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A local extension of proper holomorphic maps between some unbounded domains in [C.sup.n].

1 Introduction and results

The boundary regularity of proper holomorphic maps between smooth domains in [C.sup.n] is still an open problem in full generality. However, positive answers have been obtained in many special cases. For domains in [C.sup.n] with real-analytic smooth boundaries, the recent progress is related to the geometric reflection principle in [C.sup.n], based on the method of Segre varieties. For related results and without mentioning the entire list, we refer the reader to [24], [15], [17] with references included. Our first purpose in this paper is to prove a local holomorphic extension of proper holomorphic mappings under the assumption that the graph of the mapping extends as an analytic set (see Definition 1). More precisely, we prove the following

Theorem 1. Let D, D' be arbitrary domains in [C.sup.n], n > 1, and f : D [right arrow] D' be a proper holomorphic mapping. Let M [subset] [partial derivative]D, M' [subset] [partial derivative]D' be open pieces of the boundaries.

Suppose that [partial derivative]D is smooth real-analytic and nondegenerate in an open neighborhood of [bar.M] and [partial derivative]D' is smooth real-algebraic and nondegenerate in an open neighborhood of [bar.M]'. If the cluster set [cl.sub.f] (p) of a point p [member of] M contains a point q [member of] M' and the graph of f extends as an analytic set to a neighborhood of (p, q) [member of] [C.sup.n] x [C.sup.n] (see Definition 1), then f extends holomorphically to a neighborhood of p.

Note that we do not require pseudoconvexity of M or M', and we do not assume that [cl.sub.f] (M) [subset] M'. Therefore, a priori [cl.sub.f] (p) may contain the point at infinity or boundary points which do not lie in M'. Moreover, the following example (appeared in [15]) shows that the extension of the graph of f as an analytic set to a neighborhood of (p, q) does not imply in general that [cl.sub.f] (M) [subset] M' near p.

Example 1. Let D = {([z.sub.1], [z.sub.2]) [member of] [C.sup.2] : Re([z.sub.2]) + [[absolute value of [z.sub.1]].sup.2] < 0} and D' = {([z'.sub.1], [z'.sub.2]) [member of] [C.sup.2] : Re([z'.sub.2]) + [[absolute value of [z'.sub.1]].sup.2] [[absolute value of [z'.sub.2]].sup.2] < 0}. The map f : ([z.sub.1], [z.sub.2]) [??] ([z.sub.1]/[z.sub.2], [z.sub.2]) is a biholomorphism from D to D'. The graph of f is contained in {([z.sub.1], [z.sub.2], [z'.sub.1], [z'.sub.2]) [member of] [C.sup.2] x [C.sup.2] : [z'.sub.1] [z'.sub.2] - [z.sub.1] = 0, [z'.sub.2] = [z.sub.2]}, then f extends as an analytic set to a neighborhood of (0,0'). But 0' [member of] [cl.sub.f] (0) and [infinity] [member of] [cl.sub.f] (0). Note that in this example the boundary of D' is degenerate (since it contains the complex line [z'.sub.2] = 0).

Theorem 1 is already known if M and M' are smooth, real-analytic hypersurfaces of finite type and additionally [cl.sub.f] (M) [subset] M' (see Theorem 1.1 in [15]), or f is continuous on D [union] M (see [19]), or the graph of the mapping extends as a holomorphic correspondence (see [14]). The proof of all these results uses the method of analytic continuation along Segre varieties.

As an application of Theorem 1, we prove the following

Corollary 1. Let D and D' be smooth algebraic domains in [C.sup.n], n > 1, with nondegenerate boundaries and f : D [right arrow] D' be a proper holomorphic mapping.

a) If the cluster set [cl.sub.f] (p) of a point p [member of] [partial derivative]D contains a point q [member of] [partial derivative]D', then f extends holomorphically to a neighborhood of p and the set of holomorphic extendability of f is an open dense subset of [partial derivative]D.

b) If either D or D' has a global holomorphic peak function at infinity, then the set of holomorphic extendability of f is an open dense subset of [partial derivative]D.

By a smooth algebraic domain D in [C.sup.n] we mean a domain defined globally as {z [member of] [C.sup.n] : P(z, [bar.z]) < 0}, where P is a real polynomial in [C.sup.n] with dP [not equal to] 0 on [partial derivative]D. We say that its boundary is nondegenerate if {z [member of] [C.sup.n] : P(z, [bar.z]) = 0} contains no complex-analytic set with positive dimension. Note that these domains are not necessarily bounded. Recall that a function [phi] : D [right arrow] C is a global holomorphic peak function at infinity on D if [phi] is holomorphic in D, [absolute value of [phi](z)] < 1 for all z [member of] D and [phi](z) [right arrow] 1 as [absolute value of z] [right arrow] [infinity].

Remark 1. The existence of global holomorphic peak functions at infinity is due to Bedford-Fornaess in the case of rigid polynomial domains in [C.sup.2] (see [5]). These functions exist also in the case of unbounded convex domains in [C.sup.n], which does not contain complex affine lines. Indeed, if D is such a domain, there exist [H.sub.1], ..., [H.sub.n] linearly independent hyperplanes such that [bar.D] is on one side of each of these hyperplanes. Up to a linear change of coordinates [??] = ([[??].sub.1], ..., [[??].sub.n]), we may assume that [H.sub.j] = {[??] [member of] C[n.sup. ]: Re[[??].sub.j] = 0} and D is contained in {[??] [member of] [C.sup.n] : Re[[??].sub.1] < 0, ..., Re[[??].sub.n] < 0} (for details, see Proposition 3.5 in [6]). The image of infinity by the associated Cayley transform is contained in the zero of the function [[PI].sub.1 [less than or equal to] j [less than or equal to] n] ([[??].sub.j] - 1). According to [26] (Theorem 6.1.2 page 132), this is the peak set of a holomorphic function. This proves the existence of a global holomorphic peak function at infinity on D. Recall that if [OMEGA] is a domain in [C.sup.n] and E [subset] [partial derivative][OMEGA] is a compact subset, we say that E is a peak set for a holomorphic function f if f is holomorphic in [OMEGA], continuous on [bar.[OMEGA]], [absolute value of f([xi])] < 1 for every [xi] [member of] [bar.[OMEGA]]\E and f([xi]) = 1 for every [xi] [member of] E.

In Corollary 1, the function f is not assumed to possess any a priori regularity near p. If D or D' is pseudoconvex, then the holomorphic extendability of f near p will be a consequence of [25] and [17]. If D is a rigid polynomial domain in [C.sup.n] (n > 1), namely, D = {([z.sub.1], 'z) [member of] C x [C.sup.n-1] : 2Re([z.sub.1]) + P('z) < 0}, we can follow the proof of Coupet-Pinchuk [8], based on the construction of analytic discs attached to the boundary, to prove the existence of a point p [member of] [partial derivative]D satisfying [cl.sub.f] (p) [intersection] [partial derivative]D' [not equal to] [empty set] (see Lemma 1.3 in [8]). The proof of Corollary 1 shows the existence of an algebraic set [S.sup.[infinity]] [subset] [partial derivative]D such that f extends holomorphically to a neighborhood of any point from [partial derivative]D\[S.sup.[infinity]] and for all t [member of] [S.sup.[infinity]], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, we get the following decomposition of the boundary [partial derivative]D = [S.sup.h] [union] [S.sup.[infinity]], where [S.sup.h] is the set of holomorphic extendability of the mapping f.

Next, we propose a local version of Corollary 1.

Corollary 2. Let D, D' be arbitrary domains in [C.sup.n], n > 1, and M [subset] [partial derivative]D, M' [subset] [partial derivative]D' be open pieces of the boundaries. Suppose that [partial derivative]D (resp. [partial derivative]D') is smooth real- algebraic and nondegenerate in an open neighborhood of [bar.M] (resp. [bar.M]'). Let f : D [right arrow] D' be a proper holomorphic map such that the cluster set [cl.sub.f] (M) [subset] M'.

a) For n = 2, f extends across each point of M as a holomorphic map.

b) For n [greater than or equal to] 2, suppose that there exist a point p [member of] M and a neighborhood U of p such that f extends to a uniformly continuous mapping on a dense subset B of M [intersection] U, possibly with empty interior (see Definition 3). Then f extends across each point of M as a holomorphic map.

Corollary 2 was proved in [24] for proper holomorphic maps between bounded real-analytic domains in [C.sup.2]. When f is continuous on M, this result is due to Diederich-Pinchuk ([17]). The proof is based on the algebraicity of the mapping with a careful study of [cl.sub.f] (M), which is the crucial point of the proof. The difficulty is to show that [cl.sub.f] (M) [??] [M'.sub.0], where [M'.sub.0] denotes the set of points of M' with degenerate Levi-form. Notice that there is no assumption on the cluster set of M' under [f.sup.-1].

If D = {[rho] < 0} is a domain in [C.sup.n] and c is a real number, we denote by [partial derivative][D.sub.c] the set defined by {[rho] = c}. Based on the algebraicity result and by analyzing the order of vanishing of the Levi-determinant of the domain, we show the following

Theorem 2. Let D = {P(z, [bar.z]) < 0} and D' = {Q(z', [bar.z]') < 0} be smooth algebraic domains in [C.sup.n], n > 1, with nondegenerate boundaries. Suppose that either D or D' has a global holomorphic peak function at infinity. Suppose further that one of the following conditions is satisfied:

(A) P is plurisubharmonic on D and [partial derivative][D.sub.c] is nondegenerate for all c < 0.

(B) Q is plurisubharmonic on D' and [partial derivative][D'.sub.c] is nondegenerate for all c < 0.

