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.
Introduction
The main motivation is the study of realanalytic CR maps of the unit sphere \(S^{2n1}\) in \({{\,\mathrm{{\mathbb {C}}}\,}}^n\) for \(n\ge 2\), which is defined by
For \(n=2\) write \(z = z_1, w = z_2\). 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 \(H_n^d\) of degree d from \(S^{2n1}\) into \(S^{2K1}\) for some \(K=K(n,d)\in {{\,\mathrm{{\mathbb {N}}}\,}}\), which consists of all lexicographically ordered monomials in \(z=(z_1,\ldots ,z_n)\in {{\,\mathrm{{\mathbb {C}}}\,}}^n\) 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 Xvariety associated to H. The Xvariety was introduced and studied by Forstnerič in [19] to extend CR maps satisfying certain smoothness assumptions. In the case of realanalytic CR maps of spheres it is shown that these maps are rational.
The homogeneous sphere map \(H_n^d\) 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 \(C_H\) for a rational sphere map \(H=P/Q: S^{2n1} \rightarrow S^{2m1}\) with \(Q\ne 0\) on \(S^{2n1}\): Let \(V_H: {{\,\mathrm{{\mathbb {C}}}\,}}^m \rightarrow {{\,\mathrm{{\mathbb {C}}}\,}}^K\) be a matrix with holomorphic entries, satisfying \(V_H(X) \cdot {\bar{H}}_n^d /{\bar{Q}} = X \cdot {\bar{H}}\) on \(S^{2n1}\) for \(X\in {{\,\mathrm{{\mathbb {C}}}\,}}^m\), where \(\cdot \) denotes the euclidean inner product. The previous identity is achieved by the homogenization technique of D’Angelo [3]. \(V_H\) is referred to as reflection matrix and \(C_H(X) {:}{=}V_H(X) \cdot {\bar{H}}_n^d/{\bar{Q}}\) for \(X\in {{\,\mathrm{{\mathbb {C}}}\,}}^m\). See Sect. 2.4 below for more details.
In this article the reflection matrix will be applied in two ways. In the first case it is shown that nondegeneracy conditions of sphere maps can be rephrased in terms of rank conditions on the reflection matrix. The nondegeneracy conditions considered here were introduced in [20] and [22] respectively. In the case of a sphere map \(H: S^{2n1} \rightarrow S^{2m1}\) they are defined as follows: If \(\Gamma \) denotes the set of realanalytic CR vector fields tangent to \(S^{2n1}\), then H is called finitely nondegenerate at \(p\in S^{2n1}\), if there is an integer \(\ell \in {{\,\mathrm{{\mathbb {N}}}\,}}\), such that,
The map H is called holomorphically nondegenerate if there is no nontrivial holomorphic vector field tangent to \(S^{2m1}\) along the image of H. These notions of nondegeneracy were originally defined for submanifolds and introduced by [24] and [1] respectively, see also the survey of Lamel [21]. For more details on nondegeneracy conditions for CR maps see also Sect. 2.2 below. Then the following theorem is shown:
Theorem 1
Let \(H: S^{2n1} \rightarrow S^{2m1}\) be a rational map of degree d.

(a)
H is finitely nondegenerate at \(p \in S^{2n1}\) if and only if \(V_H\) is of rank m at \(p \in S^{2n1}\).

(b)
H is holomorphically nondegenerate if and only if \(V_H\) is generically of rank m on \(S^{2n1}\).
This has immediate consequences to show sufficient and necessary conditions in terms of nondegeneracy conditions for the Xvariety of H to be an affine bundle or that it agrees with the graph of the map, see Sect. 5 and Theorem 3 below for more details.
In the second case, applications of the reflection matrix to the study of infinitesimal deformations are provided. For \(M\subset {{\,\mathrm{{\mathbb {C}}}\,}}^N\) and \(M'\subset {{\,\mathrm{{\mathbb {C}}}\,}}^{N'}\) real submanifolds consider the set \({\mathcal {H}}(M,M')\) of all maps, which are holomorphic in a neighborhood of M and satisfying \(H(M) \subset M'\). In [14,15,16,17] locally rigid maps were studied. They correspond to isolated points in the quotient space of \({\mathcal {H}}(M,M')\) under automorphisms. A sufficient linear condition was provided for local rigidity of a given map, which is formulated in terms of infinitesimal deformations. An infinitesimal deformation of a map \(H:M \rightarrow M'\) is a holomorphic vector, defined in a neighborhood of M, whose real part is tangent to \(M'\) along the image of H. The set of infinitesimal deformations of a map H is denoted by \({{\,\mathrm{\mathfrak {hol}}\,}}(H)\). Examples of infinitesimal deformations of a map H can be obtained from smooth curves of maps \({{\,\mathrm{{\mathbb {R}}}\,}}\ni t \mapsto H(t)\), with \(H(0) = H\), since \(\frac{d H(t)}{dt}_{t=0} \in {{\,\mathrm{\mathfrak {hol}}\,}}(H)\).
The results involving infinitesimal deformations are summarized in the following theorem:
Theorem 2
Let \(H: S^{2n1} \rightarrow S^{2m1}\) be a holomorphically nondegenerate rational map of degree d. It holds that \(\dim {{\,\mathrm{\mathfrak {hol}}\,}}(H) \le \dim {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)=\left( \frac{2d+n}{d}\right) K(n,d)^2\) and if H is assumed to be polynomial, then \(\dim {{\,\mathrm{\mathfrak {hol}}\,}}(H) = \dim {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) if and only if H is unitarily equivalent to \(H^d_n\).
This result contains an alternative characterization of the homogeneous sphere map to the one given in [3, 23] or [6, Sect.5.1.4, Theorem 3] and demonstrates a new method to compute infinitesimal deformations for sphere maps. While the article [15] contains examples which required computerassistance, 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.
Preliminaries
The purpose of this section is to introduce the necessary notions and notations needed throughout the article. These are only required for maps of spheres but without any effort and no loss of clarity the general case of maps of manifolds M and \(M'\) is treated. To this end the following assumptions are made: Let M be a realanalytic generic submanifold of \({{\,\mathrm{{\mathbb {C}}}\,}}^N\) of codimension d. For a realanalytic CR submanifold \(M'\subset {{\,\mathrm{{\mathbb {C}}}\,}}^{N'}\) of codimension \(d'\), let \(p'\in M'\) and \(\rho ': V' \times {\bar{V}}' \rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}^{d'}, \rho '=(\rho _1',\ldots , \rho _{d'}')\), be a realanalytic mapping, such that \(M'\cap V'=\{z'\in V' : \rho '(z',{\bar{z}}') = 0\}\), where \(V' \subset {{\,\mathrm{{\mathbb {C}}}\,}}^{N'}\) is a neighborhood of \(p' \in M'\) and the differentials \(d\rho '_1,\ldots , d\rho '_{d'}\) are linearly independent in \(V'\). Denote \({\bar{V}}' = \{{\bar{z}}'\in {{\,\mathrm{{\mathbb {C}}}\,}}^{N'}: z' \in V'\}\). The complex gradient \({\rho '_j}_{z'}\) of \(\rho '_j\) is given by \({\rho '_j}_{z'} = \left( \frac{\partial \rho '_j}{\partial z_1'}, \ldots ,\frac{\partial \rho '_j}{\partial z_{N'}'}\right) \). The following notation is used: \(v \cdot w {:}{=}v_1 w_1 + \cdots + v_n w_n\) for vectors \(v=(v_1,\ldots , v_n) \in {{\,\mathrm{{\mathbb {C}}}\,}}^n\) and \(w =(w_1,\ldots , w_n) \in {{\,\mathrm{{\mathbb {C}}}\,}}^n\).
Infinitesimal Deformations of CR Maps
One of the main objects of this article are infinitesimal deformations of a CR map.
Definition 1
Let \(H: M \rightarrow M'\) be a realanalytic CR map. A realanalytic CR map \(X: M \rightarrow {{\,\mathrm{{\mathbb {C}}}\,}}^{N'}\) is called an infinitesimal deformation of H, if for every \(p\in M\) and every realanalytic mapping \(\rho '=(\rho '_1,\ldots , \rho '_{d'})\) defined in a neighborhood of H(p) vanishing on \(M'\), it holds that,
for some open neighborhood \(U \subset {{\,\mathrm{{\mathbb {C}}}\,}}^N\) of p. The space of infinitesimal deformations of H is denoted by \({{\,\mathrm{\mathfrak {hol}}\,}}(H)\).
For a real manifold M the space \({{\,\mathrm{\mathfrak {hol}}\,}}(M)\) of infinitesimal automorphisms of M consists of holomorphic vectors whose real part is tangent to M.
For a map \(H: M \rightarrow M'\) the subspace \({{\,\mathrm{\mathfrak {aut}}\,}}(H) {:}{=}{{\,\mathrm{\mathfrak {hol}}\,}}(M')_{H(M)} + H_*({{\,\mathrm{\mathfrak {hol}}\,}}(M)) \subset {{\,\mathrm{\mathfrak {hol}}\,}}(H)\) is referred to as the space of trivial infinitesimal deformations of H. Its complement in \({{\,\mathrm{\mathfrak {hol}}\,}}(H)\) is called the space of nontrivial infinitesimal deformations of H.
A map H is called infinitesimally rigid if \({{\,\mathrm{\mathfrak {hol}}\,}}(H) = {{\,\mathrm{\mathfrak {aut}}\,}}(H)\).
The infinitesimal stabilizer of H is given by \((S,S') \in {{\,\mathrm{\mathfrak {hol}}\,}}(M) \times {{\,\mathrm{\mathfrak {hol}}\,}}(M')\) such that \(H_*(S) =  S'_{H(M)}\). An infinitesimal automorphism \(S \in {{\,\mathrm{\mathfrak {hol}}\,}}(M)\) is said to belong to the infinitesimal stabilizer of H if there exists \(S' \in {{\,\mathrm{\mathfrak {hol}}\,}}(M')\) such that \(H_*(S) =  S'_{H(M)}\).
In the case of sphere mappings, for a realanalytic CR map \(H: S^{2k1} \rightarrow S^{2m1}\), a holomorphic map \(X: U \rightarrow {{\,\mathrm{{\mathbb {C}}}\,}}^n\), where \(U \subset {{\,\mathrm{{\mathbb {C}}}\,}}^k\) is an open neighborhood of \(S^{2k1}\), is an infinitesimal deformation of H if \({{\,\mathrm{Re}\,}}(X(z) \cdot \overline{H(z)}) = 0\) for \(z\in S^{2k1}\).
Nondegeneracy Conditions for CR Maps
The purpose of this section is to provide the definitions of finite and holomorphic nondegeneracy for CR maps introduced by Lamel [20] and Lamel–Mir [22] respectively, and study some of their properties.
Definition 2
A realanalytic CR map \(H: M \rightarrow M'\) is called holomorphically degenerate if there exists a realanalytic CR map \(Y: M \rightarrow {{\,\mathrm{{\mathbb {C}}}\,}}^{N'}\) satisfying \(Y\not \equiv 0\) and for every \(p\in M\) and every realanalytic mapping \(\rho '=(\rho '_1,\ldots , \rho '_{d'})\) defined in a neighborhood of H(p) vanishing on \(M'\), it holds that,
for some open neighborhood \(U \subset {{\,\mathrm{{\mathbb {C}}}\,}}^N\) of p. If a map is not holomorphically degenerate it is called holomorphically nondegenerate.
Simple examples of holomorphically degenerate maps are the following:
Example 1
Let \(F: S^{2n1} \rightarrow S^{2m1}\), then \(H = F \oplus 0\), where \(0 \in {{\,\mathrm{{\mathbb {C}}}\,}}^k\), is a holomorphically degenerate sphere map from \(S^{2n1}\) into \(S^{2(n+k)1}\), since \(X=0 \oplus G\) for \(0\in {{\,\mathrm{{\mathbb {C}}}\,}}^n\) and G a holomorphic function from \({{\,\mathrm{{\mathbb {C}}}\,}}^n\) into \({{\,\mathrm{{\mathbb {C}}}\,}}^k\) satisfies \(X \cdot {\bar{H}} = 0\).
Finite nondegeneracy is defined as follows:
Definition 3
Consider a realanalytic CR map \(H: M \rightarrow M'\). Let \({\bar{L}}_1, \ldots , {\bar{L}}_n\) a basis of CR vector fields of M and for a multiindex \(\alpha =(\alpha _1,\ldots , \alpha _n)\in {{\,\mathrm{{\mathbb {N}}}\,}}^n\) denote \({\bar{L}}^{\alpha } = {\bar{L}}_1^{\alpha _1} \cdots {\bar{L}}_n^{\alpha _n}\). Let \(p\in M\). For each \(k\in {{\,\mathrm{{\mathbb {N}}}\,}}\) define the following subspaces of \({{\,\mathrm{{\mathbb {C}}}\,}}^{N'}\):
for a realanalytic mapping \(\rho '=(\rho '_1,\ldots , \rho '_{d'})\) defined in a neighborhood of H(p) and vanishing on \(M'\). Define \(s(p) {:}{=}N'  \max _k \dim _{{{\,\mathrm{{\mathbb {C}}}\,}}} E_k'(p)\), which is called the degeneracy of H at p. The map H is of constant degeneracy s(q) at \(q\in M\) if \(p\mapsto s(p)\) is a constant function in a neighborhood of q. If \(s(p)=0\), then H is called finitely nondegenerate at p. Considering the smallest integer \(k_0\) such that \(E_\ell '(p) = E_{k_0}'(p)\) for all \(\ell \ge k_0\) one can say more precisely that H is \((k_0,s)\)degenerate at p. If the map is finitely nondegenerate at p one says that it is \(k_0\)nondegenerate at p.
Note that if a map is finitely nondegenerate at p, then it is also finitely nondegenerate at points in a neighborhood of p. If M is connected, by [20, Lemma 22] the set of points where the map \(H:M \rightarrow M'\) is of constant degeneracy \(s(H){:}{=}\min _{p\in M}s(p)\) is an open and dense subset of M. The number s(H) is called generic degeneracy of H.
Constant degeneracy can be phrased in terms of vector fields as follows:
Proposition 1
(Lamel [20, Proposition 18]) Let \(H: M \rightarrow M'\) be a realanalytic CR mapping of constant degeneracy s in a neighborhood of \(p\in M\). Then it holds that
These following statements are analogous to the corresponding statements for manifolds or infinitesimal automorphisms, see [2, Theorem 11.5.1] and [2, Proposition 12.5.1].
Proposition 2
Let \(H:M \rightarrow M'\) be a realanalytic CR map. Then the following statements hold:

(a)
If H is holomorphically degenerate, then \(\dim _{{{\,\mathrm{{\mathbb {R}}}\,}}} {{\,\mathrm{\mathfrak {hol}}\,}}(H) = \infty \).

(b)
If H is finitely nondegenerate at \(p\in M\), then it is holomorphically nondegenerate.

(c)
If the space of holomorphic vector fields at \(p\in M\) tangent to \(M'\) along the image of H is of complex dimension s, then, outside a proper realanalytic variety of a neighborhood of p, the map H is of constant degeneracy s.
Note that if \(s=0\), then (c) says that if the map H is holomorphically nondegenerate then H is finitely nondegenerate outside a proper realanalytic variety.
Moreover, if M is assumed to be connected, (c) yields the following statement: If at any \(p\in M\) the space of holomorphic vector fields at \(p\in M\) tangent to \(M'\) along the image of H is of complex dimension at least s, then the generic degeneracy of H is at least s.
Proof of Proposition 2
To show (a) argue as in [2, Proposition 12.5.1]: If H is holomorphically degenerate, there exists a nontrivial holomorphic map X tangent to \(M'\) along H(M). Then for each \(k\in {{\,\mathrm{{\mathbb {N}}}\,}}\) also \(Y_k {:}{=}z_1^k X\) is tangent to \(M'\) along H(M) and these maps are complexlinearly independent. Since M is generic and the real part of a nontrivial holomorphic map \({\hat{X}}\), restricted to M, cannot vanish on M (the vanishing of \({{\,\mathrm{Re}\,}}({\hat{X}})_M\) would imply that \({\hat{X}} \equiv 0\)), the vector fields \({{\,\mathrm{Re}\,}}(Y_k)\) are reallinearly independent, hence \(\dim _{{{\,\mathrm{{\mathbb {R}}}\,}}} {{\,\mathrm{\mathfrak {hol}}\,}}(H) = \infty \).
To prove (b) denote by \({\mathfrak {X}}(H)\) the set of holomorphic vector fields tangent to \(M'\) along the image of H. Consider \(X = \sum _j a_j(Z) \frac{\partial }{\partial Z_j} \in {\mathfrak {X}}(H)\) which, by the finite nondegeneracy of H and Proposition 1, satisfies \(a_j(p)=0\) and \(\sum _j a_j(Z) {\rho '_k}_{Z_j}(H(Z),\overline{H(Z)}) = 0\) for \(Z\in M, k=1,\ldots , d'\) and a realanalytic mapping \(\rho '=(\rho '_1,\ldots , \rho '_{d'})\) defined in a neighborhood of H(p) and vanishing on \(M'\). Taking derivatives w.r.t. CR vector fields L of M one gets
for multiindices \(\beta _m \in {{\,\mathrm{{\mathbb {N}}}\,}}^n\) for \(m = 1,\ldots ,N'\). Use coordinates as given in e.g., [2, Proposition 1.3.6] for M in (1), i.e. \(Z=(z,u+i\phi (z,{\bar{z}},u)) \in {{\,\mathrm{{\mathbb {C}}}\,}}^n \times {{\,\mathrm{{\mathbb {C}}}\,}}^d\), where \(\phi \) is defined near 0 in \({{\,\mathrm{{\mathbb {R}}}\,}}^{2N+d}\) with values in \({{\,\mathrm{{\mathbb {R}}}\,}}^d\) satisfying \(\phi (p)=0\) and \(d\phi (p)=0\). Taking derivatives w.r.t. z and u one gets:
where the expression “l.o.t.” stands for terms vanishing at \(Z=p\). Evaluating at \(Z=p\) one gets that \(\sum _j {a_j}_{Z_r}(p) L^{\beta _m} {\rho '_k}_{Z_j}(H(Z),\overline{H(Z)})_{Z=p} = 0\) for \(1\le k \le d'\). Thus, since H is finitely nondegenerate at p, there are multiindices \(\gamma _m\in {{\,\mathrm{{\mathbb {N}}}\,}}^n\) and integers \(k_m \in {{\,\mathrm{{\mathbb {N}}}\,}}\) with \(1\le k_m\le d'\), such that the matrix \((L^{\gamma _m} {\rho '_{k_m}}_{Z_j}(H(Z),\overline{H(Z)})_{Z=p})_{1\le j,m\le N'}\) is of full rank, which implies that \({a_j }_{Z_\ell }(p) = 0\). Proceeding inductively shows that all derivatives of \(a_j(Z)\) have to vanish at p. This means that the holomorphic vector field X vanishes in a neighborhood of p on the generic submanifold M, hence \(X \equiv 0\), which implies that H is holomorphically nondegenerate.
To show (c) let \(\{X_1,\ldots , X_s\}\) be a basis of the space of holomorphic vector fields at \(p\in M\) tangent to \(M'\) along the image of H and take \(X\in {{\,\mathrm{span}\,}}_{{{\,\mathrm{{\mathbb {C}}}\,}}}\{X_1,\ldots , X_s\}\). Consider the following equation for \(1\le k \le d'\):
Take derivatives w.r.t. L and since X is holomorphic, it holds that,
for any multiindex \(\beta \in {{\,\mathrm{{\mathbb {N}}}\,}}^n\). This implies that for any sequence \(\beta _1,\ldots , \beta _{N'}\in {{\,\mathrm{{\mathbb {N}}}\,}}^n\) of multiindices and integers \(k_m \in {{\,\mathrm{{\mathbb {N}}}\,}}\) with \(1\le k_m\le d'\) the vector field X belongs to the kernel of the matrix \((L^{\beta _m} {\rho '_{k_m}}_{Z}(H(Z),\overline{H(Z)}))_{1\le m \le N'}\). Hence outside a proper realanalytic variety Y of a neighborhood of p it holds that for any \(\ell \in {{\,\mathrm{{\mathbb {N}}}\,}}\) one has \(\dim _{{{\,\mathrm{{\mathbb {C}}}\,}}} E_\ell '(q) = N's\) for \(q \in Y\), such that the degeneracy of H is equal to s for all \(q\in Y\). \(\square \)
Infinitesimal Automorphisms of the Unit Sphere
In the following the wellknown infinitesimal automorphisms of \(S^{2n1}, n \ge 2\) are listed for later reference. For \(A=(A_1, \ldots , A_n) \in {{\,\mathrm{\mathfrak {hol}}\,}}(S^{2n1})\) the jth component is given as follows:
where \(\alpha _m, \beta _m^{\ell } \in {{\,\mathrm{{\mathbb {C}}}\,}}\) and \(s_m \in {{\,\mathrm{{\mathbb {R}}}\,}}\) and \(\dim _{{{\,\mathrm{{\mathbb {R}}}\,}}} {{\,\mathrm{\mathfrak {hol}}\,}}(S^{2n1})= n(n+2)\). The following notation is required:
For a map \(H: S^{2k1} \rightarrow S^{2m1}\) any \(T \in {{\,\mathrm{\mathfrak {aut}}\,}}(H)\) can be written as \(T = T_1 + \ldots + T_4\), such that the \(T_j\) are given as follows:
where \(\alpha ' = (\alpha '_1,\ldots , \alpha '_{m}) \in {{\,\mathrm{{\mathbb {C}}}\,}}^{m}\), \(V'\in {{\,\mathrm{{\mathbb {C}}}\,}}^{m}\times {{\,\mathrm{{\mathbb {C}}}\,}}^{m}\) is a matrix satisfying , \(\alpha = (\alpha _1,\ldots , \alpha _k) \in {{\,\mathrm{{\mathbb {C}}}\,}}^k\) and \(z=(z_1,\ldots ,z_k) \in {{\,\mathrm{{\mathbb {C}}}\,}}^k\).
The Reflection Matrix
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 Xvariety, see also [8] for the case of hyperquadric maps): Denote by \({\mathcal {H}}(n,d)\) the complex vector space of homogeneous polynomials of degree d in n holomorphic variables \(z = (z_1, \ldots , z_n)\). Write \(\bar{{\mathcal {H}}}(n,d)\) for the complex vector space with basis consisting of homogeneous polynomials of degree d in n antiholomorphic variables \({\bar{z}} = ({\bar{z}}_1, \ldots , {\bar{z}}_n)\).
Definition 4
Let \(H = \frac{P}{Q}: U \subset {{\,\mathrm{{\mathbb {C}}}\,}}^n \rightarrow {{\,\mathrm{{\mathbb {C}}}\,}}^m\) be a rational map of degree d (not necessarily a sphere map), where \(P=(P_1, \ldots , P_m)\) and \(Q: {{\,\mathrm{{\mathbb {C}}}\,}}^n \rightarrow {{\,\mathrm{{\mathbb {C}}}\,}}\) with \(Q \ne 0\) on U. Write \(H = \frac{1}{Q} \sum _{k=0}^d P^k\), where \(P^k\) is homogeneous of order k. Define the reflection map \(C_H: {{\,\mathrm{{\mathbb {C}}}\,}}^m \rightarrow \bar{{\mathcal {H}}}(n,d)\) of H by
Since \(C_H\) is linear and \(C_H(X) \in \bar{{\mathcal {H}}}(n,d)\) there exists a matrix \(V_H: {{\,\mathrm{{\mathbb {C}}}\,}}^m \rightarrow {{\,\mathrm{{\mathbb {C}}}\,}}^{K(n,d)}\) with holomorphic entries, such that
and denote \(V {:}{=}Q V_H\). The matrix V is referred to as reflection matrix of H.
Several properties of the reflection matrix and examples involving V are given in [7].
Examples and Constructions for Sphere Maps
In this section some particular examples of sphere maps and constructions of sphere maps are presented and their relation to the above nondegeneracy conditions are discussed.
The Homogeneous Sphere Maps
The purpose of this section is to show some properties of the homogeneous sphere maps defined as follows:
Definition 5
For \(d\ge 1\) and \(n\ge 2\) define \(K(n,d)=\left( {\begin{array}{c}n+d1\\ d\end{array}}\right) \) and I(n, d) as the set of all multiindices \(\alpha \in {{\,\mathrm{{\mathbb {N}}}\,}}^n\) of length d equipped with the lexicographic order. Define the homogeneous sphere map \(H_n^d\) in n variables of degree d as
A direct computation or [13, Theorem 4.2] shows that the infinitesimal stabilizer of \(H^d_n\) is given by \(S_n^2\) and \(S_n^3\). One can show that \(H^d_n\) is holomorphically nondegenerate by using the Fourier coefficient technique as in [3, Lemma 16]. Instead of showing this, it is proved that \(H_n^d\) is finitely nondegenerate on \(S^{2n1}\). Before giving a proof of this fact some preparations are needed:
For \(n\ge 3\) we define CR vector fields of \(S^{2n1}\) by \({\bar{L}}_{ij} = z_i \frac{\partial }{\partial {\bar{z}}_j}  z_j \frac{\partial }{\partial {\bar{z}}_i}\) for \(1 \le i \ne j \le n\). For \(n=2\) the CR vector field of \(S^{3}\) is given by \({\bar{L}} = z\frac{\partial }{\partial {\bar{w}}}w\frac{\partial }{\partial {\bar{z}}}\).
Let \(\{X_{ij}: 1\le i,j\le n\}\) be a collection of vector fields. In order to denote powers of such vector fields the following notation is used: Define the set \(J {:}{=}\{\alpha =(\alpha _1,\ldots , \alpha _n)\in {{\,\mathrm{{\mathbb {N}}}\,}}^{3n}: \alpha _j=(\alpha _j^1,\alpha _j^2,\alpha _j^3) \in {{\,\mathrm{{\mathbb {N}}}\,}}^3\}\) and for \(\alpha \in J\) write \(X^\alpha {:}{=}X_{\alpha _1^1\alpha _1^2}^{\alpha _1^3} \cdots X_{\alpha _n^1\alpha _n^2}^{\alpha _n^3}\). Define \(\alpha  = \sum _{j=1}^n \alpha _j^3\) for \(n\ge 3\).
For two vector fields X and Y, their Lie bracket is denoted by \([X,Y] {:}{=}X(Y)Y(X)\).
In the following lemma some basic facts about CR vector fields and their commutators are given. The proofs consist of straight forward calculations and are omitted.
Lemma 1
Assume \(n\ge 3\). In the following for \(1\le i,j,k,\ell \le n\) assume \(i\ne j\) and \(k,\ell \not \in \{i,j\}\). Define the following vector fields for \(S^{2n1}\):
Then \(T_{jk} =  {\bar{T}}_{kj}\) and \(S_{ij} = S_{ji}\) and the following commutator relations hold:

