Proper holomorphic mappings between rigid polynomial domains in Cn+1

Publ. Mat. 45 (2001), 69–77 PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN Cn+1 Bernard Coupet and Nabil Ourimi Abstract We describe the branch locus of proper holomorphic mappings between rigid polynomial domains in Cn+1. It appears, in particular, that it is controlled only by the first domain. As an application, we prove that proper holomorphic self-mappings between such domains are biholomorphic. 1. Introduction A domain D⊂Cn+1 is called rigid polynomial if D={(z0,z)∈Cn+1 :r(z0,z)=2Re(z0)+P(z, ¯z)<0} for some real polynomial P(z)=P(z, ¯z). We say that Dis nondegenerate if its boundary {(z0,z)∈Cn+1 : 2 Re(z0)+P(z)=0}contains no nontrivial complex variety. When Pis homogeneous these domains naturally appear as approximation of domains of finite type and may be considered as their homogeneous models. These ones are useful in studies of many problems for more general domains (see for instance [7]). The main result of this paper describes the branch locus of proper holomorphic mappings between rigid polynomial domains in Cn+1. Let f:D→Ω be a holomorphic mapping between domains in Cn+1.We will denote by Jf(z0,z) the Jacobian determinant of fand by Vf= {(z0,z)∈D:Jf(z0,z)=0}its branch locus. Our principal result is the following. 2000 Mathematics Subject Classification. 32H35. Key words. Proper holomorphic mappings, rigid polynomial nondegenerate pseudoconvex domains. 70 B. Coupet, N. Ourimi Theorem 1. Let Dand Ωbe rigid polynomial nondegenerate pseudoconvex domains in Cn+1. Then there exists a finite number of complex algebraic varieties ˆ B1,..., ˆ BNin Cn(irreducible) depending only on D such that the branch locus of any proper holomorphic mapping f:D→Ω satisfies: Vf⊂∪ 1≤k≤N{(z0,z)∈D:z∈ˆ Bk}. Note that the integer Nis bounded by the degree of the polynomial P. In the bounded strongly pseudoconvex case, the branch locus is empty [17], and in the real analytic case, one gives a nice description using semi-analytic stratification of the boundary (as it was observed in [9], this argument works in the smooth case as well if the set of weakly pseudoconvex boundary point admits a nice stratification). On the other hand, Rudin [18], Bedford [5], Forstneriˇc[15], BarlettaBedford [4], proved that the structure of the branch locus of a proper holomorphic mapping relies on properties of its automorphism group via factorization type theorems. As an immediate application of Theorem 1, one has the following corollary. Corollary 1. Let Dbe a rigid polynomial nondegenerate pseudoconvex domain in Cn+1. Then every proper holomorphic self-mapping f:D→ Dis a biholomorphism. For the case n= 1, this result was proved in [13] and [11]. Now, we recall some definitions and results that we will need for the proof of Theorem 1. A mapping in Cn+1 is algebraic if there exists an irreducible algebraic set of dimension n+1inCn+1 ×Cn+1 which contains the graph of the map. Thus, this map may be extended to a possibly multiple valued map defined on the complement of an algebraic set in Cn+1. Webster [19] proved that a locally biholomorphic mapping taking an algebraic nondegenerate hypersurface into another one is algebraic. Let f:D→Ω be a proper holomorphic mapping satisfying the assumption of Theorem 1. According to Coupet-Pinchuk [14], fis algebraic. Furthermore, if the cluster set of a boundary point a∈∂D contains a point b∈∂Ω, then fextends holomorphically to a neighborhood of a. Therefore, there exists an algebraic set ˆ S⊂∂D such that fextends holomorphically to a neighborhood of any point from ∂D\ˆ S and for all p∈ˆ S, limz→p|f(z)|=+∞. Then we get the following stratification of the boundary: ∂D =Sh∪ˆ S Proper Holomorphic Mappings 71 where Shis the set of points pin ∂D such that fextends holomorphically in a neighborhood of p. Note that this result of Coupet-Pinchuk does not assume the pseudoconvexity of the domains. 