Then there exists a finite number of irreducible complex-algebraic sets [[??].sub.1], ..., [[??].sub.N] in [C.sup.n] of dimension n - 1 such that the branch locus [V.sub.f] of any proper holomorphic mapping f : D [right arrow] D' satisfies :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For rigid polynomial domains, similar results were proved in [9] and [20]. The integer N is bounded by the degree of the polynomial P. The assumption on the existence of a global holomorphic peak function at infinity on D or D' assures that [cl.sub.f] ([partial derivative]D) [intersection] [partial derivative]D' is non-empty for any proper holomorphic mapping f : D [right arrow] D', and this leads to prove the algebraicity of the mapping (see the proof of Corollary 1, second part). The plurisubharmonicity of P and the nondegeneracy of the sets [partial derivative][D.sub.c] (also the plurisubharmonicity Q and the nondegeneracy of [partial derivative][D'.sub.c]) are important here to prove that the branch locus extends across the boundary of the domain. We are not able to prove this fact without these assumptions. If D is a bounded pseudoconvex smooth algebraic domain, then conditions (A) and (B) can be dropped; since the branch locus cannot be relatively compact in the domain. Moreover, the proof of Lemma 8 can be adapted in this case (in view of the existence of bounded negative plurisubharmonic exhaustion functions, see for example [13]) to prove that the branch locus extends across the boundary of the domain. Finally, note that for smooth bounded domains, we can give a nice description of the branch locus if the set of weakly pseudoconvex boundary points admits a nice stratification, as it was observed in [3] in the real-analytic case.

As examples of smooth algebraic domains in [C.sup.n] (possibly unbounded) defined by {P(z, [bar.z]) < 0} and verifying the property : [partial derivative][D.sub.c] is nondegenerate for all c [less than or equal to] 0, we propose :

1- D = {z [member of] [C.sup.n] : P(z, [bar.z]) + 1 < 0}, where P is a homogeneous polynomial such that {P = -1} is nondegenerate. Set for example, P(z, [bar.z]) = Re([z.sup.2.sub.1]) + [[absolute value of [z.sub.2]].sup.2], with z = ([z.sub.1], [z.sub.2]) [member of] [C.sup.2].

2- D = {([z.sub.1], 'z) [member of] C x [C.sup.n-1] : Re([z.sub.1]) + [phi]('z, [bar.'z], Im[z.sub.1]) < 0} (called semi-rigid domain), where [phi] is a polynomial such that {Re([z.sub.1]) + [phi]('z, [bar.'z], Im[z.sub.1]) = 0} is nondegenerate .

The following example shows that the nondegeneracy of [partial derivative]D does not imply in general the nondegeneracy of [partial derivative][D.sub.c] for all c < 0.

Example 2. Let D = {([z.sub.1], [z.sub.2]) [member of] [C.sup.2] : P([z.sub.1], [z.sub.2]) < 0}, where P([z.sub.1], [z.sub.2]) = 2Re([z.sub.2]) + [[absolute value of [z.sub.1]].sup.2] [[absolute value of [z.sub.2]].sup.2] - 1. The set {P([z.sub.1], [z.sub.2]) = -1} contains the complex line [z.sub.2] = 0 and the boundary of D is nondegenerate; since D is strictly pseudoconvex.

Remark 2. Note that any convex domain D in [C.sup.n], which does not contain complex affine lines is biholomorphic to a bounded domain in [C.sup.n]. Indeed, there exist [H.sub.1], ..., [H.sub.n] linearly independent hyperplanes such that [bar.D] is on one side of each of these hyperplanes. Up to a linear change of coordinates [??] = ([[??].sub.1], ..., [[??].sub.n]), we can assume that [H.sub.j] = {[??] [member of] [C.sup.n] : Re[[??].sub.j] = 0} and D is contained in {[??] [member of] [C.sup.n] : Re[[??].sub.1] < 0, ..., Re[[??].sub.n] < 0}. Then the map

g: ([[??].sub.1], ..., [[??].sub.n]) [??] ([[??].sub.1] + 1/[[??].sub.1] - 1 , ..., [[??].sub.n] + 1/[[??].sub.n] - 1)

maps D biholomorphically onto a bounded domain [OMEGA] in [C.sup.n]. Moreover, if in addition, D is algebraic, then [OMEGA] is also algebraic. In this case, Theorem 2 can be reformulated as follows

Theorem 2-bis. Let D be a convex smooth algebraic domain in [C.sup.n] (possibly unbounded), which does not contain complex affine lines and D' be a smooth algebraic domain in [C.sup.n] with nondegenerate boundary. Then there exists a finite number of irreducible complex-algebraic sets [[??].sub.1], ..., [[??].sub.N] in [C.sup.n] of dimension n - 1 such that the branch locus [V.sub.f] of any proper holomorphic mapping f : D [right arrow] D' satisfies :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As a conclusion from the proof of Theorem 2, one has the following

Corollary 3. Let D = {P(z, [??]) < 0} be a simply connected, smooth algebraic domain in [C.sup.n], n > 1, with nondegenerate boundary, where P is a plurisubharmonic polynomial on D and f : D [right arrow] D be a proper holomorphic self-mapping. Suppose that [partial derivative][D.sub.c] is nondegenerate for all c < 0 and [cl.sub.f] ([partial derivative]D) intersects the boundary of D. Then f is a biholomorphism.

Corollary 3 generalizes Corollary 1 in [9] for semi-rigid polynomial domains in [C.sup.n]. It can be also observed as a generalization of the well known result of Alexander [1], stating that any proper holomorphic self-map of the unit ball in [C.sup.n] is biholomorphic. A similar result was proved in [23] for bounded smooth algebraic domains in [C.sup.n] without the pseudoconvexity assumption.

Remark 3. Since any convex smooth algebraic domain D in [C.sup.n], which does not contain complex affine lines is biholomorphic to a bounded smooth algebraic domain, then according to [23], any proper holomorphic self-mapping of D is a biholomorphism.

2 Notations, definitions and preliminaries

Definition 1. ([15]) Let f : D [right arrow] D' be a holomorphic map between domains in [C.sup.n] and 0 [member of] [partial derivative]D, 0' [member of] [partial derivative]D' such that 0' [member of] [cl.sub.f] (0). If U and U' are small neighborhoods of 0 and 0', we say that the graph [[GAMMA].sub.f] of f extends as an analytic set to U x U' if there exist an irreducible (closed) complex-analytic set A [subset] U x U' of pure dimension n and a sequence [a.sub.v] [right arrow] 0 in D [intersection] U with neighborhoods [V.sup.v] [contains as member] [a.sub.v] such that f ([a.sub.v]) [right arrow] 0' and A [contains] [[GAMMA].sub.f] [intersection] ([V.sup.v] x U') for all v.

In Definition 1, note that A does not necessarily contain the whole graph of f over D [intersection] U (since f is not necessarily continuous at 0).

Now, we recall the definition of a holomorphic correspondence. Let D, D' be domains in [C.sup.n] and A be a complex purely n-dimensional subvariety contained in D x D'. We denote by [[pi].sub.1] : A [right arrow] D and [[pi].sub.2] : A [right arrow] D' the natural projections. When [[pi].sub.1] is proper, ([[pi].sub.2] o [[pi].sup.-1.sub.1])(z) is a non-empty finite subset of D' for any z [member of] D and one may therefore consider the multi-valued mapping f = [[pi].sub.2] o [[pi].sup.-1.sub.1]. Such a map is called a holomorphic correspondence between D and D'; A is said to be the graph of f. Since [[pi].sub.1] is proper, in particular it is a branched analytic covering. Then there exist an (n - 1)-dimensional complex-analytic subset [V.sub.f] of the graph of f and an integer m such that [[pi].sub.1] is an m-sheeted covering map from the set A\[[pi].sup.-1.sub.1] ([[pi].sub.1]([V.sub.f])) onto D\[[pi].sub.1]([V.sub.f]). Hence, f(z) = {[f.sup.1](z), ..., [f.sup.m](z)} for all z [member of] D\[[pi].sub.1]([V.sub.f]) and the [f.sup.j]'s are distinct holomorphic functions in a neighborhood of z [member of] D\[[pi].sub.1]([V.sub.f]). The integer m is called the multiplicity of f and [[pi].sub.1] ([V.sub.f]) is its branch locus. One says that f is irreducible if A is irreducible as an analytic set and proper if both [[pi].sub.1] and [[pi].sub.2] are proper.

Definition 2. Let f : D [right arrow] D' be a holomorphic mapping between domains in [C.sup.n], [[GAMMA].sub.f] be the graph of f and [z.sub.0] be a point in [partial derivative]D. We say that f extends as a holomorphic correspondence to a neighborhood U of [z.sub.0] if there exist an open set U' [subset] [C.sup.n] and a closed complex-analytic set A [subset] U x U' of pure dimension n, which may possibly be reducible, such that,

i) [[GAMMA].sub.f] [intersection] {(D [intersection] U) x D'} [subset] A

ii) the natural projection [pi] : A [right arrow] U is proper.

We will write z = ([z.sub.1], 'z) [member of] C x [C.sup.n-1] for a point z [member of] [C.sup.n]. Let M be a smooth real-analytic hypersurface that contains the origin. By [rho](z, [bar.z]) we denote a real-analytic defining function of M near 0. In a small neighborhood U of the origin, the complexification [rho](z, [bar.w]) of [rho] is well-defined by means of a convergent power series in U x U. For w [member of] U, the associated Segre variety is defined as

[Q.sub.w] = {z [member of] U : [rho](z, [bar.w]) = 0}.

