Contact loci of semihomogeneous singularities

TL;DR

This paper investigates the geometry of contact loci in semihomogeneous singularities, providing solutions to the arc-Floer conjecture and the embedded Nash problem, advancing understanding of singularity contact loci.

Contribution

It offers a comprehensive analysis of contact loci in semihomogeneous singularities, resolving key open problems in the field.

Findings

Affirmative answer to the arc-Floer conjecture under certain conditions.

Complete solution to the embedded Nash problem.

New geometric insights into contact loci of semihomogeneous singularities.

Abstract

Two of the main open problems in the theory of contact loci are the arc-Floer conjecture, which states that the compactly supported cohomology of the restricted $m$-contact locus and the fixed-point Floer cohomology of the $m$-th iterate of the Milnor monodromy are isomorphic up to a shift; and the embedded Nash problem, which asks for a description of the irreducible components of the unrestricted $m$-contact locus in terms of an embedded resolution of singularities. In this paper we study the geometry of contact loci of semihomogeneous singularities, and use our results to give an affirmative answer to the arc-Floer conjecture (under some conditions on the dimension and the degree) and a complete solution to the embedded Nash problem.