Complex Links and Algebraic Multiplicities

Abstract: 

In this talk I will consider two invariants associated to pairs of strata X , Y in an analytic stratification of some ambient complex projective variety W. The main result states that if the closure of X is contained in the closure of Y and both are subvarieties of W with the closure of X irreducible, then the Hilbert-Samuel (or algebraic) multiplicity of the closure of Y along the closure of X equals the Euler characteristic of the space obtained by intersecting Y with the complex link of X in W. I will also discuss the ingredients used to prove this result, and in particular will present a local Lefschetz hyperplane theorem for such complex linking spaces. Applications of this result include a new algebraic formula for MacPherson's local Euler obstruction of W. This talk is based on recent joint work with Vidit Nanda [Oxford] which is available on the arxiv (https://arxiv.org/abs/2006.10452).