By the implicit function theorem, it is possible to choose neighborhoods [U.sub.1] [subset][subset] [U.sub.2] of the origin such that for any w [member of] [U.sub.1], [Q.sub.w] is a closed, complex hypersurface in [U.sub.2] and

[Q.sub.w] = {([z.sub.1], 'z) [member of] [U.sub.2] : [z.sub.1] = h('z, [bar.w])}, (2.1)

where h('z, [bar.w]) is holomorphic in 'z and antiholomorphic in w. Following the terminology of [16], [U.sub.1] and [U.sub.2] are usually called a standard pair of neighborhoods of 0. It can be shown that [Q.sub.w] is independent of the choice of the defining function. We denote by S = S(U) the set of Segre varieties {[Q.sub.w], w [member of] U} and A the so-called Segre map defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [I.sub.w] := {z [member of] U : [Q.sub.w] = [Q.sub.z]} be the fiber of [lambda] over [Q.sub.w]. For any w [member of] M, the set [I.sub.w] is a complex variety of M. If the hypersurface M is nondegenerate (it contains no complex-analytic set with positive dimension), then for any w [member of] M there exists a neighborhood [U.sub.w] of w such that [I.sub.w] [intersection] [U.sub.w] is finite. The set S admits the structure of a complex-analytic variety of finite dimension such that the map [lambda] is a finite antiholomorphic branched covering. The set [I.sub.w] contains at most as many points as the sheet number of [lambda]. We next list some important properties of [Q.sub.w] and [I.sub.w] (see e.g. [10] and [18]).

(a) z [member of] [Q.sub.w] [??] w [member of] [Q.sub.z].

(b) z [member of] [Q.sub.z] [??] z [member of] M.

(c) [I.sub.w] = [intersection]{[Q.sub.z]: z [member of] [Q.sub.w]}.

(d) The Segre map [lambda] : w [??] [Q.sub.w] is locally one-to one near strictly pseudoconvex points of M.

Let f : D [right arrow] D be a proper holomorphic mapping between domains in [C.sup.n] with smooth real-analytic boundaries which extends as a holomorphic correspondence F to a neighborhood of a point p [member of] [partial derivative]D. Assume that p = 0, f( p) = 0' and choose standard neighborhoods [U.sub.2] [contains] [contains] [U.sub.1] [contains as member] 0 and [U'.sub.2] [contains] [contains] [U'.sub.1] [contains] 0'. Then we have the following invariance property for the Segre variety under F (see [16]) :

For all (w, w') [member of] graph(F) [intersection] ([U.sub.1] x [U'.sub.1]), F([Q.sub.w]) [subset] [Q'.sub.w']. (2.2)

This means that any branch of F maps any point from [Q.sub.w] to [Q'.sub.w'] for any point w' [member of] F(w).

Definition 3. Under the hypothesis of Corollary 2, we say that f extends to a uniformly continuous mapping on B [subset] M [intersection] U if for all [epsilon] > 0, there exists [alpha] > 0 such that for all z [member of] D and w [member of] B,

[absolute value of z - w] < [alpha] [??] [absolute value of f(z) - f(w)] < [epsilon].

Finally, recall that if f : D [right arrow] [C.sup.n] is a holomorphic mapping and z [member of] [partial derivative]D, then the cluster set [cl.sub.f] (z) is defined as :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3 Proof of theorem 1.

The proof consists of two steps : first, to show that the mapping extends as a holomorphic correspondence to a neighborhood of p by using the process of analytic continuation along paths on the boundary and finally to apply the result of [14], which shows that all extending correspondences are holomorphic mappings.

3.1 Extension as a holomorphic correspondence.

Without loss of generality we assume that p = 0, q = 0' and 0 is not in the envelope of holomorphy of D. Let U, U' be neighborhoods of 0 and 0' respectively, and A [subset] U x U' be the irreducible, closed, complex-analytic set of dimension n extending the graph [[GAMMA].sub.f] of f in the sense of Definition 1. Let [[pi].sub.1] : A [right arrow] U be the coordinate projection to the first component and let E = {z [member of] U : dim [[pi].sup.-1.sub.1](z) [greater than or equal to] 1}. We denote by F : U\E [right arrow] [C.sup.n] the multiple-valued map corresponding to A; that is,

F(w) = {w' : (w, w') [member of] A}.

We denote by [S.sub.F] its branch locus (i.e., for z [member of] U\E, z [member of] [S.sub.F] if the coordinate projection [[pi].sub.1] is not locally biholomorphic near [[pi].sup.-1.sub.1](z)). The crucial point in the proof is to show that [[pi].sup.-1.sub.1](0) [intersection] E = [empty set] (i.e., [[pi].sup.-1.sub.1] (0) is discrete). We denote [U.sup.-] = D [intersection] U and [U.sup.+] = U\[bar.D]. We need the following observation.

Lemma 1. A [intersection] ([U.sup.+] x U') [not equal to] [empty set].

Proof. We follow the ideas of [19]. By contradiction assume that A [intersection] ([U.sup.+] x U') = [empty set]. Let A be the irreducible component of A [intersection] (U x U') which contains [[GAMMA].sub.f] [intersection] ([U.sup.-] x U'). It follows that A [not subset] (M [intersection] U) x U'. Let L be a complex line in [C.sup.n] which contains 0 and is transverse to M such that [[GAMMA].sub.f] [intersection] {([U.sup.-] [intersection] L) x U'} [not equal to] [empty set]. We may choose U such that [U.sup.-] [intersection] L is connected. Let A be the irreducible component of A [intersection] {(U [intersection] L) x U'} containing [[GAMMA].sub.f] [intersection] {([U.sup.-] [intersection] L) x U'}. The analytic set [??] has pure complex dimension 1 and it contains (0,0'). Moreover, [??] [not subset] (M [intersection] U) x U'. We consider two cases:

--If [??] [intersection] {(M [intersection] U) x U'} is discrete, then by the continuity principle we deduce that (0,0') is in the envelope of holomorphy of [U.sup.-] x U'.

--If [??] [intersection] {(M [intersection] U) x U'} is not discrete, then no open subset of [??] can be contained in [??] [intersection] {(M [intersection] U) x U'}. Now, the strong disc theorem shows that (0,0') is again in the envelope of holomorphy of [U.sup.-] x U'. Hence, 0 is in the envelope of holomorphy of D. Indeed, if g [member of] O([U.sup.-]), we may regard it as a function [??] [member of] O([U.sup.-] x U'). Then [??] extends to a neighborhood of (0,0') and the uniqueness theorem shows that the extension of [??] is also independent of the variables z' [member of] U'. Hence, g extends holomorphically across 0. This is a contradiction; since 0 is not in the envelope of holomorphy of D.

As a consequence of Lemma 1, we deduce the following result due to Diederich Pinchuk [15].

Lemma 2. There exists an open set [GAMMA] [subset] M [intersection] U such that

1) f extends holomorphically to a neighborhood of [U.sup.-] [union] [GAMMA], and the graph off near any point (z, f(z)), z [member of] [GAMMA], is contained in A.

2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Let [V.sub.F] = {(z, z') [member of] A : z [member of] [S.sub.F]}. Since the complex dimension of [V.sub.F] is at most n - 1 (because A is irreducible and the projection [[pi].sub.1] is locally biholomorphic in an open set of A), then A\[V.sub.F] is connected by paths. Without loss of generality we may assume that M [intersection] U = [partial derivative]D [intersection] U. Let (a, b) [member of] [[GAMMA].sub.f] [intersection] (A\[V.sub.F]) [intersection] ([U.sup.-] x U') and (a', b') [member of] (A\[V.sub.F]) [intersection] ([U.sup.+] x U'), and connect them by a path [gamma] [subset] A\[V.sub.F]. It follows that [[pi].sub.1]([gamma]) [intersection] M [not equal to] [empty set]. Let [z.sub.0] be the point where [[pi].sub.1] ([gamma]) first intersects M. Then f extends analytically along [[pi].sub.1]([gamma]) from a to [z.sub.0] and the graph of f over this part of [[pi].sub.1]([gamma]) is contained in A\[V.sub.F]. It follows that [z.sub.0] is a point of holomorphic extendability for f. The second part follows from the fact that U and U' may be chosen arbitrarily small.

The proof of Theorem 1 uses some ideas of Shafikov developed in [22] to study the analytic continuation of holomorphic correspondences and equivalence of domains.

Lemma 3. There exists a holomorphic change of variables such that in the new coordinates [Q.sub.0] [not subset] E.

Proof. The ideas of the proof were given in [22]. Assume that [Q.sub.0] [subset] E. From Proposition 4.1 of [21] there exists a point t [member of] [GAMMA]\E such that [Q.sub.0] [intersection] [Q.sub.t] [not equal to] [empty set] ([GAMMA] is the set defined in Lemma 2). Let h : [??] [right arrow] [C.sup.n] be the germ of the mapping f defined in a neighborhood [??] of t. We shrink [??] and choose V in such a way that for any w [member of] V, the set [Q.sub.w] [intersection] [??] is connected. Observe that if V is small enough then [Q.sub.w] [intersection] [??] [not equal to] [empty set] for any w [member of] V, as w [member of] [Q.sub.t] implies t [member of] [Q.sub.w]. Note that V is a neighborhood of [Q.sub.t] [intersection] [??]; since if w [member of] Qt, then t [member of] [Q.sub.w] and [Q.sub.w] [intersection] [??] [not equal to] [empty set]. Following the ideas in [10] and [16], we define

X = {(w, w') [member of] V x [C.sup.n] : h([Q.sub.w] [intersection] [??]) [subset] [Q'.sub.w']}.

