Fourier-Laplace transforms and related questions on certain analytic varieties in C2

For an analytic variety V-psi = {(z(1), z(2)) is an element of C-2: psi(z(1), z(2)) = 0}, defined by a holomorphic function psi, we assume that the point 0 is an element of V-psi and that V-psi - {0} is smooth. In this setting, we construct holomorphic differentials theta, on V-psi - {0}, with prescribed certain of the values of the integrals integral z(1)(k)z(2)(1)theta(z(1), z(2)), taken over closed curves on V-psi which surround 0. The construction is quite explicit and is based on a residue process. We also study similar questions with specific choices of psi, in which cases we obtain more complete results. (c) 2005 Elsevier Inc. All rights reserved.