(a)
\([T_{k\ell },{\bar{L}}_{ij}] = [T_{kj},{\bar{L}}_{ij}] = [T_{ki},{\bar{L}}_{ij}] = [T_{ij},{\bar{L}}_{ij}] = [T_{ji},{\bar{L}}_{ij}] = 0\), \([T_{jk},{\bar{L}}_{ij}]= {\bar{L}}_{ik}, [T_{ki},{\bar{L}}_{kj}] = {\bar{L}}_{i j}\).

(b)
\([S_{k\ell },{\bar{L}}_{ij}] = 0\), \([S_{kj},{\bar{L}}_{ij}] =[S_{ki},{\bar{L}}_{ij}] =  {\bar{L}}_{ij}, [S_{ij},{\bar{L}}_{ij}] =  2 {\bar{L}}_{ij}\).

(c)
\([T_{k\ell }, L_{ij}] = [T_{jk}, L_{ij}] = [T_{ki},{\bar{L}}_{kj}] = [T_{ij}, L_{ij}] = [T_{ji}, L_{ij}] = 0\), \([T_{kj}, L_{ij}] = L_{ki}, [T_{ki}, L_{ij}] = L_{jk}\).

(d)
\([S_{k\ell },L_{ij}] = 0\), \([S_{kj},L_{ij}] =[S_{ki},L_{ij}] = L_{ij}\), \([S_{ij}, L_{ij}] = 2 L_{ij}\).

(e)
\([S_{ij},T_{k\ell }] = [S_{ij},T_{ij}] = 0\), \([S_{ij},T_{j\ell }] = T_{j\ell }\), \([S_{ij},T_{i\ell }] = T_{i\ell }\).
For \(n=2\) define the following vector field for \(S^{3}\):
Then it holds that \([S,{\bar{L}}] = 2 {\bar{L}}\).
Lemma 2
The map \(H^d_n: S^{2n1} \rightarrow S^{2K(n,d)1}\) is dnondegenerate at each point of \(S^{2n1}\).
Proof
Set \(H{:}{=}H^d_n\), fix \(1\le m \le n\) and define the following set of multiindices
where the index set J from the beginning of Sect. 3.1 is used. It will be shown that the \(K(n,d) \times K(n,d)\)matrix
is of full rank if \(z_m \ne 0\). This implies that H is dnondegenerate at each point of \(S^{2n1}\). The proof consists of two steps:
 (A):

It is proved that
$$\begin{aligned} L^\alpha H \cdot {\bar{L}}^\beta T^\gamma S^\delta {\bar{H}} = 0, \end{aligned}$$(3)on \(S^{2n1}\), for all multiindices \(\alpha ,\beta \in J_m\) with \(\alpha <\beta \) and all \(\gamma \in J,\delta \in J_m\), where J is defined in the beginning of Sect. 3.1.
 (B):