We would like to have [Q.sub.w] [intersection] [??] connected for any w [member of] V to avoid ambiguity in the condition h([Q.sub.w] [subset] [??]) [subset] [Q'.sub.w']; since different components of [Q.sub.w] [intersection] [??] could be mapped a priori to different Segre varieties. Let Q(w', [bar.w']) be a defining polynomial function of M'. Let z [member of] [??] and z' = h(z). The inclusion h([Q.sub.w] [intersection] U) [subset] [Q'.sub.w'] can be expressed as

Q(h(z), [bar.w']) = 0 for any z [member of] [Q.sub.w] [intersection] [??]. (3.1)

Therefore by property (2.1) of Segre varieties we can choose [??] in the form [??] = [[??].sub.1] x '[??] [subset] C x [C.sup.n-1] such that [Q.sub.w] = {(k('z, [bar.w]), 'z), 'z [member of] '[??]}, and (3.1) is equivalent to

Q(h(k('z, [bar.w]), 'z), [bar.w']) = 0, for any 'z [member of] '[??]. (3.2)

Thus, X is defined by an infinite system of holomorphic equations in ([bar.w], [bar.w']). By the Noetherian property of the ring of holomorphic functions, we can choose finitely many points ['z.sup.1], ..., ['z.sup.m] so that (3.2) can be written as a finite system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where k = 1, ..., m, d' is the degree of Q in w' and [[alpha].sup.k.sub.J] are holomorphic functions in w. We define the closure of X in V x [P.sup.n] in the following way. Let [??] = ([t.sub.0], [t.sub.1], ..., [t.sub.n]) be homogeneous coordinates in [P.sup.n], and let [w'.sub.j] = [t.sub.j]/[t.sub.0] and t = ([t.sub.1], ..., [t.sub.n]). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a system of equations homogeneous in [??] that defines an analytic function in V x [P.sup.n]. We denote this variety by [??]. Clearly, the restriction of [??] to V x ([P.sup.n]\[H.sub.0]) = V x [C.sup.n] coincides with the set defined by (3.2). Here [H.sub.0] = {[t.sub.0] = 0} is the hyperplane at infinity. Let [pi] : [bar.X] [right arrow] V and [pi]' : [bar.X] [right arrow] [P.sup.n] be the natural projections. According to [22] (Lemma 3), h extends as a holomorphic correspondence to V\([[LAMBDA].sub.1] [union] [[LAMBDA].sub.2]), where [[LAMBDA].sub.1] = [pi]([[pi]'.sup.-1]([H.sub.0])) and [[LAMBDA].sub.2] = [pi]{(w, w') [member of] [bar.X] : dim [[pi].sup.-1](w) [greater than or equal to] 1}. It is easy to see that [[LAMBDA].sub.1] is a complex manifold of dimension at most n - 1, and according to [21] (Proposition 3.3), [[LAMBDA.sub.2] is a complex-analytic set of dimension at most n - 2. By considering dimension, we may assume that [Q.sub.0] [intersection] V [not subset] [[LAMBDA].sub.2]. Also, we may assume that [Q.sub.0] [intersection] V [not subset] [[LAMBDA].sub.1]; since otherwise we can perform a linear fractional transformation such that [H.sub.0] is mapped onto another complex hyperplane H [subset] [P.sup.n] with H [intersection] M' = [empty set]. Thus, by the holomorphic extension along [Q.sub.t] we can find points in [Q.sub.0] where h extends as a holomorphic correspondence. This implies that in the new coordinates [Q.sub.0] [not subset] E.

As a consequence, we deduce the following

Lemma 4. [[pi].sup.-1.sub.1] (0) is discrete.

Proof. It suffices to show that 0 [not member of] E. In view of Lemma 3, we may assume that [Q.sub.0] [not subset] E. By contradiction, suppose that 0 [member of] E. It follows that there exist a point b [member of] [Q.sub.0] and a small neighborhood [U.sub.b] [contains as member] b such that [U.sub.b] [intersection] E = [empty set]. As in the proof of Lemma 3, we may choose U and [U.sub.b] so small such that for any z [member of] U, the set [Q.sub.z] [intersection] [U.sub.b] is non-empty and connected. Let [summation] = {z [member of] U : [Q.sub.z] [intersection] [U.sub.b] [subset] [S.sub.F]}. We define

X = {(w, w') [member of] (U\[summation]) x [C.sup.n] : F([Q.sub.w] [intersection] [U.sub.b]) [subset] [Q'.sub.w']}.

We follow the convention of using the right prime to denote the objects in the target domain. For instance, [Q'.sub.w'] will mean the Segre variety of w' with the respect to the hypersurface M'.

We prove the following properties of X.

Claim 1.

i) X is not empty;

ii) X is a complex-analytic set in (U\[summation]) x [C.sup.n];

iii) X is closed in (U\[summation]) x [C.sup.n];

iv) [summation] x [C.sup.n] is a removable singularity for X.

Proof. i) In view of Lemma 2, there exists a sequence {[a.sub.j]} [subset] [GAMMA]\(E [union] [summation]) such that [a.sub.j] [right arrow] 0 as j [right arrow] [infinity], f extends holomorphically across [a.sub.j] and the graph of f near ([a.sub.j], f([a.sub.j])) is contained in A. It follows by the invariance property of Segre varieties (see (2.2)) that ([a.sub.j], f([a.sub.j])) [member of] X and so X [not equal to] [empty set].

ii) Let (w, w') [member of] X. Consider an open simply connected set [OMEGA] [subset] [U.sub.b]\[S.sub.F] such that [Q.sub.w] [intersection] [OMEGA] [not equal to] [empty set]. The branches of F are globally defined in [OMEGA]. Since [Q.sub.w] [intersection] [U.sub.b] is connected, the inclusion F([Q.sub.w] [intersection] [U.sub.b]) [subset] [Q'.sub.w'] is equivalent to

[F.sup.j] ([Q.sub.w] [intersection] [OMEGA]) [subset] [Q'.sub.w'], j = 1, ..., m,

where the [F.sup.j] denote the branches of F in [OMEGA]. Recall that Q(w', [bar.w']) denotes a defining polynomial function of M'. The inclusion [F.sup.j]([Q.sub.w] [intersection] [omega]) [subset] [Q'.sub.w'], j = 1, ..., m can be expressed as

Q([F.sup.j](z), [bar.w']) = 0 for any z [member of] [Q.sub.w] [intersection] [OMEGA], j = 1, ..., m.

As in the proof of Lemma 3, we can choose [OMEGA] in the form [OMEGA] = [[OMEGA].sub.1] x '[OMEGA] [subset] C x [C.sup.n-1] such that [Q.sub.w] = {(k('z, [bar.w]), 'z), 'z [member of] '[OMEGA]}, and

Q([F.sup.j](k('z, [bar.w]), 'z), [bar.w']) = 0, for any 'z [member of] '[OMEGA]. (3.3)

Thus, X is defined by an infinite system of holomorphic equations in ([bar.w], [bar.w']). By the Noetherian property of the ring of holomorphic functions, we can choose finitely many points ['z.sup.1], ..., ['z.sup.m] so that (3.3) can be written as a finite system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where k = 1, ..., m, d' is the degree of Q in w' and [[alpha].sup.k.sub.J] are holomorphic functions in w. Thus, X is a complex-analytic set in (U\[summation]) x [C.sup.n].

iii) The set X is closed in (U\[summation]) x [C.sup.n]. Indeed; let ([w.sup.j], [w'.sup.j]) be a sequence in X that converges to ([w.sub.0], [w'.sub.0]) [member of] (U\[summation]) x [C.sup.n], as j [right arrow] [infinity]. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which implies that ([w.sub.0], [w'.sub.0]) [member of] X and thus, X is a closed set.

iv) Now, let us show that [summation] x [C.sup.n] is a removable singularity for X. Let t [member of] [summation]. It follows that [bar.X] [intersection] ({t} x [C.sup.n]) [subset] {t} x {z' : F([Q.sub.t] [intersection] [U.sub.b]) [subset] [Q'.sub.z']}. If w' [member of] F([Q.sub.t] [intersection] [U.sub.b]) [subset] [Q'.sub.z'],, then z' [member of] [Q'.sub.w']. Since [dim.sub.C][Q'.sub.w'] = n - 1, then {z' : F([Q.sub.t] [intersection] [U.sub.b]) [subset] [Q'.sub.z']} has dimension at most 2n - 2 and [bar.X] [intersection] ([summation] x [C.sup.n]) has 2n-dimensional measure zero. Now, Bishop's theorem can be applied to conclude that [summation] x [C.sup.n] is a removable singularity for X.

Now, we continue with the proof of Lemma 4. Let {[a.sub.j]} be a sequence in [GAMMA]\(E [union] [summation]) such that [a.sub.j] [right arrow] 0 as j [right arrow] [infinity]. Then, f extends holomorphically across [a.sub.j] and the graph of f near ([a.sub.j], f([a.sub.j])) is contained in A. Moreover, for small neighborhoods [U.sub.j] [contains as member] [a.sub.j] we have :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.4)

We denote by [bar.X] the closure of X in U x [C.sup.n]. Without loss of generality we may assume that [bar.X] is irreducible. Then in view of (3.4) and by the uniqueness theorem (see for instance [7]) we deduce that [bar.X] = A.

Let [??] be the multiple-valued mapping corresponding to [bar.X]. By construction, for any a' [member of] [??](0), [??](0) = [I'.sub.a']. Since 0' [member of] [??](0) [intersection] M', it follows that [??](0) [subset] M' and so [??](0) is a finite set. Thus, if V is a bounded open neighborhood of [??](0), we may choose U such that [bar.X] [intersection] (U x [partial derivative]V) = [empty set]. Otherwise; there exists a sequence [([z.sub.j], [z'.sub.j]).sub.j] in X such that [([z.sub.j]).sub.j] converges to 0 and [([z'.sub.j]).sub.j] converges to [z'.sub.0] [member of] [partial derivative]V as j [right arrow] [infinity]. This implies that (0, [z'.sub.0]) [member of] [bar.X] and [z'.sub.0] [not member of] [??](0): a contradiction. Then [??] : U [right arrow] V defines a holomorphic correspondence extending f. This contradicts the fact that 0 [member of] E and completes the proof of Lemma 4.

