DSpace at VNU: Extensions of holomorphic maps through hypersurfaces and relations to the Hartogs extensions in infinite dimension

Extensions of Holomorphic Maps through Hypersurfaces and Relations to the Hartogs Extensions in Infinite Dimension Author(s): Do Duc Thai and Nguyen Thai Son Source: Proceedings of the American Mathematical Society, Vol 128, No (Mar., 2000), pp 745754 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/119736 Accessed: 02-02-2016 19:40 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/119736?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive We use information technology and tools to increase productivity and facilitate new forms of scholarship For more information about JSTOR, please contact support@jstor.org American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of the American Mathematical Society http://www.jstor.org This content downloaded from 204.235.148.92 on Tue, 02 Feb 2016 19:40:18 UTC All use subject to JSTOR Terms and Conditions PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 3, Pages 745 754 S 0002-9939(99)05033-9 Article electronically published on July 27, 1999 EXTENSIONS OF HOLOMORPHIC MAPS THROUGH HYPERSURFACES AND RELATIONS TO THE HARTOGS EXTENSIONS IN INFINITE DIMENSION DO DUC THAI AND NGUYEN THAI SON (Communicated by Steven R Bell) ABSTRACT A generalization of Kwack's theorem to the infinite dimensional case is obtained We consider a holomorphic map f from Z \ H into Y, where H is a hypersurface in a complex Banach manifold Z and Y is a hyperbolic Banach space Under various assumptions on Z, H and Y we show that f can be extended to a holomorphic map from Z into Y Moreover, it is proved that an increasing union of pseudoconvex domains containing no complex lines has the Hartogs extension property INTRODUCTION We discuss the problems of extending holomorphic maps through hypersurfaces and extending to the envelope of holomorphy, i.e the Hartogs holomorphic extension These two problems are related In [10], Kwack showed that if f is a holomorphic map from the punctured unit disc A* into a hyperbolic manifold X such that, for a suitable sequence of points Zk E A* converging to the origin, f(zk) converges to a point po EX, then f extends to a holomorphic map from the unit disc A into X The above-mentioned theorem of Kwack plays an essential role and has strongly motivated the study of the extension problem of holomorphic maps Thus this was generalized in many directions In particular, it also generalized to the problem of extending holomorphic maps through hypersurfaces; see, for instance, the survey [11] Therefore proving Kwack type theorems in the infinite dimensional case is really necessary and motivates the study of the extension problem of holomorphic maps in infinite dimension In Sect we discuss the problem of determining when a holomorphic map can be extended through hypersurfaces in complex Banach manifolds We prove the following theorems Theorem 1.1 Let X be a hyperbolicBanach analytic space and f: Z \ H -* X a holomorphic map, where H is a hypersurface in a complex Banach manifold Z Assume that for every z E H there exists a sequence {zn fE Z\H converging to z such Received by the editors May 27, 1997 and, in revised form, April 20, 1998 1991 Mathematics Subject Classification Primary 32E05, 32H20; Secondary 32F05, 58B12 Supported by the State Program for Fundamental Research in Natural Science (?1999 American Mathematical 745 This content downloaded from 204.235.148.92 on Tue, 02 Feb 2016 19:40:18 UTC All use subject to JSTOR Terms and Conditions Society DO DUC THAI AND NGUYEN THAI SON 746 that the sequence {f(zk)} converges to xz Z E X Then f extends holomorphically to Theorem 1.2 Let X be a hyperbolic Banach analytic space which is complete in the Cauchy mean and f: Z\H -* X