Some compactness theorems of families of proper holomorphic correspondences

Publ. Mat. 47 (2003), 31–43 SOME COMPACTNESS THEOREMS OF FAMILIES OF PROPER HOLOMORPHIC CORRESPONDENCES Nabil Ourimi∗ Abstract In this paper we prove some compactness theorems of families of proper holomorphic correspondences. In particular we extend the well known Wong-Rosay’s theorem to proper holomorphic correspondences. This work generalizes some recent results proved in [17]. 1. Introduction and results In [29], B. Wong gave a characterization of the unit ball in Cnby its automorphism group, namely, if Dis a smooth strongly pseudoconvex bounded domain in Cnwith noncompact automorphism group, then D is biholomorphic to the unit ball. Later, J.-P. Rosay [21] proved that the same conclusion holds under considerably weaker hypotheses on the boundary of the domain. S. Pinchuk [19]gave alocal version of this theorem with an elementary proof by using the scaling technique: the unit ball is a model for the class of C2strongly pseudoconvex domains at an accumulation point. E. B. Lin and B. Wong [14] observed that this result (termed “the Wong-Rosay theorem”) is interesting only when the domain Dis an Eilenberg-Maclane space (i.e. πk(D)=0for all k≥1); since a smooth bounded domain Din Cnwith noncompact automorphism group and nontrivial πk(D) for some k≥1 admits a complex analytic variety in the boundary. In particular, they proved that the set of proper holomorphic mappings between bounded strongly pseudoconvex domains in Cnis noncompact if both of the domains are biholomorphic to the unit ball. In [17], the author showed a local version of this result: if Dis a bounded domain in Cnand there exist a point p∈Dand a sequence of proper holomorphic self-mappings fk:D→Dof uniformly 2000 Mathematics Subject Classification. 32H35. Key words. Proper holomorphic correspondences, scaling technique. ∗The research was supported by the ICTP, Trieste, Italy. 32 N. Ourimi bounded multiplicity such that {fk(p)}kconverges to a strongly pseudoconvex boundary point, then Dis biholomorphic to the unit ball in Cn. Our aim in this paper is to prove a suitable version of the Wong-Rosay theorem for families of proper holomorphic correspondences. The notion of holomorphic correspondence is very interesting. It is a generalization to several complex variables of the classical global analytic function of one complex variables. More precisely, let Dand Gbe two domains in Cn.Aholomorphic correspondence is a closed complex analytic subset A⊂(D×G)ofpure dimension nwith A∩(D×∂G)=∅. We can regard Aas the graph of the multivalued mapping f:= π2◦π−1 1, where π1:A→Dand π2:A→Gdenote the natural projections. It follows from the definition that the projection π1:A→Dis proper. Then there exists an n−1-dimensional analytic subset V⊂graph f and an integer m, such that π1is an m-sheeted covering map from the set A\π−1 1(π1(V)) onto D\π1(V). Hence f(z)={f1(z),...,fm(z)}for all z∈D\π1(V) and the fj’s are distinct holomorphic functions in a neighborhood of z∈D\π1(V). The integer mis called the multiplicity of fand π1(V)isits branch locus. If both π1and π2are proper then Ais a proper holomorphic correspondence. If Ais irreducible as an analytic set, then it is called an irreducible holomorphic correspondence. For the basic topic on holomorphic correspondences, we refer the reader to the work of K. Stein [25], [26] and for its boundary behavior to [1], [6] and [27], where the phenomena of continuous and holomorphic extension for correspondences were studied with local boundary assumptions. We denote by Cor(D, G, m) the set of all µ-valued holomorphic mappings from Donto Gfor µ=1,...,m and Cor(D,G, m, l) the set of correspondences f∈Cor(D, G, m) for which f−1∈Cor(G, D, l). Our main result can be stated as follows: Theorem 1. Let