3.2 Conclusion of the proof of Theorem 1.

In view of Lemma 4, f extends as a holomorphic correspondence to a neighborhood U of 0. Then f extends to U as an algebroid m-valued mapping [??] = ([[??].sup.1], ..., [[??].sup.n]) whose components [w.sub.v] = [[??].sup.v](z) satisfy polynomial equations

[w.sup.m.sub.v] + [a.sub.1v] (z)[w.sub.v.sup.m-1] + ... + [a.sub.mv](z) = 0, v = 1, ..., n

with holomorphic coefficients [a.sub.[mu]v] [member of] O(U). In particular, the map f extends continuously to [bar.D] [intersection] U. Then after an appropriate shrinking of U and U' the map f : D [intersection] U [right arrow] D' [intersection] U' defines a proper holomorphic mapping that extends as a holomorphic correspondence in a neighborhood of 0. According to [14], this extension is in fact a holomorphic mapping.

4 Proof of corollary 1 and 2

4.1 Proof of Corollary 1.

a) First, we prove that the mapping f is algebraic (i.e., the graph of the mapping is contained in an irreducible complex n-dimensional algebraic set in [C.sup.n] x [C.sup.n]). If D is not pseudoconvex, there exist [??] [member of] [partial derivative]D and a neighborhood U of [??] such that f extends holomorphically to U. By moving slightly [??], we may assume that f extends to a biholomorphic mapping in a neighborhood of [??]. The classical Webster's theorem ([27]) implies that f extends to an algebraic mapping. Assume now that D is pseudoconvex, which implies that D' is also pseudoconvex. In view of [25] and [17], f extends holomorphically to a neighborhood of p. Then, as above we can conclude that f is algebraic by using Webster's theorem. Now, it follows from Theorem 1 that f extends holomorphically to a neighborhood of p. To finish the proof, we have to show that the set of holomorphic extendability of f is an open dense subset of [partial derivative]D. Let [S.sup.h] = {z [member of] [partial derivative]D : [cl.sub.f] (z) [intersection] [partial derivative]D' [not equal to] [empty set]} and [S.sup.[infinity]] = [partial derivative]D\[S.sup.h].

Claim 2. [S.sup.h] is a dense subset of [partial derivative]D.

Proof. Since f is algebraic, its components [f.sup.j], j = 1, ..., n are also algebraic. Then there exist polynomials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are holomorphic polynomials for all [k.sub.j] [member of] {0, ..., [m.sub.j]} such that

[P.sub.j](z, [f.sup.j](z)) = 0, for all z [member of] D and j = 1, ..., n.

Without loss of generality, we may assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all j = 1, ..., n. If [??] [member of] [S.sup.[infinity]], there exists j [member of] {1, ..., n} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It follows that the polynomial function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] vanishes identically on [S.sup.[infinity]]. If [S.sup.[infinity]] has an interior point, then by the boundary uniqueness theorem (see for instance [7]) the polynomial [??] vanishes identically on [C.sup.n], which implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on [C.sup.n] for some j. This contradiction completes the proof of Claim 2.

Now, the assertion follows from Theorem 1, Claim 2 and the algebraicity of the mapping.

b) First, assume that D' has a global holomorphic peak function at infinity. It suffices to prove that [S.sup.[infinity]] has no interior point. There exists a holomorphic function [phi] on D' satisfying [absolute value of [phi](w)] < 1 on D' and [phi](w) [right arrow] 1 as [absolute value of w] [right arrow] [infinity]. Set G(z) = [phi] [omicron] f (z) - 1. If [S.sup.[infinity]] has an interior point [??] [member of] [partial derivative]D, then the function G(z) [right arrow] 0 as z tends to a boundary point close to [??]. By the boundary uniqueness theorem we get that f [equivalent to] [infinity] on D : a contradiction.

Assume now, that D has a global holomorphic peak function at infinity. Consider the proper holomorphic correspondence [f.sup.-1] : D' [right arrow] D.

Claim 3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is nowhere dense.

Proof. Suppose that [S'.sup.[infinity]] has an interior point [q.sub.0] [member of] [partial derivative]D'. There exists a holomorphic function [phi] on D satisfying [absolute value of [phi](w)] < 1 on D and [phi](w) [right arrow] 1 as [absolute value of w] [right arrow] [infinity]. The function G(z') = [[PI].sub.1 [less than or equal to] j [less than or equal to] m] [phi] [omicron] [g.sup.j](z') - 1] is holomorphic in D'\[[sigma]', [sigma]' [subset] D' is a complex-analytic set of dimension [less than or equal to] n - 1 and [g.sup.1], ..., [g.sup.m] are the branches of [f.sup.-1]. Since G(z') is bounded ([absolute value of G(z')] [less than or equal to] [2.sup.m]), then it extends as a holomorphic function on D'. The function G(z') [right arrow] 0 as z' tends to a boundary point close to [q.sub.0]. By the boundary uniqueness theorem, we get that one of the branch [g.sup.j] [equivalent to] [infinity] on D'. This contradiction completes the proof of the claim.

Claim 3 shows in particular, that [cl.sub.f] ([partial derivative]D) [intersection] [partial derivative]D' is not empty. Then the assertion follows from the result of Corollary 1-a).

4.2 Proof of Corollary 2.

The idea of the proof is as in Corollary 1, but the method of proof is different; since here the domains are only algebraic in a neighborhood of an open piece of the boundary. In view of Theorem 1, it suffices to prove that f is algebraic. Note that here [cl.sub.f] (M) [subset] M.

a) First, assume that n = 2. Without loss of generality we may assume that M is connected. Let U [subset] [C.sup.2] be an open neighborhood such that M = [partial derivative]D [intersection] U = {z [member of] U : P(z, [bar.z]) = 0} with P a real polynomial and dP [not equal to] 0 on [bar.M]. We denote by [M.sup.+.sub.s] the set of all strictly pseudoconvex points of M and [M.sup.-.sub.s] the set of all strictly pseudoconcave points of M. Let [M.sup.+] (resp. [M.sup.-]) denote the interior of [bar.[M.sup.+.sub.s]] (resp. [bar.[M.sup.-.sub.s]]). The set [M.sup.+] is the pseudoconvex part of M and [M.sup.-] is the pseudoconcave part of M. It is known that [M.sup.-] [subset] [??], where [??] denotes always the envelope of holomorphy of D. Let T = {z [member of] M : [L.sub.p](z) = 0}, where [L.sub.p] is the Levi-form of the boundary restricted to [T.sup.c.sub.z]M, the complex tangent space at z to M. The set T is a real-algebraic set in M of dimension at most 2. It admits a locally finite semi-algebraic stratification as, T = [T.sub.0] [union] [T.sub.1] [union] [T.sub.2], where [T.sub.i] for i = 0, 1, 2 are disjoint union of connected real-algebraic submanifolds of M of real dimension i. The set [M.sub.b] := M\{[M.sup.+] [union] [M.sup.-]} is called the border set in M. It is a closed semi-algebraic subset of M and [M.sub.b] [subset] T. Let [M.sub.e] := [M.sub.b] [intersection] ([T.sub.0] [union] [T.sub.1]), be called the exceptional set. The set [M.sub.e] is a pluripolar set. It was shown in [12] that [M.sub.b]\[M.sub.e] [subset] [??].

Let U' [subset] [C.sup.2] be an open neighborhood such that M' = [partial derivative]D' [intersection] U' = {z [member of] U' : Q(z', [bar.z']) = 0} with Q a real polynomial and dQ [not equal to] 0 on [bar.M']. We follow the same notations as above by using the right prime to denote the objects in the target domain. Let p [member of] M, then there exists a point q [member of] M' such q [member of] [cl.sub.f](p). We consider several cases :

(1)--If p [member of] [M.sup.-], then there exists a neighborhood V of p such that f extends holomorphically to V. Hence, by moving slightly p (if necessary), we may assume that f extends biholomorphically near p. Now, the classical Webster's theorem implies that f extends to an algebraic mapping.

(2)--If p [member of] [M.sup.+] and q [member of] [M'.sup.+], then in view of [25] and [17], f extends holomorphically to a neighborhood of p. Hence, we can conclude as above that f is algebraic.

(3)--If p [member of] [M.sup.+] and q [member of] [M'.sup.-] [union] [M'.sub.b]. Let Y := {z [member of] M : [cl.sub.f](z) [subset] [M'.sub.e]}. Since [M'.sub.e] is pluripolar, there exists [phi] [member of] PSH([C.sup.2]), [phi] [not equivalent to] -[infinity] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [psi] = [phi] [omicron] f [member of] PSH(D) and [phi](z) [right arrow] -[infinity] as z [right arrow] [z.sup.0] [member of] Y. The set Y has no interior point, as otherwise [psi] [equivalent to] -[infinity] and this is a contradiction. Hence, by moving slightly p, we may assume that q [member of] [M'.sup.-] [union] ([M'.sub.b]\[M'.sub.e]). We need the following observation due to Diederich-Pinchuk [16].

Lemma 5. Let f : D [right arrow] D' be a proper holomorphic mapping. Assume that there exist a [member of] [partial derivative]D and a' [member of] [partial derivative]D' such that a' [member of] [cl.sub.f](a). Then a [member of] [??], if a' [member of] [??]'.