a holomorphic map, where H is a hypersurface in a complex Banach manifold Z Assume that for every branch Hogof H there exist C Z \ H converging to za, such that the z,, E Reg(HO) and a sequence {zc}?l sequence {f (Zn)} converges to pc, E X Then f extends holomorphically to Z We would like to emphasize here that the local compactness of (finite dimensional) complex manifolds plays an essential role in proving the finite dimensional Kwack theorems Since complex Banach manifolds not have the local compactness property, the technique for proving Kwack type theorems in the infinite dimensional case required substantial changes The proofs of the above-mentioned theorems are based on the maximum principle for plurisubharmonic functions In Sect we go further We would like to investigate deep interactions between the Brody hyperbolicity of Banach analytic spaces and the Hartogs extension property First we recall the definition of a (finite-dimensional) complex space having the Hartogs extension property (HEP for short) which was introduced by Ivashkovicz [7] A complex space X is said to have HEP if every holomorphic map f from Q into X, where Q is a Riemann domain over Cn, can be extended holomorphically to the envelope of holomorphy Q of Q This definition can be generalized naturally to the case of Banach analytic spaces by replacing Cn by an arbitrary Banach space B However, for technical reasons we need the statement that every pseudoconvex Riemann domain over a Banach space B is a domain of existence One only was proved in the case that the Banach space B has a Schauder basis (see Mujica [13, Theorem 54.12, p 390]) Thus in the infinite-dimension case we give the following definition A Banach analytic space X is said to have the Hartogs extension property (HEP) if every holomorphic map from a Riemann domain Q over a Banach space B with a Schauder basis into X can be extended holomorphically to Q, the envelope of holomorphy of Q We prove the following Theorem 2.1 Let X be a Banach analytic space which is an increasing union of pseudoconvex domains Assume that X contains no complex lines Then X has the HEP Finally in this note we frequently make use of the definition and properties of the Kobayashi pseudodistance in Banach analytic spaces as in [18], [19] ?1 EXTENDING HOLOMORPHIC MAPS THROUGH HYPERSURFACES IN COMPLEX BANACH MANIFOLDS First we prove the Kwack theorem [10] in infinite dimensions 1.1 Theorem Let X be a hyperbolicBanach space and f: Z\ H -* X a holomorphic map, where Z is a Banach manifold and H is a hypersurface in Z Assume that for every z E H there exists a sequence {zn} C Z\ H converging to z such that the sequence {f(zn)} converges to xz E X Then f extends holomorphically to Z This content downloaded from 204.235.148.92 on Tue, 02 Feb 2016 19:40:18 UTC All use subject to JSTOR Terms and Conditions HYPERSURFACES AND HARTOGS EXTENSIONS 747 Proof (i) First consider the case where Z = A and H = {O} Choose a pseudoconvex coordinate neighbourhood W of xo in X which is isomorphic to an analytic subset of an open ball in a Banach space B Let V = W/2 The problem is to show that, for a suitable positive number a, the small punctured disc {z E A: < lzl < a} is mapped into W by f By taking a subsequence of {Zn} if necessary, we may assume that the sequence {IZ} is monotone decreasing Consider the set of integers n such that the image of the annulus IZn+lI < IzI < IznI by f is not entirely contained in V If this set of integers is finite, then f maps a small punctured disc < lzl < a*into V Assuming that this set of integers is infinite, we shall obtain a contradiction By taking a subsequence, we may assume also that, for every n, the image of the annulus 1zn+I1< IZj < lznlby f is not entirely contained in V For each n, put rn Sn= = inf{r < lznl f(r < IzI < lzn) C