Dand Gbe bounded domains in Cn. Suppose that there exist a point p∈D,asequence of proper holomorphic irreducible correspondences {fk}kin Cor(D,G, m, l)and a sequence {qk}k,qk∈ fk(p),converging to a strongly pseudoconvex boundary point q∈∂G. Then there exists a proper holomorphic correspondence in Cor(D, B,m, l), where Bdenotes the unit ball in Cn. In the case m=1,wefind the result of [17] for proper holomorphic mappings of uniformly bounded multiplicity. If the domain Dis a pseudoconvex, simply connected with a C∞boundary and of finite type (in the sense of J. P. D’Angelo [10]), then the correspondence f:D→B defined by Theorem 1 splits at each point z∈D, i.e. f=f1,...,f m, where the f jsare distinct holomorphic functions in a neighborhood of z Some Compactness Theorems for Correspondences 33 (see [3]). In view of the simple connectedness we have a global splitting. So each branch of fdefines a proper holomorphic mapping from D onto B.Inparticular any proper holomorphic self-mapping of Dis biholomorphic. Moreover the Lie groups Aut(D) and Aut(B)havethe same dimension (see also [3]). Note that the same conclusions hold if we assume that Gis pseudoconvex, simply connected with C∞boundary and of finite type. As an application of Theorem 1, we have the following version of the Wong-Rosay theorem for families of proper holomorphic correspondences. Theorem 2. Let Dand Gbe bounded domains in Cn. Assume that Gis strongly pseudoconvex, simply connected with a C∞boundary. Suppose that there exist a point p∈D,asequence of proper holomorphic irreducible correspondences {fk}kin Cor(D,G, m, l)and a sequence {qk}k, qk∈fk(p),converging to a boundary point q∈∂G. Then Gis biholomorphic to the unit ball. Example 1. The following example due to E. Bedford and S. Bell [2], shows that the strong pseudoconvexity in Theorem 2 is necessary. Let D={(z,w)∈C2:|z|4+|w|2<1}and let us consider the proper holomorphic mapping f:D→B (z,w)→ (z2,w). If {ϕk}kis a sequence of automorphisms of the unit ball converging to (1,0) ∈∂Bthen {f−1◦ϕk◦f}kis a sequence of self-correspondences of Dconverging to the points (±1,0) ∈∂D. In [12], W. Klingenberg and S. Pinchuk proved that the set of proper holomorphic correspondences of uniformly bounded multiplicity between bounded domains is normal. In strongly pseudoconvex case we get more information on the convergence of such correspondences as follows: Corollary 1. Let Dand Gbe bounded domains in Cn. Assume that at least one of them is strongly pseudoconvex, simply connected with aC∞boundary and not biholomorphic to the unit ball. Then for any sequence {fk}kof proper holomorphic irreducible correspondences in Cor(D,G,m, l),wemay extract a subsequence converging to a proper holomorphic correspondence in Cor(D,G, m, l). This corollary generalizes the result of E. B. Lin and B. Wong [14] mentioned above even for proper holomorphic mappings. 34 N. Ourimi We can also use Theorem 1 to complete the result of [17], concerning homogeneous complex manifolds. Corollary 2. Let Mbe an n-dimensional complex homogeneous manifold and Gabounded domain in Cnthat possesses strong pseudoconvexity boundary points. Then (1) There is no proper holomorphic mapping between Mand G. (2) If, in addition, Gis a strongly pseudoconvex, simply connected domain with a C∞boundary and there exists a proper holomorphic correspondence between Mand G, then Gis biholomorphic to the unit ball. 2. Basic facts about convergence of holomorphic correspondences In this section we recall some definitions on the notion of convergence of holomorphic correspondence. Let zobe a point in Dand let {z1,...,zm}be a set in G.Wesay that f(z)={f(z1),...,f(zm)}converges to {z1,...,zm}when ztends to zoif after possible renumeration of fj, one has limz→zofj(z)=zj. Equivalently, f(z) tends to {z1,...,zm}in the