SALOMON OFMAN

Formule de transformation pour les courants-résidus

Pages 259-279
Received: 8 November 1999
Mathematics Subject Classification (2000): 32A27 - 32A40

Abstract In this article, we give a "transformation" formula for the Residue-currents, i.e. we compute the Residue-current 1/(P^l_j=1(å^l_k=1 u^j_k f_k)^a_j )   (u^j_k and a_j are respectively holomorphic functions and natural integers) with respect to the Residues-currents of the powers of 1/f_k. This is obtained through the use of both the construction of Leray's Residues and the theory of Coleff-Herrera's Residue-currents. This transformation formula has several applications, in particular it is used in [O2], in the form of the corollary 4.5, to generalize to any dimension the results on the analytic Radon Transformation obtained in [O1]. In the first paragraph, we give the properties of the Residue-currents we need later. In the second one, we give an abstract of the classical theory of cohomological residues and we generalize a formula of Leray on the division of the differential forms, useful for the computations of some integral Transformations. In the third one, we prove a continuity formula for Residue-currents depending of a parameter when the differential form is d-closed (otherwise, it is easy to get counter-examples, cf. remark II.6 in [O1]). In the last one, we prove the transformation formula (proposition 4.2 and theorem 4.3).