On a quadratic form associated with a surface automorphism and its applications to Singularity Theory

TL;DR

This paper investigates a quadratic form linked to surface automorphisms, exploring its properties, computability, and applications to singularity theory, especially in distinguishing plane curve singularities and analyzing Milnor fibers.

Contribution

It introduces a new quadratic form associated with surface automorphisms, proves its positivity and evenness under certain conditions, and applies these results to singularity theory and Milnor fibers.

Findings

The quadratic form is positive definite when certain screw numbers are positive.

The form is even on the absolute homology if the quotient graph is a tree.

The form can distinguish singularities with identical spectral pairs.

Abstract

We study the nilpotent part $N'$ of a pseudo-periodic automorphism $h$ of a real oriented surface with boundary $\Sigma$. We associate a quadratic form $Q$ defined on the first homology group (relative to the boundary) of the surface $\Sigma$. Using the twist formula and techniques from mapping class group theory, we prove that the form $\tilde{Q}$ obtained after killing ${\ker N}$ is positive definite if all the screw numbers associated with certain orbits of annuli are positive. We also prove that the restriction of $\tilde Q$ to the absolute homology group of $\Sigma$ is even whenever the quotient of the Nielsen-Thurston graph under the action of the automorphism is a tree. The case of monodromy automorphisms of Milnor fibers $\Sigma=F$ of germs of curves on normal surface singularities is discussed in detail, and the aforementioned results are specialized to such situation. Moreover, the form $\tilde{Q}$ is computable in terms of the dual resolution or semistable reduction graph, as illustrated with several examples. Numerical invariants associated with $\tilde{Q}$ are able to distinguish plane curve singularities with different topological types but same spectral pairs. Finally, we discuss a generic linear germ defined on a superisolated surface. In this case the plumbing graph is not a tree and the restriction of $\tilde Q$ to the absolute monodromy of $\Sigma=F$ is not even.