sense of the Hausdorff convergence of sets. We can also define a distance in Cnm×Cnmto study the problem of convergence of correspondences (see [5]). Let f∈Cor(D,G,m)beanirreducible correspondence, a∈A⊂D and b∈f(a). We define f(a,b) A∈Cor( ◦ A,G,m), where ◦ A refers to the interior of A as follows: Consider all irreducible germs of branches of fat (a, b). Analytic continuation of each of these irreducible germs along all possible paths in A define f(a,b) A∈Cor( ◦ A,G,m). Equivalently, graph f(a,b) Ais the union of those irreducible components of graph f∩{A×G}, which contains (a, b). Let {fk}k⊂Cor(D,G,m). We say that {fk}kis compactly divergent if ∀K1⊂⊂ D,K2⊂⊂ G,∃jo, such that ∀j≥jo: fk(K1)∩K2=∅. If the fkare irreducible, we say that fkconverge to f∈Cor(D, G,m) if ∃(a, bk)∈graph fkwith bk→b∈Gand for all K⊂⊂ Dwith a∈K: f(a,b) k,K →fKfor some fK∈Cor(D, G, m) and ∪K⊂⊂Dgraph fK= graph f. Some Compactness Theorems for Correspondences 35 3. Proof of results Proof of Theorem 1: Our basic tool is the scaling method, successfully applied in different problems for holomorphic and CR mappings by several authors (see for instance [18], [4], [11], [8], [9]). It is worth to remark that here we adapt the scaling technique for proper holomorphic correspondences. We believe that this technique will be useful to deal with other problems as well. We write z∈Cnas z=( z,zn) where zdenotes the first n−1coordinates of z. Let Vbeaneighborhood of qin Cnwhich does not intersect the set of weakly pseudoconvex points of ∂G.Forall w∈∂G ∩V,we consider the change of variables hwdefined by:        z∗ j=∂r ∂¯zn (w)(zj−wj)−∂r ∂¯zj (w)(zn−wn),1≤j≤n−1 z∗ n=1≤j≤n ∂r ∂zj (w)(zj−wj) where ris a defining function of G. The mapping hwmaps wonto 0 and the real normal to ∂G at wonto the line {z=0,y n=0}. Let wkbe the point in ∂G closest to qkand hk=hwkbe the mapping as above. We denote by Gk=hk(G) and γk= dist(hk(qk),∂G k). We introduce the dilatation of the coordinates as follows: βk(w,wn)= (w √γk ,wn γk). Set ˆ Gk=βk◦hk(G) and consider the holomorphic correspondence ˆ fk=βk◦hk◦fk∈Cor(D, ˆ Gk,m). Each ˆ fkis a proper holomorphic correspondence of multiplicity at most mand satisfies s= (0,−1) ∈ˆ fk(p). Let rkbeadefining function of Gk. Without loss of generality, we may assume that q=0and in a neighborhood of the origin we have r(w)=2Rewn+|w|2+R(w) with R(w)=o(|w|2). By Taylor’s formula, we get the estimate rk(w)=2Rewn+Hk(w)+Bk(w)+Rk(w), where Hkis hermitian, Bkis bilinear and Rk(w)iso(|w|2) uniformly in a neighborhood of the origin. As k→∞, the limit of the matrix Hk is the identity and the limit of Bkis 0. Consequently, there exists a neighborhood Uof 0 such that for every kand w∈U,wehave rk(w)≥2Rewn+1 2|w|2.(∗) Let ˆrk=1 γkrk◦βk−1beadefining function of the domain ˆ Gk.Itis well known that the sequence {ˆrk(w)}kconverges uniformly on compact subsets of Cnto ˆϕ(w)=2Re(wn)+|w|2. Let Kbe an arbitrary compact 36 N. Ourimi subset of Dcontaining the point p.Forz∈K, let wk∈ˆ f(p,s) k,K (z). Starting with some ko=ko(K), we have 0>ˆrk(wk)= 1 γk rk(√γkwk,γ kwk n). To prove the convergence of the correspondence {ˆ fk}k,weshall prove that for large k’s, βk−1◦ˆ f(p,s) k,K (K)⊂U.Weneed the following important statement on the localization of holomorphic correspondences. It is the crucial point of our scaling construction. First of all, we recall that a point a∈∂D is a local plurisubharmonic peak point if there is a neighborhood Vof ain Cnand ψ∈PSH(D∩ V)∩C(D∩V) such that ψ(a)=1and ψ<1on(D∩U)\{a}.IfV∩∂D is a strongly pseudoconvex hypersurface of class C2, then ais a local plurisubharmonic peak point (see [22] and [24]). Proposition 1. Let D⊂⊂ Cnand D⊂CN,N≥1be domains. Fix R>0and zo∈D. Then for all compact L⊂Dcontaining zo, there exists δ=δ(L)>0such if a∈∂D be alocal plurisubharmonic peakpoint of ∂D,F∈Cor(D,D,m)and