then by definition, F_{j−2}(Q_{w} ∩ U_{j−2}) ⊂ Q′_{w′}. From property (i), which by the induction hypothesis holds for F_{j−1}, the correspondences F_{j−2} and F_{j−1} agree in U_{j−2}, and therefore we have

(46)  F_{j−1}(Q_{w} ∩ U_{j−2}) ⊂ Q′_{w′}.

From (iii), F_{j−1} maps all components of Q_{w} ∩ U_{j−1} into the same Segre variety. Therefore (46) implies F_{j−1}(Q_{w} ∩ U_{j−1}) ⊂ Q′_{w′}, which by definition means that (w, w′) ∈ A_{j}. In particular, the set A_{j} is non-empty. Condition (i) for A_{j} will be completely proved, once we know that A_{j} is a complex-analytic set, dim A_{j} ≡ n, and select only the irreducible components of A_{j} which have intersection with A_{j−1} of dimension n.

Proof of (ii). Let (w^{0}, w′^{0}) ∈ A_{j}. If Q_{w} ∩ U_{j−1} is connected for all w sufficiently close to w^{0}, then the proof of the fact that A_{j} is complex-analytic near (w^{0}, w′^{0}) ∈ A_{j} is the same as for A_{1} in Step 2. Let Ũ be the largest connected relatively open subset of U_{j} such that 0 ∈ Ũ and for all z ∈ Ũ, F_{j−1} maps Q_{z} ∩ U_{j−1} into the same Segre variety. From property (iii) for A_{j−1} we have U_{j−1} ⊂ Ũ. Then (45) defines a holomorphic correspondence Ã ⊂ Ũ × ℙ^{N}. The proof is the same as in Step 2 for A_{1}. Denote by F̃ the multiple valued map associated with Ã. By repeating the argument used for A_{1}, we can show that for any w ∈ Ũ, F̃ maps all connected components of Q_{w} ∩ Ũ into the same Segre variety.

For each point p ∈ Ũ we may define now the following set

(47)  A_{p} = {(w, w′) ∈ U(Q_{p}) × ℙ^{N} : F̃(Q_{w} ∩ U_{p}) ⊂ Q′_{w′}},

where the neighborhoods U(Q_{p}) of Q_{p} and U_{p} of p are chosen as in the construction of the set defined by (43). Let F_{p} be the map associated with A_{p}. Clearly, F_{p} coincides with F_{j−1} for p sufficiently close to the origin. We claim that the map F_{p} agrees with F_{j−1} in U(Q_{p})∩U_{j−1} for all p ∈ Ũ. Indeed, suppose that w ∈ U(Q_{p})∩U_{j−1} and (w, w′) ∈ A_{j−1}. Then F_{j−2}(Q_{w} ∩U_{j−2}) ⊂ Q′_{w′}. To prove the assertion we need to show that

(48)  F̃(Q_{w} ∩ U_{p}) ⊂ Q′_{w′}.

Let z ∈ Q_{w} ∩ U_{p} be an arbitrary point, and z′ ∈ F̃(z). Then F_{j−1}(Q_{z} ∩ U_{j−1}) ⊂ Q′_{z′}. Since F̃ and F_{j−1} agree in U_{j−1}, it follows that F̃(Q_{z} ∩ U_{j−1}) ⊂ Q′_{z′}. For z ∈ Ũ, F̃ maps different components of Q_{z} ∩ Ũ into the same Segre varieties, and therefore we have F̃(Q_{z} ∩ Ũ) ⊂ Q′_{z′}. In particular, w′ ∈ Q′_{z′}, which implies z′ ∈ Q′_{w′}. Since z was arbitrary, (48) holds.

HOLOMORPHIC CORRESPONDENCES BETWEEN CR MANIFOLDS. 13

By the construction, F_{p} maps Q_{p} into the same Segre variety for all p ∈ Ũ. By analyticity this means that for any point w in ∂Ũ ∩ U_{j}, F_{j−1} maps Q_{w} ∩ U_{j−1} into the same Segre variety. Therefore, Ũ = U_{j}. We choose only irreducible components of A_{j} of dimension n which contain A_{j−1}. This proves (ii) and also completes the proof of (i).

Finally, property (iii) can be shown the same way as it was done for A_{1}.