Since [M'.sup.-] [union] ([M'.sub.b]\[M'.sub.e]) [subset] [??]', then in view of Lemma 5 we deduce that p [member of] [??]. Hence, again Webster's theorem implies that f extends to an algebraic mapping.

b) Next, assume that n [greater than or equal to] 2. We denote by [M.sup.+.sub.s] the set of strictly pseudoconvex points of M and [M.sup.-.sub.s] the set of strictly pseudoconcave points of M. The set of points where the Levi-form [L.sub.p] has eigenvalues of both signs on [T.sup.c] (M) and no zero will be denoted by [M.sup.[+ or -]] and by [M.sup.0] we mean the set of points of M where [L.sub.p] has at least one eigenvalue 0 on [T.sup.c] (M). Note that [M.sup.0] is a closed real-algebraic set of dimension at most 2n - 2. We have

M = [M.sup.+.sub.s] [union] [M.sup.-.sub.s] [union] [M.sup.[+ or -]] [union] [M.sup.0].

It is well known that [M.sup.-.sub.s] [union] [M.sup.[+ or -]] [subset] [??]. Then if M is not pseudoconvex, the same argument used in a)-(1) shows that the mapping f is algebraic. For the rest of the proof we may suppose that M is pseudoconvex. Let p [member of] B [subset] M [intersection] U be a strictly pseudoconvex boundary point (such a point exists; since M is pseudoconvex and B is dense). It suffices to prove that f extends holomorphically to a neighborhood of p. We consider several cases:

--Assume, that q = f(p) [member of] [M'.sup.+.sub.s]. Then in view of [25], f extends continuously to a neighborhood of p and in view of [17], f extends holomorphically to a neighborhood of p.

--Assume next, that q = f(p) [member of] [M'.sup.[+ or -]] [union] [M'.sup.-.sub.s]. Then by Lemma 5, we deduce that p [member of] [??]. Hence, f extends holomorphically to a neighborhood of p.

--Finally, assume that q = f(p) [member of] [M'.sub.0]. We need the following lemma (appeared in [17] in the case of continuous CR-mapping between real-analytic hypersurfaces in [C.sup.n]).

Lemma 6. Let N' [subset] M' be a real [C.sup.2]-smooth generic manifold of real dimension at most 2n - 2 that contains q and let V be a neighborhood of p [member of] [M.sup.+.sub.s]. Then [cl.sub.f] (M [intersection] V) [??] N'.

Proof. We follow the ideas of [17] with some minor modifications. If [dim.sub.R]N' < 2n - 2, we can always find a generic manifold in M' of dimension 2n - 2 which contains N'. Then without loss of generality, we may assume that [dim.sub.R]N' = 2n - 2 and q = 0'. There exists a complex plane L' [contains as member] 0' such that L' [intersection] N' is a totally real-manifold of real dimension 2 near 0'. For a' [member of] [C.sup.n], let [L.sub.a'] be the complex plane parallel to L' and passing through a'. For a small neighborhood V' of 0' in [C.sup.n] the intersection [L.sub.a'] [intersection] N' [intersection] V' is a totally real-manifold of real dimension 2. There exists a strictly plurisubharmonic function [[phi].sub.a' on V' such that:

* [[phi].sub.a'] [greater than or equal to] 0

* [[phi].sub.a'] = 0 on [L.sub.a'] [intersection] N' [intersection] V'.

Since p is a strictly pseudoconvex point, then in a new coordinates we may assume that p = 0 and the defining function r of D can be written near 0 as

r(z, [bar.z]) = 2Re([z.sub.1]) + [[absolute value of 'z].sup.2] + o([[absolute value of z].sup.2]).

Let a [member of] D [intersection] V be a point closed to 0 such that f(a) [member of] V'. Set [H.sub.a] := {z [member of] D [intersection] V : [z.sub.1] = [a.sub.1]}. Notice that [H.sub.a] is a complex-manifold of dimension n - 1 and [H.sub.a] [subset] [subset] V. Set [A.sub.a] := [H.sub.a] [intersection] [f.sup.-1] ([L.sub.f(a)] [intersection] V'), which is a complex-analytic set in D [intersection] V. Since f is proper and [dim.sub.C][L.sub.f(a)] = 2, the complex dimension of [A.sub.a] is at least 1. If [cl.sub.f] (M [intersection] V) [subset] N', we would have [cl.sub.f] ([partial derivative][H.sub.a]) [subset] N'. The function [g.sub.a] = [[phi].sub.f(a)] [omicron] f is plurisubharmonic and positive on [A.sub.a] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It follows by the maximum principle that [g.sub.a] [equivalent to] 0 on [A.sub.a]. But [[phi].sub.f(a)] is strictly plurisubharmonic, then f is constant on [A.sub.a] with image in N'. This is a contradiction; since f(a) [member of] D'.

Now, we continue with the proof of Corollary 2. We want to show that there exists an open set [U.sup.1] [subset] U such that [cl.sub.f] (M [intersection] [U.sup.1]) [??] [M'.sub.0].

The set [M'.sub.0] can be stratified as [M'.sub.0] = [[union].sub.k][N'.sub.k] by smooth generic manifolds [N'.sub.k] of dimension k less or equal to 2n - 2. Let [j.sub.0] be the largest index such that [cl.sub.f] (M [intersection] U) [intersection] [N'.sub.j] [not equal to] [empty set]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [cl.sub.f] (M [intersection] U) [intersection] [N'.sub.j] = [empty set] for all j > [j.sub.0]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, there exists a sequence {[p.sub.k]} in D such that [p.sub.k] [right arrow] p and f([p.sub.k]) [right arrow] b'. Let [epsilon] be a positive real number so that D(b', 2[epsilon]) [intersection] [N'.sub.j] = [empty set] for all j < [j.sub.0] and let [k.sub.1] = [k.sub.1]([epsilon]) be an integer such that for all k > [k.sub.1], [absolute value of f([p.sub.k]) - b'] < [epsilon]

Claim 4. There exists a point a [member of] [M.sup.+.sub.s] [intersection] U with the following properties: f is continuous at a and f(a) [member of] D(b', 2[epsilon]).

Proof. Let {[a.sub.k]} be a sequence in B with [a.sub.k] [right arrow] p. Since f extends to a uniformly continuous function on B, there exist a real number [alpha] > 0 and an integer [k.sub.2] such for all z [member of] D and k > [k.sub.2], [absolute value of z - [a.sub.k]] < [alpha] [??] [absolute value of f(z) - f([a.sub.k])] < [epsilon]. Starting with some integer [k.sub.3], we have [absolute value of [a.sub.k] - [p.sub.k]] < [alpha]. Hence, for all k [greater than or equal to] max([k.sub.1], [k.sub.2], [k.sub.3]), [absolute value of f([a.sub.k]) - b'] < 2[epsilon].

By Claim 4, there exists [U.sup.1] [subset] U, a neighborhood of a, such that f(D [intersection] [U.sup.1]) [subset] D(b', 2[epsilon]). It follows that [cl.sub.f] (M [intersection] [U.sup.1]) [subset] D(b', 2[epsilon]). But [cl.sub.f] (M [intersection] [U.sup.1]) [intersection] [N'.sub.j] = [empty set] for all j > [j.sub.0] and D(b', 2[epsilon]) [intersection] [N'.sub.j] = [empty set] for all j < [j.sub.0]. Hence, if [cl.sub.f] (M [intersection] [U.sup.1]) [subset or equal to] [M'.sub.0], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]: a contradiction with Lemma 6. Now, as in the previous cases we can show that f is algebraic. This finishes the proof of Corollary 2.

5 Proof of theorem 2 and 2-bis, and corollary 3

5.1 Proof of Theorem 2

Let f : D [right arrow] D' be a proper holomorphic mapping as in Theorem 2. Following the proof of Corollary 1-b), it is clear that if D or D' has a global holomorphic peak function at infinity, then [cl.sub.f] ([partial derivative]D) intersects [partial derivative]D'. Now, by repeating the argument used in the proof of Corollary 1-a), we may show that f is algebraic.

(A) First, assume that P is plurisubharmonic on D and [partial derivative][D.sub.c] is nondegenerate for all c < 0. We denote by [J.sub.f] the Jacobi determinant of f and by [V.sub.f] = {z [member of] D : [J.sub.f](z) = 0} its branch locus. Following [3] and [4], we consider the Levi-determinant of D defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that [[LAMBDA].sub.[partial derivative]D](z) [greater than or equal to] 0 for all z [member of] [partial derivative]D and the set

[omega]([partial derivative]D) = {z [member of] [partial derivative]D : [[GAMMA].sub.[partial derivative]D](z) = 0}

is precisely the set of weakly pseudoconvex boundary points. For any point p [member of] [partial derivative]D we consider also the order of vanishing of the Levi-determinant at p denoted by [tau](p), which is defined as follows: we choose smooth coordinates x = ([x.sub.1], ..., [x.sub.2n-1]) on [partial derivative]D such that p corresponds to x = 0, and the formal power series

[[GAMMA].sub.[partial derivative]D](x) = [[infinity].summation over (j=0)] [summation over ([absolute value of [alpha]] = j)] [a.sub.[alpha]][x.sup.[alpha]],

where [alpha] = ([[alpha].sub.1], ..., [[alpha].sub.2n-1]) is a multi-index,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [absolute value of [alpha]] = [[alpha].sub.1] + ... + [[alpha].sub.2n-1]. We set

[tau](p) = min{[absolute value of [alpha]] : [a.sub.[alpha]] [not equal to] 0}.

The definition does not depend on the choice of the coordinates. The number [tau](p) can be also defined as the smallest nonnegative integer m such that there is a tangential differential operator T of order m on [partial derivative]D (i.e., T is a differential operator of order m satisfying TP = 0) such that T[[GAMMA].sub.[partial derivative]D](p) [not equal to] 0. The function [tau] is upper-semicontinuous. Indeed, for each [lambda] > 0 the set {z [member of] [partial derivative]D : [tau](z) < [lambda]} is open; since its complement {z [member of] [partial derivative]D : T[[GAMMA].sub.[partial derivative]D] (z) = 0 for all T, with order < [lambda]} is closed, being the intersection over all T of closed zero sets of the smooth functions T [[GAMMA].sub.[partial derivative]D]. Note that {p [member of] [partial derivative]D : [tau](p) = 0} is the set of strictly pseudoconvex boundary points and {p [member of] [partial derivative]D : [tau](p) [greater than or equal to] 1} is the set of weakly pseudoconvex boundary points.

