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d’d” et d”-cohomologies d’une variété compacte privée d’un point. Application à l’intégration sur les cycles. (French) Zbl 0586.32009

The author continues his study of \(\partial {\bar \partial}\)-cohomology of a compact complex manifold minus a point, \(Y=Z\setminus point\). This subject is closely related to work by A. Andreotti and F. Norguet [Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 21, 31- 82 (1967; Zbl 0176.040), ibid. 25, 59-114 (1971; Zbl 0212.537)]. Several geometric questions are formulated and answered by properly choosing various cohomology sequences and studying the exactness of such sequences. Let \(C_{n-1}(Y)\) denote the space of n-1 dimensional analytic cycles in Y and \(\Omega^ k\) the sheaf of germs of holomorphic k-forms. It is shown that the obstruction to exactitude of \(H^{n- 1}(Y,\Omega^{n-2})\to^{d}H^{n-1}(Y,\Omega^{n-1})\to^{\rho_ 0}H^ 0(C_{n-1}(Y),{\mathcal O})\) is precisely a vector space of dimension \(\dim_ CH^ 0(Z,\Omega^ 2)\). Here \(\rho_ 0\) is induced by integration. Chapter 1 treats surfaces \((n=2)\); Chapter 2 treats higher dimensions.
Reviewer: E.J.Akutowicz

MSC:

32Q99 Complex manifolds
32C30 Integration on analytic sets and spaces, currents
32S05 Local complex singularities
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C36 Local cohomology of analytic spaces
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References:

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