By the construction, from minimality of M and from [BER1], for some j_{0} > 1, the set A_{j_{0}} defines a holomorphic correspondence F_{j_{0}} in a neighborhood Ω_{0} ⊂ U_{j_{0}} of the origin. Note that the size of this neighborhood depends only on the geometry of M and is independent of U_{0}, where f was originally defined. It remains to show now that F_{j_{0}} satisfies

(49)  F_{j_{0}}(M ∩ Ω_{0}) ⊂ M′.

If (z, z′) ∈ A_{j_{0}}, then F_{j_{0}−1}(Q_{z} ∩ U_{j_{0}−1}) ⊂ Q′_{z′}. From property (i), F_{j_{0}}(Q_{z} ∩ U_{j_{0}−1}) ⊂ Q′_{z′}, and from (iii) it follows that

(50)  F_{j_{0}}(Q_{z} ∩ U_{j_{0}}) ⊂ Q′_{z′}.

Suppose now that for some z^{0} ∈ M ∩ Ω_{0}, F_{j_{0}}(z^{0}) ⊄ M′. Then there exists z′ ∈ F_{j_{0}}(z^{0}) \ M′. Note that F_{j_{0}}(M ∩ U_{0}) ⊂ M′, and therefore by continuity we may find z^{0} and z′ such that z′ is close to M′. From (50) we have F_{j_{0}}(Q_{z^{0}} ∩ U_{j_{0}}) ⊂ Q′_{z′}. Since z^{0} ∈ Q_{z^{0}}, we have F_{j_{0}}(z^{0}) ⊂ Q′_{z′}, in particular, z′ ∈ Q′_{z′}. But from (15), z′ ∉ Q′_{z′}, since z′ ∉ M′. This contradiction proves (49).

Theorem 2.1 is proved.

5. Extension as a finite correspondence along M.

In this section we give the proof of Theorem 2.2. One difficulty in the proof of analytic continuation of correspondences lies in the fact that the continued correspondence may acquire additional branches. To deal with this we define the notion of a complete correspondence as follows.

Definition 5.1. Let M ⊂ ℂ^{n} be a smooth real-analytic essentially finite generic CR submanifold, and let M′ ⊂ ℙ^{N} be a smooth compact real-algebraic essentially finite generic submanifold. Let F : U → ℙ^{N} be a holomorphic correspondence such that F (U ∩ M) ⊂ M′. Then F is called complete if for every z ∈ M, we have F (z) = Ĩ′_{z′} for some (and therefore for any) z′ ∈ F (z). Here Ĩ′_{z′} is defined as in (29).

Note that since M′ is essentially finite, a complete correspondence is finite-valued near M, but in general it may be reducible, even if defined on all of M.

Assume that M is locally, near p ∈ M, generically embedded into an open set in ℂ^{n}, so n = m + d. Let f be a holomorphic map defined in a neighborhood U_{p} ⊂ ℂ^{n} of p ∈ M, of maximal rank and such that f(U_{p} ∩M) ⊂ M′. We replace f with a complete correspondence. For that we choose a small neighborhood U_{0} of p and shrink U_{p} in such a way, that for any w ∈ U_{0}, the set Q_{w} ∩ U_{p} is non-empty and connected. We define A_{0} as in (34). Denote the corresponding multiple valued map by F_{0}. Then F_{0} is a complete holomorphic correspondence. Indeed, since f is of maximal rank and d′ ≥ d, for z ∈ U_{0}, dim f(Q_{z} ∩ U_{p}) = dim Q_{z} = m. For z sufficiently close to p, f(z) is close to M′, and since M′ is essentially finite, the only points whose Segre varieties can contain f(Q_{z} ∩ U_{p}) are in I′_{f(z)}. Furthermore, F_{0} is a finite correspondence. To see this we observe that if E ⊂ U_{0} is a positive dimensional set such that F_{0}(E) is discrete, then by the construction f(Q_{z} ∩ U_{p}) ⊂ Q′_{z′} for all z ∈ E. But since ⋃_{z∈E} Q_{z} ∩ U_{p} has dimension bigger than m, this contradicts the fact that f is of maximal rank in U_{p}. Therefore the pre-image of any point under F_{0} is finite, and thus F_{0} is a finite correspondence.