It is shown that for each \(0\le k \le d\) the set \(D_m^k{:}{=}\{L^\alpha H_{S^{2n1}}: \alpha \in J_m, \alpha =k\}\) consists of linearly independent vectors in \({{\,\mathrm{{\mathbb {C}}}\,}}^{K(n,d)}\) if \(z_m\ne 0\).
In order to show (A) one proceeds by induction on the length of \(\alpha \) in (3): For \(\alpha =0\) one needs to argue as follows. Since \(H \cdot {\bar{H}} = 1\) on \(S^{2n1}\), it follows that \(p_{H,\beta }{:}{=}H \cdot {\bar{L}}^\beta {\bar{H}} = 0\) on \(S^{2n1}\) for all \(\beta \in J_m\). This means that \(p_{H,\beta }\) is a homogeneous polynomial vanishing on \(S^{2n1}\), hence \(p_{H,\beta }\) vanishes in \({{\,\mathrm{{\mathbb {C}}}\,}}^n\), see [5, Sect. II] or [6, Sect. 5.1.4]. Applying \({\bar{z}}_k\frac{\partial }{{\bar{z}}_j}\) to \(p_{H,\beta } \equiv 0\), implies that \(H \cdot {\bar{L}}^\beta T^\gamma S^\delta {\bar{H}}= 0\) for all \(\gamma \in J\) and \(\delta \in J_m\).
Assume that (3) holds for \(\alpha =k\) and \(\alpha +1 < \beta \). If one applies \(L_{mj}\) to (3) one obtains:
on \(S^{2n1}\). If one can show that
the induction is completed and (A) is proved. In order to show (4), use the identities from Lemma 1, which imply that the expression of the lefthand side of (4) can be rewritten as a sum of terms of the form \(L^\alpha H \cdot {\bar{L}}^{\beta '} T^{\gamma '} S^{\delta '} L_{mj} {\bar{H}}\) and \(L^\alpha H \cdot {\bar{L}}^{\beta '} T^{\gamma '} S^{\delta '} {\bar{H}}\), where \(\beta '\ge \beta 1, \gamma '\in J\) and \(\delta '\in J_m\). Hence using the induction hypothesis proves (4).
To prove (B) fix \(0\le k \le d\) and assume that the set \(D_m^k\) consists of vectors which are not linearly independent. By setting \(K{:}{=}\{\alpha \in J_m: \alpha =k\}\), the linear dependence says that there are \(c_\alpha \in {{\,\mathrm{{\mathbb {C}}}\,}}\), not all of them are zero, such that
on \(S^{2n1}\). Since the lefthand side is a homogeneous polynomial, (5) holds in \({{\,\mathrm{{\mathbb {C}}}\,}}^n\). For \(\alpha \in K\) define the multiindex
such that \(z^{r(\alpha )}{:}{=}z_1^{k_1} \dots z_{m1}^{k_{m1}} z_{m+1}^{k_{m+1}} \cdots z_n^{k_n}\). Note that \(L^\alpha = {\bar{z}}_m^k \frac{\partial ^k}{\partial z^{r(\alpha )}} + \ldots \), where \(\ldots \) stands for derivatives of order k with coefficients being monomials of degree k containing \({\bar{z}}_m^\ell \) for \(\ell < k\). Define \(R_\alpha {:}{=}\frac{\partial ^k}{\partial {\bar{z}}_m^k} L^\alpha \). Assume \(z_m\ne 0\). Expanding (5) as a power series in \({\bar{z}}\) with vectorvalued coefficients, the coefficient of \({\bar{z}}_m^k\) is given by
Note that in the vector \(R_\alpha H\in {{\,\mathrm{{\mathbb {C}}}\,}}^{K(n,d)}\) each monomial of \(H^{dk}\) appears in exactly one component and it is of the following form:
where \(1\le j_1(\alpha )\le K(n,d)\) and \(m^1_{t(\alpha ),j_1(\alpha )} = c_{t(\alpha )} z^{t(\alpha )} \ne 0\) such that \(t(\alpha ) \in {{\,\mathrm{{\mathbb {N}}}\,}}^n\) with \(t(\alpha ) = K(n,dk)\).
Consider the minimal \(\alpha _0 \in K\) w.r.t. the lexicographic order. If \(m=1\), then \(\alpha _0 = (2,\ldots , 2)\), and otherwise \(\alpha _0 = (1,\ldots , 1)\). This implies that \(j_1(\alpha )\ge j_1(\alpha _0)\) for all \(\alpha >\alpha _0\). Moreover \(t(\alpha )> t(\alpha _0)\) for all \(\alpha >\alpha _0\), which can be seen as follows: Denote the monomial in \(H^d_n\) at the kth position by \(h_{s(k),k}=a_k z^{s(k)}\), where \(s(k) \in {{\,\mathrm{{\mathbb {N}}}\,}}^n\) with \(s(k) = K(n,d)\). Note that \(s(j_1(\alpha )) = \alpha + t(\alpha )\) and for \(\alpha >\alpha _0\) if \(j_1(\alpha _0)\le j_1(\alpha )\), then \(s(j_1(\alpha _0)) \le s(j_1(\alpha ))\). Hence
which implies that \(t(\alpha _0) < t(\alpha )\). Thus, in (6), considering the coefficient of \(z^{t(\alpha _0)}\) shows that \(c_{\alpha _0}=0\).
Proceed inductively and assume \(c_\alpha = 0\) for all \(\alpha <\alpha _1, \alpha \in K\). Define \(K_1 = K\setminus \{\alpha \in K: \alpha < \alpha _1\}\). Argue as above, since \(\alpha _1\) is the minimal index in \(K_1\) it holds that \(j_1(\alpha )\ge j_1(\alpha _1)\) for all \(\alpha >\alpha _1\) and the same argument as above shows that \(t(\alpha )>t(\alpha _1)\). Thus \(c_{\alpha _1}=0\), which proves (B) and completes the proof. \(\square \)
The Group Invariant Sphere Maps
Another important class of sphere maps are the following, first introduced in [3]:
Definition 6
Define \(G^\ell : S^{3} \rightarrow S^{2\ell +3}\) for \(\ell \ge 0\) by
where \(c_k^{\ell }\ge 0\) for \(1\le k \le \ell +2\) is given in [3] or [6, Sect. 5.2.2, Theorem 9].
The infinitesimal stabilizer of \(G^\ell \) consists of the vector field \(S^3_2\). The maps \(G^\ell \) are invariant under a fixedpointfree finite unitary group and appear in [12] as socalled sharp polynomials in the study of degree bounds for monomial maps.
The Tensor Product for Infinitesimal Deformations
Similar to the case of sphere maps ( [3, 5, Definition 4]) one can introduce a tensor operation for infinitesimal deformations.
Let \(A \subseteq {{\,\mathrm{{\mathbb {C}}}\,}}^n\) be a linear subspace such that \({{\,\mathrm{{\mathbb {C}}}\,}}^n = A \oplus A^{\perp }\) is an orthogonal decomposition. For \(v\in {{\,\mathrm{{\mathbb {C}}}\,}}^n\) write \(v = v_A \oplus v_{A^\perp }\in A \oplus A^{\perp }\). Similarly one can decompose the image of a map \(F:S^{2n1}\rightarrow S^{2m1}\) w.r.t. A and write \(F = F_A \oplus F_{A^{\perp }} \in A \oplus A^{\perp }\).
For vectors \(v=(v_1,\ldots , v_n) \in {{\,\mathrm{{\mathbb {C}}}\,}}^{n}\) and \(w=(w_1,\ldots , w_m) \in {{\,\mathrm{{\mathbb {C}}}\,}}^{m}\) the usual tensor product of v and w is denoted by
Definition 7
Let \(H: S^{2n1} \rightarrow S^{2m1}\) and \(G: S^{2n1} \rightarrow S^{2\ell 1}\) be CR maps, \(X \in {{\,\mathrm{\mathfrak {hol}}\,}}(H)\) and \(A \subseteq {{\,\mathrm{{\mathbb {C}}}\,}}^m\) be a linear subspace, then
is called the tensor product of X by G on A.
We recall that the tensor product of mappings of spheres was introduced in [3] and [5, Definition 4]: For \(f: S^{2n1} \rightarrow S^{2m1}\) and \(g: S^{2n1} \rightarrow S^{2\ell 1}\) CR maps and \(A \subseteq {{\,\mathrm{{\mathbb {C}}}\,}}^m\) a linear subspace the tensor product of f by g on A given by \(E_{(A,g)} f = (f_A \otimes g) \oplus f_{A^{\perp }}\) is a mapping of spheres, see [5, Lemma 5]. An analogous result holds for infinitesimal deformations:
Lemma 3
Let \(H: S^{2n1} \rightarrow S^{2m1}\) and \(G: S^{2n1} \rightarrow S^{2\ell 1}\) be CR maps, \(X\in {{\,\mathrm{\mathfrak {hol}}\,}}(H)\) and \(A \subseteq {{\,\mathrm{{\mathbb {C}}}\,}}^m\) be a linear subspace. Then \(T_{(A,G)} X \in {{\,\mathrm{\mathfrak {hol}}\,}}(E_{(A,G)} H)\).
Proof
Set \(Y = T_{(A,G)} X\) and \(F = E_{(A,G)} H\). By orthogonality it holds that on \(S^{2n1}\),
since \(\Vert G \Vert ^2 = 1\) on \(S^{2n1}\), hence \(Y \in {{\,\mathrm{\mathfrak {hol}}\,}}(F)\). \(\square \)
The next result shows that holomorphic degeneracy is preserved by tensoring.
Lemma 4
Let \(H: S^{2n1} \rightarrow S^{2m1}\) be a CR map, \(A \subseteq {{\,\mathrm{{\mathbb {C}}}\,}}^m\) a linear subspace and \(G: S^{2n1}\rightarrow S^{2\ell 1}\) a CR map. If H is holomorphically degenerate then \(E_{(A,G)} H\) is holomorphically degenerate.
Proof
Since H is holomorphically degenerate there exists a nontrivial holomorphic map \(W: {{\,\mathrm{{\mathbb {C}}}\,}}^n \rightarrow {{\,\mathrm{{\mathbb {C}}}\,}}^m\) such that \(W \cdot {\bar{H}} =0\) on \(S^{2n1}\). Write \(H' = E_{(A,G)}H\) and consider \(W'=T_{(A,G)} W\), which is a nontrivial holomorphic vector. Then the same computation (without taking the real part) as in the proof of Lemma 3 shows that \(W' \cdot {\bar{H}}'=0\) on \(S^{2n1}\), i.e., \(H'\) is holomorphically degenerate. \(\square \)
Example 2
The converse of Lemma 4 is not true in general: Consider the holomorphically nondegenerate maps \(H:S^{3}\rightarrow S^{7}, H(z,w)=(z, z w, z^2 w, w^3)\) and \(G:S^{3}\rightarrow S^{5}, G(z,w)=(z,zw,w^2)\). Tensoring H at the first component with G one obtains the holomorphically degenerate map \(F'(z,w) = (z^2, z^2 w, z w^2, z w, z^2 w, w^3)\). Moreover, if one applies a unitary change of coordinates and a projection \({{\,\mathrm{{\mathbb {C}}}\,}}^6 \rightarrow {{\,\mathrm{{\mathbb {C}}}\,}}^5\) to \(F'\), one obtains \(F(z,w) = (z^2, z w, \sqrt{2} z^2 w, z w^2, w^3)\), which still is holomorphically degenerate: the holomorphic vector \(X=(0,1,z /\sqrt{2}, w ,0)\) satisfies \(X \cdot {\bar{F}} = 0\) on \(S^{3}\).
Example 3
For a sphere map H its trivial infinitesimal deformations may give rise to nontrivial infinitesimal deformations of tensors of H: Let H be the map
and A be the complex subspace spanned by \((1,0\ldots , 0)\in {{\,\mathrm{{\mathbb {C}}}\,}}^4\). Write \(H=(H_1,\ldots , H_4)\) and define \(X = (a,0,\ldots , 0)  ({\bar{a}} H_1) H \in {{\,\mathrm{\mathfrak {aut}}\,}}(H), a \in {{\,\mathrm{{\mathbb {C}}}\,}}\setminus \{0\}\). Then it is straightforward to show that \(T_{(A,H^1)} X \not \in {{\,\mathrm{\mathfrak {aut}}\,}}(E_{(A,H^1)} H)\).
Example 4
Following [11, Proposition 2.4] one can combine two sphere maps to construct a new sphere map into a higher dimensional sphere: For \(H: S^{2n1} \rightarrow S^{2m1}\) and \(G: S^{2n1} \rightarrow S^{2\ell 1}\) realanalytic CR maps define the juxtaposition of H and G with parameter \(t\in [0,1]\) as the map \(j_t(H,G):S^{2n1}\rightarrow S^{2(m+\ell )1}\) given by
Note that \(j_t(H,G)\) is holomorphically degenerate: The holomorphic vector field \(X = t H \oplus \sqrt{1t^2} G\) satisfies \(X \cdot {\bar{H}} = 0\).
Nondegeneracy Conditions for Sphere Maps
In this section it is shown that holomorphic and finite degeneracy can be expressed in terms of rank conditions of the reflection matrix.
Holomorphic nondegeneracy of a sphere map is equivalent to a generic rank condition of V:
Proposition 3
Let \(H: S^{2n1} \rightarrow S^{2m1}\) be a rational map of degree d. Then the following statements are equivalent:

(a)
H is holomorphically nondegenerate.

(b)
There is no nontrivial holomorphic map \(Y: U \rightarrow {{\,\mathrm{{\mathbb {C}}}\,}}^m\), where U is a neighborhood of \(S^{2n1}\), such that \(VY = 0\) on \(S^{2n1}\).

(c)
The matrix V is generically of rank m on \(S^{2n1}\).
Proof
For a map \(H = \frac{P}{Q}\) the following equation holds on \(S^{2n1}\):
for any vector \(X\in {{\,\mathrm{{\mathbb {C}}}\,}}^m\). Let \(Y: U \rightarrow {{\,\mathrm{{\mathbb {C}}}\,}}^m\), where U is a neighborhood of \(S^{2n1}\), be a nontrivial holomorphic map such that \(Y \cdot {\bar{H}} = 0\) on \(S^{2n1}\). By (7), this is equivalent to \(VY \cdot {\bar{H}}^d_n = 0\) on \(S^{2n1}\). Since \(H^d_n\) is holomorphically nondegenerate on \(S^{2n1}\), the last equation is equivalent to \(VY = 0\) on \(S^{2n1}\). From this consideration the equivalence of (a) and (b) follows.
The equivalence of (b) and (c) holds, since (b) is equivalent to the fact that V is injective on an open, dense subset of \(S^{2n1}\), which is equivalent to (c). \(\square \)
The following proposition shows that finite nondegeneracy of a map H is equivalent to a pointwise rank condition of \(V_H\):
Proposition 4
Let \(H: S^{2n1} \rightarrow S^{2m1}\) be a rational map of degree d. Then the following statements are equivalent:

(a)
H is of degeneracy s at \(p \in S^{2n1}\).

