Embeddings for 3-dimensional CR-manifolds Charles L. Epstein and Gennadi M. Henkin Department of Mathematics, University of Pennsylvania and University of Paris, VI Abstract. We consider the problem of projectively embedding strictly pseudocon- cave surfaces, X− containing a positive divisor, Z and affinely embedding its 3- dimensional, strictly pseudoconvex boundary, M = −bX−. We show that embed- dability of M in affine space is equivalent to the embeddability of X− or of appropriate neighborhoods of Z inside X− in projective space. Under the cohomological hypothe- 2 1 ses: Hcomp(X−, Θ)=0andH (Z, NZ ) = 0 these embedding properties are shown to be preserved under convergence of the complex structures in the C∞-topology. §1. Conditions for embeddability (geometric approach). Let M denote a smooth compact, strictly pseudoconvex 3-dimensional CR- manifold. Such a structure is induced on a strictly pseudoconvex, real compact hypersurface in a complex space. The CR-manifold, M is called fillable if M can be realized as the boundary of a 2-dimensional (Stein) space. From the results of H.Grauert, 1958, J.J.Kohn, 1963, H.Rossi, 1965, it follows that M is fillable iff M is embeddable by a CR-mapping into affine complex space. It is now well known that the generic CR-structure on M is not fillable or embeddable (H.Rossi,1965, A.Andreotti-Yum Tong Siu,1970, L.Nirenberg, 1974, D.Burns,1979, H.Jacobovitz, F.Treves,1982, D.Burns, C.Epstein,1990.) On the other hand L.Lempert,1995, has proved that any embeddable strictly pseudoconvex CR-manifold M can be realized as a separating hypersurface in a projective variety X. This means, that if a strongly pseudoconvex, compact 3- dimensional CR-manifold M bounds a strongly pseudoconvex surface, X+ then −M also bounds a strongly pseudoconcave complex surface, X− containing a smooth holomorphic curve, Z with positive normal bundle, NZ . Suppose that the CR-manifold, −M is the boundary of a two-dimensional strictly pseudoconcave manifold, X− which contains a smooth curve, Z with positive nor- mal bundle, NZ . It is quite possible that this assumption is valid for any strictly pseudoconvex compact CR-manifold, M. N Definition. X− will be called weakly embeddable in CP if there exists a holo- N morphic map ϕ : X− → CP injective in some neighborhood of Z in X−.A Key words and phrases. deformation, CR–structure, embeddability, pseudoconcave manifold, stability. Research supported in part by the University of Paris, VI and NSF grant DMS96–23040 TtbAMS TX 2 CHARLES L. EPSTEIN AND GENNADI M. HENKIN N weakly embeddable X− will be called almost embeddable in CP if there exists an N almost injective holomorphic map ϕ : X− → CP , i.e. a map which embeds the complement of a proper analytic subset of X−. The next result (Epstein-Henkin,1998) shows that the embeddability of M in affine space depends on the embeddability of X− in projective space. Theorem 1. The following properties are equivalent: i)M is embeddable in complex affine space ii)X− is embeddable in complex projective space. iii)X− is almost embeddable in complex projective space. Sketch of the proof. The implication i) ⇒ ii) follows from the classical results. Indeed, if M is embeddable in affine space then by H.Rossi,1965, we have M = bX+,whereX+ is strictly pseudoconvex Stein space. Hence, X = X+ tM X− is compact complex space, satisfying condition of projectivity of K.Kodaira, 1960 and H.Grauert,1962. The implication ii) ⇒ i) is proved by the following scheme. If X− is embeddable N N in projective space CP (by the holomorphic mapping, ϕ : X− → CP )then by A.Andreotti, 1963, there exists an irreducible compact projective 2-dimensional N variety W ⊂ CP such that W− = ϕ(X−) ⊂ W . Let p0 be a non-negative defining function for bX− which is strictly plurisubhar- monic in a neighborhood of bX−. For sufficiently small ε>0 the sets Xε,− = {x ∈ X− : p0(x) >ε} are domains in X− with smooth strictly pseudoconcave boundaries. We let Wε,− = ¯ ϕ(Xε,−)andWε,+ = W \Wε,−. By H.Grauert,1958, for any small ε>0theCR- manifold bWε,+ is embeddable in complex affine space. Hence the CR-manifold, bXε,− is also embeddable in complex affine space for any small ε>0. The embed- dability of bX− = limε→0 bXε,− follows from here, by applying the relative index theory for CR-structures from C.Epstein,1998. The proof of implication iii) ⇒ ii) is based on Andreotti,1963, theorem. We show N that image, ϕ(X−) is an open subset of an irreducible algebraic variety W ⊂ CP and that the image ϕ(G) of the analytic subset, G ⊂ X−,whereϕ fails to be an embedding, is a discrete set in ϕ(X−). Using Hironaka’s 1964 theorem as well as a germ version of the Castelnuovo criterion proved in Epstein, Henkin 1998 we show that there exists a desingularizationπ ˜ : W˜ → W and a meromorphic map ˜ ϕ˜ : X− → W , which is a biholomorphism on the complement of a discrete set and such that ϕ =˜π◦ϕ˜. From this and from Andreotti-Tomassini,1970, theorem follows N the embeddability of X− in CP . 