388 research outputs found
Volume form on moduli spaces of d-differentials
Given , , and an integral vector
such that and , let
denote the moduli space of meromorphic
-differentials on Riemann surfaces of genus whose zeros and poles have
orders prescribed by . We show that
carries a canonical volume form that is parallel with respect to its affine
complex manifold structure, and that the total volume of
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 be a (topological) compact closed surface of genus two. We associate
to each translation surface a subgraph
of the curve graph of . 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 . The subgraph is by
definition -invariant. Hence, it may be seen as
the image of the corresponding Teichm\"uller disk in the curve graph. We will
show that is always connected and has infinite
diameter. The group of affine automorphisms of
preserves naturally , we show that
is precisely the stabilizer of in . We also prove that is
Gromov-hyperbolic if is completely periodic in the sense of Calta.
It turns out that the quotient of by is closely related to McMullen's prototypes in the case
is a Veech surface in . We finally show that this
quotient graph has finitely many vertices if and only if is a
Veech surface for in both strata and
.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 are singular K\"ahler-Einstein metrics when
is embedded in the Deligne-Mumford-Knudsen
compactification . As a consequence, we obtain a
formula computing the volumes of with respect to these
metrics using intersection of boundary divisors of
. 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 , 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
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