Triangulations and Volume Form on Moduli Spaces of Flat Surfaces

In this paper, we study the moduli spaces of flat surfaces with cone singularities verifying the following property: there exists a union of disjoint geodesic tree on the surface such that the complement is a translation surface. Those spaces can be viewed as deformations of the moduli spaces of translation surfaces in the space of flat surfaces. We prove that such spaces are quotients of flat complex affine manifolds by a group acting properly discontinuously, and preserving a parallel volume form. Translation surfaces can be considered as a special case of flat surfaces with erasing forest, in this case, it turns out that our volume form coincides with the usual volume form (which are defined via the period mapping) up to a multiplicative constant. We also prove similar results for the moduli space of flat metric structures on the n-punctured sphere with prescribed cone angles up to homothety. When all the angles are smaller than 2π, it is known (cf. [T]) that this moduli space is a complex hyperbolic orbifold. In this particular case, we prove that our volume form induces a volume form which is equal to the complex hyperbolic volume form up to a multiplicative constant.