Lemma 7. Let f : D [right arrow] D' be a proper holomorphic mapping as in Theorem 2. Then for all p [member of] [S.sup.h], the set of holomorphic extendability of f, [tau](p) [greater than or equal to] [tau](f(p)) and the inequality holds if and only if f is branched at p.

Proof. Let p [member of] [S.sup.h] [intersection] {[J.sub.f] [not equal to] 0}. Since Q is a defining function for D', [nabla](Q [omicron] f)(p) [not equal to] 0. Then Q [omicron] f is a local defining function of D in a neighborhood of p, and by the chain rule we have:

[[LAMBDA].sub.Q [omicron] f](p) = [[absolute value of [J.sub.f](p)].sup.2] [[LAMBDA].sub.Q] (f(p)).

Hence, we are able to deduce the lemma.

Following [3] (or [9]), there is a semi-algebraic stratification for [omega]([partial derivative]D) given by

[omega]([partial derivative]D) = [A.sub.1] [union] [A.sub.2] [union] [A.sub.3] [union] [A.sub.4]

where [A.sub.4] is a closed, real-algebraic set of dimension at most 2n - 4 and [A.sub.2] [union] [A.sub.3] [union] [A.sub.4] is also a closed, real algebraic set of dimension at most 2n - 3. Further, [A.sub.1], [A.sub.2] and [A.sub.3] are either empty or smooth, real-algebraic manifolds; [A.sub.2] and [A.sub.3] have dimension 2n - 3, and [A.sub.1] has dimension 2n - 2. When [A.sub.2] and [A.sub.3] are non-empty, [A.sub.2] and [A.sub.3] are CR manifolds with

[dim.sub.C][T.sup.c] [A.sub.2] = n - 2 and [dim.sub.C][T.sup.c][A.sub.3] = n - 3.

Recall that [dim.sub.C][T.sup.c][A.sub.j] denotes the complex dimension of the complex tangent space to [A.sub.j] (j [member of] {2,3}). Finally, the function [tau] is constant on every connected component of [A.sub.1].

Lemma 8. Let W be an irreducible component of [V.sub.f] and [[epsilon].sub.W] := [bar.W] [intersection] [partial derivative]D.

1) There exists an open dense subset [O.sub.W] of [[epsilon].sub.W] such that for all p [member of] [O.sub.W]:

i) W extends across the boundary of D as a pure (n - 1)-dimensional polynomial variety in [C.sup.n] and [[epsilon].sub.W] is a polynomial submanifold of dimension 2n - 3 in a neighborhood of p.

ii) f extends holomorphically in a neighborhood of p.

2) [bar.W] does not intersect the set [partial derivative]D\w([partial derivative]D) of strictly pseudoconvex boundary points.

Proof. 1) -i) Since W is an irreducible algebraic set in D of dimension n - 1, there exists an irreducible polynomial h in [C.sup.n] such that W = {z [member of] D : h(z) = 0}. If W does not extend across [partial derivative]D, the polynomial P will be negative on [??] = {z [member of] [C.sup.n] : h(z) = 0}. According to [7] (Proposition 2, page 76), there exists an analytic cover [pi] : [??] [right arrow] [C.sup.n-1]. Let [g.sup.1], ..., [g.sup.k] be the branches of [[pi].sup.-1] which are locally defined and holomorphic in [C.sup.n-1]\[sigma], with [sigma] [subset] [C.sup.n-1] a complex-analytic set of dimension at most n - 2. Consider the function [??](w) = sup{P [omicron] [g.sup.1](w), ..., P [omicron] [g.sup.k] (w)}. Since [pi] is an analytic cover, [??] extends as a plurisubharmonic function to [C.sup.n-1]. Then there exists a negative constant c such that [??] = c; since [??] is negative. It follows that for all [w.sub.0] [member of] [C.sup.n-1]\[sigma], there exist a neighborhood [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and an integer s [member of] {1, ..., k} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This contradicts the nondegeneracy of [partial derivative][D.sub.c].

Let us verify now that there exists an open dense subset [O.sub.w] of [[epsilon].sub.w] such that for all p [member of] [O.sub.w], [[epsilon].sub.W] is a polynomial submanifold of dimension 2n - 3 in a neighborhood of p. We may choose a multi-index [alpha] = ([[alpha].sub.1], ..., [[alpha].sub.n]) such that v := [[partial derivative].sup.[alpha]]h/[partial derivative][z.sup.[alpha]] vanishes on W; but [nabla]v is not identically zero on W. We need the following observation (appeared in [2] in the strictly pseudoconvex case).

Claim 5. Let [OMEGA] be a pseudoconvex domain in [C.sup.n] with real-analytic and nondegenerate boundary. For any point p [member of] [partial derivative][OMEGA], there exists [[epsilon].sub.0] > 0 so that for any [epsilon] [member of] (0, [[epsilon].sub.0]), there is some [delta] [member of] (0, [member of]) with the following property : for any point q [member of] [OMEGA] [intersection] B( p, [delta]), there exists a plurisubharmonic function [[phi].sub.q] on [OMEGA] [intersection] B(p, [[epsilon].sub.0]), continuous on [bar.[OMEGA]] [intersection] B(p, [[epsilon].sub.0]) such that [[phi].sub.q] (q) = 1 and 0 < [[phi].sub.q] (z) < 1/2 for all z [member of] D [intersection] [partial derivative]B (p, [epsilon]).

Proof. According to [11], there exists a local plurisubharmonic peak function at p (i.e., there exist a small [[epsilon].sub.0] > 0 and a function [[psi].sub.p] [member of] PSH ([OMEGA] [intersection] B(p, [[epsilon].sub.0])) [intersection] [C.sup.0] ([bar.[OMEGA]] [intersection] B(p, [[epsilon].sub.0])) such that [[psi].sub.p](p) = 1 and [[phi].sub.p](z) < 1 on ([bar.[OMEGA]] [intersection] B(p, [[epsilon].sub.0]))\{p}). Set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and for 0 < [epsilon] < [[epsilon].sub.0], let M = max{[[phi].sub.p](z) : z [member of] [bar.[OMEGA]] [intersection] [partial derivative]B(p, [epsilon])}. Note that 0 < M < 1. The set {z [member of] [bar.[OMEGA]] [intersection] B(p, [epsilon]) : [[phi].sub.p](z) > M} is an open neighborhood of p in [bar.[OMEGA]] [intersection] B(p, [epsilon]), so there is some [delta] > 0 so that 0 < M < [[phi].sub.p](z) for all z [member of] [OMEGA] [intersection] B(p, [delta]). For any q [member of] [OMEGA] [intersection] B(p, [delta]), let [h.sub.q](z) = [[phi].sub.p](z)/[[phi].sub.p](q). Then [h.sub.q](q) = 1 and for all z [member of] [bar.[OMEGA]] [intersection] [partial derivative]B(p, [epsilon]), 0 < [h.sub.q](z) < M/[[phi].sub.p](q) < 1. It follows that for N > 0 large enough, the function [[phi].sub.q](z) = [([h.sub.q](z)).sup.N] has the properties given by the claim.

Let [W.sup.1] = {z [member of] W : ([partial derivative]v/[partial derivative][z.sub.1])(z) = 0}. Using the irreducibility assumption, [W.sub.1] is a nowhere dense subvariety of W. First, we show that for p [member of]e [[epsilon].sub.W], ([partial derivative]v/[partial derivative][z.sub.1]) can not vanish everywhere on {[absolute value of z - p] [less than or equal to] [[epsilon].sub.0]} [intersection] [[epsilon].sub.W] for any [[epsilon].sub.0] > 0. We may assume that [[epsilon].sub.0] is the real number given by Claim 5 corresponding to p. Let [epsilon] [member of] (0, [[epsilon].sub.0]) and a corresponding [delta] > 0. Let [z.sup.1] [member of] (W\[W.sub.1]) [intersection] B(p, [delta]). By Claim 5, there exists a function [phi] [member of] PSH(D [intersection] B(p, [[epsilon].sub.0])) [intersection] [C.sup.0]([bar.D] [intersection] B(p, [[epsilon].sub.0])) such that [phi]([z.sup.1]) = 1 and 0 < [phi](z) < 1/2 for all z [member of] D [intersection] [partial derivative]B(p, [epsilon]). Note that for any natural number N, the function [q.sup.N] is plurisubharmonic; since by construction, [phi] is the exponential of a plurisubharmonic function. Assume that ([partial derivative]v/[partial derivative][z.sub.1]) vanishes everywhere on {[absolute value of z - p] [less than or equal to] [[epsilon].sub.0]} [intersection] [[epsilon].sub.W]. There exists a constant C > 0 such that [absolute value of ([partial derivative]v/[partial derivative][z.sub.1])(z)] < C for all z [member of] B(p, [epsilon].sub.0]). By the maximum principle, 0 < [absolute value of ([partial derivative]v/[partial derivative][z.sub.1])(z)] ([[phi].sup.N](z)) < C/[2.sup.N] for all integer N and for all z [member of] W [intersection] B(p, [epsilon]). Letting N [right arrow] [infinity], we conclude that ([partial derivative]v/[partial derivative][z.sub.1])([z.sup.1]) = 0, which is a contradiction. Then there exists an open dense subset [O.sub.w] of [[epsilon].sub.W] such that for any q [member of] [O.sub.w], ([partial derivative]v/[partial derivative][z.sub.1])(q) [not equal to] 0. For a fixed q [member of] [O.sub.W], there exists a neighborhood U of q in [C.sup.n] such that ([partial derivative]v/[partial derivative][z.sub.1]) vanishes nowhere on U. Then [??] = {z [member of] U : v(z) = 0} is a polynomial submanifold of U. Since W extends across the boundary of D as a variety, a useful consequence of this fact is that WW has dimension 2n - 3. Otherwise, the Hausdorff dimension of [??] will be less than or equal to 2n - 4. Then [??]\([??] [intersection] [partial derivative]D) will be connected (see [7]). This implies that [??] cannot be separated by [partial derivative]D and contradicts the first statement of this lemma.

