The Reflection Map and Infinitesimal Deformations of Sphere Mappings
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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.
Keywords
CR geometry CR mapping Infinitesimal deformations Reflection map Unit sphereMathematics Subject Classification
32V40 32V301 Introduction
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.
Theorem 1
 (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.
2 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 \(w =(w_1,\ldots , w_n) \in {{\,\mathrm{{\mathbb {C}}}\,}}^n\).
2.1 Infinitesimal Deformations of CR Maps
One of the main objects of this article are infinitesimal deformations of a CR map.
Definition 1
For a real manifold M the space \({{\,\mathrm{\mathfrak {hol}}\,}}(M)\) of infinitesimal automorphisms ofM 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 \(z\in S^{2k1}\).
2.2 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
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
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 ofH.
Constant degeneracy can be phrased in terms of vector fields as follows:
Proposition 1
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
 (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 \(\dim _{{{\,\mathrm{{\mathbb {R}}}\,}}} {{\,\mathrm{\mathfrak {hol}}\,}}(H) = \infty \).
2.3 Infinitesimal Automorphisms of the Unit Sphere
2.4 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
Several properties of the reflection matrix and examples involving V are given in [7].
3 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.
3.1 The Homogeneous Sphere Maps
The purpose of this section is to show some properties of the homogeneous sphere maps defined as follows:
Definition 5
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
 (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)
\([S_{ij},T_{k\ell }] = [S_{ij},T_{ij}] = 0\), \([S_{ij},T_{j\ell }] = T_{j\ell }\), \([S_{ij},T_{i\ell }] = T_{i\ell }\).
Lemma 2
The map \(H^d_n: S^{2n1} \rightarrow S^{2K(n,d)1}\) is dnondegenerate at each point of \(S^{2n1}\).
Proof
 (A)
 It is proved thaton \(S^{2n1}\), 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.$$\begin{aligned} L^\alpha H \cdot {\bar{L}}^\beta T^\gamma S^\delta {\bar{H}} = 0, \end{aligned}$$(3)
 (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 \(\delta \in J_m\).
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. \(\square \)
3.2 The Group Invariant Sphere Maps
Another important class of sphere maps are the following, first introduced in [3]:
Definition 6
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.
3.3 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 \(F = F_A \oplus F_{A^{\perp }} \in A \oplus A^{\perp }\).
Definition 7
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 fby gon 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
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
Example 4
4 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
 (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
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
 (a)
H is of degeneracy s at \(p \in S^{2n1}\).
 (b)
The kernel of the matrix V is of dimension s at \(p \in S^{2n1}\).
Theorem 1 follows from Propositions 3 and 4.
Proof
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
Example 6
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 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 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.
5 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
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.
 (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.
Theorem 3
 (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. \(\square \)
6 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
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
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 Open image in new window on \(S^{2n1}\).
Proof
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
Lemma 6
 (a)
It holds that \(X\in {{\,\mathrm{\mathfrak {hol}}\,}}(H)\) if and only if Open image in new window 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 Open image in new window 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
Proposition 6
 (a)
If the map H is infinitesimally rigid then Open image in new window .
 (b)
Assume that the map H is holomorphically nondegenerate. If Open image in new window , then H is infinitesimally rigid.
Proof
To prove (a), assume Open image in new window such that there exists \(Y \in {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) with Open image in new window . Thus, Open image in new window (in particular Open image in new window 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 Open image in new window . For the other implication, note that if Open image in new window , then, since \(V_H{{\,\mathrm{\mathfrak {aut}}\,}}(H) \subset {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\), it follows that Open image in new window .
For (b) let \(X \in {{\,\mathrm{\mathfrak {hol}}\,}}(H)\), then \(V_H X \in {{\,\mathrm{\mathfrak {hol}}\,}}(H^d_n)\) and hence Open image in new window . Thus there exists \(T \in {{\,\mathrm{\mathfrak {aut}}\,}}(H)\), such that \(A X = A T\) for Open image in new window . 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 \(X \in {{\,\mathrm{\mathfrak {aut}}\,}}(H)\). \(\square \)
7 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
Theorem 2 is an immediate consequence of Theorems 4 and 5. Some examples illustrating Theorem 5 are provided in the following:
Example 10
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
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
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.
Notes
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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