An approach to Lagrangian specialisation through MacPherson's graph construction

Abstract

Let $f:M\to N$ be a holomorphic map between two complex manifolds. Assume $f$ is flat and sans éclatement en codimension 0 (no blowup in codimension 0). We study the theory of Lagrangian specialisation for such $f$, and prove a González-Sprinberg type formula for the local Euler obstruction relative to $f$. With the help of this formula and MacPherson's graph construction for the vector bundle map $f^{\ast }{T}^{\ast }N\to T^{\ast }M$, we find the Lagrangian cycle of the Milnor number constructible function $\mu$. As an application, we study the Chern class transformation of $\mu$ when $f$ has finite contact type.