Logarithmic vector fields and multiplication table

Susumu TANABÉ Abstract. This is a review article on the Gauss-Manin system associated to the complete intersection singularities of projection. We show how the logarithmic vector fields appear as coefficients to the Gauss-Manin system (Theorem 2.7). We examine further how the multiplication table on the Jacobian quotient module calculates the logarithmic vector fields tangent to the discriminant and the bifurcation set (Proposition 3.3, Proposition 5.3). As applications, we establish signature formulae for Euler characteristics of real hypersurfaces (Theorem 4.2) and real complete intersections (Theorem 5.2) by means of these fields.