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SEMI-GLOBAL EXTENSION OF MAXIMALLY COMPLEX SUBMANIFOLDS GIUSEPPE DELLA SALA, ALBERTO SARACCO 7 0 0 Abstract. LetAbeadomainoftheboundaryofastrictlypseu- 2 doconvex domain Ω of Cn and M a smooth, closed, maximally n complex submanifold of A. We find a subdomain A of Ω, depend- a ing only on Ω and A, and a complex variety W ⊂ A such that e J bW ∩A=M. Moreover,a generalizationto analytic sets of depth e 9 at least 4 is given. 1 ] V C 1. Introduction . h In the last fifty years, the boundary problem, i.e. the problem of t a characterizing real submanifolds which are boundaries of “something” m analytic, has been widely treaten. [ The first result of this kind is due to Wermer [19]: compact real 2 curves in Cn are boundaries of complex varieties if and only if they v satisfy a global integral condition, the moments condition. For greater 7 4 dimension the problem was solved, by Harvey and Lawson [8], prov- 7 ing that an obviously necessary condition (maximal complexity) is also 7 sufficient for compact manifolds in Cn. Later on, characterizations for 0 6 closed (non necessarily compact) submanifolds in q-concave open sub- 0 sets of CPn were provided by Dolbeault-Henkin and Dihn in [5, 6, 4]. / h A new approach to the problem in CPn has been recently set forth by t a Harvey-Lawson [10, 11, 12, 13]. m Our goal is to drop the compactness hypothesis. The results in [3] : v deal with the global situation of submanifolds contained in the bound- i X ary of a special class of strongly pseudoconvex unbounded domains in r Cn. In this paper we deal with the boundary problem for complex a analytic varieties in a “semi-global” setting. More precisely, let Ω ⊂ Cn be a strongly pseudoconvex open domain in Cn, and bΩ its boundary. Let M be a maximally complex (2m+1)- dimensional real closed submanifold (m ≥ 1) of some open domain A ⊂ bΩ, and let K be its boundary. We want to find a domain A in Ω, independent of M, and a complex subvariety W of A such that: e Date: February 2, 2008. e 2000 Mathematics Subject Classification. Primary 32V25, Secondary 32T15. Key words and phrases. boundary problem · pseudoconvexity · maximally complex submanifold · CR geometry. 1 2 GIUSEPPE DELLA SALA,ALBERTO SARACCO (i) bA∩bΩ = A; (ii) bW ∩bΩ = M, e In this paper we show that, if A ⋐ bΩ, the problem we are dealing with has a solution (A,W) whose A can be determined in terms of the envelope K of K with respect to the algebra of functions holomorphic e e in a neighbourhood of Ω, i.e. b For any maximally complex (2m + 1)-dimensional closed real submanifold M of A, m ≥ 1, there exists an (m+1)-dimensional complex variety W in Ω \ K, with isolated singularities, such that bW ∩(A\K) = M ∩(A\K). b This result echoes that obf Lupacciolu onbthe extension of CR-functions (see [15, Theorem 2]) If A is not relatively compact, this result can be restated in terms of “principal divisors hull”, leading to a global result for unbounded strictly pseudoconvex domains, different fromtheresults in[3]. Indeed, this method of proof allows us to drop the Lupacciolu hypothesis in [3] and extend the maximally complex submanifold to a domain, which can anyhow not be the whole of Ω. If the Lupacciolu hypothesis holds, thenthedomainofextension isinfact allofΩ. So thisresultisactually a generalization of the one in [3]. The crucial question of the maximality of the domain A we construct is not answered; in some simple cases the domain is indeed maximal e (see Example 4.1). In the last section, by the same methods, the extension result is proved for analytic sets (see Theorem 5.1). It worths noticing that in [17] related results are obtained via a bump Lemma and cohomological methods. That approach may be generalized to complex spaces. We wish to thank Giuseppe Tomassini for suggesting us the problem in the first place and for useful discussions. 