Holomorphic arcs on analytic spaces

TL;DR

This paper studies the structure of holomorphic arcs on complex analytic spaces, revealing their connected components and linking them to fundamental group conjugacy classes, with applications to surface singularities and the McKay correspondence.

Contribution

It characterizes the connected components of the moduli space of short analytic arcs on surface singularities and relates them to fundamental group conjugacy classes, providing new insights into singularity theory.

Findings

Connected components described via conjugacy classes of the fundamental group

Concrete realization of the McKay correspondence for quotient singularities

New connections between surface cusp singularities, their duals, and Inoue surfaces

Abstract

Let X be a complex analytic space. A short analytic arc is a holomorphic map of the closed unit disc to X such that only the origin is mapped to a singular point. In contrast with the space of formal arcs studied by Nash, the moduli space of short analytic arcs usually has infinitely many connected components. We describe these for surface singularities, in terms of certain conjugacy classes of the fundamental group of the link. For quotient singularities (in any dimension), this gives a concrete realization of the McKay correspondence. Our results also give new connections between a surface cusp singularity, its dual and hyperbolic Inoue surfaces. version 2: References added.