V}, sup{r > lznI f(IznI < IZI< r) C V}l an= {z E A IZI AYn= {z E A IZ: On = {z E Since da* (an) +dA* (-Yn)+dA* (/3n) it follows that A rn}, lznl= IZI = Sn}l and by the distance decreasing principle, dx f(n) + dx f(yn) + dx f((n) as n x0 00 Put K= U f(yn Uyn+i) n=I By the maximum principle, we have KpSH(w) D U f(an U/n) n>2 U f(an U o3n)is relatively compact in V By the relative compactness of n>2 U f(an UOn) and since dx f (an) and dx f(on) -+ 0, without loss of generality n>2 Hence we may assume that {f(an)} -xi and {f(13n)}I- X2 By the definition of rn and Sn it follows that xI, x2 E OV and hence xI, x2 xo Choose a continuous linear functional u on B such that u(xI), u(x2) 74u(xo) = C Since f(Arnsn) C V C W, there exist in < rn < Sn < ?n such that f = {Zz E C: W, where Arnsn = {Z E C: rn < IZI< Sn} and i? n < IzI < s4n Consider the holomorphic function on = u o f * Since {fun(3n) } U(X2) - U(X2)1 < Applying the maximum we have VE> 0,3N, Vn > N, VO: I mn(sne90) principle to the function z a-4 Un(Z) - U(X2)on the annulus {z E (C: in < IZI< Sn}, : n < IZl < Sn}, in particular the circle {z E C : IZI = lZnl} C r = {z E C ? I it implies that oun(1zn1ei0) - u(x2)1 < c for every Thus u(xo) = u(X2) This is to A Hence f extends holomorphically impossible H contains no (ii) Assume singular points loss of generality we may assume that the manifold Z has the form Without U x A, where U is an open subset of a Banach space, and H = U x {0} This content downloaded from 204.235.148.92 on Tue, 02 Feb 2016 19:40:18 UTC All use subject to JSTOR Terms and Conditions DO DUC THAI AND NGUYEN 748 For each z E U consider fZ THAI SON /A*- X given by fZ(A) = f(z,A) for each A E A* Since (z, 0) E H there exists {(Zn, An)} C U x A*, {(Zn, An)} (z, 0) such that the distance for the holomorx0 {f(zn, An)} By applying decreasing principle Iphic map f: U x A* - X, we have - dx(f (zIAn), f (Zn An)) < It follows that {f (z, An)}IxO duxA* ((Zn, An), (z, An)) = du(z, Zn) 0? By (i), fZ extends to a holomorphic map fz A - X Define the mapf U x A X by f(z,A) = fz(A) for (z,A) E U x A Since X is hyperbolic, f is continuous Indeed, let (z, 0) E H and { (zn, An)} C U x A be such that { (zn A)} (z ?0) An Choose {A0} C A* such that {An} We have dx(f(znl An)i f(zi 0)) < dx(f(znl An)i f(zni An)) + dx(f(zni An)i f(zi An)) + dx(f (z, An), f (z, 0)) = fzn(An)) + dx(f(zn,An), dx(fzn(An), f(z,A\n)) + dx(fz(An), fz(o)) > O as n +oo X, where S(H) denotes the singular locus of H Since [3] dBxA(U, v) = inf {dBnFxA(U, V): F u, v and dimF < oo}, we have dx(f(u), f(v)) < inf {dBnFxA(U,vV) F u,v and dimF < oo} Hence f is continuous This yields the holomorphicity of f = dBxA(u,v) Q.E.D 1.3 Remark Theorem 1.2 was proved by Fujimoto [5] when Z is a finite-dimensional complex space and X is a taut complex space Howeover, since tautness is not defined in the infinite-dimensional case, the assumption on the complete hyperbolicity of X is a natural substitute ?2 HARTOGS HOLOMORPHIC EXTENSION We now prove the following 2.1 Theorem Let X be a Banach analytic space which is an increasing union of pseudoconvex domains Assume that X contains no complex lines Then X has the HEP This content downloaded from 204.235.148.92 on Tue, 02 Feb 2016 19:40:18 UTC All use subject to JSTOR Terms and Conditions HYPERSURFACES AND HARTOGS EXTENSIONS Proof (i) First we assume that X is pseudoconvex Let f: Q phic map Consider the commutative diagram X be a holomor- Qf XQ >r 751 X f B where Qf is the domain of existence of f with the canonical extension f: Qf and e, -y,7rare locally biholomorphic canonical maps X We need the following 2.2 Lemma The map f: Qf -i X is locally pseudoconvex, i.e for every x E X there exists a pseudoconvex neighbourhood U of x in X such that f (U) is pseudoconvex Proof Given x E X Choose a neighbourhood V of x in X