The moduli of substructures of a compact complex space

We construct a space of fine moduli for the substructures of an arbitrary compact complex space . A substructure of is given by a subalgebra of the structure sheaf with the additional feature that is also a complex space; and are called equivalent if and only if and are isomorphic as subalgebras of .

Since substructures are quotients, it is only natural to start with the fine moduli space of all complex-analytic quotients of . In order to obtain a representable moduli functor of substructures, we are forced to concentrate on families of quotients which satisfy some flatness condition for relative differential modules of higher order. Considering the corresponding flatification of , we realize that its open subset consisting of all substructures turns out to be a complex space which has the required universal property.

     

Projective structures on moduli spaces of compact complex hypersurfaces

It is shown that moduli spaces of complete families of compact complex hypersurfaces in complex manifolds often come equipped canonically with projective structures satisfying some natural integrability conditions.

     

The moduli space of the modular group in complex hyperbolic geometry

Inventiones mathematicae - We construct the space of discrete, faithful, type-preserving representations of the modular group into the isometry group of complex hyperbolic 2-space up to conjugacy....     

On the moduli space of quadruples of points in the boundary of complex hyperbolic space

We consider the space $ \mathcal{M} $ of ordered quadruples of distinct points in the boundary of complex hyperbolic n-space, $ H_\mathbb{C}^n $ , up to its holomorphic isometry group PU(n, 1): One of the important problems in complex hyperbolic geometry is to construct and describe a moduli space for $ \mathcal{M} $ . For n = 2, this problem was considered by Falbel, Parker, and Platis. The main purpose of this paper is to construct a moduli space for $ \mathcal{M} $ for any dimension n ≥ 1.     

Holomorphic immersions between compact hyperbolic space forms

Research partially supported by an NSF grant     

Canonical measures on the moduli spaces of compact Riemann surfaces

We study some explicit relations between the canonical line bundle and the Hodge bundle over moduli spaces for low genus. This leads to a natural measure on the moduli space of every genus which is related to the Siegel symplectic metric on Siegel upper half-space as well as to the Hodge metric on the Hodge bundle.     

The topology of moduli spaces of group representations: The case of compact surface

Let G be a connected complex semisimple affine algebraic group, and let K be a maximal compact subgroup of G. Let X be a noncompact oriented surface. The main theorem of Florentino and Lawton (2009) [3] says that the moduli space of flat K-connections on X is a strong deformation retraction of the moduli space of flat G-connections on X. We prove that this statement fails whenever X is compact of genus at least two.     

A complex hyperbolic structure for moduli of cubic surfaces

We show that the moduli space M of marked cubic surfaces is biholomorphic to (B⁴ − H)/Г, where B⁴ is complex hyperbolic four-space, Γ is a specific group generated by complex reflections, and H is the union of reflection hyperplanes for Γ. Thus M has a complex hyperbolic structure, i.e., an (incomplete) metric of constant negative holomorphic sectional curvature.     

Homogeneous hypersurfaces in complex hyperbolic spaces

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.      

Target space duality of Calabi-Yau spaces with two moduli

In the context of superstring compactifications on Calabi-Yau threefolds, we consider the Picard-Fuchs equations that are obeyed by the periods of the holomorphic three-form. We review, focusing on an example with two moduli, some powerful algebro-geometric techniques for computing the monodromy group of these equations, which is closely related to the target space duality group. For the example investigated, the latter is shown to be given by a three-dimensional representation of a central extension ofB ₅, the braid group on five strands.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98. No. 3, pp. 358–374, March, 1994.     

Complete minimal hypersurfaces in complex hyperbolic space

We construct new examples of embedded, complete, minimal hypersurfaces in complex hyperbolic space, including deformations of bisectors and some minimal foliations. Received: 20 March 2000 / Revised version: 21 July 2000     

On hyperbolic summation and hyperbolic moduli of smoothness

This paper deals with the method of hyperbolic summation of tensor product orthogonal spline functions onI ^d. The spaces, defined in terms of the order of the best approximation by the elements of the space spanned by the tensor product functions with indices from a given hyperbolic set, are described both in terms of the coefficients in some basis and as interpolation spaces. Moreover, the hyperbolic modulus of smoothness is studied, and some relations between hyperbolic summation and hyperbolic modulus of smoothness are established.     

Non-hyperbolic complex space with a hyperbolic normalization

We construct an example of a non-hyperbolic singular projective surface whose normalization is the square of a genus 3 curve and hence, hyperbolic.