1) -ii) Since f is algebraic, its components [f.sup.j], j = 1, ..., n are also algebraic. Then there exist polynomials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are holomorphic polynomials for all [k.sub.j] [member of] {0, ..., [m.sub.j]} such that

[P.sub.j] (z, [f.sup.j] (z)) = 0, [for all]z [member of] D and j = 1, ..., n. (5.1)

Consider (5.1) only for z [member of] W. First, note that we may assume that for all j = 1, ..., n, there exists [k.sub.j] [member of] {1, ..., [m.sub.j]} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Without loss of generality, assume that [k.sub.j] = [m.sub.j] for all j. If p [member of] [S.sup.[infinity]] [intersection] [[epsilon].sub.W] (recall that [S.sup.[infinity]] = [partial derivative]D\[S.sup.h]), then there exists j [member of] {1, ..., n} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So the polynomial function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] vanishes identically on [S.sup.[infinity]] [intersection] [[epsilon].sub.W]. To Show that [S.sup.h] [intersection] [O.sub.w] is dense in [O.sub.W], suppose by contradiction that [S.sup.[infinity]] [intersection] [O.sub.W] has an interior point. Then by the boundary uniqueness theorem the polynomial [??] vanishes identically on W. As h is irreducible, h divides [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some j, contradicting the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on W for all j.

This contradiction shows that [O.sub.W] = [S.sup.h] [intersection] [O.sub.W] has the properties claimed by the lemma.

2) Now, let prove that [bar.W] does not intersect the set of strictly pseudoconvex boundary points. Let p [member of] [O.sub.W]. As in the proof of Lemma 7, we have

[[GAMMA].sub.Q [omicron] f](p) = [[absolute value of [J.sub.f](p)].sup.2] [[GAMMA].sub.Q](f(p)).

So [O.sub.W] [subset] w([partial derivative]D), which implies that [[epsilon].sub.W] [subset] w([partial derivative]D).

Conclusion of the proof of Theorem 2-(A). The conclusion is similar to [3] and [9]. For completeness we add it here. The algebraic set [A.sub.2] contains finitely many components, which we will denote as [[sigma].sub.1], ..., [[sigma].sub.N]. Since [dim.sub.R][[sigma].sub.j] = [dim.sub.R][T.sup.c][[sigma].sub.j], then [[sigma].sub.j] is a complex-manifold of dimension n - 1. Let [[??].sub.j] be the complex-algebraic set in [C.sup.n] such that Reg([[??].sub.j]) = [[sigma].sub.j]. By considering dimension and CR dimension, we see that [A.sub.3] [intersection] [O.sub.W] and [A.sub.4] [intersection] [O.sub.W] are nowhere dense in [O.sub.W]. Next we claim that [A.sub.1] [intersection] [O.sub.W] cannot contain an open subset of [O.sub.W]. By contradiction, let suppose p [member of] [O.sub.W] [subset] [A.sub.1]. We may choose a sequence [{[q.sub.k]}.sub.k] [subset] [A.sub.1] [intersection] {[J.sub.f] [not equal to] 0} such that [q.sub.k] [right arrow] p. The mapping f is a local diffeomorphism in a neighborhood of all points [q.sub.k] and the function [tau] is constant on every connected component of [A.sub.1], then for all k,

[tau](p) = [tau]([q.sub.k]) = [tau](f([q.sub.k])).

On the other hand, by Lemma 7,

[tau](p) > [tau](f(p)).

This is a contradiction; since [tau] is upper-semicontinuous. We conclude that [A.sub.2] [intersection] [O.sub.W] contains an open subset of [A.sub.2]. Thus, it contains an open subset of [[sigma].sub.j] for some j. Applying the maximum principle, we conclude that W [subset] [[??].sub.j]. This completes the proof of Theorem 2-(A).

(B) In this case, suppose that Q is plurisubharmonic on D' and for all c < 0, [partial derivative][D'.sub.c] is nondegenerate. The only crucial point here is to show that the branch locus of f extends across the boundary of D and the rest of the proof is as in the case (A). The set W denotes always an irreducible component of [V.sub.f]. The set f(W) is an irreducible algebraic set of dimension n - 1 in D', then there exists an irreducible polynomial [??] in [C.sup.n] such that f(W) = {z' [member of] D' : [??](z') = 0}. If W does not extend across [partial derivative]D, then Q(z') [less than or equal to] 0 for all z' [member of] W' = {z' [member of] [C.sup.n] : [??](z') = 0}. By repeating the argument used in the proof of Lemma 8 (first part), we may show that there exists a negative constant c such that {Q = c'} contains a complex-analytic set with positive dimension. This contradicts the nondegeneracy of [partial derivative][D'.sub.c'].

Proof of Theorem 2-bis. Let f : D [right arrow] D' be a proper holomorphic mapping as in Theorem 2-bis. Then f is algebraic (the proof of this fact can be deduced easily from the proof of Corollary 1 if D is bounded and from Remark 1 and the proof of Corollary 1 if D is unbounded). It suffices to show that the branch locus extends across the boundary of the domain and the rest of the proof is as in the proof of Theorem 2-(A). According to Remark 2, D is biholomorphic to a bounded pseudoconvex smooth algebraic domain [OMEGA] in [C.sup.n]. (Recall that the boundary of any bounded real-analytic domain in [C.sup.n] is nondegenerate). Let g : D [right arrow] [OMEGA] be such a biholomorphism. Note that g is an algebraic mapping. Let G = f [omicron] [g.sup.-1] : [OMEGA] [right arrow] D', W be an irreducible component of [V.sub.f] and W = g(W). According to [13], [OMEGA] has a bounded negative plurisubharmonic exhaustion function [rho] (i.e., a continuous real negative plurisubharmonic function on [OMEGA] such that {z [member of] [OMEGA] : [rho](z) < c} is a compact subset of [OMEGA] for each constant c < 0 and [partial derivative][OMEGA] = {[rho] = 0}). It suffices to prove that W extends across the boundary of [OMEGA]. Let us assume that [rho](z) [less than or equal to] 0 for all z [member of] W and argue by contradiction. Since W is an algebraic set of dimension n - 1, then again by repeating the argument used in the proof of Lemma 8 (first part) we may show that there exists a negative constant c such that {z [member of] [OMEGA] : [rho](z) = c} contains a complex-analytic set with positive dimension : a contradiction; since {z [member of] [OMEGA] : [rho](z) = c} is a compact.

5.2 Proof of Corollary 3

In view of the simple connectedness of D, it suffices to prove that [V.sub.f] is empty. Since [cl.sub.f]([partial derivative]D) [intersection] [partial derivative]D [not equal to] [empty set], by repeating the arguments used in the proof of Corollary 1, we may show that f is algebraic and [S.sup.[infinity]] has no interior point in [partial derivative]D. Let [z.sub.0] [member of] [S.sup.h] [intersection] {[J.sub.f] [not equal to] 0}. There exists a neighborhood U of [z.sub.0] such that f is a diffeomorphism from U [intersection] [partial derivative]D onto f(U [intersection] [partial derivative]D). Since [S.sup.[infinity]] has no interior point and [partial derivative]D = [S.sup.h] [union] [S.sup.[infinity], there exists [[??].sub.0] [member of] U [intersection] [partial derivative]D such that f([[??].sub.0]) [member of] [S.sup.h]. This proves that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The same argument shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all N, where [f.sup.N] denotes the N-th iteration of f. This leads to prove that [f.sup.N] is algebraic for all N (see the proof of Corollary 1). Now, the proof of Theorem 2-(A) shows that for all N the variety [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has a finite number of irreducible components independent of [f.sup.N]. Then there exists an integer s such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We may assume s = 1, that is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it follows that [V.sub.f] [subset or equal to] f([V.sub.f]), where f([V.sub.f]) is a complex-analytic variety of D by a theorem of Remmert (see [7]). Hence, we have [V.sub.f] = f([V.sub.f]) because [V.sub.f] has finitely many components. Assume that [V.sub.f] is not empty. According to Lemma 8, there exists a boundary point p [member of] [[bar.V].sub.f] [intersection] [partial derivative]D such that f extends holomorphically in a neighborhood of p. Note that for all N, [f.sup.N](p) [member of] [[bar.V].sub.f]; since [V.sub.f] = f([V.sub.f]). The sequence of numbers [tau]([f.sup.N](p)) is strictly decreasing and [tau](p) is a finite integer, so there exists an integer [N.sub.0] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a strictly pseudoconvex boundary point, contradicting the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This proves that [V.sub.f] = [empty set] and completes the proof.

Acknowledgements. The authors are very grateful for the referee's comments and suggestions which improves the paper greatly.

References

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Faculte des Sciences de Monastir, (Monastir), 5019, Tunisia

email:ayed_besma@y ahoo.fr

King Saud University, Department of mathematic, P. O. Box 2455, Riyadh 1145 1, King Saudi Arabia

Current address: Faculte des Sciences de Bizerte, Jarzouna, 7021, Tunisia

email: ourimin@yahoo.com

* Supported by the King Saud University D.S.F.P. program.

Received by the editors April 2008--In revised form in February 2009.

Communicated by F. Brackx.
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