arXiv research

We construct a triangulation of a compactification of the Moduli space of a surface with at least one puncture that is closely related to the Deligne-Mumford compactification. Specifically, there is a surjective map from the compactification we construct to the Deligne-Mumford compactification so that the inverse image…

Let $S$ be a closed, oriented surface with a finite (possibly empty) set of points removed. In this paper we relate two important but disparate topics in the study of the moduli space $\M(S)$ of Riemann surfaces: Teichmüller geometry and the Deligne-Mumford compactification. We reconstruct the Deligne-Mumford compactif…

In 1969, P. Deligne and D. Mumford compactified the moduli space of curves. Their compactification is a projective algebraic variety, and as such, it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmueller space by the action of the mapping class group gives a compactification of…

The systole function has a universal index gap on moduli spaces.

New homotopy theory reveals the structure of stable curves.

Study rational homology of moduli space via Morse functions, proving stability phenomena.

In arXiv:1503.08402v2 Gelander described a new compactification of the moduli space of finite area hyperbolic surfaces using invariant random subgroups. The goal of this paper is to relate this compactification to the classical augmented moduli space, also known as the Deligne-Mumford compactification. We define a cont…

In the moduli space M_g of genus g Riemann surfaces, consider the locus RM_O of Riemann surfaces whose Jacobians have real multiplication by the order O in a totally real number field F of degree g. If g = 2 or 3, we compute the closure of RM_O in the Deligne-Mumford compactification of M_g and the closure of the locus…

Study the boundary of Riemann surfaces with abelian automorphisms.

The Kalinin effectivity is studied and applied to compactifications and Hilbert squares.

Fixing a closed hyperbolic surface S, we define a moduli space AI(S) of unmarked hyperbolic 3-manifolds homotopy equivalent to S. This 3-dimensional analogue of the moduli space M(S) of unmarked hyperbolic surfaces homeomorphic to S has bizarre local topology, possessing many points that are not closed. There is, howev…

The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of Abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of Abelian differentials. The main ingredients to study the compactifications of the strata ar…

This note is devoted to the definition of moduli spaces of rational tropical curves with n marked points. We show that this space has a structure of a smooth tropical variety of dimension n-3. We define the Deligne-Mumford compactification of this space and tropical $ψ$-class divisors.

Using the the theory of FS^op modules, we study the asymptotic behavior of the homology of $\overline M_{g,n}$, the Deligne--Mumford compactification of the moduli space of curves, for $n >> 0$. An FS^op module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via maps that g…

We prove that a certain series defines a constant function using Wolpert's formula for the variation of the length of a geodesic along a Fenchel Nielsen twist. Subsequently we determine the value viewing it as function on the the Deligne Mumford compactification and evaluating it at the stable curve at infinity.

Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne-Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected c…

We construct a compactification of the moduli spaces of abelian differentials on Riemann surfaces with prescribed zeroes and poles. This compactification, called the moduli space of multi-scale differentials, is a complex orbifold with normal crossing boundary. Locally, our compactification can be described as the norm…

We consider the Riemann moduli space $\mathcal M_γ$ of conformal structures on a compact surface of genus $γ>1$ together with its Weil-Petersson metric $g_{\mathrm{WP}}$. Our main result is that $g_{\mathrm{WP}}$ admits a complete polyhomogeneous expansion in powers of the lengths of the short geodesics up to the singu…

Constructs a family to handle unstable fibers on complex surfaces.

There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface. The Deligne-Mumford compactification of the moduli space of curves then suggests…

New method proves achiral Lefschetz fibrations exist using Riemannian geometry.

We describe the closure of the strata of abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne-Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, bot…

Let S be a closed oriented surface of genus at least two. Labourie and the author have independently used the theory of hyperbolic affine spheres to find a natural correspondence between convex RP^2 structures on S and pairs (Σ,U) consisting of a conformal structure Σon S and a holomorphic cubic differential U over Σ. …

In this paper, we determine the distribution of the length partition of a random multicurve of fixed topological type on a closed hyperbolic surface using the methods of Margulis' thesis and Mirzakhani's equidistribution theorem for horospheres. This distribution admits a polynomial density, whose coefficients can be e…

We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise as the functor of points of such objects. We choo…

We show that the complex hyperbolic metrics defined by Deligne-Mostow and Thurston on ${\mathcal{M}}_{0,n}$ are singular Kähler-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 volu…

The paper examines the behavior of Weierstrass measures on stable curves as they approach a nodal stable curve.

Let S be an orientable, finite type surface with negative Euler characteristic. The augmented moduli space of convex real projective structures on S was first defined and topologized by the first author. In this article, we give an explicit description of this topology using explicit coordinates. More precisely, given …

The Weil-Petersson and Takhtajan-Zograf metrics on the Riemann moduli spaces of complex structures for an $n$-fold punctured oriented surface of genus $g,$ in the stable range $g+2n>2,$ are shown here to have complete asymptotic expansions in terms of Fenchel-Nielsen coordinates at the exceptional divisors of the Knuds…

Calculates volumes of linear subvarieties in moduli spaces of Abelian differentials.

The second author and Norbury initiated the enumeration of lattice points in the Deligne-Mumford compactifications of moduli spaces of curves. They showed that the enumeration may be expressed in terms of polynomials, whose top and bottom degree coefficients store psi-class intersection numbers and orbifold Euler chara…

Stability in homology of moduli spaces of admissible covers.

The study proves semisimplicity of totally geodesic subvarieties in moduli spaces of Riemann surfaces.

We consider (local) parametrizations of Teichmuller space $T_{g,n}$ (of genus $g$ hyperbolic surfaces with $n$ boundary components) by lengths of $6g-6+3n$ geodesics. We find a large family of suitable sets of $6g-6+3n$ geodesics, each set forming a special structure called "admissible double pants decomposition". For …

This paper begins the study of Morse theory for orbifolds, or more precisely for differentiable Deligne-Mumford stacks. The main result is an analogue of the Morse inequalities that relates the orbifold Betti numbers of an almost-complex orbifold to the critical points of a Morse function on the orbifold. We also show …

This work characterizes global quotient stacks---smooth stacks associated to a finite group acting a manifold---among smooth quotient stacks $[M/G]$, where $M$ is a smooth manifold equipped with a smooth proper action by a Lie group $G$. The characterization is described in terms of the action of the connected componen…

Computes infinitesimal automorphisms for $L$-valued Higgs bundles, leading to DM stacks.

For G a complex reductive group and X a smooth projective or convex quasi-projective polarized G-variety we construct a formal map in quantum K-theory from the equivariant quantum K-theory $QK^G(X)$ to the quantum K-theory of the git quotient $QK(X//G)$ assuming the quotient $X//G$ is a smooth Deligne-Mumford stack wit…

The paper provides a uniform lower bound for intersection numbers of psi-classes on moduli spaces.

Extends harmonic maps compactification to punctured Riemann surfaces.

We consider horofunction compactifications of symmetric spaces with respect to invariant Finsler metrics. We show that any (generalized) Satake compactification can be realized as a horofunction compactification with respect to a polyhedral Finsler metric.

The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.

Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.

Satake has constructed compactifications of symmetric spaces D=G/K which (under a condition called geometric rationality by Casselman) yield compactifications of the corresponding locally symmetric spaces. The different compactifications depend on the choice of a representation of G. One example is the Baily-Borel-Sata…

The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.

New coordinates for Teichmüller space compactification.

New compactification for character varieties with good topological properties.

Embeds Higson compactification into adelic solenoids.