Abstract:
We study 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. 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 the quotient of $\mathbb R^d$ by a discrete group. We also discuss applications to real forms of Brieskorn–Pham singularities.
Key words and phrases:higher spin bundles, real forms, Riemann surfaces, Klein surfaces, Arf functions, lifts of Fuchsian groups.
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