Milnor classes of local complete intersections

Let V be a compact local complete intersection defined as the zero set of a section of a holomorphic vector bundle over the ambient space. For each connected component S of the singular set Sing(V) of V, we define the Milnor class mu(*) (V, S) in the homology of S. The difference between the Schwartz-MacPherson class and the Fulton-Johnson class of V is shown to be equal to the sum of mu(*) (V, S) over the connected components S of Sing(V). This is done by proving Poincare-Hopf type theorems for these classes with respect to suitable tangent frames. The 0-degree component mu(0)(V, S) coincides with the Milnor numbers already defined by various authors in particular situations. We also give an explicit formula for mu(*) (V, S) when S is a non-singular component and V satisfies the Whitney condition along S.