The following structure theorems are proved: An orbit of a real solvable Lie group in projective space that is a complex submanifold is isomorphic to C x (C*)m x Í2 , where Í2 is an open orbit of a real solvable Lie group in a projective rational variety. Also, any homogeneous space of a complex Lie group that is isomorphic to C" can be realized as an orbit in some projective space. As a conseq...

We study the GIT-quotient of the Cartesian ppower of a projective space modulo the projective orthogonal group. A classical isomorphism of this group with the Inversive group of birational transformations of the projective space of one dimension less allows one to interpret these spaces as configuration spaces of complex or real spheres.

These are notes for a talk at a topology seminar at ND. 1. GENERAL FACTS In the sequel, for simplicity we denote the complex projective space CPN by PN , and we denote the projective coordinates by [~z] = [z0, . . . , zN ]. A smooth projective variety is a connected compact complex submanifold of some projective space PN . All the smooth projective varieties are Kähler manifolds, but the conver...

The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be regarded as the unique complex orbits of the (projectivized) isotropy representa...

We exhibit upper bounds for the probability distribution of the generalized condition number of singular complex matrices. To this end, we develop a new technique to study volumes of tubes about projective varieties in the complex projective space. As a main outcome, we show an upper bound estimate for the volume of the intersection of a tube with an equi–dimensional projective algebraic variety.

The space of marked n distinct points on the complex projective line up to projective transformations will be called a configuration space in this paper. There are two families of complex hyperbolic structures on the configuration space constructed by Deligne-Mostow and by Thurston. We first confirm that these families are the same. Then in view of the deformation theory for real hyperbolic con...

Spaces in the genus of infinite quaternionic projective space which admit essential maps from infinite complex projective space are classified. In these cases the sets of homotopy classes of maps are described explicitly. These results strengthen the classical theorem of McGibbon and Rector on maximal torus admissibility for spaces in the genus of infinite quaternionic projective space. An inte...

A complex projective (or CP ) structure on a compact surface S is an atlas of charts with values in CP and Möbius transition functions. Let P.S/ denote the space of (isotopy classes of) marked complex projective structures on S , and let T .S/ be the Teichmüller space of (isotopy classes of) marked complex structures on S . Because Möbius maps are holomorphic, there is a forgetful projection W ...