2. Definitions and notations In all the paper we will always consider, unless otherwise stated, Cn withcoordinatesz = x +iy ,...,z = x +iy ,x ,...,x ,y ,...,y ∈ 1 1 1 n n n 1 n 1 n R. A smooth real (2m+1)-dimensional submanifold M of Cn is said to be a CR manifold if its complex tangent H M has constant dimension p at each point p. If m > 0 and dimCHpM = m, i.e. it is the maximal possible, M is said to be maximally complex. Observe that a smooth hypersurface of Cn is always maximally complex. SEMI-GLOBAL EXTENSION 3 If m = 0 and M = γ is a compact curve, we say that γ satisfies the moments condition if ω = 0, Z γ for any holomorphic (1,0)-form ω. Itiseasytoobserve thatthe(smooth)boundaryofacomplex variety of Cn of dimension m+1 is maximally complex if m > 0 (respectively satisfies the moments condition if m = 0). A domain Ω ⊆ Cn is called strongly pseudoconvex if there is a neighbourhood U of its boundary such that Ω∩U = {z ∈ U : ρ(z) < 0}, where ρ is strictly plurisubharmonic in U. 3. Main result Let Ω ⊂ Cn be a strongly pseudoconvex open domain in Cn. Let A be a subdomain of bΩ, and K = bA. For any Stein neighborhood Ω of Ω we set K to be the hull of K with respect to the algebra of α α holomorphic functions of Ω , i.e. α b K = {x ∈ Ω : |f(x)| ≤ kfk ∀f ∈ O(Ω )}. α α K α We define Kb as the intersection of the K when Ω varies through α α the family of all Stein neighborhoods of Ω. Observe that, since Ω b b is strongly pseudoconvex (and thus admits a fundamental system of Stein neighborhoods, see [18]), K coincides with the hull of K with respect to the algebra of the functions which are holomorphic in some b neighborhood of Ω. We claim that the following result holds: Theorem 3.1. Foranymaximallycomplex(2m+1)-dimensionalclosed real submanifold M of A, m ≥ 1, there exists an (m+1)-dimensional complex variety W in Ω \ K, with isolated singularities, such that bW ∩(A\K) = M ∩(A\K). b Followingbthe same stratbegy as in [3] we first have a semi-local extension result (see Lemma 3.2 below). In order to “ globalize” the extension the main differences with respect to [3] are due to the fact that we have to cut Ω with level-sets of holomorphic functions instead of hyperplanes. This creates some additional difficulties: first of all it is no longer possible to use the parameter which defines the level-sets as a coordinate; secondly the intersections between tubular domains (see Lemmas 3.7, 3.9 and 3.10) may not be connected. With the same proof as in [3] we have Lemma 3.2. There exist a tubular neighborhood I of A in Ω and an (m+ 1)-dimensional complex submanifold with boundary W ⊂ Ω ∩I I such that S ∩bW = M I 4 GIUSEPPE DELLA SALA,ALBERTO SARACCO Now, the hypothesis on the hull of K allows us to prove the following Lemma 3.3. Let z0 ∈ Ω\ K. Then there exist an open Stein neighborhood Ω ⊃ Ω and f ∈ O(Ω ) such that α α b 1) f(z0) = 0; 2) {f = 0} is a regular complex hypersurface of Ω \K; α 3) {f = 0} intersects M transversally in a compact manifold. b Remark 3.1. If f is such a function for z0, for any point z′ sufficiently near to z0, f(z)−f(z′) satisfies conditions 1), 2) and 3) for z′. Proof. By definition of K, since z0 ∈ Ω\K there is a Stein neighborhood Ω such that z0 6∈ K . So we can find a holomorphic function g α bα b in Ω such that g(z0) = 1 and kgk < 1; h(z) = g(z)−1 is a holomor- α K b phic function whose zero set does not intersect K. Since regular level sets are dense, by choosing a suitable small vector v and redefining h b as h(z + v) − h(z0 + v) we can safely assume that h satisfies both 1) and 2). We remark that {h = 0} ∩ bΩ ⋐ A by Alexander’s Theorem (see [1, Theorem 3]), and this shows compactness. Then, we may suppose that M is not contained in {z = z0} and, for ε small enough, we 1 1 consider the function f(z) = h(z)+ε(z −z0). It’s not difficult to see 1 1 (by applying Sard’s Lemma) that 3) holds for generic ε. 2 Now, we divide the proof of Theorem 3.1 in two cases: m ≥ 2 and m = 1. This is due to the fact that in the latter case proving that we can apply Harvey-Lawson to {f = 0}∩M is not automatic. 