SIMULTANEOUS LOCAL CLASSIFICATION OF 2 CO MPATIBLE SYMPLECTIC FORMS

Consider a symplectic form omega and a closed 2-form omega1 on a real or complex manifold. Suppose that the Nijenhuis torsion of the tensor field J defined by omega1(X, Y) = omega(JX, Y) vanishes. In this paper we give the complete local classification of the couple {omega, omega 1} on a dense open set, defined by some minor conditions of regularity . Around each point of this open set we can find coordinates on wich o mega is written with constant coefficients and omega1 with affine ones .