Milnor Number and Milnor Classes

Abstract

Both Schwartz–MacPherson and Fulton–Johnson classes generalize Chern classes to the case of singular varieties. It is known that for local complete intersections with isolated singularities, the 0-degree SM and FJ classes differ by the local Milnor numbers [149] and all other classes coincide [155]. As we explain in the sequel, is V has nonisolated singularities, the difference $C_i^{SM} (V) - C_i^{FJ} (V)$ of the SM and FJ classes is, for each i, a homology class with support in the homology H_2i(Sing(V)) of the singular set of V. That is the reason for which their difference was called in [30,31] the Milnor class of degree i. These classes have been also considered, from different viewpoints, by other authors, most notably by P. Aluffi, T. Ohmoto, A. Parusiński, P. Pragacz, J. Schürmann, S. Yokura.

In this chapter we introduce the Milnor classes of a local complete intersection V of dimension n ≥ 1 in a complex manifold M, defined by a regular section s of a holomorphic bundle N over M. The aim of this chapter is to show that, as mentioned above, the Milnor classes are localized at the connected components of the singular set of V : If S is such a component then one has Milnor classes µ_i of V at S in degrees i = 0, … , dim S. The 0–degree class coincides with the generalized Milnor number of V at S, introduced by Parusiński in [127] (if V is a hypersurface in M). The sum of all the Milnor classes over the connected components of Sing(V) gives the global Milnor classes studied in [8, 126, 131, 169]. See [28] for another presentation.

The method we use for constructing the localized Milnor classes comes from [31] and uses Chern–Weil theory. The idea is to use stratified frames to localize at the singular set the Schwartz–MacPherson and the FultonJohnson classes, in such a way that the difference of these localizations canonical.