Inspired by his vanishing results of tautological classes and by Harer’s computation of the virtual cohomological dimension of the mapping class group, Looijenga conjectured that the moduli space of smooth Riemann surfaces admits a stratification by affine subsets with a certain number of layers. Similarly, Roth and Vakil extended the conjecture to the moduli spaces of Riemann surfaces of compa...

1. Differentiable Manifolds 2 2. Complex Manifolds 3 2.1. Riemann Surfaces of Genus One 4 2.2. Constructing Riemann Surfaces as Curves in P 6 2.3. Constructing Riemann Surfaces as Covers 9 2.4. Constructing Riemann Surfaces by Glueing 10 3. Topological Vector Bundles 11 3.1. The Tangent and Cotangent Bundles 13 3.2. Interlude: Categories, Complexes and Exact Sequences 14 3.3. Metrics on Vector ...

We study that subset of the moduli space M g of stable genus g, g > 1, Riemann surfaces which consists of such stable Riemann surfaces on which a given nite group F acts. We show rst that this subset is compact. It turns out that, for general nite groups F, the above subset is not connected. We show, however, that for Z 2 actions this subset is connected. Finally, we show that even in the modul...

We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichmüller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices. We show that the only Riemann surface in that family admitting a non-hyperel...

We provide an intrinsic description of N -super Riemann surfaces and TN -semirigid surfaces. Semirigid surfaces occur naturally in the description of topological gravity as well as topological supergravity. We show that such surfaces are obtained by an integrable reduction of the structure group of a complex supermanifold. We also discuss the supermoduli spaces of TN -semirigid surfaces and the...