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## Moduli Spaces of Higher Spin Klein Surfaces

Moscow Mathematical Journal. 2017. Vol. 17. No. 2. P. 327-349.
Natanzon S. M., Pratoussevitch A.

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 R^d by a discrete group. We also
discuss applications to real forms of Brieskorn{Pham singularities.