3.1. Dimension of M greater than or equal to 5: m ≥ 2. For any z0 ∈ Ω\K, Lemma 3.3 provides a holomorphic function such that the level-set f = {f = 0} contains z0 and intersects M transversally in 0 b a compact manifold M . The intersection is again maximally complex 0 (it is the intersection of a complex manifold and a maximally complex manifold, see [8]), so we can apply Harvey-Lawson Theorem to obtain a holomorphic chain W such that bW = M . For τ in a small neigh- 0 0 0 borhood U of 0 in C, the hypersurface f = {f −τ = 0} intersects M τ transversally alongacompact submanifoldM which, againbyHarvey- τ Lawson Theorem, bounds a holomorphic chain W . Observe that since τ M ⊂ f , W ⊂ f . τ τ τ τ We claim the following proposition holds: Proposition 3.4. The union W = W is a complex variety U τ∈U τ contained in the open set U = f .S τ∈U τ S We need some intermedeiate results. Let us consider a generic projection π : U → Cm and set Cn = Cm+1 ×Cn−m−1, with holomorphic e SEMI-GLOBAL EXTENSION 5 coordinates (w′,w), w′ ∈ Cm+1, w = (w ,...,w ) ∈ Cn−m−1. Let 1 n−m−1 V = Cm+1 \π(M ). τ τ For τ ∈ U, w′ ∈ Cm+1 \π(M ) and α ∈ Nn−m−1, we define τ Iα(w′,τ) + ηαω (η′ −w′), BM Z(η′,η)∈Mτ where ω is the Bochner-Martinelli kernel. BM In [3] the following was proved Lemma 3.5. Let F(w′,τ) be the multiple-valued function which repre- sents W on Cm+1\π(M ) and denote by Pα(F(w′,τ)) the sum of the τ τ αth powers of the values of F(w′,τ). Then Pα(F(w′,τ)) = Iα(w′,τ). In particular, the cardinality P0(F(w′,τ)) of F(w′,τ) is finite. Remark 3.2. Lemma 3.5 implies, in particular, that the functions Pα(F(w′,τ)) are continuous in τ. Indeed, they are represented as inte- grals of a fixed form over a submanifold M which varies continuously τ with the parameter τ. Lemma 3.6. Pα(F(w′,τ)) is holomorphic in the variable τ ∈ U ⊂ C, for each α ∈ Nn−m−1. Proof. Let us fix a point (w′,τ) such that w′ ∈/ M (this condition τ remains true for τ ∈ B (τ)). Consider as domain of Pα(F) the set ǫ {w′}×B (τ). In view of Morera’s Theorem, we need to prove that for ǫ any simple curve γ ⊂ B (τ), ǫ Pα(F(w′,τ))dτ = 0. Z γ Let Γ ⊂ B (τ) be an open set such that bΓ = γ. By γ ∗M (Γ∗M ) ǫ τ τ we mean the union of M along γ (along Γ). Note that these sets are τ submanifolds of C×Cn. The projection π : Γ∗M → Cn on the second τ factor is injective and π(Γ ∗ M ) is an open subset of M bounded by τ 6 GIUSEPPE DELLA SALA,ALBERTO SARACCO π(bΓ∗M ) = π(γ ∗M ). By Lemma 3.5 and Stokes Theorem τ τ Pα(F(w′,τ))dτ = Iα(w′,τ)dτ = Z Z γ γ = ηαω (η′ −w′) dτ = BM Zγ(cid:18)Z(η′,η)∈Mτ (cid:19) = ηαω (η′ −w′)∧dτ = BM ZZ γ∗Mτ = d(ηαω (η′ −w′)∧dτ) = BM ZZ Γ∗Mτ = dηα ∧ω (η′ −w′)∧dτ = BM ZZ Γ∗Mτ = dηα ∧ω (η′ −w′)∧π dτ = BM ∗ ZZ π(Γ∗Mτ) = 0. The last equality follows from the fact that in dηα appear only holomorphic differentials, ηα being holomorphic. But since all the holomorphic differentials supported by π(Γ ∗M ) ⊂ M already appear in τ ω (η′−w′)∧π dτ (due to the fact that M is maximally complex and BM ∗ contains only m+1 holomorphic differentials) the integral is zero. 2 Proof of Proposition 3.4. From [9] it follows that each W has τ isolated singularities1. So, let us fix a regular point (w′,w ) ∈ f ⊂ U. 0 0 τ0 In a neighborhood of this point W = W is a manifold, since the U e construction depends continuously on the initial data. We want to show that W is indeed analytic in U. Let us fix j ∈ {1,...,n−m−1} and consider multiindexes α of the form (0,...,0,α ,0,...,0); let Pα bee the corresponding Pα(F(w′,τ)). j j Observe that for any j we can consider a finite number of Pα (it suffices j to use h = P0(F(w′,τ)) of them; not that h is independent of j). By