14 C. DENSON HILL AND RASUL SHAFIKOV

We now show that F_{0} extends as a finite correspondence along any path on M_{1}. Our construction of the analytic continuation of F_{0} will preserve completeness, and therefore from Lemmas 3.1 and 3.3 and we conclude that such analytic continuation will have the same number of branches near M_{1}. The problem can be localized as follows. Let γ : [0, 1] → M_{1} be the given path, γ(0) = p, and assume that for t_{0} ≤ 1, q = γ(t_{0}) is the first point on γ to which F_{0} does not extend as a finite correspondence. We choose t_{1} ∈ [0, t_{0}) so close to t_{0}, that for a = γ(t_{1}) we have q ∈ Ω_{a}, where Ω_{a} is defined as in (21). Then by Theorem 2.1, F_{0} extends as a holomorphic correspondence A ⊂ Ω_{a} × ℙ^{n}. Thus we only need to show that after possibly shrinking the neighborhood Ω_{a}, the set A is a finite correspondence. Let π : A → Ω_{a} and π′ : A → ℙ^{N} be the natural projections, and set F := π′ ∘ π^{−1}.

From (49) there exists a neighborhood Ũ ⊂ Ω_{a} of M ∩Ω_{a} such that F (Ũ) ⊂ U′, where U′ ⊂ ℙ^{N} is a neighborhood of M′ as in Lemma 3.1. We now repeat the argument of analytic continuation of A_{0} along γ by constructing the sets A_{j}^{∗}, j = 1, 2, …, with the only difference that the standard neighborhoods of the form (17) are chosen so small that they are contained in Ũ. Since the new set Ω_{a} may be smaller than Ũ, this continuation may require more than one step. More precisely, we choose a sequence of points {a_{ν}}_{ν=0}^{l} such that a_{ν} ∈ γ, a_{0} = a, a_{l} = q and a_{ν} ∈ Ω_{a_{ν−1}} for 0 < ν ≤ l. For each a_{ν} starting with a_{0} we use Theorem 2.1 to extend a finite correspondence F_{ν} defined in a neighborhood of a_{ν} to a holomorphic correspondence F_{ν+1} defined in Ω_{a_{ν}} ⊂ Ũ. This time we show in addition that at each step the extension is a finite correspondence. Since a_{ν+1} ∈ Ω_{a_{ν}}, the process can be continued until we reach the point q.

Suppose that F_{ν+1} : Ω_{a_{ν}} → ℙ^{N} is a holomorphic correspondence, which is obtained from the finite correspondence F_{ν} defined in a small neighborhood of the point a_{ν} for some ν by the inductive construction of the sets A_{j}^{∗}; that is

(51)  A_{j}^{∗} = {(w, w′) ∈ U_{j}^{∗} × ℙ^{N} : F_{j−1}^{∗}(Q_{w} ∩ U_{j−1}^{∗}) ⊂ Q′_{w′}} ,

where the Segre set Q_{a_{ν}}^{j} is contained in U_{j}^{∗}, the map F_{ν} is associated with the set A_{0}^{∗}, and F_{ν+1} corresponds to the set A_{j_{0}}^{∗} with U_{j_{0}}^{∗} = Ω_{a_{ν}}. Let F_{j}^{∗} be the map associated with A_{j}^{∗}.

Clearly, the sets A_{j}^{∗} are contained in the set A, which defines a correspondence F : Ũ → U′. We claim that F_{ν+1} is a finite correspondence. To prove this assertion we let k be the smallest integer such that F_{k}^{∗} is not a finite correspondence. By assumption, k > 0. Suppose that there exists a point w′ ∈ U′ such that the analytic set F_{k}^{∗}^{−1}(w′) ⊂ U_{k}^{∗} has positive dimension. By the construction we have

(52)  F_{k−1}^{∗}(Q_{z} ∩ U_{k−1}^{∗}) ⊂ Q′_{w′}, for all z ∈ F_{k}^{∗}^{−1}(w′).

Since M is essentially finite, there exists a set E ⊂ F_{k}^{∗}^{−1}(w′) such that dim ∪_{z∈E}Q_{z} = m + 1. It follows from (52) that dim F_{k−1}^{∗}(∪_{z∈E}Q_{z}) ≤ m. But this contradicts the induction hypothesis that F_{k−1}^{∗} is finite.

Suppose now that there exists a point w ∈ U_{k}^{∗} such that F_{k}^{∗}(w) is not discrete. By the construction this means that F_{k−1}^{∗}(Q_{w} ∩ U_{k−1}^{∗}) ⊂ Q′_{z′}, where z′ belongs to the non-discrete set. Since F_{k−1}^{∗} is finite, F_{k−1}^{∗}(Q_{w} ∩ U_{k−1}^{∗}) has dimension m. But then there can be only finitely many z′ whose Segre varieties can contain F_{k−1}^{∗}(Q_{w} ∩ U_{k−1}^{∗}). Therefore F_{k}^{∗} is also finite for any k ≥ 0.

