The Euler characteristic of Milnor fibers over 2-generic symmetric determinantal varieties

TL;DR

This paper derives a formula for the Euler characteristic of Milnor fibers over 2-generic symmetric determinantal varieties using toric geometry and Newton polyhedra, with applications to local Euler obstructions.

Contribution

It provides the first explicit formula for Milnor fiber Euler characteristics in this setting, connecting toric structures and Newton polyhedra computations.

Findings

Explicit formula for Euler characteristic of Milnor fibers over symmetric determinantal varieties.

Computed local Euler obstructions at the origin and for the function.

Related Euler obstruction of the function to a Milnor number of an associated polynomial.

Abstract

In this work we present a formula for the Euler characteristic of the Milnor fiber of non-degenerate functions $f: X \to \mathbb{C}$ with isolated critical set relative to a stratification, where $X$ is a $2$-generic symmetric determinantal variety. The formula is obtained in two steps. Firstly, we explicitly describe the toric structure of those varieties. Secondly, we compute volumes of Newton polyhedra arising from the toric structure. The result then follows from Matsui-Takeuchi's formula for Milnor fibers over toric varieties. As an application, we compute the local Euler obstruction of $X$ at the origin and the local Euler obstruction of $f$. We also relate the Euler obstruction of $f$ to the Milnor number of a certain polynomial associated to $f$.