Volume form on moduli spaces of d-differentials

Given $d\in \mathbb{N}$, $g\in \mathbb{N} \cup\{0\}$, and an integral vector $\kappa=(k_1,\dots,k_n)$ such that $k_i>-d$ and $k_1+\dots+k_n=d(2g-2)$, let $\Omega^d\mathcal{M}_{g,n}(\kappa)$ denote the moduli space of meromorphic $d$-differentials on Riemann surfaces of genus $g$ whose zeros and poles have orders prescribed by $\kappa$. We show that $\Omega^d\mathcal{M}_{g,n}(\kappa)$ carries a canonical volume form that is parallel with respect to its affine complex manifold structure, and that the total volume of $\mathbb{P}\Omega^d\mathcal{M}_{g,n}(\kappa)=\Omega^d\mathcal{M}_{g,n}/\mathbb{C}^*$ with respect to the measure induced by this volume form is finite.Comment: Streamlined, minor corrections added, definition of the volume form independent of the choice of a d-th root of unit

Translation surfaces and the curve graph in genus two

Let $S$ be a (topological) compact closed surface of genus two. We associate to each translation surface $(X,\omega) \in \mathcal{H}(2)\sqcup\mathcal{H}(1,1)$ a subgraph $\hat{\mathcal{C}}_{\rm cyl}$ of the curve graph of $S$. The vertices of this subgraph are free homotopy classes of curves which can be represented either by a simple closed geodesic, or by a concatenation of two parallel saddle connections (satisfying some additional properties) on $X$. The subgraph $\hat{\mathcal{C}}_{\rm cyl}$ is by definition $\mathrm{GL}^+(2,\mathbb{R})$-invariant. Hence, it may be seen as the image of the corresponding Teichm\"uller disk in the curve graph. We will show that $\hat{\mathcal{C}}_{\rm cyl}$ is always connected and has infinite diameter. The group ${\rm Aff}^+(X,\omega)$ of affine automorphisms of $(X,\omega)$ preserves naturally $\hat{\mathcal{C}}_{\rm cyl}$, we show that ${\rm Aff}^+(X,\omega)$ is precisely the stabilizer of $\hat{\mathcal{C}}_{\rm cyl}$ in ${\rm Mod}(S)$. We also prove that $\hat{\mathcal{C}}_{\rm cyl}$ is Gromov-hyperbolic if $(X,\omega)$ is completely periodic in the sense of Calta. It turns out that the quotient of $\hat{\mathcal{C}}_{\rm cyl}$ by ${\rm Aff}^+(X,\omega)$ is closely related to McMullen's prototypes in the case $(X,\omega)$ is a Veech surface in $\mathcal{H}(2)$. We finally show that this quotient graph has finitely many vertices if and only if $(X,\omega)$ is a Veech surface for $(X,\omega)$ in both strata $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$.Comment: 47 pages, 17 figures. Minor changes, some proofs improved. Comments welcome

Complete periodicity of Prym eigenforms

This paper deals with Prym eigenforms which are introduced previously by McMullen. We prove several results on the directional flow on those surfaces, related to complete periodicity (introduced by Calta). More precisely we show that any homological direction is algebraically periodic, and any direction of a regular closed geodesic is a completely periodic direction. As a consequence we draw that the limit set of the Veech group of every Prym eigenform in some Prym loci of genus 3,4, and 5 is either empty, one point, or the full circle at infinity. We also construct new examples of translation surfaces satisfying the topological Veech dichotomy. As a corollary we obtain new translation surfaces whose Veech group is infinitely generated and of the first kind.Comment: 35 page

Complex hyperbolic volume and intersection of boundary divisors in moduli spaces of genus zero curves

We show that the complex hyperbolic metrics defined by Deligne-Mostow and Thurston on ${\mathcal{M}}_{0,n}$ are singular K\"ahler-Einstein metrics when ${\mathcal{M}}_{0,n}$ is embedded in the Deligne-Mumford-Knudsen compactification $\overline{\mathcal{M}}_{0,n}$. As a consequence, we obtain a formula computing the volumes of ${\mathcal{M}}_{0,n}$ with respect to these metrics using intersection of boundary divisors of $\overline{\mathcal{M}}_{0,n}$. In the case of rational weights, following an idea of Y. Kawamata, we show that these metrics actually represent the first Chern class of some line bundles on $\overline{\mathcal{M}}_{0,n}$, from which other formulas computing the same volumes are derived.Comment: Added a new expression of the divisor whose self-intersection computes the volume in Theorem 1.1. Exposition improve