2. Behavior of the mapping and its branch locus on the boundary For an irreducible component Wof Vf, we define EW:= W∩∂D. Lemma 1. (1) Wextends across the boundary of Das a pure n-dimensional polynomial variety in Cn+1. (2) There exists an open dense subset OW⊂EWsuch that for each p∈OW: (i) EWis a polynomial submanifold in a neighborhood of pof dimension 2n−1. (ii) fis holomorphic in a neighborhood p. Proof: (1) Since Wis an irreducible algebraic set in Dof dimension n, there exists an irreducible polynomial hin Cn+1 such that W={Z= (z0,z)∈D:h(Z)=0}.IfWdoes not extend across ∂D, the defining function rwill be negative on ˆ W={Z∈Cn+1 :h(Z)=0}. According to [12] (see Proposition 2, p. 76), there exists an analytic cover π:ˆ W→ Cn. Let g1,...,g kbe the branches of π−1which are locally defined and holomorphic on Cn\σ, with σ⊂Cnan analytic set of dimension at most n−1. Consider the function ˆr(w) = sup{r◦g1(w),...,r◦gk(w)}. Since πis an analytic cover, ˆrextends as a plurisubharmonic on Cn. Then it is constant; since it is negative. This contradicts the fact that the domain Dis nondegenerate. (2-i) We may assume that ∇his not identically zero on W. Thus, his a defining function of W. Let for example ∂h ∂z1(p)= 0 for some point p∈ W. Applying the maximum principle to W, then there exists an open dense subset OWof EWsuch that for any q∈OW,∂h ∂z1(q)= 0. For a fixed q∈OW, there exists a neighborhood Uin Cn+1 of qsuch that ∂h ∂z1 vanishes nowhere on U. Then ˜ W={z∈U:h(z)=0}is a polynomial submanifold of U. Since Wextends across the boundary of Das a variety, a useful consequence of this fact is that ˜ Whas dimension 2n−1. Otherwise, the Hausdorff dimension of ˜ Wwill be less or equal to 2n−2. Then ˜ W\˜ W∩∂D will be connected (see [12, p. 347]). This implies that ˜ Wcannot be separated by ∂D and contradicts (i). 72 B. Coupet, N. Ourimi (2-ii) Since fis algebraic, all its components fjare also algebraic. Then there exist n+ 1 polynomial equations Pj(z,w) = 0 satisfied by wj=fj(z). Let be Pj(z,fj(z)) = amj j(z)fj(z)mj+···+a1 j(z)fj(z)+a0 j(z), where mj∈Nand ak jare holomorphic polynomials for all k∈{0,...,m j} and for all j∈{1,...,n+1}. We may assume that for all j,amj j≡ 0on W. Since ∂D =Sh∪ˆ S, for all p∈ˆ Sthere exists j∈{1,...,n +1} such that amj j(p) = 0. Then the polynomial function a=1≤j≤n+1 amj j vanishes identically on ˆ S. Now, we prove that Sh∩OWis a dense subset in OW. Suppose by contradiction that ˆ S∩OWhas an interior point. The uniqueness theorem implies that a≡0onCn+1. This implies that amj j≡0 for a certain j∈{1,...,n+1}: a contradiction. This completes the proof of the lemma. The Levi determinant of Dis defined by: Λr:Cn+1 →Rvia −det 0rzj rzjrzjzj. The set of weakly pseudoconvex points in ∂D is ω(∂D)={(z0,z)∈Cn+1 : 2 Re(z0)=−P(z) and Λr(z0,z)=0}. Since Dis a rigid polynomial domain, Λr(z0,z) depends only on z.We write Λr(z0,z)asΛ r(z0,z)=L(z)=Lα1 1...L αs s(z), where the Ljdenote the irreducible components of the polynomial Land αj∈Nfor j= 1,...,s. If pis a boundary point of D, we define the member τ(p), the vanishing order of Λr, to be the smallest nonnegative integer msuch that there is a tangential differential operator Tof order mon ∂D such that TΛr(p)= 0. It can easily be checked that τ(p) is independent of the choice of the defining function r. Note that the set {p∈∂D :τ(p)=0}is the set of strongly pseudoconvex boundary points. The function τis uppersemicontinuous. In our case, it is bounded by the degree of the polynomial P. We need the following important statement. Lemma 2. Let f:D→Ωbe a proper holomorphic mapping as in Theorem 1. Then for all p∈Sh,τ(p)≥τ(f(p)) and the inequality holds if and only if fis branched at p. Proper Holomorphic Mappings 73 Proof: Let p∈Sh. Then fextends holomorphically to a neighborhood of p. By the Hopf lemma, ∇(ρ◦f)(p)= 0. Then ρ◦fis a local defining function of Din a neighborhood of p, and by the chain rule we have: Λρ◦f(p)=|Jf(p)|2Λρ(f(p)). Hence, we are able to deduce the lemma (see [10]). Remark 1.Note that the lemma above still remains true, if the domains are not pseudoconvex. The proof is as in [16]. It uses some results of Baouendi-Rothschild [1] and Baouendi-Jacobwitz-Treves [3] to show that the transversal component f0of fsatisfies ∂f0 ∂z0(p)= 0 for all p∈Sh. Proposition 1. The closure Vfdoes not intersect the set ∂D\ω(∂D) of strongly pseudoconvex points in ∂D. Proof: As in the proof of Lemma 2, we have: Λρ◦f(z)=|Jf(z)|2Λρ(f(z)),∀z∈Sh so OW⊂ω(∂D), which implies that EW⊂ω(∂D). 