w∈F(zo)∩B(a, δ)then F(zo,w) L(L)⊂B(a, R). Proof: This proposition was proved by K. Verma in [27]. For the sake of completeness, we include a brief proof. By contradiction, assume that the proposition is not true. Then there exist a compact L⊂D,asequences {ak}k⊂∂D and {Fk}k⊂Cor(D,D,m) with wk∈Fk(zo)∩ B(ak,δ k) and δk→0 such that for all k,F(zo,wk) k,L (L)⊂ B(ak,R). Since the domain Dis bounded, we can assume (after taking a subsequence) that {Fk}kconverges uniformly on compact subsets of Dto a holomorphic correspondence F∞∈Cor(D,D, m) and {ak}kconverges to a∈ ∂D. Without loss of generality, we may assume that zo=0 ∈CNand a=0  ∈Cn. Then we have (0,0)∈(D×Cn)∩graph F∞.Byusing [7, pp. 36 and 46], it easy to see that the graph F∞is a pure N-dimensional analytic set in D×Cn. Let V1be an irreducible component of graph F∞ containing (0,0), then again according to [7] there exist small neighborhoods U0,U 0,sothe projection π:V1∩(U×U)→Uis proper. Let g1,...,g kbe the branches of π−1which are locally defined and holomorphic on U\σ, with σan analytic set of dimension at most N−1. Since a=0  is a local plurisubharmonic peak point, there exist an >0 and a local plurisubharmonic peak function ψ∈PSH(B(0,)∩ D)∩C(B(0,)∩D). Consider ρ(z)=max(ψ◦g1(z),...,ψ◦gk(z)) ∈ Some Compactness Theorems for Correspondences 37 PSH(U\σ). Since ρ(z)isbounded, so it extends as a plurisubharmonic function on U. But ρ(0)=1,then by the maximum principle, ρ(z)≡1onU. This implies that, one of the branches gi≡0 on U. Hence U×{0}⊂V1.Itfollows by irreducibility that D×{0}=V1. Since V1is an arbitrary component containing (0,0), it follows that the only one component of graph F∞containing (0,0)isD×{0}. The same argument as above shows that for all z∈D, the only component of graph F∞containing (z,0)isD×{0}. Therefore, if Vjis some component of graph F∞distinct from D×{0}, then Vj∩(D×{0})=∅. This shows that for all z∈D,π2◦π1−1(z)={f∞ 1(z),...,f∞ m−1(z),0} where π1and π2are the canonical projections of graph F∞respectively on Dand Cn.Forall kthe branches of Fkand F∞are contained in a bounded domain; hence by the continuity of the root of canonical functions (see [7, pp. 45 and 46]), m−1 branches of F∞are a subset of D\B(0, o) while the mth branch has the constant 0, for all z∈Land for some constant o>0. Choose ˆo= min(R 2,o 2). Since {Fk}kconverges uniformly to F∞on L, then for k≥koand z∈Lall the branches of Fkare contained in the disjoint union B(0,ˆo)∪D\B(0, o−ˆo). Since wk→0, it follows that for k≥kowe have F(0,wk) k,L (L)⊂B(0,ˆo). But the sequence {ak}kconverges to 0, then we may assume that for k≥ko,|ak|<R 2.Sowehave B(0,ˆo)⊂B(ak,R) for all k≥ko. This contradicts the fact that F(0,wk) k,L (L)⊂ B(ak,R). We continue with the proof of Theorem 1. Let α>0 such that B(0,α)⊂U(Uis the open set defined in (∗)) and Ka compact in D. Let us consider the correspondence fk,K ∈Cor(Ko,G,m). Fix 0<R<min(1,α 3) and set γ k= dist(qk,∂D)=|wk−qk|. Then qk∈fk(p)∩B(wk,2γ k) and for ksufficiently large one has 2γ k<δ(K), where δ(K)isaconstant from Proposition 1. Thus Proposition 1 implies that f(p,qk) k,K (K)⊂B(wk,R). Since hk→id uniformly on compact subsets of Cnand wk→0ask→∞, then for large k’s, we may assume that B(wk,R)⊂B(0,2) and supz∈B(0,2) ||Dhk(z)|| <3. Hence hk◦f(p,qk) k,K (L)⊂B(0,α). Then for z∈Kand wk∈ˆ f(p,s) k,K (z), we have for large k’s: ˆrk(wk)>2Re(wk n)+1 2|wk|2.