Hence, the obtained extension F_{ν+1} is a finite correspondence. We can repeat the same argument for extending F_{0} along γ ∩ Ũ until we reach the point q.

Thus we have proved that F_{0} extends along any path on M_{1} as a finite correspondence F. It follows from the construction that the number of branches of F coincides with the number R defined in Lemma 3.1 for almost all points on M_{1}. It remains to observe now that from the simple connectivity of M_{1} and the monodromy theorem, the extension of F_{0} along homotopically

HOLOMORPHIC CORRESPONDENCES BETWEEN CR MANIFOLDS. 15

equivalent paths gives the same result. The version of the monodromy theorem for finite-valued correspondences can be found in [Sh2].

Part (2) of the statement of the theorem follows from the construction of the extension F. The proof of Theorem 2.2 is now complete.

6. Pseudoconcave submanifolds in ℙ^{n}.

In this section we prove the rest of the results stated in Section 2.

Proof of Theorem 2.4.

(a) According to Thm 5.2 of [HN5], there exists an m + d dimensional irreducible algebraic subvariety Y of ℙ^{n} such that M is a generic CR submanifold of the regular part of Y, reg Y. Because of the pseudoconcavity of M (or because of Property E), the continuous CR map f is smooth and has a unique holomorphic extension to an open neighborhood Ω of M in reg Y. Thus f can be regarded as a holomorphic map from Ω to ℙ^{N}. This means that f may be given by N meromorphic functions f_{1}, f_{2}, … f_{N} in Ω. To see this, we choose homogeneous coordinates [z_{0} : z_{1} : ⋯: z_{N}] in ℙ^{N} such that the hyperplane {z_{0} = 0} is in general position with respect to f(Ω), and set Ω_{j} = f^{−1}(V_{j}), where V_{j} = ℙ^{N} \ {z_{j} = 0}, j = 0, 1, …, N. The Ω_{j} give an open cover of Ω, in each V_{j} we have the inhomogeneous coordinates (w_{1j}, w_{2j}, …, w_{Nj}), where w_{ij} = z_{i}/z_{j}, and f|_{Ω_{j}} is given by holomorphic functions

(53)  Ω_{j} ∋ t → (w_{1j}(t), w_{2j}(t), …, w_{Nj}(t)).

We define the meromorphic functions f_{1}, f_{2}, …, f_{N} by f_{k}(t) = w_{k0}(t) in Ω_{0}, and by f_{k}(t) = w_{kj}(t)/w_{0j}(t) in Ω_{j} for j, k = 1, 2, …, N. Note that these definitions are consistent on the overlaps.

By Thm 5.2 of [HN5], each f_{k} is the restriction to M of a rational function on Y, and hence can be regarded as a rational function on ℙ^{n}. This gives the desired rational map from ℙ^{n} to ℙ^{N}.

(b) Since M is generic in ℙ^{n}, and since f is a local CR diffeomorphism, the Jacobian det J_{f} of the extension of f to ℙ^{n} is not identically zero. Hence, the set Σ = {z ∈ ℙ^{n} : det J_{f}(z) = 0} if non-empty, is a subvariety of ℙ^{n} \ Λ of complex codimension one, where Λ is the indeterminacy locus of f. Suppose Σ ≠ ∅. Then, since f is locally biholomorphic near any point on M, M ∩ Σ = ∅. On the other hand, by the Remmert-Stein theorem, the closure of Σ is a subvariety of ℙ^{n} of codimension one. It is well known that its complement in ℙ^{n} is therefore a Stein manifold. But a pseudoconcave M (or an M satisfying Property E) has no CR embedding into a Stein manifold (see [HN1] and [HN5]). Thus Σ = ∅.