3. Stratification of the weakly pseudoconvex set Here, we give a real analytic stratification of the weakly pseudoconvex set. For bounded pseudoconvex domains with real analytic boundary, Bedford [6] obtained a similar stratification. Lemma 3. There exists an algebraic stratification of ω(∂D)as follows: ω(∂D)={(z0,z)∈∂D :z∈A1∪A2∪A3∪A4} with the following properties. (a) A4is an algebraic set of dimension ≤2n−3. (b) A1,A2and A3are either empty or algebraic manifolds; A2and A3 have dimension 2n−2and A1has dimension 2n−1. (c) A2and A3are CR manifolds with dimCHA2=n−1 and dimCHA3=n−2. (d) τis constant on every component of {(z0,z)∈∂D :z∈A1}. Proof: Let A={z∈Cn:L(z)=0}and let ˆ A1be the union of all components of Awith dimension 2n−1 (if there are any). We consider A1= Reg( ˆ A1)=∪k{Lk= 0 and Lj= 0 for j=k}. 74 B. Coupet, N. Ourimi Next we let ˆ A2be the union of all 2n−2-dimensional components of A\A1. We see that we may write Reg( ˆ A2)=A2∪A3∪ˆ A3 where A2and A3are an open subsets of ˆ A2with dimCHA2=n−1 dimCHA3=n−2 and dimRˆ A3≤2n−3. Now, let A4=A\(A1∪A2∪A3) then, we have the desired stratification. To show (d), we consider the complex tangential derivative along the boundary of D, i.e., Tj=∂ ∂zj −1 2 ∂P ∂zj ∂ ∂z0 ,1≤j≤n. For example, we prove that τ≡α1on C1=∂D ∩{L1= 0 and Lk= 0,k=1}. Let (z0,z)∈C1, we have Tm jΛr(z0,z)=Tm jL(z)=∂mL ∂zm j (z) =α1...(α1−m+1)∂L ∂zjm Lα1−m 1.Lα2 2...L αs s(z). Since L1is irreducible, D(L1)(z)= 0. Then there exists jsuch that Tm jL(z) = 0 for all m<α 1and Tα1 jL(z)= 0. This finishes the proof of the lemma. Proof of Theorem 1: The analytic set A2contains finitely many components which we will denote by B1,B 2,...,B N. Since dimRBj= dimRHBj, then for each j,Bjis an n−1-dimensional complex manifold. We denote by Γj={(z0,z)∈∂D :z∈Aj}for j=1,...,4. By considering dimension and CR dimension, we see that Γ3∩OWand Γ4∩OWare nowhere dense in OW. Next, we prove that Γ1∩OWcannot contain an open subset of OW. By contradiction, let suppose p∈OW⊂Γ1. We may choose a sequence {qk}k⊂Γ1∩{Jf=0}such that qk→p. The mapping fis a local diffeomorphism in a neighborhood of all points qkand the function τis constant on Γ1. Then, we have for all k τ(p)=τ(qk)=τ(f(qk)).(1) Proper Holomorphic Mappings 75 On the other hand, by Lemma 2, τ(p)>τ(f(p)).(2) Since τis uppersemicontinuous, then (1) and (2) together give a contradiction. We mention that the same argument has appeared in [6]. We conclude that Γ2∩OWcontains an open subset of Γ2. Thus it contains an open subset of {(z0,z)∈∂D :z∈Bj}for some j.Fork=1,...,N, let ˆ Bjbe the complex variety in Cnsuch that Reg ˆ Bk=Bk.Applying the maximum principle, we conclude that W⊂{(z0,z)∈D:z∈ˆ Bj}, and by irreducibility, W={(z0,z)∈D:z∈ˆ Bj}. This completes the proof of Theorem 1. Remark 2.(i) Using the same argument of Bedford [6] (appeared also in [16]), we can prove that the branching multiplicity of the mapping fis bounded by a constant independent of f. (ii) For a holomorphic function Hbetween algebraic hypersurface M and M(Mis essentially finite at p0), Baouendi-Rothschlid [2] showed that the multiplicity of its components is bounded by a constant depending only on Mand Mand the points p0and H(p0). 4. Proper self-mappings Here, we give the proof of Corollary 1. Since Dis simply connected, it suffices to prove that Vfis empty. The variety Vfhas a finite number of connected components independent of the mapping f, then there exists an integer ksuch that Vfk=Vfk+1 . We may assume k= 1, that is Vf=V2 f. Since Vf2=Vf∪f−1(Vf), it follows that Vf⊆f(Vf), where f(Vf) is a complex analytic variety of Dby a theorem of Remmert. Hence, we have Vf=f(Vf) because Vfhas finitely many components. Assume that Vfis not empty. 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