(b)
The kernel of the matrix V is of dimension s at \(p \in S^{2n1}\).
In particular, the map H is finitely nondegenerate at \(p \in S^{2n1}\) if and only if the matrix V is of rank m at \(p \in S^{2n1}\).
Theorem 1 follows from Propositions 3 and 4.
Proof
Since \(X \cdot {\bar{H}} = V_H X \cdot {\bar{H}}_n^d/{\bar{Q}}\) on \(S^{2n1}\) for any \(X\in {{\,\mathrm{{\mathbb {C}}}\,}}^m\) and V is holomorphic, it follows that
on \(S^{2n1}\). For any sequence of multiindices \(\alpha =(\alpha _1,\ldots , \alpha _\ell ), \alpha _j \in {{\,\mathrm{{\mathbb {N}}}\,}}^{n}\) and a sphere map \(F:S^{2n1} \rightarrow S^{2k1}\) define the \(\ell \times k\)matrix \(A^\alpha _q({\bar{F}}) {:}{=}\left( {\bar{L}}^{\alpha _j}{\bar{F}}_q\right) _{1\le j \le \ell }\) for \(q\in S^{2n1}\). Then it holds that
for any \(p\in S^{2n1}\), any sequence of multiindices \(\beta =(\beta _1,\ldots , \beta _r), \beta _j \in {{\,\mathrm{{\mathbb {N}}}\,}}^n\) and \(r\in {{\,\mathrm{{\mathbb {N}}}\,}}\).
Observe that \({{\,\mathrm{rk}\,}}A^\gamma _p({\bar{H}}^d_n/{\bar{Q}}) = {{\,\mathrm{rk}\,}}A^\gamma _p({\bar{H}}^d_n)\), for multiindices \(\gamma =(\gamma _1,\ldots ,\gamma _{K(n,d)}), \gamma _j\in {{\,\mathrm{{\mathbb {N}}}\,}}^n\), chosen according to the finite nondegeneracy of \(H^d_n\) given in the proof of Lemma 2, which can be seen as follows: One has that
where \(c_\varepsilon ^{\gamma _j} \in {{\,\mathrm{{\mathbb {C}}}\,}}\) involves some constants, derivatives of \({\bar{Q}}\) and terms of the form \({\bar{Q}}^{m_{\gamma _j,\varepsilon }}\) for some \(m_{\gamma _j,\varepsilon } \in {{\,\mathrm{{\mathbb {N}}}\,}}\). Since the first row of \(A^\gamma _p({\bar{H}}^d_n /{\bar{Q}})\) consists of \({\bar{H}}^d_n/{\bar{Q}}\), using elementary row operations one can see that the rank of \(A^\gamma _p({\bar{H}}^d_n/{\bar{Q}})\) agrees with the rank of \(A^\gamma _p ({\bar{H}}^d_n)\).
Now it is possible to prove the equivalence of (a) and (b). Assume (a), then there is \(k_0\in {{\,\mathrm{{\mathbb {N}}}\,}}\), such that H is \((k_0,s)\)degenerate at p. This means that \(\dim _{{{\,\mathrm{{\mathbb {C}}}\,}}} E_{k_0}'(p) = N's\) and hence, for any \(\beta =(\beta _1,\ldots , \beta _r), \beta _j\in {{\,\mathrm{{\mathbb {N}}}\,}}^n\) and \(r\in {{\,\mathrm{{\mathbb {N}}}\,}}\), the kernel of the matrix \(A^\beta _p({\bar{H}})\) is at least of dimension s.
Consider \(\gamma =(\gamma _1,\ldots ,\gamma _{K(n,d)}), \gamma _j\in {{\,\mathrm{{\mathbb {N}}}\,}}^n\) according to the finite nondegeneracy of \(H^d_n\) given in the proof of Lemma 2. Let \(X_j\) for \(1\le j \le s\) be linearly independent vectors in the kernel of \(A^\gamma _p({\bar{H}})\). By taking \(\beta =\gamma \) in (8), it follows that \(V(p) X_j \in \ker A^\gamma _p({\bar{H}}^d_n/{\bar{Q}})\). Since \(H_n^d\) is finitely nondegenerate in \(S^{2n1}\) and by the fact that \({{\,\mathrm{rk}\,}}A^\gamma _p({\bar{H}}^d_n/{\bar{Q}}) = {{\,\mathrm{rk}\,}}A^\gamma _p({\bar{H}}^d_n)\), it follows that \(X_j \in \ker V(p)\). Hence \(\dim \ker V(p) \ge s\).
Assume that \(\dim \ker V(p) = s'>s\), i.e., there are linearly independent vectors \(Y_j \in \ker V\) for \(1\le j \le s'\). Since H is of degeneracy s there exists a sequence of multiindices \(\delta =(\delta _1,\ldots , \delta _q), \delta _j \in {{\,\mathrm{{\mathbb {N}}}\,}}^n\) and \(q\in {{\,\mathrm{{\mathbb {N}}}\,}}\), such that the kernel of \(A^\delta _p({\bar{H}})\) is precisely of dimension s. Then \(A^\delta _p({\bar{H}}^d_n/{\bar{Q}}) V(p) Y_j = 0\) and using (8) with \(\beta =\delta \), it follows that \(A^\delta _p({\bar{H}})Y_j = 0\), i.e., \(Y_j \in \ker A^\delta _p({\bar{H}})\) for \(1\le j \le s'\), which is a contradiction to \(\dim \ker A^\delta _p({\bar{H}}) = s\).
For the other direction, assume (b) and argue similarly: If \(\dim \ker V(p) = s\), consider any sequence of multiindices \(\epsilon =(\epsilon _1, \ldots , \epsilon _t)\) for \(\epsilon _j \in {{\,\mathrm{{\mathbb {N}}}\,}}^n\) and \(t\in {{\,\mathrm{{\mathbb {N}}}\,}}\). Let \(X_j\) for \(1\le j \le s\) be linearly independent vectors belonging to \(\ker V(p)\). By (8) it follows that \(X_j \in \ker A^\epsilon _p({\bar{H}})\). Thus, the degeneracy of H is at least s.
Assume the degeneracy of H is equal to \(s'>s\). Argue as in the proof of the sufficient direction to conclude that \(\dim \ker V(p) \ge s\), which is a contradiction.
The last statement follows immediately from the above shown equivalence. \(\square \)
Example 5
For each \(\ell \ge 0\) the map \(G^\ell \) is finitely nondegenerate at \(p \in S^{3}\): In this case the reflection matrix V is the following \((2\ell +2) \times (\ell +2)\)matrix:
where , blank spaces are filled up with zeros, \(D_1\) is the \((2\ell +2)\times (2\ell +2)\)diagonal matrix whose nonzero entries are the reciprocals of the coefficients of \(H^{2\ell +2}\) and \(D_2\) is the \((\ell +2) \times (\ell +2)\)diagonal matrix which consists of the coefficients of \(G^\ell \) on the diagonal. It follows that V is of full rank on \(S^{3}\).
Example 6
The map \(H(z,w)=(z^4,z^3 w,\sqrt{3} z w, w^3)\), which sends \(S^{3}\) into \(S^{7}\), is listed in [3]. The reflection matrix is given by
which is of full rank if and only if \(w\ne 0\) and if \(w=0\) the kernel is of dimension 1, hence by Proposition 4 the map is finitely nondegenerate for \(w\ne 0\) and of degeneracy 1 when \(w=0\). A direct computation (as in Definition 3) shows that the map is 3nondegenerate at points \(\{z\ne 0,w\ne 0\}\cap S^{3}\) and 4nondegenerate when \(\{z=0,w = 1\}\). If \(w=0\) and \(z=1\), the map is (3, 1)degenerate.
The following example gives a map, for which the set of points in \(S^{3}\) where the map is 2degenerate consists of one isolated point.
Example 7
The map \(H(z,w)= \bigl ((azb zw)z,(a z  b zw) w, {\bar{b}} z + {\bar{a}} z w, w^2\bigr )\), for \(a,b\in {{\,\mathrm{{\mathbb {C}}}\,}}\) satisfying \(a^2+b^2=1\), sends \(S^{3}\) into \(S^{7}\). The matrix V is given by
where D is the \(4\times 4\)diagonal matrix whose nonzero entries are the reciprocals of the coefficients of \(H^3\). Assuming \(a=b=\frac{1}{\sqrt{2}}\), it holds that V is of full rank if \(Y_1 {:}{=}\{z\ne 0,w\ne 0\} \cap S^{3}\) and the complex dimension of the kernel is 1 if \(Y_2{:}{=}\bigl \{\{z=0,w\ne 1\} \cup \{z\ne 0,w=0\} \bigr \}\cap S^{3}\). The kernel of V is of complex dimension 2 at \(p_0{:}{=}(0,1)\in S^{3}\). It can be shown by a direct computation (as in Definition 3) that H is 3nondegenerate in \(Y_1\), (2, 1)degenerate in \(Y_2\) and (1, 2)degenerate at \(p_0\).
The following result gives conditions to guarantee that a sphere map is finitely degenerate:
Corollary 1
If a rational sphere map \(H: S^{2n1} \rightarrow S^{2m1}\) of degree d satisfies \(K(n,d) < m\), then H is finitely degenerate at any \(p \in S^{2n1}\). In particular the map is holomorphically degenerate.
Proof
The \(K(n,d)\times m\)matrix V satisfies \({{\,\mathrm{rk}\,}}V \le \min (K(n,d),m)=K(n,d)\) on \(S^{2n1}\). If H would be finitely nondegenerate at \(p \in S^{2n1}\), by Proposition 4, V would be injective at p, hence \({{\,\mathrm{rk}\,}}V = m\) at p, a contradiction. By Proposition 3 it follows that H is holomorphically degenerate. \(\square \)
Example 8
The map \(H(z,w) = (z, \cos (t)w, \sin (t)zw, \sin (t) w^2), t\in [0,2\pi )\), sends \(S^{3}\) to \(S^{7}\) and is holomorphically degenerate by Corollary 1. The reflection matrix V is given as follows:
If \(\sin (t) \ne 0\), then \(X(z,w)=(0,1, \cot (t) z, \cot (t) w)\) is a holomorphic vector field tangent to \(S^{7}\) along the image of H. One can check that if \(\cos (t),\sin (t)\ne 0\) the map is of degeneracy 1 for \(z\ne 0\) and of degeneracy 2 if \(z=0\). If \(\cos (t) \ne 0\) and \(\sin (t) = 0\) the map is of degeneracy 2 and when \(\cos (t)=0\) the map is 1degenerate.
The set of points where the map is finitely degenerate can be described by using Proposition 4:
Corollary 2
Let \(H: S^{2k1} \rightarrow S^{2m1}\) be of generic degeneracy s in \(S^{2k1}\). The set of points in \(S^{2k1}\), where H is of degeneracy \(s'>s\) is contained in a complex algebraic variety intersecting \(S^{2k1}\).
Proof
The set D of points where H is of degeneracy \(s'> s\) is the complement Y of the set where H is of generic degeneracy s, which is given by the union of the zero sets of any minor of V of size strictly less than \({{\,\mathrm{rk}\,}}V\). Since V consists of holomorphic polynomial entries, Y is a complex algebraic variety and, by Proposition 4, agrees with D. \(\square \)
Note that Proposition 4 shows that Corollaries 1 and 2 are equivalent to [7, Corollary 4.4] and [7, Corollary 4.2] respectively.
The Xvariety of a Sphere Map
In this section sufficient and necessary conditions in terms of nondegeneracy conditions are provided to guarantee that the Xvariety of a sphere map satisfies certain properties, such as agreeing with the graph of the map or being an affine bundle.
First, the general definition of the Xvariety of a map is repeated for the reader’s convenience, see [19] and [7]:
Definition 8
Let \(M\subset {{\,\mathrm{{\mathbb {C}}}\,}}^N\) and \(M'\subset {{\,\mathrm{{\mathbb {C}}}\,}}^{N'}\) be realanalytic hypersurfaces and \(H: M \rightarrow M'\) be a realanalytic CR map. Let \(p\in M\) and \(p' = H(p)\). Assume \(M \cap U = \{Z \in U: \rho (Z,{\bar{Z}}) = 0\}\) and \(M'\cap U' = \{Z' \in U': \rho '(Z',{\bar{Z}}') = 0\}\), where \(U \subset {{\,\mathrm{{\mathbb {C}}}\,}}^N\) and \(U'\subset {{\,\mathrm{{\mathbb {C}}}\,}}^{N'}\) are neighborhoods of p and \(p'\) and \(\rho \) and \(\rho '\) are realanalytic defining functions for M and \(M'\) defined in U and \(U'\) respectively. Define the following set
which is called the Xvariety of H near \(p \in M\).
Since H maps M into \(M'\) it follows that \((Z,H(Z)) \in X_H\), i.e., the graph of H is contained in \(X_H\). In [7, Theorem 4.1] it is shown in the case when \(M \subset {{\,\mathrm{{\mathbb {C}}}\,}}^n\) and \(M'\subset {{\,\mathrm{{\mathbb {C}}}\,}}^m\) are unit spheres that for any \(z\ne 0\) it holds that \((z,z') \in X_H\) if and only if \(z'H(z) \in \ker V(z)\). \(X_H\) has an exceptional fiber at \(p\in S^{2n1}\) if the dimension of the fiber \(\{p' \in {{\,\mathrm{{\mathbb {C}}}\,}}^{N'}: (p,p')\in X_H\}\) exceeds its generic value. In [7, Corollary 4.2] it is argued that the set of points over which \(X_H\) has an exceptional fiber agrees with the set of points \(p\in S^{2n1}\) where the rank of V(p) drops.
Moreover, in [7, Theorem 4.1] the following properties of \(X_H\) are proved:

(a)
\(X_H\) is an affine bundle over \({{\,\mathrm{{\mathbb {C}}}\,}}^n \setminus \{0\}\) if and only if the rank of V(z) is constant for each \(z\ne 0\) in the domain of H.

(b)
\(X_H\) equals the graph of H if and only if, for each \(z\ne 0\) in the domain of H, the null space of V(z) is trivial.
Using the facts from Sect. 4, relating nondegeneracy conditions and rank conditions of the reflection matrix, the following characterizations hold:
Theorem 3
Let \(H: S^{2n1} \rightarrow S^{2m1}\) be a rational map. Then the following statements hold:

(a)
\(X_H\) is an affine bundle over \({{\,\mathrm{{\mathbb {C}}}\,}}^n\setminus \{0\}\) if and only if H is of finite degeneracy s at any point of \(S^{2n1}\).

(b)
\(X_H\) equals the graph of H if and only if H is finitely nondegenerate at any point of \(S^{2n1}\).

(c)
\(X_H\) has an exceptional fiber at \(p\in S^{2n1}\) if and only if H is not of generic degeneracy s(H) at \(p\in S^{2n1}\).
Proof
The proofs of (a) and (b) follow from Proposition 4 and the characterizations from [7, Theorem 4.1] stated above. Since the rank conditions involved are constant on \(S^{2n1}\), they also hold in a neighborhood of \(S^{2n1}\), to which H extends. For (c) note that the points where H is of generic degeneracy s(H) (see the remark after Definition 3) form an open dense subset S of \(S^{2n1}\). Hence in the complement of S the degeneracy of H is strictly bigger and by Proposition 4 the rank of the reflection matrix is strictly smaller. Thus, the complement of S is precisely the set where \(X_H\) possesses an exceptional fiber. \(\square \)
Infinitesimal Deformations of Sphere Maps
In this section infinitesimal deformations of rational sphere maps are studied. It turns out that similarly as in the case of sphere maps, where each sphere map is related to the homogeneous sphere map by tensoring, infinitesimal deformations of a sphere map are related to infinitesimal deformations of the homogeneous sphere map by the reflection matrix.
Lemma 5
Let \(H = \frac{P}{Q}: U \rightarrow S^{2m1}\) be a holomorphically nondegenerate rational sphere map of degree d, where U is a neighborhood of \(S^{2k1}\). Then each \(X \in {{\,\mathrm{\mathfrak {hol}}\,}}(H)\) is of the form \(X = \frac{X'}{Q}\), where \(X'\) is a holomorphic polynomial of degree at most 2d satisfying \({{\,\mathrm{Re}\,}}(X' \cdot {\bar{P}}) = 0\) on \(S^{2k1}\).
Proof
Let H be given as in the assumption of the lemma, where \(P=(P_1, \ldots , P_m)\) and \(Q: U \rightarrow {{\,\mathrm{{\mathbb {C}}}\,}}\) with \(Q \ne 0\) on U, a neighborhood of \(S^{2k1}\). Then \(Q^2{{\,\mathrm{Re}\,}}(X \cdot {\bar{H}}) = {{\,\mathrm{Re}\,}}(Q X \cdot {\bar{P}})\). Set \(X' {:}{=}Q X\). Considering homogeneous expansions of \(X' = \sum _{\ell \ge 0} {X'}^\ell \) and \(P = \sum _{j=0}^d P^j\) one obtains the following equation:
After setting \(Z\mapsto Z e^{i t}\) for \(t\in {{\,\mathrm{{\mathbb {R}}}\,}}\) collect the Fourier coefficient of degree \(d+\ell _0\) for \(\ell _0 \ge 1\) to get:
By the holomorphic nondegeneracy of H this implies that \(X'^{\ell } \equiv 0\) for \(\ell \ge 2 d+1\), i.e., \(\deg X' \le 2 d\). \(\square \)
Denote by \({\mathcal {P}}^d(k,m)\) the space of complex polynomial maps from \({{\,\mathrm{{\mathbb {C}}}\,}}^k\) to \({{\,\mathrm{{\mathbb {C}}}\,}}^m\) of degree d with \(\dim _{{{\,\mathrm{{\mathbb {R}}}\,}}} {\mathcal {P}}^d(k,m) = 2 m \sum _{\ell =0}^d\left( {\begin{array}{c}\ell + k 1\\ \ell \end{array}}\right) \). The following definition is justified by the previous Lemma 5 and in fact \({{\,\mathrm{\mathfrak {hol}}\,}}(H) \) can be identified with a space of polynomial maps.
Definition 9
Let \(H = \frac{P}{Q}: U \rightarrow S^{2m1}\) be a holomorphically nondegenerate rational sphere map of degree d, where U is a neighborhood of \(S^{2k1}\). Define \(\dim {{\,\mathrm{\mathfrak {hol}}\,}}(H) {:}{=}\dim _{{{\,\mathrm{{\mathbb {R}}}\,}}} \{X' \in {\mathcal {P}}^{2d}(k,m): \frac{X'}{Q} \in {{\,\mathrm{\mathfrak {hol}}\,}}(H)\}\).
In [5] and [6, Sect. 5.1.4, Theorem 4] it is shown that for any polynomial sphere map of degree d if one applies finitely many tensoring operations to it one obtains the homogeneous sphere map of degree d. Moreover in [6, Sect. 5.1.4, Theorem 3] it is shown that the homogeneous sphere map is up to a unitary transformation unique among all polynomial and homogeneous sphere maps. The following theorem gives the corresponding results in terms of infinitesimal deformations.
Theorem 4
Let \(H: S^{2n1} \rightarrow S^{2m1}\) be a holomorphically nondegenerate rational map of degree d, then \(\dim {{\,\mathrm{\mathfrak {hol}}\,}}(H) \le \dim {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\).
If \(H: S^{2n1} \rightarrow S^{2m1}\) is a polynomial map of degree d, it holds that \(\dim {{\,\mathrm{\mathfrak {hol}}\,}}(H) = \dim {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) if and only if H is unitarily equivalent to \(H^d_n\).
Proof
Let \(H = \frac{P}{Q}: S^{2n1} \rightarrow S^{2m1}\) be a rational map with \(Q\ne 0\) on \(S^{2n1}\). Consider as in Definition 4 the matrix \(V: {{\,\mathrm{{\mathbb {C}}}\,}}^m \rightarrow {{\,\mathrm{{\mathbb {C}}}\,}}^{K(n,d)}\) whose entries are holomorphic polynomials in \(z\in {{\,\mathrm{{\mathbb {C}}}\,}}^n\). Then it holds on \(S^{2n1}\) that \(Q^2 X \cdot {\bar{H}} = VX \cdot {\bar{H}}^d_n\) for \(X\in {{\,\mathrm{{\mathbb {C}}}\,}}^m\) as in (7). Thus on \(S^{2n1}\) one obtains,
By Lemma 5 there are polynomials \(X_1', \ldots , X_k' \in {\mathcal {P}}^{2d}(n,m)\) such that \(\{X_j = \frac{X'_j}{Q}:1 \le j \le k\}\) is a basis of \({{\,\mathrm{\mathfrak {hol}}\,}}(H)\). From Proposition 3 it follows that \(\{VX_j: 1\le j \le k\}\) is a set of linearly independent polynomials in \({{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) which implies \(k \le \dim {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\).
To show the nontrivial implication of the second claim, assume that \(H=P\) is polynomial of degree d and \(\dim {{\,\mathrm{\mathfrak {hol}}\,}}(P) = \dim {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\). By Proposition 3, since the reflection matrix V is injective on a dense, open subset S of \(S^{2n1}\) as a map from \({{\,\mathrm{\mathfrak {hol}}\,}}(P)\) to \({{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\), it follows by the rank theorem that \(\dim V({{\,\mathrm{\mathfrak {hol}}\,}}(P))= \dim {{\,\mathrm{\mathfrak {hol}}\,}}(P)\). Using the assumption \(\dim {{\,\mathrm{\mathfrak {hol}}\,}}(P) = \dim {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) this implies that V is invertible as a map from \({{\,\mathrm{\mathfrak {hol}}\,}}(P)\) to \({{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\), for \(z\in S\). Thus, for any \(Y\in {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) there exists \(X\in {{\,\mathrm{\mathfrak {hol}}\,}}(P)\) with \(VX = Y\), such that on S, the following equation holds:
Choosing \(Y = i H^d_n \in {{\,\mathrm{\mathfrak {aut}}\,}}(H^d_n) \subset {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) in the previous equation, it becomes after using \(H^d_n \cdot {\bar{H}}^d_n = 1\) on \(S^{2n1}\):
Note that the matrix \(B {:}{=}V^{1}\) depends holomorphically on \(z \in U\), where U is an open set in \({{\,\mathrm{{\mathbb {C}}}\,}}^n\), such that \(U \cap S^{2n1} = S\). Consider the homogeneous expansion of \(B = \sum _{k\ge 0}B_k\) and \(P=\sum _{\ell =0}^d P_\ell \). Take \(z \mapsto e^{i t} z\) in (10), and collect Fourier coefficients. Looking at the constant Fourier coefficient we see that
on S. This equation can be rewritten as in the proof of [6, Sect. 5.1.4, Theorem 3] as follows,
such that the holomorphic nondegeneracy of \(H^d_n\) implies that . By some linear algebra this shows that \(P_d = U H^d_n\), where U is a unitary matrix. Write \(P = U H^d_n + F\), where F is a holomorphic polynomial of degree \(d1\). Since P maps \(S^{2n1}\) to \(S^{2K(n,d)1}\) it holds on \(S^{2n1}\) that,
hence after setting , one obtains
Consider \(z\mapsto e^{i t} z\) and a homogeneous expansion of \(F'=\sum _{j=0}^{d1} F_j'\) and collect the coefficient of \(e^{i d t}\) to get that \(H^d_n \cdot {\bar{F}}'_0 = 0\), hence by the holomorphic nondegeneracy of \(H^d_n\) one obtains \(F_0' = 0\). Proceed inductively to show that \(F_k' = 0\) for \(k \le d1\). Assume that \(F'_\ell = 0\) for all \(0\le \ell \le k1\). Collect the coefficient of \(e^{i(dk)t}\) in (11) to obtain that,
which, by using the induction hypothesis and the holomorphic nondegeneracy of \(H^d_n\), implies that \(F'_{k} = 0\). In total one obtains that P is unitarily equivalent to \(H^d_n\) in S, hence they are equivalent everywhere on \(S^{2n1}\). \(\square \)
One has the following inequality for the dimension of the space of infinitesimal deformations, when the tensor product is involved.
Corollary 3
Let \(A \subseteq {{\,\mathrm{{\mathbb {C}}}\,}}^m\) be a complex subspace, \(H: S^{2n1} \rightarrow S^{2m1}\) and \(G: S^{2n1} \rightarrow S^{2\ell 1}\) be nonconstant realanalytic CR maps. Assume \(F = E_{(A,G)}H\). Then \(\dim {{\,\mathrm{\mathfrak {hol}}\,}}(H) \le \dim {{\,\mathrm{\mathfrak {hol}}\,}}(F)\) and if H is holomorphically nondegenerate and equality holds if and only if \(F = H\).
Proof
If F is holomorphically degenerate, by Proposition 2, the inequality is satisfied. Assume that F is holomorphically nondegenerate, then the same holds for H by Lemma 4. Instead of using V as in the proof of Theorem 4, one considers \({\widetilde{V}}\) a linear map defined by \({\widetilde{V}}(X) {:}{=}T_{(A,G)} X\). Then \(X \cdot {\bar{H}} = {\widetilde{V}}(X) \cdot {\bar{F}}\), i.e., \(X \in {{\,\mathrm{\mathfrak {hol}}\,}}(H) \Leftrightarrow {\widetilde{V}}(X) \in {{\,\mathrm{\mathfrak {hol}}\,}}(F)\).
It holds that if there exists \(Y \in {{\,\mathrm{\mathfrak {hol}}\,}}(H)\) with \({\widetilde{V}}(Y) = 0\) on \(S^{2n1}\), then \(0 = {\widetilde{V}}(Y) \cdot {\bar{F}} = Y \cdot {\bar{H}}\) on \(S^{2n1}\), which implies, since H is holomorphically nondegenerate, that \(Y\equiv 0\). From this it follows that the set \(\{{\widetilde{V}}(X_j): 1\le j \le k\}\), for \(X_1,\ldots , X_k\) a basis of \({{\,\mathrm{\mathfrak {hol}}\,}}(H)\), is linearly independent in \({{\,\mathrm{\mathfrak {hol}}\,}}(F)\), which gives the claimed inequality.
For the equality, assume that \(\dim {{\,\mathrm{\mathfrak {hol}}\,}}(F) = \dim {{\,\mathrm{\mathfrak {hol}}\,}}(H) < \infty \). Then \(\dim {{\,\mathrm{\mathfrak {hol}}\,}}(H) = \dim {\widetilde{V}}({{\,\mathrm{\mathfrak {hol}}\,}}(H)) \le \dim {{\,\mathrm{\mathfrak {hol}}\,}}(F) = \dim {{\,\mathrm{\mathfrak {hol}}\,}}(H)\), which implies, as in the proof of Theorem 4, that on \(S^{2n1}\) the map \({\widetilde{V}}\), as a map from \({{\,\mathrm{\mathfrak {hol}}\,}}(H)\) to \({{\,\mathrm{\mathfrak {hol}}\,}}(F)\), is invertible. Using a similar argument as in the proof of Theorem 4 (replacing V by \({\tilde{V}}\), \(H^d_n\) by F and P by H and using \(X \cdot {\bar{H}} = {\tilde{V}}(X) \cdot {\bar{F}}\)) it follows that H and F are unitarily equivalent, which can only happen, when the complex subspace A is trivial. This concludes the proof. \(\square \)
The remainder of this section is a collection of lemmas concerning some properties of \(V_H\) and its transpose and provide sufficient and necessary conditions for infinitesimal rigidity in terms of \(V_H\) and its adjoint.
Proposition 5
For any polynomial map \(H:S^{2n1} \rightarrow S^{2m1}\) of degree d one has \(V_H H = H^d_n\) and on \(S^{2n1}\).
Proof
Using the matrix \(V_H\) from Definition 4 the following holds on \(S^{2n1}\):
for all \(X\in {{\,\mathrm{{\mathbb {C}}}\,}}^m\). Taking \(X = H\) in the above equation and using \(H \cdot {\bar{H}} = 1 = H^d_n \cdot {\bar{H}}^d_n\) on \(S^{2n1}\), it holds that,
on \(S^{2n1}\). Note that \(V_H H\) has holomorphic components such that the holomorphic nondegeneracy of \(H^d_n\) implies that \(V_H H = H^d_n\).
For the other identity one has,
for all \(X \in {{\,\mathrm{{\mathbb {C}}}\,}}^n\), which concludes the proof. \(\square \)
The following example shows that a similar relation as the second identity in Proposition 5 does not hold for infinitesimal deformations in general:
Example 9
Let X be of the form as \(T_1\) in Sect. 2.3 with H being the map
and \(\alpha ' = (0,0,0,a,0) \in {{\,\mathrm{{\mathbb {C}}}\,}}^5\). Then, on \(S^{3}\), one has:
which does not extend holomorphically to a neighborhood of \(S^{3}\).
Lemma 6
Let \(H:S^{2n1} \rightarrow S^{2m1}\) be a polynomial map of degree d.

