Moduli Spaces of Higher Spin Klein Surfaces

TL;DR

This paper classifies the connected components of higher spin bundle spaces on hyperbolic Klein surfaces, revealing their topological structure and applications in real algebraic geometry and singularity theory.

Contribution

It provides a complete description of the connected components of higher spin bundles on Klein surfaces using topological invariants, linking them to Euclidean quotients.

Findings

Connected components characterized by topological invariants.

Each component homeomorphic to a Euclidean space quotient.

Applications to real forms of Gorenstein surface singularities.

Abstract

We study the connected components of the space of higher spin bundles on hyperbolic Klein surfaces. A Klein surface is a generalisation of a Riemann surface to the case of non-orientable surfaces or surfaces with boundary. The category of Klein surfaces is isomorphic to the category of real algebraic curves. An m-spin bundle on a Klein surface is a complex line bundle whose m-th tensor power is the cotangent bundle. The spaces of higher spin bundles on Klein surfaces are important because of their applications in singularity theory and real algebraic geometry, in particular for the study of real forms of Gorenstein quasi-homogeneous surface singularities. In this paper we describe all connected components of the space of higher spin bundles on hyperbolic Klein surfaces in terms of their topological invariants and prove that any connected component is homeomorphic to a quotient of a Euclidean space by a discrete group. We also discuss applications to real forms of Brieskorn-Pham singularities.