Abstract
The reflection map introduced by D’Angelo is applied to deduce simpler descriptions of nondegeneracy conditions for sphere maps and to the study of infinitesimal deformations of sphere maps. It is shown that the dimension of the space of infinitesimal deformations of a nondegenerate sphere map is bounded from above by the explicitly computed dimension of the space of infinitesimal deformations of the homogeneous sphere map. Moreover a characterization of the homogeneous sphere map in terms of infinitesimal deformations is provided.
Highlights
The main motivation is the study of real-analytic CR maps of the unit sphere S2n−1 in Cn for n ≥ 2, which is defined byS2n−1 = {z = (z1, . . . , zn) ∈ Cn : z 2 = |z1|2 + . . . + |zn|2 = 1}
The homogeneous sphere map Hnd plays a crucial role in the classification of polynomial maps, see the works of D’Angelo [3,5] and [10] for rational sphere maps
The homogeneous sphere map appears in the definition of the reflection map CH for a rational sphere map H = P/Q : S2n−1 → S2m−1 with Q = 0 on S2n−1: Let VH : Cm → CK be a matrix with holomorphic entries, satisfying VH (X ) · Hnd /Q = X · Hon S2n−1 for X ∈ Cm, where · denotes the euclidean inner product
Summary
The main motivation is the study of real-analytic CR maps of the unit sphere S2n−1 in Cn for n ≥ 2, which is defined by. (b) H is holomorphically nondegenerate if and only if VH is generically of rank m on S2n−1 This has immediate consequences to show sufficient and necessary conditions in terms of nondegeneracy conditions for the X-variety of H to be an affine bundle or that it agrees with the graph of the map, see Sect. The results involving infinitesimal deformations are summarized in the following theorem: Theorem 2 Let H : S2n−1 → S2m−1 be a holomorphically nondegenerate rational map of degree d. It holds that dim hol(H ) ≤ dim hol(Hnd ) =. While the article [15] contains examples which required computer-assistance, it is shown in several examples in this article that the reflection matrix allows for explicit and effective computations of infinitesimal deformations of sphere maps
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