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

Thaís M. Dalbelo, Daniel Duarte, Danilo da Nóbrega Santos

TL;DR

The paper computes the Euler characteristic of Milnor fibers for non-degenerate functions with isolated critical sets on 2-generic symmetric determinantal varieties by first giving a toric description of these varieties and then applying Matsui–Takeuchi’s toric Milnor-fiber formula via Newton polyhedra. It derives an explicit closed-form expression for the Milnor-fiber Euler characteristic and obtains the local Euler obstruction, enabling the construction of counterexamples to Matsui–Takeuchi’s conjecture on smoothness characterization in odd dimensions. The work connects toric geometry, Milnor fiber topology, and singularity invariants, yielding concrete relations between local invariants and Milnor numbers in a natural, normal toric setting. It also highlights how the combinatorics of the toric semigroup and its cone control both the Milnor-fiber topology and obstructions, providing a framework for further study of singular toric determinantal varieties.

Abstract

In this work we present a formula to compute the Euler characteristic of the Milnor fiber of non-degenerate functions $f: X \to \mathbb{C}$ with isolated critical set, 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 the Milnor fibers over toric varieties. As an application, we compute the local Euler obstruction of $X$ at the origin. This, in turn, allow us to provide a family of odd-dimensional normal toric varieties with isolated singular point not satisfying Matsui-Takeuchi's conjecture of the characterization of smoothness through the local Euler obstruction.

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

TL;DR

The paper computes the Euler characteristic of Milnor fibers for non-degenerate functions with isolated critical sets on 2-generic symmetric determinantal varieties by first giving a toric description of these varieties and then applying Matsui–Takeuchi’s toric Milnor-fiber formula via Newton polyhedra. It derives an explicit closed-form expression for the Milnor-fiber Euler characteristic and obtains the local Euler obstruction, enabling the construction of counterexamples to Matsui–Takeuchi’s conjecture on smoothness characterization in odd dimensions. The work connects toric geometry, Milnor fiber topology, and singularity invariants, yielding concrete relations between local invariants and Milnor numbers in a natural, normal toric setting. It also highlights how the combinatorics of the toric semigroup and its cone control both the Milnor-fiber topology and obstructions, providing a framework for further study of singular toric determinantal varieties.

Abstract

In this work we present a formula to compute the Euler characteristic of the Milnor fiber of non-degenerate functions with isolated critical set, where is a -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 the Milnor fibers over toric varieties. As an application, we compute the local Euler obstruction of at the origin. This, in turn, allow us to provide a family of odd-dimensional normal toric varieties with isolated singular point not satisfying Matsui-Takeuchi's conjecture of the characterization of smoothness through the local Euler obstruction.

Paper Structure

This paper contains 9 sections, 23 theorems, 77 equations, 1 figure.

Table of Contents

  1. The local Euler obstruction
  2. Toric structure of 2-generic symmetric determinantal varieties
  3. A semigroup defining the 2-generic symmetric determinantal varieties
  4. Some properties of the cone generated by $\mathcal{S}$
  5. The Euler characteristic of the Milnor fiber
  6. A formula for the Euler characteristic of the Milnor fiber on $S^2_n$
  7. The local Euler obstruction
  8. Counterexamples to Matsui-Takeuchi conjecture
  9. Relation to Milnor number

Key Result

Lemma 1

For sufficiently small $\epsilon >0$, there exists $\eta_0 >0$ with $0<\eta_0 \ll \epsilon$ such that for all $0 < \eta <\eta_0$ the restriction of $f$ to is a topological fiber bundle over the punctured disk $D_{\eta}^*:=\{ z \in \mathbb{C} \ |\ 0<|z|<\eta\}$, where $B(0,\epsilon)$ is the open ball in $\mathbb{C}^N$ with radius $\epsilon$ centered at the origin.

Figures (1)

  • Figure 1: Newton polyhedron of $f$.

Theorems & Definitions (47)

  • Lemma 1
  • Theorem A: see Theorem \ref{['esttoric']}
  • Theorem B: see Theorem \ref{['differentds']}
  • Theorem C: see Corollary \ref{['corobs']}
  • Definition 2
  • Lemma 3
  • Definition 4
  • Theorem 5
  • Definition 6
  • Definition 7
  • ...and 37 more