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Duality and Distribution Cohomology of CR Manifolds

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Duality and Distribution Cohomology of CR Manifolds ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze C. DENSON HILL M. NACINOVICH Duality and distribution cohomology ofCR manifolds Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 22, no 2 (1995), p. 315-339 <http://www.numdam.org/item?id=ASNSP_1995_4_22_2_315_0> © Scuola Normale Superiore, Pisa, 1995, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Duality and Distribution Cohomology of CR Manifolds C. DENSON HILL - M. NACINOVICH Our aim here is to investigate the am-complexes on currents, in the sense of de Rham [deR], associated to a CR manifold M of arbitrary CR dimension and codimension. This brings into play global distribution aM-cohomology on M, and involves its relationship to the smooth 5m-cohomology on M, as well as its connection with the classical Dolbeault cohomology of an ambient complex manifold X, in the case where M is assumed to be embeddable. We also discuss compact abstract CR manifolds M, and achieve a duality formula which is in the spirit of de Rham [deR] and Serre [S] for real and complex manifolds respectively, and which is related to the work [AK], [AB]. Actually the bridge between the intrinsic and extrinsic notions of distribution m-cohomology on M is inspired by the work of Martineau [M] on extendable distributions and boundary values of holomorphic functions, and is also related to the residues of Leray [L] and Grothendiek [Gr]. The first applications of these ideas to CR manifolds can be found in [AHLM], [NV]. To simplify the exposition we always use the term abstract CR manifold to indicate one that is not assumed to be locally embeddable. Otherwise, by the term CR-manifold we mean one that is assumed to be locally embeddable. In particular we use the term pseudoconcave CR manifold for one that is locally embeddable and at least I-pseudoconcave. We rely on the previous work [NV], [HN1], [HN2], [HN3]. Contents § 1 Preliminaries. §2 Distribution cohomology and currents. §3 Interpretation of the distribution cohomology on M n U. Pervenuto alla Redazione il 24 Gennaio 1994. 316 1 §4 Residues and extendable distributions. §5 Some applications. §6 Intrinsic definiton of the aM-complexes on smooth forms. §7 Intrinsic definition of the aM-complexes on currents. §8 A remark on the 5m-cohomology for nonorientable CR manifolds. §9 Duality on compact abstract CR manifolds. §10 Distribution cohomology for CR manifolds that are q-pseudoconcave at infinity. § 11 By-passes for pseudoconcave CR manifolds and pseudoconvexity at infinity. 1. - Preliminaries Abstract CR manifolds. Let M be a smooth real manifold of dimension m, countable at infinity. A partial complex structure of type (n, k) on M is the pair consisting of a vector subbundle HM of the tangent bundle TM and a smooth vector bundle isomorphism J : HM --~ HM such that We say that the partial complex structure (HM, J) is formally integrable if We denote respectively by and the complex vector subbundles of the complexiiication CRM of HM corresponding to the eigenvalues i and -i of J. Then the condition of formal integrability can also be expressed by or, equivalently, by 317 We note that the following relations hold: An abstract CR manifold (M, HM, J) of type (n, k) is the triple consisting of a smooth paracompact manifold M of dimension m = 2n + k and a formally integrable partial complex structure (HM, J) of type (n, k) on it. We call n the CR dimension and k the CR codimension of (M, HM, J). In the following, we shall write for simplicity M instead of (M, HM, J). The characteristic bundle and the Levi form. Let M be a CR manifold of type (n, k). The characteristic bundle H°M is the annihilator of HM in T*M : it is a rank k subbundle of T*M: Given a e X, Y E let us choose a E r(M, H°M), X, such that a(x) = a, Y(x) = Y. Then we have and hence the two sides of this equation only depend on a, X, Y. In this way we associate to