REAL ANALOG OF RAMADANOVS CONJECTURE

Let X and Y be two smooth real n-manifolds. To each smooth hypersurfac e M of their product satisfying a suitable convexity condition, one ca n associate a real microlocal analogue of the Bergman kernel. The real incidence relation in the real projective space is the real analogue of the complexe sphere. We give, when n is greater than or equal to 3, an example of a hypersurface M not equivalent to the model, whose Ber gman kernel B has no logarithmic term. In the 2-dimensional case, we s how that if the coefficient of the logarithmic term of B vanishes of o rder 4 near a point of M, then M is equivalent to the model.