We give an abstract definition, similar to the axioms of a Stein manifold, of
a class of complex Banach manifolds in such a way that a manifold belongs to
the class if and only if it is biholomorphic to a closed split complex Banach
submanifold of a separable Banach space.Comment: 5 page

Patyi, Imre

English

arXiv

Imre Patyi

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 349 (2011) 43–45Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comComplex Analysis/Analytic GeometryOn complex Banach manifolds similar to Stein manifoldsSur une classe de variétés banachiques complexes analogues aux variétés de SteinImre Patyi 1Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303-3083, USAa r t i c l e i n f o a b s t r a c tArticle history:Received 22 October 2010Accepted after revision 19 November 2010Available online 18 December 2010Presented by Jean-Pierre DemaillyWe give an abstract definition, similar to the axioms of a Stein manifold, of a class ofcomplex Banach manifolds in such a way that a manifold belongs to the class if and onlyif it is biholomorphic to a closed split complex Banach submanifold of a separable Banachspace.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éOn donne ici une définition d’une classe de variétés banachiques complexes analoguesaux variétés de Stein par leur définition et leurs propriétés. On montre qu’une variétébanachique quelconque de cette classe est biholomorphe à une sous-variété banachiquecomplexe fermée et scindée dans un espace de Banach complexe séparable, et réciproque-ment, qu’une telle sous-variété est bien toujours membre de la classe.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Stein manifolds can be characterized among complex manifolds in various ways, including the two ways (I) and (II)below. A paracompact second countable Hausdorff complex manifold M of pure dimension is a Stein manifold if and only ifone and hence both of the following equivalent conditions (I) and (II) below hold.(I) (a) M is holomorphically convex, i.e., if K ⊂ M is compact, then its O(M) holomorphic hull K̂ is compact in M . (b) Ifx = y in M , then there is an f ∈ O(M) with f (x) = f (y). (c) If x ∈ M , then there are an integer n  1 and a holomorphicfunction g ∈ O(M,Cn) that is a biholomorphism from an open neighborhood W of x in M to an open neighborhood g(W )of g(x) in Cn .(II) There is an n  1 such that M is biholomorphic to a closed complex submanifold M ′ of Cn .Let X be a separable Banach space, and M a paracompact second countable Hausdorff complex Banach manifold mod-elled on X . We call M a Stein Banach manifold modelled on X if (i)–(iv) below hold.(i) Holomorphic domination is possible in M , i.e., if u : M → R is any locally upper bounded function, then there are aBanach space Z and a holomorphic function h : M → Z with u(x) < ‖h(x)‖ for all x ∈ M .(ii) There are open sets Un, Vn ⊂ M , and holomorphic functions fn ∈ O(M), n  1, such that ⋃∞n=1(Un × Vn) = (M × M) \M , where M = {(x, x): x ∈ M} is the diagonal of M × M , and fn(Un) and fn(Vn) are disjoint sets in C for all n  1.(iii) There are open sets Wn ⊂ M and holomorphic maps gn ∈ O(M, X) for n  1 such that ⋃∞n=1 Wn = M and gn|Wn isa biholomorphism from Wn onto an open set gn(Wn) in X .E-mail address: ipatyi@gsu.edu.1 Supported in part by NSF grant DMS 0600059.1631-073X/$ – see front matter © 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2010.11.02044 I. Patyi / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 43–45(iv) There are open sets Gk ⊂ M , k  1, with ⋃∞k=1 Gk = M and supx∈Gk (| fn(x)| + ‖gn(x)‖) < ∞ for all k,n  1, where( fn) and (gn) are as in (ii) and (iii).If M is finite dimensional, then it is easy to see that (i)–(iii) together are equivalent to (I), and (iv) is vacuous, since if Gk ,k  1, is an exhaustion of M by precompact open sets Gk , then any continuous function | fn(x)| + ‖gn(x)‖ on M is boundedon Gk for k,n  1. Thus if M is finite dimensional, then (i)–(iv) together are equivalent to M being a Stein manifold.Theorem 1. Let X be a separable Banach space, and M a paracompact second countable Hausdorff complex Banach manifold modelledon X. Then M is a Stein Banach manifold modelled on X if and only if there is a separable Banach space X ′ such that M is biholomorphicto a closed split