a C H°M a quadratic form on This form is hermitian for the complex structure of HxM defined by J. Indeed, we have We denote by u (a) = (u+(a), u-(a)) the signature of L(a,.) as a hermitian form for the complex structure of defined by J: the numbers u+(a) and Q-(a) are respectively the number of positive and negative eigenvalues of L(a, ~). We say that M is E M if for every a E H°M with we have u - (a) &#x3E; q and that M is q-pseudoconcave if it is q-pseudoconcave at all points. The hermitian form L(a,.) is called the Levi form of M at a E H°M. 318 Local embeddability of abstract CR manifolds. A CR map of a CR manifold (Ml , Hl , Jl ) into a CR manifold (M2, H2, J2) is a differentiable map 0 : Mi - M2 such that C H2 and J2§*(X) for every X E H, - A CR embedding ~ of an abstract CR manifold M into a complex manifold X is a CR map which is an embedding. We say that a CR embedding 0 : M -&#x3E; X is generic if the complex dimension of X is n + k, where (n, k) is the type of M. In this case we say that X is a tubular neighborhood of M. We say that a CR manifold M is a CR-submanifold of a CR-manifold N if M is a differentiable submanifold of N and the inclusion map t : M ~--~ N is CR. Pseudoconcavity and pseudoconvexity at in, finity. We recall the definiton of pseudoconvexity and pseudoconcavity given in [AG], [A] for the case of complex manifolds. An N-dimensional complex manifold X is called r-pseudoconvex (r-pseudoconcave) if there is a real valued smooth function 0 on X, a compact subset K of X and a constant co E R U { oo }, such that (1) 0 co on M; (2) for every c co, the set {x E c} is compact in X; (3) the complex hessian of 0 has at least (N - r) positive ((r + 1) negative) eigenvalues at each point of X - K. If, in the definition of r-pseudoconvexity, we can choose K = 0, the manifold X is called r-complete. In [HNl] these notions have been extended to CR manifolds, in the following way. : _ Let M be an abstract CR manifold of type (n, k). We denote by ~ (.~) the sheaf of germs of complex valued smooth differential 1-forms on M, which vanish on T°,1 (T1,0). Let e(1) be the sheaf of germs of complex valued smooth 1-forms on M. Then we can define a sheaf homomorphism (called a CR gauge: cf. [MN]) such that Having fixed a CR gauge on M, a real transversal 1 -jet o on M is the pair (~, a) of a smooth real valued function § : M - R and of a smooth section a E r(M, HM). Its complex hessian at x E M is the hermitian form on HxM defined by - In [MN] it was shown that the complex hessian of a transversal 1-jet is invariant with respect to the choice of a CR-gauge and that, when M is a generic CR submanifold of a complex manifold X, there is a natural correspondence between 319 real transversal 1-jets 0 on M and 1-jets on M of real valued smooth functions p defined in X, in such a way that the complex hessian of 0 at x E M is the restriction to HxM of the complex hessian of p in X. A q-pseudoconcave CR manifold M is said to be q-pseudoconcave at infinity if there is a real valued transversal 1-jet o = (0, a) on M, a compact subset K of M and a constant co E R U f ool such that ( 1 ) (2) fx E c} is compact in M for every c co; (3) for every x E M - K and # E the hermitian form on has at least q negative eigenvalues. A q-pseudoconcave CR manifold M is said to be (n - q)-pseudoconvex at infinity if there is a real valued transversal 1-jet 0 = (~, a) on M, a compact subset K of M and a constant co E R U {oo}, such that (1) 4J co on M; (2) {x E c} is compact in M for every c co; (3) for every x E M - K and # E HxM, the hermitian form on has at least q positive eigenvalues. If, in the definition above, we can take K = 0, then M is said to be (n - q)-complete. 2. - Distribution cohomology and currents First we discuss distribution cohomology in the embeddable case. Let X be a complex manifold of complex dimension N. On X we have the Dolbeault complexes on the sheaves of germs of smooth forms: Here denotes the sheaf of germs of complex valued C°° forms of bidegree (p, j) on X.
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