On the mixed Hodge structures of the intersection cohomology stalks of complex hypersurfaces

TL;DR

This paper investigates the mixed Hodge structures of intersection cohomology stalks at isolated singular points of complex hypersurfaces, linking their graded pieces to Milnor monodromy Jordan block counts.

Contribution

It provides a formula connecting the dimensions of graded pieces of the mixed Hodge structure to the Jordan block counts of the Milnor monodromy for hypersurfaces with isolated singularities.

Findings

Expressed the dimension of each graded piece in terms of Jordan blocks.

Linked mixed Hodge structures to Milnor monodromy data.

Enhanced understanding of the topology of complex hypersurfaces.

Abstract

We consider a hypersurface in $\mathbb{C}^n$ with an isolated singular point at the origin, and study the mixed Hodge structure of the stalk of its intersection cohomology complex at the origin. In particular we express the dimension of each graded piece of the weight filtration of this mixed Hodge structure in terms of the numbers of the Jordan blocks in the Milnor monodromy.