0 Let H (X−, [d·Z]) denotes the space of holomorphic sections of the line bundle [d· Z], defined by the divisor Z. The following result (Epstein-Henkin,1998) shows that weak embeddability of X− in complex projective space depends on the asymptotic 0 behavior of dim H (X−, [d · Z]) as d →∞. EMBEDDINGS FOR 3-DIMENSIONAL CR-MANIFOLDS 3 Theorem 2. Let X− be strictly pseudoconcave surface, which contains a smooth curve, Z with positive normal bundle, NZ .Let d(d +1) M(d)=(deg N ) +(1− g)(d +1), Z 2 where g is a genus of Z. Then the following properties are equivalent i) X− is weakly embeddable in complex projective space, 0 ii) M(d) ≤ dim H (X−, [d · Z]) ≤ M(d)+C, where C is a constant, depending only on Z and NZ . Proof. The proof of implication i) ⇒ ii) is based on the L2- Kodaira vanishing theorem of Pardon-Stern,1991, for a projective variety with singularities and on the Riemann-Roch theorem for line bundles, NZ over smooth curves, Z. The proof of implication ii) ⇒ i) uses a refinement of the Nakai- Moishezon theorem (see N Hartshorne, 1977). We show that ii) implies that there is a map, ϕ : X− → CP , which is an embedding of a neighborhood, U of Z and such that ϕ(Z)=ϕ(U) ∩ H, where H is a hyperplane in CP N . Problem. Is a weakly embeddable strictly pseudoconcave surface X− ⊃ Z embed- dable? §2. Conditions for embeddability (analytic approach). ¯ Weighted L2-estimates for the ∂-equation on appropriate singular domains give a different approach to the results above and also lead to more general conditions of embeddability of strictly pseudoconvex CR-varieties. Definition. A compact subset M of the almost complex manifold X is called a strictly pseudoconvex compact CR-variety if M is a compact level set of a C∞ strictly plurisubharmonic function (with possible critical points). Such a CR- N N variety, M is called embeddable in C if there exists a real embedding Φ : X → C ¯ with the property ∂Φ M = 0. A smooth CR-variety is called CR-manifold. Definition. Two compact CR-varieties M0 and M1 are called strictly CR-cobordant if there exists a complex manifold X, embedded as an open subset in some almost complex manifold X˜,andaC∞ strictly plurisubharmonic function ρ on X˜ such that bX = M1 − M0,0<ρ(x) < 1, x ∈ X, ρ = 0 and ρ =1. M0 M1 Theorem 3. Let M be a compact, 3-dimensional, strictly pseudoconvex CR- 0 manifold. Then M0 is embeddable iff there exists a complex space X with bX = M1 − M0,whereM1 is strictly pseudoconvex CR-manifold, embeddable by a CR- N mapping ϕ : M1 → C which admits a holomorphic almost injective extension to X. Sketch of the proof. The proof of necessity is obtained by the following arguments. If M0 is em- beddable in affine space, then by Rossi, 1965, and Harvey, Lawson, 1975, results 0 0 there exists a Stein space with isolated singularities, X such that bX = M0.Ap- plying the results of Ohsawa, 1984, and Heuneman, 1986, and (or) the technique of §3 below we show that the space X0 admits a proper embedding into a bigger Stein space X1 with a strictly pseudoconvex smooth boundary bX1 M From 4 CHARLES L. EPSTEIN AND GENNADI M. HENKIN Kohn, 1964, result it follows that the strictly pseudoconvex CR-manifold, M1 is embeddable in affine space by a mapping holomorphic on X and smooth on X¯. The proof of sufficiency contains several steps. Step 1. The proof of embed- dability of X in projective space and the reduction of the sufficiency to a special case. N Let ϕ : X → C be the holomorphic almost injective mapping which admits N a smooth extension to M1 and realizes an embedding of M1 into C .Fromthe concavity of X near M0 and from the Cauchy formula on analytic discs embedded in X it follows that ϕ has also a smooth extension to M0. One can show further that for set, G ⊂ X¯,whereϕ fails to be an embedding, the subset, G ∩ X is at most a 1-dimensional analytic set in X with boundary b(G ∩ X) ⊂ M0. Besides, ϕ(G ∩ X) is a discrete subset in ϕ(X). By results of Rossi, 1965, or Harvey, Lawson, 1975, there exists a Stein space with isolated singularities, W embedded N ¯ in C and such that bW = ϕ(M1).