Related papers: Teichmueller space via Kuranishi families

We show that the deformation space of complex parallelisable nilmanifolds can be described by polynomial equations but is almost never smooth. This is remarkable since these manifolds have trivial canonical bundle and are holomorphic…

The orbit polytope for a finite group G acting linearly and freely on a sphere S is used to construct a cellularized fundamental domain for the action. A resolution of the integers over G results from the associated G-equivariant…

We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disc removed. We define a refined Teichmueller space of such Riemann surfaces and demonstrate that in…

We prove that the mapping class group $\Gamma_{g,n}$ for surfaces of negative Euler characteristic has a cofinite universal space $\E$ for proper actions (the resulting quotient is a finite $CW$-complex). The approach is to construct a…

We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $\frac{1}{2}(3^{10}-1)$ cover of the 9-dimensional Deligne-Mostow…

We show that the classes of partial isometries in finite-dimensional polyhedral spaces and in finite-dimensional rational polyhedral spaces do not have the weak amalgamation property. This implies that the linear isometry group of the…

In the first part of the paper we describe the complex geometry of the universal Teichm\"uller space $\mathcal T$, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc.…

We generalize a new class of cluster type mutations for which exchange transformations are given by reciprocal polynomials. In the case of second-order polynomials of the form $x+2\cos{\pi/n_o}+x^{-1}$ these transformations are related to…

Geometric structures on manifolds became popular when Thurston used them in his work on the geometrization conjecture. They were studied by many people and they play an important role in higher Teichm\"uller theory. Geometric structures on…

We construct a modular desingularisation of $\overline{\mathcal{M}}_{2,n}(\mathbb{P}^r,d)^{\text{main}}$. The geometry of Gorenstein singularities of genus two leads us to consider maps from prestable admissible covers: with this enhanced…

In this paper we explore the idea that Teichm\"uller space is hyperbolic "on average." Our approach focuses on studying the geometry of geodesics which spend a definite proportion of time in some thick part of Teichm\"uller space. We…

We prove that the Teichm\"uller space of surfaces of genus $\mathbf{g}$ with $\mathbf{p}$ punctures contains balls which are not convex in the Teichm\"uller metric whenever $3\mathbf{g}-3+\mathbf{p} > 1$.

In this paper, we present a unified study of the moduli space of tropical curves and Outer space which we link via period maps to the moduli space of tropical abelian varieties and the space of positive definite quadratic forms. Our work is…

We continue the study of linear families of contact forms on 3-manifolds begun in our paper `Contact geometry and complex surfaces'. The present paper introduces Teichmuller and moduli spaces for so-called taut contact circles. By…

We present a natural family of Hilbert function spaces on the d-dimensional complex unit ball and classify which of them satisfy that subsets of the ball yield isometrically isomorphic subspaces if and only if there is an analytic…

We describe in elementary geometrical terms Teichm\" uller spaces of decorated and holed surfaces. We construct explicit global coordinates on them as well as on the spaces of measured laminations with compact and closed support…

We consider Riemann surfaces $\Sigma$ with $n$ borders homeomorphic to $\mathbb{S}^1$ and no handles. Using generalized Grunsky operators, we define a period mapping from the infinite-dimensional Teichm\"uller space of surfaces of this type…

We construct explicit dominant, rational morphisms from projective bundles over rational varieties to relevant moduli spaces, showing their unirationality. These constructions work for $U_{r,d,g}$; for all ranks, degrees and genus $2\leq g…

We build compact moduli spaces of Grassmannian framed bundles over a Riemann surface, essentially replacing a group by its bi-invariant compactification. We do this both in the algebraic and symplectic settings, and prove a…

We prove that the rational cohomology $H^i(\mathcal{T}_g;\mathbf{Q})$ of the moduli space of trigonal curves of genus $g$ is independent of $g$ in degree $i<\lfloor g/4\rfloor.$ This makes possible to define the stable cohomology ring as…