The $X$-varieties for CR mappings between hyperquadrics

The Reflection Map and Infinitesimal Deformations of Sphere Mappings

“…The following definition is a summary of [ 7 , Definition 2.1, 2.2] introducing the homogenization and reflection map (which appears in the study of the X -variety, see also [ 8 ] for the case of hyperquadric maps): Denote by the complex vector space of homogeneous polynomials of degree d in n holomorphic variables . Write for the complex vector space with basis consisting of homogeneous polynomials of degree d in n anti-holomorphic variables .…”

“…A lot is known about mappings of spheres, see the survey by D’Angelo [ 9 ] and the references therein. A prominent example of a sphere map is the homogeneous sphere map of degree d from into for some , which consists of all lexicographically ordered monomials in of degree d and is given by The purpose of this article is to study the reflection map , which was introduced by D’Angelo [ 7 ] in the case of sphere mappings and further investigated by the same author in [ 8 ] in the case of maps of hyperquadrics. The reflection map of a mapping H allows to effectively compute and deduce several properties of the X-variety associated to H .…”