arXiv research

We develop a systematic method for classifying supersymmetric orbifold compactifications of M-theory. By restricting our attention to abelian orbifolds with low order, in the special cases where elements do not include coordinate shifts, we construct a "periodic table" of such compactifications, organized according to …

Study Kähler groups from orbifold compactifications of curve moduli.

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…

Study on moduli spaces of sextic curves with simple singularities and their compactifications.

The paper classifies and proves properties of ALE manifolds and orbifolds.

The paper studies Kähler compactifications of C^n and their applications in geometry.

We describe a class of compact $G_2$ orbifolds constructed from non-symplectic involutions of K3 surfaces. Within this class, we identify a model for which there are infinitely many associative submanifolds contributing to the effective superpotential of M-theory compactifications. Under a chain of dualities, these can…

In this technical note we give a purely geometric understanding of discrete torsion, as an analogue of orbifold Wilson lines for two-form tensor field potentials. In order to introduce discrete torsion in this context, we describe gerbes and the description of certain type II supergravity tensor field potentials as con…

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X=G/K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$-invariant Finsler metric on $X$. As an application, we establish the existence of natural bordifications, as orbifold…

In this paper, we consider solutions and spectral functions of M-theory from Milne spaces with extra free dimensions. Conformal deformations to the metric associated with the real hyperbolic space forms are derived. For the three-dimensional case, the orbifold identifications $SL(2,{\mathbb Z}+i{\mathbb Z})/\{\pm Id\}$…

Constructs a family to handle unstable fibers on complex surfaces.

We develop methods to study the singularities of certain $G_2$ cones related to toric hyperkahler spaces and Einstein selfdual orbifolds. This allows us to determine the low energy gauge groups of chiral N=1 compactifications of M-theory on a large family of such backgrounds, which includes the models recently studied …

We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension 2 provided the corresponding lattice is arithmetic and small enough. The method is to show that the A…

We give an explicit local classification of conformally equivalent but oppositely oriented Kaehler metrics on a 4-manifold which are toric with respect to a common 2-torus action. In the generic case, these structures have an intriguing local geometry depending on a quadratic polynomial and two arbitrary functions of o…

Orbifold uniformization of complex algebraic variety via polystable parabolic Higgs bundle

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

We find boundaries of Borel-Serre compactifications of locally symmetric spaces, for which any filling is incompressible. We prove this result by showing that these boundaries have small singular models and using these models to obstruct compressions. We also show that small singular models of boundaries obstruct $S^1$…

The paper proves properties of quantum representations and their Toledo invariants.

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 …

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 M=P(E) be a ruled surface. We introduce metrics of finite volume on M whose singularities are parametrized by a parabolic structure over E. Then, we generalise results of Burns--de Bartolomeis and LeBrun, by showing that the existence of a singular Kahler metric of finite volume and constant non positive scalar cur…

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.

We discuss the `hd-compactification' of a semi-simple Lie group to a manifold with corners; it is the real analog of the wonderful compactification of deConcini and Procesi. There is a 1-1 correspondence between the boundary faces of the compactification and conjugacy classes of parabolic subgroups with the boundary fa…

We show that the horofunction compactification of Teichmüller space with the Teichmüller metric is homeomorphic to the Gardiner-Masur compactification.

We define a compactification of symmetric spaces of noncompact type, seen as spaces of isometry classes of marked lattices, analogous to the Thurston compactification of the Teichmüller space, and we show that it is equivariantly isomorphic to a Satake compactification. We then use it to define a new compactification o…

We construct several examples of compactifications of Einstein metrics. We show that the Eguchi--Hanson instanton admits a projective compactification which is non--metric, and that a metric cone over any (pseudo)--Riemannian manifolds admits a metric projective compactification. We construct a para--$c$--projective co…

Researchers create a new compactification of character varieties using geometric and algebraic methods.

Paper studies compactifications of Higgs bundles and self-duality equations.

No natural topological compactification for Fulton-MacPherson.

Paper shows geometric properties preserved by compactifications in relation to coarse structures and group actions.

The paper classifies and computes limits of equivariant compactifications of groups.

The study finds only finitely many Kähler-Einstein compactifications for semisimple groups.

The paper extends end concepts to arbitrary groups and spaces.

In this paper we present a topological way of building a compactification of a symmetric space from a compactification of a Weyl Chamber.

To study a noncompact Riemannian manifold, it is often useful to find a compactification. We discuss several common compactifications and survey some recent results.

The group of direct isometries of the real n-dimensional hyperbolic space is G=SOo(n,1). This isometric action admits many differentiable compactifications into an action on the closed ball. We prove that all such compactifications are topologically conjugate but not necessarily differentiably conjugate. We give the cl…

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…

Consider a finite dimensional (generally reducible) polynomial representation ρof GL_n. A projective compactification of GL_n is the closure of ρ(GL_n) in the space of all operators defined up to a factor (this class of spaces can be characterized as equivariant projective normal compactifications of GL_n). We give an …

Schwartz functions smoothly extend to real projective spaces.

We define a new compactification of outer space $CV_N$ (the \emph{Pacman compactification}) which is an absolute retract, for which the boundary is a $Z$-set. The classical compactification $\overline{CV_N}$ made of very small $F_N$-actions on $\mathbb{R}$-trees, however, fails to be locally $4$-connected as soon as $N…