(∗∗) Let πn:Cn→Cbe the nth projection and set ˆ fn k,K =πn◦ˆ f(p,s) k,K ∈ Cor( ◦ K,H,m), where H={z∈C:Re(z)<0}denotes the half-plane 38 N. Ourimi in C. Let T:H→∆ z→ z+1 z−1, be the biholomorphic map transforming Honto the unit disc. Consider now the correspondence Tk,K =T◦ˆ fn k,K ∈Cor( ◦ K,∆,m). For all k,we have (s, 0) ∈graph Tk,K. The following statement due to S. Pinchuk [20] is important to prove the convergence of the sequence {ˆ fn k,K }k. Proposition 2. Let Dbeadomain in Cn,n≥1and f∈Cor(D,∆,m) with (p, 0) ∈graph f. Then for all compact L⊂D, there exists a constant r=r(m, L)<1not depending on f, such that f(p,0) L(L)⊂∆r. According to Proposition 2 and by using Montel’s theorem and the diagonal process, we may assume (after taking a subsequence) that {Tk,K}k converges to TK∈Cor( ◦ K,∆,m). Hence ˆ fn k,K →ˆ fn K∈Cor( ◦ K,H,m). It follows from (∗∗) that the correspondence ˆ f(p,s) k,K is uniformly bounded. Then we can extract a subsequence converging to ˆ fK∈Cor( ◦ K,Cn,m). By exhausting Dwith an increasing sequence of compacts containing p, we show that ˆ fkconverges to ˆ f∈Cor(D,Cn,m). Passing to the limit in (∗∗), we conclude that ϕ(ˆ f)(z)≤0 for all z∈Σ. Hence ˆ f∈ Cor(D,Σ,m). Now we consider the correspondence ˆgk=ˆ f−1 k, the inverse of the correspondence ˆ fk.Forall k,ˆgk∈Cor( ˆ Gk,D,l) and satisfies (s, p)∈ graph(ˆgk). Let L⊂Σacompact containing s, and consider the correspondence ˆgk,L ∈Cor(◦ L, D, l). Since Dis bounded, there exists a subsequence of ˆgk,L converging to ˆg∈Cor(◦ L, D,l). By exhausting Σ with compacts and passing to the diagonal subsequence, we obtain a limit ˆg∈Cor(Σ,D, l). To prove that ˆ f∈Cor(D,Σ,m) and ˆg∈Cor(Σ,D,l), we need the following statement, which follows from the Schwarz lemma for proper holomorphic correspondences (see [12]). Lemma 1. Let Dand Gbe bounded domains in Cnand (a, b)∈D×G. Then for any neighborhood U2bin Gthere exists a neighborhood U1 ain Dsuch that if h∈Cor(D,G,m)with b∈h(a)then h(a,b) U1(U1)⊂U2. Some Compactness Theorems for Correspondences 39 Recall that the branches {ˆ f1,..., ˆ fm}of ˆ fare locally defined and holomorphic on D\π1(V). Then the Jacobian of ˆ fiinduce in a natural manner a holomorphic function on the graph ˆ f\Vas follows: if z∈graph ˆ f\V; there exists only one i∈{1,...,m}such that z∈ graph ˆ fi\V.Wedefine Jac ˆ f(z)=Jac ˆ fi(π1z). Claim 1. ˆ f∈Cor(D,Σ,m). Proof: Since for any compact Kp, the correspondence ˆ f(p,s) k,K is uniformly bounded, so in view of Lemma 1 there exist neighborhoods U2s and U1psuch that for all k,ˆ f(p,s) k,U1(U1)⊂U2. Then we have z∈ ˆg(s,p) k,U2◦ˆ f(p,s) k,U1(z) for all z∈U1.Passing to a convergent subsequence, we get z∈ˆgU2◦ˆ fU1(z) for all z∈U, which implies that Jac ˆ fU1≡ 0. Now assume that there exist points a∈Dand b∈∂Σ such that b∈f(a). Let A1be an irreducible component of graph ˆ fcontaining (a, b). According to [7], there exist small neighborhoods Ua, U b,sothe projection π:A1∩(U×U)→Uis proper. Let h1,...,h kbe the branches of π−1which are locally defined and holomorphic on U\σ, with σan analytic set of dimension at most n−1. Since bis a strong pseudoconvexity point, there exists a local plurisubharmonic peak function ψdefined in a neighborhood of b. Without loss of generality, we may assume that ψ∈PSH(U ∩Σ) ∩C(U ∩Σ). Consider ρ(z)=max(ψ◦h1(z),...,ψ ◦hk(z)). It is clear that ρis plurisubharmonic in U\σ, since it is bounded (ψ≤1), so it extends as a plurisubharmonic function on U. But ρ(a)=1,then by the maximum principle, ρ(z)≡1onU. This implies that one of the branches hi≡bon U. Hence U×{b}⊂A1.Itfollows by irreducibility that D×{b}=A1. Since A1is an arbitrary component containing (a, b), it follows that the only one component of graph ˆ fcontaining (a, b)isD×{b}.Asimilar argument shows that for all z∈D, the only component of graph ˆ fcontaining (z,b)isD×{b}. Therefore, if Ajis some component of graph ˆ f distinct from D×{b}, then Aj∩(D×{b})=∅. This shows that for all z∈D,ˆ f(z)={ˆ f1(z),..., ˆ fm−1(z),b}. But the branches of ˆ fare locally open on U1\(π1{V∪W}∩U1), since Jac ˆ fU1≡ 0. This contradiction completes the proof of Claim 1. Claim 2. ˆg∈Cor(Σ,D,l). Proof: Since the domain Dis bounded, we can repeat the same proof of W. Klingenberg and S. Pinchuk [12] (see also [17]) to show that