ON THE GAUSS-MANIN CONNECTION FOR REAL SINGULARITIES

We prove that to each real singularity f : (R n+1 , 0) → (R, 0) one can associate two systems of differential equations g k± f which are pushforwards in the category of D-modules over R ± , of the sheaf of real analytic functions on the total space of the positive, respectively negative, Milnor fibration. We prove that for k = 0 if f is an isolated singularity then g ± determines the the n-th homology groups of the positive, respectively negative, Milnor fibre. We then calculate g + for ordinary quadratic singularities and prove that under certain conditions on the choice of morsification, one recovers the top homology groups of the Milnor fibers of any isolated singularity f