j a linear combination of the Pα with rational coefficients, we obtain j the elementary symmetric functions S0(w′,k),...,Sh(w′,τ) in such a j j way that for any point (w′,w) ∈ W there exists τ ∈ U such that (w′,w) ∈ W ; thus, defining τ Q (w′,w,τ) = Sh(w′,τ)+Sh−1(w′,τ)w +···+S0(w′,τ)wh = 0, j j j j j j 1There could be singularities coming up from intersections of the solutions rela- tive to different connected components of Mτ. These singularities are analytic sets andthereforeshouldintersectthe boundary. This cannothappenandsoalsothese singularities are isolated. SEMI-GLOBAL EXTENSION 7 we have, in other words, n−m−1 ′ W ⊂ V = {Q (w ,w,τ) = 0}. j τ[∈U j\=1 Define V ⊂ Cn(w′,w)×C(τ) as e n−m−1 ′ V = {Q (w ,w,τ) = 0} j j\=1 e and W = W ∗U ⊂ V. τ Observe that, since the functions Sα are holomorphic, V is a complex f j e subvariety of Cn ×U. Since V and W have the same dimension, in a e neighborhood of (w′,w ,τ) W is an open subset of the regular part of 0 0 e f V,thusacomplexsubmanifold. WedenotebyReg(W)thesetofpoints f z ∈ W such that W ∩U is a complex submanifold in a neighborhood e f U of z. It is easily seen that Reg (W) is an open and closed subset f f of Reg (V), so a connected component. Observing that the closure of f a connected component of the regular part of a complex variety is a e complex variety we obtain the that W is a complex variety, W being the closure of Reg (W) in V. f f Finally, since the projection π : W → W is a homeomorphism and f e so proper, it follows that W is a complex subvariety as well. 2 f Now we prove that the varieties W that we have found — which are U defined in the open subsets of type U (see Proposition 3.4) — patch f together in such a way to define a complex variety on the whole of e Ω\K. Lembma 3.7. Let U and U be two open subsets as in Proposition 3.4 f g and let W and W be the corresponding varieties. Let z1 ∈ U ∩U . f eg e f g Then W and W coincide in a neighborhood of z1. f g e e Proof. Let λ = f(z1) and τ = g(z1) and consider ′ ′ ′ ′ L(λ,τ ) = {f = λ}∩{g = τ } ⊂ Ω for (λ′,τ′) in a neighborhood of (λ,τ). Note that for almost every (λ′,τ′) L(λ′,τ′) is a complex submanifold of codimension 2 of U ∩ f U . Moreover, W ∩L(λ′,τ′) and W ∩L(λ′,τ′) are both solutions of g f g e the Harvey-Lawson problem for M ∩L(λ′,τ′), consequently they must ceoincide. Since the complex subvarieties L(λ′,τ′) which are regular form a dense subset, W and W coincide on the connected component f g of U ∩U containing z1. 2 f g e e 8 GIUSEPPE DELLA SALA,ALBERTO SARACCO Remark 3.3. The above proof does not work in the case m = 1 since M ∩L(λ′,τ′) is generically empty. In order to end the proof of Theorem 3.1, we have to show that the set S of the singular points of W is a discrete subset of Ω \ K. Let z1 ∈ Ω\ K, and choose a function h, holomorphic in a neighborhood b of Ω such that h(z1) = 1 and K ⊂ {|h| ≤ 1} and consider f = h− 3. b 2 4 Observe that z1 ∈ {Ref > 0} and K ⊂ {Ref < 0}. Choose a defining function ϕ for bΩ, strongly psh in a neighborhood of Ω and let us consider the family (φ = λ(ϕ)+(1−λ)Ref) λ λ∈[0,1] of strongly plurisubharmonic functions. For λ near to 1, {φ = 0} λ does not intersect the singular locus. Let λ be the biggest value of λ for which {φ = 0} ∩ S 6= ∅. Then the analytic set S touches the λ boundary of the Stein domain {φ < 0}∩Ω ⊂ Ω. λ So {φ = 0}∩S is a set of isolated points in S. By repeating the same λ argument, we conclude that S is made up by isolated points. 