which is isomorphic to an analytic set in an open ball of a Banach space -+ V be a holomorphic Consider the restriction fIf 1(V) Let g: Af-1(V) extension of flf- 1(V) to the envelope of holomorphy Af-1 (V) of f-1 (V) Since Qf is the domain of existence of f, it follows that Afi-(V)cQf On the other hand, from the relation f(Af-1(V)) = g(Afil(V))CV we have f-1(V) = Af-1(V) Consequently f: QfJ> X is locally pseudoconvex In order to show that Qf = AQ it remains to check that Qf satisfies the weakly disc-convex condition Indeed, let {Jk} C Hol(A, Qf) be such that the sequence {akI*A} converges to a in Hol (A*,Qf) Since X is pseudoconvex and X contains no complex lines, X satisfies the weakly disc-convex condition (see [17, Proposition 2.3]) Thus the sequence {f o ok} C Hol(A, X) convergesto f o a in Hol(A, X) Choose a pseudoconvex neighbourhood V of f o a(O) which is isomorphic to an analytic set in an open ball of a Banach space and fl (V) is pseudoconvex Since f -(V) is a pseudoconvex Riemann domain over a Banach space B with a Schauder basis, f-1 (V) is a domain of holomorphy It is easy to see that there exist ko and C V, where E > such that (fo(Jk) (A,6) C V for every k > ko and fooa(A,) /\ = {z E C: izl < E} Hence ak(A6:) C f'-(V) for every k > ko It follows that {oJkIzyj -+ (J in Hol(AL, f-1(V)) (see [6, Theorem and Lemma 6] Thus the sequence {ak } is convergent in Hol(A, Qf) 00 (ii) Assume that X = U Xn where Xn are pseudoconvex domains and X1 C n=1 X2 C Put Qn = f'(Xn) for each n > This content downloaded from 204.235.148.92 on Tue, 02 Feb 2016 19:40:18 UTC All use subject to JSTOR Terms and Conditions DO DUC THAI AND NGUYEN THAI SON 752 By (i), for each n > 1, the map fn = fl,,, fn : extends to a holomorphic map Qn -+Xn- It is easy to see that for each n > there exists a unique locally biholomorphic map en :A Qn _A Qn+1 such that the following diagram is commutative: AQn 7T{7rn AQn+1 + la n+1 B B defines AQn as a Riemann and fn+len = fn for n > 1, where irn :A Qn -* Qlim indAQn -* X and 7r Q-+ B domain over B Thus we can define maps f: A r for all n > Since irn is a local by f|AQn = fn for all n > and 1rF/AQ > homeomorphism for n 1, it follows that 7ris also a local homeomorphism Moreover, we have dn(z) < dn+1(en(z)) for all z E AQn and n > 1, where dn denotes the boundary distance with respect to 7rn: AQn+ B for each n > Since AQn is pseudoconvex, -log dn is plurisubharmonic for all n > Hence the function - log d(z) = lim - log dn(Z), for every z E Q, is plurisubharmonic This n-oo means that Q is pseudoconvex and hence Q is a domain of holomorphy This yields Q = AQ The theorem is proved Q.E.D 2.3 Remark There exists a complex manifold X which is not pseudoconvex such 00 that X U Xn, where Xn is Stein n=1 Indeed, as in [4] for each n we put Xn= {(Z7w) C E3 f j (z :wz =pn(z),p(z) - ) } Obviously, Xn are closed submanifolds of C3 and hence, Xn are Stein For each consider the map n: Xn- Xn+j defined as follows: n, W7 77)= (Z7W777(Z AYn(Z) n+1) Clearly, an is biholomorphic from Xn onto Xn+j \ {f x C2 } Thus we can define X = lim (Xn7 -yn) We shall prove that X is not pseudoconvex For the converse case, we assume that X is pseudoconvex and hence, in our case X satisfies the weakly disc-convex condition Let {ffn} C Hol(A, X) be a sequense of maps defined by fn(A) = (p7A n(A)) + Then fn(A) C Xn+1 We prove that {fn} is uniformly convergent in Hol(A*, X) definedby For each k, consider fn E Hol(\ 1,X) k (A)>1 fk(A(P1 fn (v n + 1A- k \+1 1+ ) ,7 where Akal,1 {z E : k? < Z| k, {f1} converges in Hol((Al A On the other hand, since PkAA)) (A) = (A, "YnOYn-1o *.