On the Milnor classes of local complete intersections

Abstract

In this work we study algebraic, geometric and topological properties of the Milnor classes of local complete intersections with arbitrary singularities. We describe first the Milnor class of the intersection of a finite number of hypersurfaces, under certain conditions of transversality, in terms of the Milnor classes of the hypersurfaces. Using this description we obtain a Parusiński-Pragacz type formula, an Aluffi type formula and a description of the Milnor class of the local complete intersection in terms of the global Lê cycles of the hypersurfaces that define it. We consider next the general case of a local complete intersection $Z(s)$ defined by a regular section $s$ of a rank $r$ holomorphic bundle $E$ over a compact manifold $M$, $r \geq 2$. We notice that $s$ determines a hypersurface $Z(\tilde s)$ in the total space of the projectivization $\mathbb{P}(E^{\vee})$ of the dual bundle $E^{\vee}$, and we give a formula expressing the total Milnor class of the local complete intersection $Z(s)$ in terms of the Milnor classes of the hypersurface $Z(\tilde s)$.