3.2. Dimension of M equal to 3: m = 1. The first goal is to show that when we slice transversally M with complex hypersurfaces, we obtain 1-dimensional real submanifolds which satisfy the moments condition. Again, we fix our attention to a neighborhood of the form U = g . τ τ[∈U e Let us choose an arbitrary holomorphic (1,0)-form ω in Cn. Lemma 3.8. The function Φ (τ) = ω ω Z Mτ is holomorphic in U. Proof. Using again Morera’s Theorem, we need to prove that for any simple curve γ ⊂ U, γ = bΓ, Φ (τ)dτ = 0. ω Z γ SEMI-GLOBAL EXTENSION 9 By Stokes Theorem, we have Φ (τ)dτ = ω dτ = ω Z Z (cid:18)Z (cid:19) γ γ Mk = ω ∧dτ = ZZ γ∗Mτ = d(ω ∧dτ) = ZZ Γ∗Mτ = ∂ω ∧dτ = ZZ Γ∗Mτ = ∂ω ∧π dτ = ∗ ZZ π(Γ∗Mτ) = 0. The last equality is due to the fact that π(Γ∗M ) ⊂ M is maximally τ complex and thus supports only (2,1) and (1,2)-forms, while ∂ω∧π dτ ∗ is a (3,0)-form. 2 Lemma 3.9. Let g be a holomorphic function on a neighborhood of Ω, and suppose {|g| > 1} ∩ K = ∅. Then there exists a variety W on g Ω∩{|g| > 1} such that bW ∩bΩ = M ∩{|g| > 1}. g b Lemma 3.10. Given two functions g and g as above, then W and 1 2 g1 W agree on {|g | > 1}∩{|g | > 1}. g2 1 2 Proof of Lemma 3.9. We are going to use several times open subsets of the type U as in Proposition 3.4, so we need to fix some notations. Given an open subset U ⊂ C, define U by e U = {fe= τ}. τ[∈U e From now on we use open subsets of the form U = B(τ,δ), where B(τ,δ) is the disc centered at τ of radius δ. We say that {f = τ} is the core of U and δ is its amplitude. For a fixed d > 1 consider the compact set H = Ω ∩ {|g| ≥ d}; d e we show that W is well defined on H . Let us fix also a compact set g d C ⊂ Ω such that W (see Lemma 3.2) is a closed submanifold in H \C. I d Consider all the open subsets V = U ∩ Ω, constructed using only α α the function f = g − 1 up to addition of the function ε(z − z0) (see e j j Lemma 3.3). If we do not allow ε to be greater than a fixed ε > 0, then by a standard argument of semicontinuity and compactness we may suppose that the amplitude of each U is greater than a positive δ. We claim that it is possible to find a countable covering of H made d e by a countable sequence V of those V in such a way to have i α (1) V ⊂ H \C; 0 d 10 GIUSEPPE DELLA SALA,ALBERTO SARACCO (2) if l B = V l i i[=1 then V ∩B ∩Ω 6= ∅. l+1 l The only thing we have to prove is the existence of V , since the 0 second statement follows by a standard compactness argument. Set L = max Reg. Since Reg is a non constant pluriharmonic Hd function, {Reg = L} is a compact subset of bΩ ∩ H . Then we can d choose η > 0 such that {Reg = L−η}∩Ω is contained in H \C, and d this allows to define V . 0 Let U and U be two such open sets and z0 ∈ U ∩ U . We can 1 2 1 2 suppose that the cores of U and U contain z0. They are of the form e e 1 2 e e f +ε (z −z0) =eτ(ε ) aend f +ε (z −z0) = τ(ε ). 1 j j 1 2 j j 2 For ε ∈ (ε ,ε ), we consider the open sets U whose core, passing by 1 2 ε z0, is f +ε(z −z0) = τ(ε). We must show that the set j j e Λ = ε ∈ (ε ,ε ) : ∃W s.t. W ∩(U ∩U ) = W ∩(U ∩U ) 1 2 ε ε 1 ε 1 1 ε n o e e e e is open and closed, where W is a variety in U . ε ε Λisopen. Indeed, ifε ∈ Λ, thenforε′ inaneighborhoodofεthecore e of Uε′ is contained in Uε and so its intersection with M is maximally complex. BecauseofLemma3.8theconditionholdsalsoforallthelevel e e setsinUε′ andthenwecanapplyagaintheHarvey-Lawson Theorem[8] and the arguments of Proposition 3.4 in order to obtain Wε′. Moreover, e there is a connected component of Uε ∩ Uε′ which contains z0 and touches the boundary of Ω, where the Wε and Wε′ both coincide with W (see Lemma 3.2). By virtue of the analytic continuation principle, I they must coincide in the whole connected component. Λ is closed. Indeed, since each U has an amplitude of at least δ, we again have that, for ε ∈ Λ, the intersection of U and U must include e ε ε (for ε ∈ Λ, |ε−ε| sufficiently small) a connected component containing e e z0 and touching the boundary. We then conclude as in the previous 2 case. Proof of Lemma 3.10. Let us consider the connected components of W ∩{|g | > 1}. For each connected component W two cases are g1 2 1 possible: (1) W touches the boundary of Ω: W ∩bΩ 6= ∅; 1 1 (2) the boundary of W is inside Ω: 1 bW ⋐ {|g | = 1}∪{|g | = 1} ⊂ Ω 1 1 2