complex Banach submanifold M ′ of X ′ .Here Banach manifolds and Banach submanifolds are understood in terms of biholomorphically related charts, and aBanach submanifold is called split if each of its tangent spaces has a direct complement in the ambient Banach space.Clearly, a complex Banach submanifold M of X is split if and only if near each point x0 ∈ M it is possible to split X as adirect sum X = X ′ × X ′′ of closed linear subspaces X ′, X ′′ of X such that with x0 = (x′0, x′′0) and x = (x′, x′′) we can write Mas the graph x′′ = m(x′) of a holomorphic function m from an open neighborhood of x′0 in X ′ to X ′′ , where x′′0 = m(x′0).Proof. Suppose first that M is biholomorphic to an M ′ and verify that M satisfies (i)–(iv). It is enough to show that M ′does.As holomorphic domination is possible in X ′ by [3], and thus also in M ′ , since M ′ is closed in X ′ , (i) is true. We definesome linear functions fn : X ′ → C and gn : X ′ → X for n  1 whose restrictions to M ′ will do the job. For linear functions(iv) is automatic: we can let Gk be the intersection of M ′ with the open ball ‖x‖ < k in X ′ and write | fn(x)| + ‖gn(x)‖ (‖ fn‖ + ‖gn‖)‖x‖  (‖ fn‖ + ‖gn‖)k < ∞ for x ∈ Gk and n,k  1.If x = y in M ′ , then x − y = 0 in X ′ and the Hahn–Banach theorem gives us a complex linear functional fxy ∈ (X ′)∗of norm 1 with Re fxy(x − y) = ‖x − y‖ > 0. Let Uxy = {z ∈ X ′: − 12 ‖x − y‖ + Re fxy(x) < Re fxy(z)} and V xy = {z ∈X ′: Re fxy(z) < 12 ‖x − y‖ + Re fxy(y)}. Then x ∈ Uxy , y ∈ V xy , and their images fxy(Uxy) and fxy(V xy) are disjoint sincethey are the half planes − 12 ‖x − y‖ + Re fxy(x) < Re w , Re w < 12 ‖x − y‖ + Re fxy(y), which are clearly disjoint since− 12 ‖x − y‖ + Re fxy(x) = 12 ‖x − y‖ + Re fxy(y).Fix any point x0 ∈ M ′ and denote its complex tangent space Tx0 M ′ by X and regard it as a closed linear subspace of X ′ .If x ∈ M ′ , then the complex tangent space Tx M ′ and X are linearly isomorphic via a bounded linear map ix : Tx M ′ → X , andthere is a bounded linear projection px : X ′ = Tx X ′ → Tx M ′ . Thus the linear map gx : X ′ → X given by gx(y) = ix(px(y))for y ∈ X ′ satisfies that (dgx)(x)y = ix(y) for y ∈ Tx M ′ , i.e., (dgx)(x) is a linear isomorphism from Tx M ′ onto X . By theinverse function theorem gx is biholomorphic from an open neighborhood W x of x in M ′ to an open neighborhood gx(W x)of gx(x) = 0 in X .By Lindelöf’s theorem in the second countable (separable metric) spaces (M ′ × M ′) \ M′ and M ′ the open coveringsUxy × V xy , (x, y) ∈ (M ′ × M ′) \ M′ , and W x , x ∈ M ′ , can be reduced to countable subcoverings Un × Vn , Wn , where Un =Uxn yn , Vn = V xn yn , and Wn = W x′n for n  1. Thus the functions fn = fxn yn , gn = gx′n , n  1, do the job.Conversely, assume that M satisfies (i)–(iv) and embed M biholomorphically as M ′ into a separable Banach space X ′ .If i  1, then let Ci = 1 + sup{| fn(x)| + ‖gn(x)‖: 1  k,n  i, x ∈ Gk}. So if k,n  1, and x ∈ Gk , then | fn(x)| + ‖gn(x)‖ CkCn . Thus upon replacing fn by fn/(Cn2n) and gn by gn/(Cn2n), we obtain new functions again to be called fn , gn thatsatisfy (ii), (iii), and the slightly strengthened version supx∈Gk (| fn(x)| + ‖gn(x)‖) < Ck/2n , k,n  1, of (iv).The covering Wn , n  1, of the paracompact space M has a locally finite refinement, which by Lindelöf’s theorem can betaken to be countable, and can be shrunk since a paracompact Hausdorff space M is normal. There are open sets Mn ⊂ M ,n  1, with⋃∞n=1 Mn = M , and for each n  1 there is an index j(n)  1 with the closure Mn ⊂ W j(n) . Define u : M → R byu(x) = inf{n  1: x ∈ Mn}. Then u is locally upper bounded on M since u  n on the open set Mn .By assumption (i) on holomorphic domination there are a Banach space Z and a holomorphic function h ∈ O(M, Z) withu(x) < ‖h(x)‖ for x ∈ M . As Z ′ = span{h(x): x ∈ M} is a separable Banach space, and as any separable Banach space can beembedded into C[0,1] we can replace the Banach space Z by the separable space Z = C[0,1] endowed with the sup norm.Define a Banach space X ′ by X ′ = Z × 1 × 1(X), where 1 and 1(X) denote the spaces of summable sequences in Cand in X . Let us write the variable y in X ′ as y = (y′, y′′, y′′′), where y′′ = (y′′n) ∈ 1 and y′′′ = (y′′′n ) ∈ 1(X). Clearly, X ′ isa separable Banach space, being the product of