Let F : ℂ^{n+1} → ℂ^{n+1} be a polynomial map such that f ∘π = π∘F, where π : ℂ^{n+1} → ℙ^{n} is the canonical projection. Without loss of generality assume that the components of F are homogeneous polynomials of degree k without common factors. We claim that det J_{F} (z) ≠ 0 for any point z ∈ ℂ^{n+1}. Indeed, suppose on the contrary that

(54)  E = {z ∈ ℂ^{n+1} : det J_{F} (z) = 0}

is a non-empty subvariety of complex codimension one. Then F (E) ≠ {0}, and therefore there exists a point p ∈ E such that F (p) ≠ 0. For z ∈ ℂ^{n+1}, let L_{z} := {λz : λ ∈ ℂ} be the complex line passing through the point z and the origin. Since the Jacobian of f does not vanish outside the indeterminacy locus, and F (L_{p}) ≠ {0}, there exists a small neighborhood U of p such that for all z and w in U, z ≠ w,

(55)  F (L_{z}) ∩ F (L_{w}) = {0}.

Furthermore, F |_{U∩L_{z}} = λ^{k}z, and after shrinking U if necessary, we may assume that F |_{U∩L_{z}} is an injective function for all z ∈ U. From this and (55) we conclude that F is injective in U, which contradicts the assumption that p ∈ E. Thus det J_{F} ≠ 0 and therefore is a constant.

Finally, observe that det J_{F} (z) is a homogeneous polynomial of degree (k − 1)(n + 1), and being constant means that k = 1, i.e. F is a linear automorphism.

□

Proof of Theorem 2.5 and Corollary 2.6. We may regard M as being a generic CR submanifold of a complex manifold X. Note that the pseudoconcavity of M (or Property E) implies that M is minimal, because minimality is well-known to be equivalent to wedge extendability. There is a neighborhood V_{p} of p in X such that f extends to a holomorphic mapping f : V_{p} → ℙ^{N}. It is easy to check that, by possibly shrinking V_{p}, the extended map f has maximal rank in V_{p}. Hence we may conclude from Theorem 2.2 that f extends to a finite holomorphic correspondence F : V → ℙ^{N}, where V ⊂ X is some neighborhood of M.

Since the Segre map associated with M′ is injective, F splits at every point of M, and every map F^{j} of the splitting is a CR map from M to M′. One of them, say F^{1} is the extension of f. By Theorem 2.4(a), F^{1} extends to a rational map from ℙ^{n} to ℙ^{N}. Moreover, if n = N and M is generic in ℙ^{n}, then F^{1} is locally biholomorphic, because the Segre map associated with M is injective. Thus by Theorem 2.4(b) F^{1} extends to a linear automorphism of ℙ^{n}. Rationality of F^{1} implies that M must also be algebraic, which proves Corollary 2.6. □

Proof of Theorem 2.3. Since M and M′ are Levi non-degenerate, the associated Segre maps are locally injective, and M and M′ satisfy the conditions of Theorem 2.2. If M and M′ are strictly pseudoconvex then f extends as a locally biholomorphic map to a neighborhood of p by [PT]. If M is pseudoconcave, the result follows from [HN2]. Therefore the map f defined near p also satisfies the conditions of Theorem 2.2. Thus f extends as a finite correspondence along M. Since the set Σ′, where the Segre map associated with M′ branches, is empty, the extended correspondence is single-valued. Since the Segre map associated with M is injective, the extension f : M → M′ is a locally biholomorphic map.

We now show that the extension is globally biholomorphic in a neighborhood of M. For that we note that M′ is simply connected. Indeed, if k = n − 1 or k = 0, then M′ = S^{2n−1} which is simply connected. If 0 < k < n − 1, then we choose an affine patch V′_{n} in ℙ^{n} where z′_{n} ≠ 0. Then

(56)  M′ ∩ V′_{n} = {|w′_{0}|^{2} + ⋯ + |w′_{k}|^{2} − |w′_{k+1}|^{2} − ⋯ − |w′_{n−1}|^{2} = 1},

where w′_{j} = z′_{j}/z′_{n}. Let π be the projection from M′ to the coordinates (w′_{k+1}, …, w′_{n−1}). Then π is onto, and for any w = (w′_{k+1}, …, w′_{n−1}), π^{−1}(w) ≅ S^{2k+1}, which is simply connected. Therefore, M′ is also simply connected. Because of that the germ of a map f^{−1} extends holomorphically along any path on M′ to a holomorphic map f^{−1} : M′ → M. Thus f|_{M} maps M one-to-one and onto M′, and f is globally biholomorphic.

If M′ is pseudoconcave, then by the invariance of the Levi form, so is M. Thus by Theorem 2.4(b) f extends to a linear automorphism of ℙ^{n}. □

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