(a)
It holds that \(X\in {{\,\mathrm{\mathfrak {hol}}\,}}(H)\) if and only if satisfies \({{\,\mathrm{Re}\,}}(X' \cdot {\bar{H}}) = 0\) on \(S^{2n1}\).

(b)
If \(Y \in {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) has the property that is holomorphic, then \(Y \in V_H({{\,\mathrm{\mathfrak {hol}}\,}}(H))\).
In (b) the necessary direction need not be true as Example 9 shows.
Proof
By Proposition 5 the following holds on \(S^{2n1}\):
which shows (a). In (b) assume is holomorphic. By Proposition 5 one has,
on \(S^{2n1}\) and taking the real part shows that \(X \in {{\,\mathrm{\mathfrak {hol}}\,}}(H)\). Consider the above equation and note that one has \( X \cdot {\bar{H}} = V_H X \cdot {\bar{H}}^d_n\), such that, since \(V_H X\) is holomorphic, the holomorphic nondegeneracy of \(H^d_n\) implies \(Y=V_H X \in V_H({{\,\mathrm{\mathfrak {hol}}\,}}(H))\). \(\square \)
Proposition 6
Let \(H:S^{2n1} \rightarrow S^{2m1}\) be a polynomial map of degree d.

(a)
If the map H is infinitesimally rigid then .