three such spaces.Define the map Φ : M → X ′ given by y = Φ(x), where⎧⎨⎩y′ = h(x),y′′n = fn(x),y′′′n = gn(x),n  1.If k  1 and x ∈ Gk , then we have ∑∞n=1(| fn(x)| + ‖gn(x)‖) ∑∞n=1 Ck/2n  Ck < ∞. Thus Φ is holomorphic.Our Φ is injective, since if x = y in M , then there is an index n  1 with x ∈ Un and y ∈ Vn , so fn(x) = fn(y), and evenmore so Φ(x) = Φ(y).We claim that the set M ′ = Φ(M) is closed in X ′ . Indeed, suppose that yi = Φ(xi), xi ∈ M , converges yi → y in the normin X ′ as i → ∞ to an element y ∈ X ′ , we must show that there is an x ∈ M with y = Φ(x), i.e., y ∈ M ′ . As y′ = h(xi), i  1, isiI. Patyi / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 43–45 45bounded, being convergent, there is an index b  1, with ‖h(xi)‖  b for i  1, i.e., u(xi) < ‖h(xi)‖  b, or, xi ∈ M1 ∪ · · · ∪ Mbfor i  1. By the pigeon hole principle there are an index a with 1  a  b and an infinite set I of indices i such that xi ∈ Mafor all i ∈ I . As Ma ⊂ W j(a) , and g j(a) is biholomorphic on W j(a) , we see that Φ(xi), i ∈ I , may converge only if the xi , i ∈ I ,converge in Ma to one of its elements x ∈ Ma ⊂ W j(a) . Thus yi = Φ(xi) → Φ(x) = y as i → ∞ in I . As M ′ contains y, it isclosed.If y0 = Φ(x0) in M ′ , then there is an index n  1 with x0 ∈ Wn . So gn is biholomorphic from a connected open setW ′n with x0 ∈ W ′n ⊂ W ′n ⊂ Wn to a connected open set gn(W ′n) in X . Then the connected component of the set M ′ ∩{(y′, y′′, y′′′): y′′′n ∈ gn(W ′n)} that contains the point y0 equals the graph⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩y′′′n = y′′′n ,y′ = h(g−1n (y′′′n )),y′′n = fn(g−1n (y′′′n )),y′′ν = fν(g−1n (y′′′n )),y′′′ν = gν(g−1n (y′′′n )),ν = n,of a holomorphic map y′′′n → (y′, y′′, y′′′1 , y′′′2 , . . . , y′′′n−1,0, y′′′n+1, y′′′n+2, . . .) from W ′n to the Banach space X ′ ∩ {(y′, y′′, y′′′):y′′′n = 0}.Thus M ′ is a closed split complex Banach submanifold of X ′ and Φ : M → M ′ is a biholomorphism. The most substantial part of the above proof is to show that holomorphic domination is possible on a separable Banachspace. That was done in Ref. [3] based upon the work of Lempert in [1]. It might be possible to weaken the axioms (i)–(iv)perhaps by dropping (iv) and replacing (i), that stands in for holomorphic convexity, by plurisubharmonic domination, i.e.,by requiring a continuous plurisubharmonic function ψ : M → R that dominates the given locally upper bounded functionu : M → R. Nevertheless, axioms (i)–(iv) represent perhaps the ultimate axioms for “Stein Banach manifolds” since any othersystem for which the desirable Theorem 1 holds must be equivalent with (i)–(iv). Most known methods of plurisubharmonicdomination also yield holomorphic domination, and a ‘constructive’ procedure for building the functions fn, gn in (ii) and(iii) is likely to produce functions that also satisfy (iv). The author doubts whether a successful “Stein theory” could bebuilt up for nonseparable Banach spaces and Banach manifolds. Even for separable Banach manifolds it would be better torestrict attention to the ones modelled on separable Banach spaces with the bounded approximation property (there arevirtually no practical separable Banach spaces that do not satisfy the bounded approximation property). If M is a SteinBanach manifold modelled on such a Banach space, then the sheaf cohomology group Hq(M, S) vanishes when q  1 andS → M is a so-called cohesive sheaf defined in [2] by Lempert and Patyi. The question arises whether M is a Stein Banachmanifold if Hq(M, S) = 0 for all q  1 and all cohesive sheaves S → M . If M is an open subset of a separable Banach spacewith the bounded approximation property, then the answer is Yes.The author is grateful to the referees for their helpful comments.References[1] L. Lempert, Plurisubharmonic domination, J. Amer. Math. Soc. 17 (2004) 361–372.[2] L. Lempert, I. Patyi, Analytic sheaves in Banach spaces, Ann. Sci. École Norm. Sup. Sér. 4 40 (2007) 453–486.[3] I. Patyi, On holomorphic domination, I, arXiv:0910.0476, Bull. Sci. Math., doi:10.1016/j.bulsci.2010.11.003, in press.

On complex Banach manifolds similar to Stein manifolds

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On complex Banach manifolds similar to Stein manifolds

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