×

Integral transforms for divisors in \(\mathbf P_n(\mathbb C)\) and solutions of systems of PDE’s. (English) Zbl 1079.53116

The paper is devoted to the study of two integral transforms in complex analysis, namely the analytic Radon transform and the Andreotti-Norguet transform. There is already a whole series of papers investigating these transforms, but most of them deal only with linear cycles. In this situation the description of both the transforms (kernel, image) is relatively simple. The aim of the authors is more difficult. They characterize the image of these transforms essentially in the case od compact hypersurfaces (not necessarily smooth) of arbitrary fixed degree which are contained in a linearly concave open subset of \(\mathbf P_n(\mathbb C)\). They show that the image is the set of all solutions of a certain system of partial differential equations. In fact, they generalize in two directions: instead of hyperplanes they consider hypersurfaces, and they do not assume that the hypersurfaces are smooth. In the end they even extend their results from the analytic category to the \(C^{\infty }\)-category.

MSC:

53C65 Integral geometry
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Andreotti and F. Norguet: Problème de Levi et convexité holomorphe pour les classes de cohomologie. Ann. Sc. Norm. Sup. Pisa 20 (1966), 197-241. · Zbl 0154.33504
[2] A. Andreotti and F. Norguet: Cycles of algebraic manifolds and \(\partial \bar{\partial }\)-cohomology. Ann. Sc. Norm. Sup. Pisa 25 (1971), 59-114. · Zbl 0212.53701
[3] D. Barlet: Espace des cycles et \(d^{\prime }d^{\prime \prime }\)-cohomologie de \({\mathbb{P}_{n}(C)} \setminus {\mathbb{P}_{k}(C)}\). Fonctions de Plusieurs Variables Complexes, Sém. F. Norguet, 1970-1973, Lecture Notes in Math., 409, 1974, pp. 98-123. · Zbl 0307.32008
[4] R. Bott and L. Tu: Differential Forms in Algebraic Topology. GTM 82, Springer Verlag, 1982. · Zbl 0496.55001
[5] N. Coleff and M. Herrera: Les Courants Résiduels Associés à une Forme Méromorphe. Lecture Notes in Math., 633, Springer Verlag, 1978.
[6] R. Delanghe, F. Sommen and V. Souček: Clifford Algebra and Spinor Valued Functions, A Function Theory for the Dirac Operator. Mathematics and Its Applications, Kluwer Academic Publishers, 1992.. · Zbl 0747.53001
[7] G. Gindikin and M. Henkin: Integral geometry for \(\bar{\partial }\)-cohomology in \(q\)-linear concave domains in \({CP}^n\). Funct. Anal. Appl. 12 (1978), 6-23. · Zbl 0409.32020
[8] M. Henkin: The Abel-Radon transform and several complex variables. Ann. Math. Stud. 137, T. Bloom (ed.), Princeton University Press, 1995, pp. 223-275. · Zbl 0848.32012
[9] D.Husemoller: Fibre Bundles. McGraw-Hill, 1966. · Zbl 0356.55007
[10] J. Leray: Le calcul différentiel et intégral sur une variété analytique complexe, probléme de Cauchy III. Bull. Soc. Math. France 87 (1959), 81-180. · Zbl 0199.41203
[11] A. Martineau: Équations différentielles d’ordre infini. Bull. Soc. Math. France 95 (1967), 109-154. · Zbl 0167.44202
[12] S. Ofman: \(d^{\prime }d^{\prime \prime }\) et \(d^{\prime \prime }\)-cohomologie d’une variété compacte privée d’un point. Bull. Soc. Math. France 113 (1985), 241-254. · Zbl 0588.32012
[13] S. Ofman: Intégration des classes de \(d^{\prime }d^{\prime \prime }\)-cohomologie sur les cycles analytiques. Solution du problème de l’injectivité. J. Math. Pures Appl. 68 (1989), 73-94. · Zbl 0621.32020
[14] S. Ofman: \(d^{\prime }d^{\prime \prime }\), \(d^{\prime \prime }\)-cohomologies et intégration sur les cycles analytiques. Invent. Math. 92 (1988), 389-402. · Zbl 0689.32006 · doi:10.1007/BF01404459
[15] S. Ofman: La transformation de Radon analytique en codimension 1. Ann. Sc. Norm. Sup. Pisa 20 (1993), 415-459. · Zbl 0819.53035
[16] S. Ofman: Résidus méromorphes et Résidus de Martinelli, Seminari di Geometria 1994-1996. (1996), Università degli Studi di Bologna, 181-193. · Zbl 0855.44001
[17] S. Ofman: Une généralisation d’une transformation intégrale de \(R^{2}\). Géométrie analytique, F. Norguet, S. Ofman, J.J. Szczeciniarz (eds.), Actualités scientifiques et industrielles, 1438, Hermann, 1996.
[18] S. Ofman: la Transformation de Radon analytique en dimension quelconque, preprint University Paris 7.
[19] R. Ward and R. Wells: Twistor Geometry and Field Theory. Cambridge University Press, 1990. · Zbl 0729.53068
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.