(b)
Assume that the map H is holomorphically nondegenerate. If , then H is infinitesimally rigid.
Proof
To prove (a), assume such that there exists \(Y \in {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) with . Thus, (in particular is holomorphic) and by Lemma 6 (b) it holds that \(Y \in V_H {{\,\mathrm{\mathfrak {hol}}\,}}(H) = V_H {{\,\mathrm{\mathfrak {aut}}\,}}(H)\) by the infinitesimal rigidity of H. In total this shows that . For the other implication, note that if , then, since \(V_H{{\,\mathrm{\mathfrak {aut}}\,}}(H) \subset {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\), it follows that .
For (b) let \(X \in {{\,\mathrm{\mathfrak {hol}}\,}}(H)\), then \(V_H X \in {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) and hence . Thus there exists \(T \in {{\,\mathrm{\mathfrak {aut}}\,}}(H)\), such that \(A X = A T\) for . Since H is holomorphically nondegenerate it holds that the \(K(n,d) \times m\)matrix \(V_H\) is injective on a dense open subset S of \(S^{2n1}\) (see Proposition 3), hence one has \({{\,\mathrm{rk}\,}}V_H = m\) in S. Since \(V_H\) consists of holomorphic entries in z, this means that V is of rank m in an open set U such that \(U\cap S^{2n1}=S\). It follows that the \(m\times m\)matrix A is of full rank m in U, and thus \(X = T\) in U. Since X and T are holomorphic they agree in \({{\,\mathrm{{\mathbb {C}}}\,}}^n\). This shows that \(X \in {{\,\mathrm{\mathfrak {aut}}\,}}(H)\). \(\square \)
Infinitesimal Deformations of the Homogeneous Sphere Map
In this section the dimension of the space of infinitesimal deformations of the homogeneous sphere map \(H^d_n\) (see Definition 5) is computed.
Theorem 5
The real dimension of the space of infinitesimal deformations of \(H^d_n\) is given by \(\left( \frac{2d+n}{d}\right) K(n,d)^2\).
Proof
Define the map \({\hat{H}}_n^d(z)=(z^\alpha )_{\alpha =d} \in {{\,\mathrm{{\mathbb {C}}}\,}}^{K(n,d)}\) and write \(Y = D_n^d X \in {{\,\mathrm{\mathfrak {hol}}\,}}(H_n^d)\), where \(D_n^d\) is given as in the hypothesis. Then X has to satisfy the following equation on \(S^{2n1}\),
In the above equation consider the homogeneous expansion of \(X = \sum _{k\ge 0} X^k\), where \(X^k\in {{\,\mathrm{{\mathbb {C}}}\,}}^{K(n,d)}\) is a homogeneous polynomial in z of order \(k \ge 0\). Change coordinates via \(z \mapsto e^{i \theta }z\) for \(\theta \in {{\,\mathrm{{\mathbb {R}}}\,}}\) as in [3, Lemma 16] to obtain after shifting indices, on \(S^{2n1}\):
This implies \(X^{\ell } \equiv 0\) for \(\ell \ge 2 d + 1\), such that \(d \le s,t\le d\). Collect Fourier coefficients of \(e^{i r \theta }\) for \(d \le r \le d\) in (12) such that for \(X \in {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) it is enough to study the solutions of the following equations:
on \(S^{2n1}\). For \(r=0\) the equation is real and will be treated below. Fix \(1\le r \le d\) and homogenize (13) as in [5, Sect. II] by multiplying the second term with \(\Vert z\Vert ^{2r}\), such that the following equation holds for all \(z\in {{\,\mathrm{{\mathbb {C}}}\,}}^n\):
Write \(X^{dr} = B^{d,r}_n {\hat{H}}^{dr}_n\), where \(B_n^{d,r}\) is a \(K(n,d)\times K(n,dr)\)matrix and write \(z^\alpha {\hat{H}}^d_n = C_n^{d,\alpha } {\hat{H}}^{d+r}_n\), where \(C_n^{d,\alpha }\) is a \(K(n,d) \times K(n,d+r)\)matrix whose entries consist of 0’s or 1’s. Rewrite the second term of (14) as
Write \(z^\alpha {\hat{H}}^{dr}_n = D_n^{d,\alpha } {\hat{H}}^{dr}_n\), where \(D_n^{d,\alpha }\) is a \(K(n,d) \times K(n,dr)\)matrix whose entries consist of 0’s or 1’s, such that using the holomorphic nondegeneracy of \(H^d_n\), one obtains
Setting \(X^{d+r} = A_n^{d,r} {\hat{H}}_n^{d+r}\), where \(A_n^{d,r}\) is an \(K(n,d) \times K(n,d+r)\)matrix, (15) gives:
This implies that the solutions of (13) depend on the entries of \(B_n^{d,r}\) for \(1\le r \le d\). The number of real entries in \(B_n^{d,r}\) for \(1\le r \le d\) is given by
For \(r=0\) in (13) one has
which is a homogeneous equation and hence holds on \({{\,\mathrm{{\mathbb {C}}}\,}}^n\). Writing \(X^d=B_n^{d,0} {\hat{H}}^d_n\), where \(B_n^{d,0}\) is a \(K(n,d) \times K(n,d)\)matrix, in (17) shows that . Hence \(B_n^{d,0}\) depends on \(K(n,d)^2\) real parameters. Moreover the stabilizer of \(H^d_n\) consists of \(S_n^2\) and \(S_n^3\) and the remaining elements of \({{\,\mathrm{\mathfrak {aut}}\,}}(H_n^d)\) coming from \({{\,\mathrm{\mathfrak {hol}}\,}}(S^{2n1})\) appear in (13) for \(r=1\) and have been taken into account in (16). In total, adding (16) and the number of real solutions of (13) for \(r=0\) gives the claimed dimension of \({{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\), which finishes the proof. \(\square \)
Theorem 2 is an immediate consequence of Theorems 4 and 5. Some examples illustrating Theorem 5 are provided in the following:
Example 10
For \(n=2\) write \(H_2^d(z,w)=(a_1^d z^d, a_2^d z^{d1}w, \ldots , a_{d+1}^d w^d)\), where \((a_k^d)^2 = \left( {\begin{array}{c}d\\ k1\end{array}}\right) \) for \(1\le k \le d+1\), such that \(\dim _{{{\,\mathrm{{\mathbb {R}}}\,}}} {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_2) = (d+1)^3\) and any \(X\in {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_2)\) is a real linear combination of infinitesimal deformations \(N^{m,k}_{d,r}\) of the following form:
where \(0\le r\le d\), \(D_{d+1}\) is the \((d+1)\times (d+1)\)diagonal matrix with entries \(1/a_k^d\) for \(1\le k \le d+1\), \(C^{m,k}_{d,r} =(c_{ij})\) is a \((d+1) \times (d+1r)\)matrix with \(c_{mk} \in \{1,i\}\) for \(1\le m \le d+1\) and \(1\le k \le d+1r\) and all other entries are 0 and \(0_j\) denotes the zerovector in \({{\,\mathrm{{\mathbb {C}}}\,}}^j\). To illustrate which nontrivial infinitesimal deformations for \(H^d_2\) appear, a list for \(d=2,3\) is given. Note that if N is a nontrivial infinitesimal deformation of \(H^d_2\), then \({\tilde{N}} = \phi ' \circ N \circ \phi \) is again a nontrivial infinitesimal deformation of \(H^d_2\) different from N, where \(\phi (z,w) = (w,z)\) and
Below half of all nontrivial infinitesimal deformations of \(H^d_2\) are listed as rows of the matrix \(Y^d_2\). The remaining nontrivial elements of \({{\,\mathrm{\mathfrak {hol}}\,}}(H^d_2)\) can be deduced by applying \(\phi \) and \(\phi '\) to each row of \(Y^d_2\).
where \(a,b\in {{\,\mathrm{{\mathbb {C}}}\,}}\), which are also listed in [17, Example 1].
where \(a,b,c,d,e,f,g, j, k\in {{\,\mathrm{{\mathbb {C}}}\,}}\).
Using the fact that for a sphere map H and any \(X\in {{\,\mathrm{\mathfrak {hol}}\,}}(H)\) one has \(V_HX \in {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) (see (9) in the proof of Theorem 4) it is possible to compute \({{\,\mathrm{\mathfrak {hol}}\,}}(H)\) from a description of \({{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\): By considering each element of \({{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) one needs to check if it can be written as \(V_H Y\), where Y is a holomorphic vector field, such that \(Y \in {{\,\mathrm{\mathfrak {hol}}\,}}(H)\). The vector fields obtained in this way span \({{\,\mathrm{\mathfrak {hol}}\,}}(H)\).
Example 11
The map \(H(z,w) = (z,zw,w^2)\) from \(S^{3}\) to \(S^{5}\) is of finite degeneracy 1 in \(S^{3} \cap \{z=0\}\) and 2nondegenerate otherwise. Its infinitesimal stabilizer consists of \(S^3_2\).
It can be checked that H is infinitesimally rigid.
Example 12
For the family of sphere maps \(G^\ell \) for \(\ell \ge 1\) there are infinitesimal rigid maps and some maps which admit nontrivial infinitesimal deformations: One can compute that for \(G^1: S^{3} \rightarrow S^{5}\) the space \({{\,\mathrm{\mathfrak {hol}}\,}}(G^1)\) only consists of trivial infinitesimal deformations, see also [7]. The map \(G^4: S^{3} \rightarrow S^{11}\) given by
where \(c_1 = c_4 = 3, c_2 = 3 \sqrt{3}\) and \(c_3 = \sqrt{30}\) is not infinitesimally rigid, since the vector
corresponds to a nontrivial infinitesimal deformation of \(G^4\). This can be verified by showing that X is not of the form as the trivial infinitesimal deformation given in Sect. 2.3.
Some of the nontrivial infinitesimal deformations of \(H^d_n\) originate from curves passing through the map as the following example shows: In the following a family of finitely nondegenerate rational sphere maps is constructed, which contains the homogeneous sphere map \(H^d_n\) for d odd. It is wellknown that families of sphere maps exist, see the examples in [4] and [18, Examples 4.1, 4.2], which motivated the construction. See also [11] for a study of homotopies of sphere maps.
Theorem 6
For \(k\ge 1\) the map \(H^{2 k + 1}_2: S^{3} \rightarrow S^{2 k + 2}\) is not locally rigid. More precisely, there exists a family \(F^{k}_s: S^{3} \rightarrow S^{2k +2}\) of \((2 k + 1)\)nondegenerate rational maps, where \(s \in {{\,\mathrm{{\mathbb {R}}}\,}}\) is sufficiently close to 0, with \(F^k_0 = H^{2 k + 1}_2\) and each \(F^k_s\) is not equivalent to \(H^{2 k +1}_2\) for \(s\ne 0\).
Proof
Consider for \(s\in {{\,\mathrm{{\mathbb {R}}}\,}}\) the map \(T_s = (T_s^1,T_s^2): S^{3} \rightarrow S^{3}\) given by
such that \(T_0(z,w) = (z,w)\), which is an automorphism of \(S^{3}\) when \(s\ne 1\). Define \(c_\ell {:}{=}\sqrt{\left( {\begin{array}{c}2 k + 1\\ \ell \end{array}}\right) }\) and note that \(c_k = c_{k+1}\). Set
It holds for all s that \(F^k_s\) maps \(S^{3}\) into \(S^{2 k + 2}\), since it originates from \(H^{2k+1}_2\) by applying the inverse of tensoring to the \((k+1)\)th and \((k+2)\)th component using the map \((z,w) \mapsto (z,w)\), and then tensoring the \((k+1)\)th component of the resulting map with \(T_s\). Furthermore \(F^k_0 = H^{2k+1}_2\) and since \(F^k_0\) is \((2k+1)\)nondegenerate by Lemma 2, the same holds for each map \(F^k_s\) for s sufficiently close to 0.
The remaining step is to show that for s sufficiently close to 0, when \(s \ne 0\) the map \(F^k_s\) is not equivalent to a polynomial sphere map (in particular \(H^{2k+1}_2\)). To see this apply the polynomiality criterion of Faran–Huang–Ji–Zhang [18, Remark 2.3 (A)]. More precisely, write \(F^k_s = \frac{P}{Q}\) with \(P = (P_1,\ldots , P_{2k+2})\) and Q being the denominator of \(T_s\), and \(d {:}{=}\deg F^k_s = \max \{\deg P, \deg Q\} = 2k+1\). Consider the map
where \(r_1 {:}{=}s(s  2)\) and \(r_2 {:}{=}r_1+2\). Write \({\hat{F}} = ({\hat{F}}_1, \ldots , {\hat{F}}_d, {\hat{Q}})\), then \(F^k_s\) is equivalent to a polynomial map if and only if there exist \(a_1,a_2, A_1, \ldots , A_d \in {{\,\mathrm{{\mathbb {C}}}\,}}\) and \(C \in {{\,\mathrm{{\mathbb {C}}}\,}}\setminus \{0\}\) such that
The claim is that under the assumption \(s\ne 0\) the Eq. (18) has no solution. Comparing the coefficient of \(z^2 t^{d2}\) one obtains that \(a_1 = 0\). Then the coefficient of \(z t^{d1}\) gives \(r_1 = 0\), which cannot be satisfied for \(0\ne s < 2\). This finishes the proof. \(\square \)
Note that for \(k=1\) the vector \(\frac{d}{d s}_{s=0} F_s^1\) is a nontrivial infinitesimal deformation of \(H^3_2\) from Example 10, when the parameter \(k \in {{\,\mathrm{{\mathbb {C}}}\,}}\) used there is taken to be real.
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Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). The author would like to thank Bernhard Lamel, John P. D’Angelo, Giuseppe della Sala and Ilya Kossovskiy for several helpful discussions and remarks.
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Reiter, M. The Reflection Map and Infinitesimal Deformations of Sphere Mappings. J Geom Anal 31, 772–800 (2021). https://doi.org/10.1007/s1222001900298w
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Keywords
 CR geometry
 CR mapping
 Infinitesimal deformations
 Reflection map
 Unit sphere
Mathematics Subject Classification
 32V40
 32V30