* o Yk fnk = fn 1,X) to the map fk given by for every n, k > 1, Pf q , where p, q are natural numbers with p < q Thus we we have ,q p can define a map f: /\* -* X by setting f(z) = fk(z) for every z E A/ 1 Since k+1 -0,the sequence {fn} convergesto f in Hol(A*,X) By hypothesis, {fn} converges to f in Hol(A,X) Consider A, {z = E C: zl < },e E (0, 1) Since 00 {fn} is uniformly convergent on A., it follows that U fn(A,) is compact Since n=1 00 X = U Xk, Xk C Xk+j and Xk is open in X for every k > 1, there exists k0 such n=1 that U fn A() C Xko Hence n=1 f ko (A) (A, APko (A)) for all AeA AE* Thus f(ko) (A) can be extended holomorphically to A,, This is impossible, because Pko (0) ?0 Hence X is not pseudoconvex REFERENCES T Barth, The Kobayashi distance induces the standard topology, Proc Amer Math Soc 35 (1972), No 2, 439-441 MR 46:5668 R Brody, Compact manifolds and hyperbolicity, Trans Amer Math Soc 235 (1978), 213-219 MR 57:10010 S Dineen, R Timoney and J P Vigue, Pseudodistances invariantes sur les domaines d'un espace localement convexe, Ann Nor Sup Pisa 12 (1985), 515-529 MR 88b:32054 J E Fornaess, An increasing sequence of Stein manifolds whose limit is not Stein, Math Ann 223 (1976), 275-277 MR 54:5498 H Fujimoto, On holomorphic maps into a taut complex space, Nagoya Math J 46 (1972), 49-61 MR 46:9375 Y Hervier, On the Weierstrass problem in Banach spaces, Proc on Infinite Dimensional Holomorphy, Lecture Notes in Math 364 (1974), 157-167 MR 53:1266 S M Ivashkovicz, The Hartogs phenomenon for holomorphically convex Kaihler manifolds, English transl : Math USSR Izvestiya 29 (1987), 225-232 P Kiernan, Extensions of holomorphic maps, Trans Amer Math Soc 172 (1972), 347-355 MR 47:7066 S Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Marcel Dekker, New York 1970 MR 43:3503 10 M Kwack, Generalizations of the big Picard theorem, Ann Math 90 (1969), No 2, 9-22 MR 39:4445 11 S Lang, Introduction to Complex Hyperbolic Spaces, Springer - Verlag, 1987 MR 88f:32065 12 P Mazet, Analytic Sets in Locally Convex Spaces, Math Studies, North - Holland, v 121, 1987 MR 86i:32012 13 J Mujica, Complex Analysis in Banach Spaces, Math Studies, North - Holland, v 120, 1986 MR 88d:46084 14 J.P Ramis, Sous-ensembles Analytiques d'une Variete Banachique Complexe, SpringerVerlag, 1970 MR 45:2205 This content downloaded from 204.235.148.92 on Tue, 02 Feb 2016 19:40:18 UTC All use subject to JSTOR Terms and Conditions 754 DO DUC THAI AND NGUYEN THAI SON 15 B Shiffman, Extension of holomorphic maps into Hermitian manifolds, Math Ann 194 (1971), 249-258 MR 45:598 16 B D Tac, Extending holomorphic maps in infinite dimension, Ann Polon Math 54 (1991), 241-253 17 Do Duc Thai and Nguyen Le Huong, On the disc - convexity of Banach analytic manifolds, Ann Polon Math 69 (1998), 1-11 CMP 98:14 18 E Vesentini and T Franzoni, Holomorphic Maps and Invariant Distances, North - Holland, Math Studies 40, Amsterdam 1980 MR 82a:32032 19 E Vesentini, Invariant distances and invariant differential metric in locally convex spaces, Spectral theory Banach centre Publication U.8 P.W.N., Polish Sci Publisher Warsaw 1982, 493-512 MR 85d:32049 DEPARTMENT OF MATHEMATICS, VIETNAM NATIONAL UNIVERSITY, INSTITUTE OF PEDAGOGY, CAU GIAY - Tu LIEM, HANOI, VIETNAM E-mail address: ddthaiInetnamn org This content downloaded from 204.235.148.92 on Tue, 02 Feb 2016 19:40:18 UTC All use subject to JSTOR Terms and Conditions ... HYPERSURFACES AND RELATIONS TO THE HARTOGS EXTENSIONS IN INFINITE DIMENSION DO DUC THAI AND NGUYEN THAI SON (Communicated by Steven R Bell) ABSTRACT A generalization of Kwack's theorem to the infinite dimensional... for instance, the survey [11] Therefore proving Kwack type theorems in the infinite dimensional case is really necessary and motivates the study of the extension problem of holomorphic maps in infinite. .. property INTRODUCTION We discuss the problems of extending holomorphic maps through hypersurfaces and extending to the envelope of holomorphy, i.e the Hartogs holomorphic extension These two