Blow-ups and infinitesimal automorphisms of CR-manifolds

For a real-analytic connected CR-hypersurface M of CR-dimension n⩾1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 1$$\end{document} having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra s=s(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {s}={\mathfrak {s}}(M)$$\end{document}: (i) either dims=n2+4n+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dim {\mathfrak {s}}=n^2+4n+3$$\end{document} and M is spherical everywhere; (ii) or dims⩽n2+2n+2+δ2,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dim {\mathfrak {s}}\leqslant n^2+2n+2+\delta _{2,n}$$\end{document} and in the case of equality M is spherical and has fixed signature of the Levi form in the complement to its Levi-degeneracy locus. A version of this result is proved for the Lie group of global automorphisms of M. Explicit examples of CR-hypersurfaces and their infinitesimal and global automorphisms realizing the bound in (ii) are constructed. We provide many other models with large symmetry using the technique of blow-up, in particular we realize all maximal parabolic subalgebras of the pseudo-unitary algebras as a symmetry.


Formulation of the problem
Investigation of symmetry is a classical problem in geometry. For a class C of manifolds endowed with particular geometric structures, denote by s(M) the Lie algebra of vector fields on M preserving the structure (infinitesimal automorphisms). It is important to determine the maximal value D max of the symmetry dimension dim s(M) over all M ∈ C.
Often the values immediately below D max are not realizabile as dim s(M) for any M ∈ C, which is known as the gap phenomenon. One then searches for the next realizable value, the submaximal dimension D smax , thus obtaining the interval (D smax , D max ) called the first gap (or lacuna) for the symmetry dimension.
The first and next gaps were successfully identified in Riemannian geometry, both in the global and infinitesimal settings [22,23], see also [13,18]. A large number of other situations where the gap phenomenon has been extensively studied falls in the framework of parabolic geometry [6], see the results and historical discussion in [31].
This article concerns symmetry in CR-geometry. While there was a considerable progress for Levi-nondegenerate CR-manifolds, in which case the geometry is parabolic, the problem of bounding symmetry dimension in general has been wide open.

The status of knowledge
Recall that an almost CR-structure on a smooth manifold M is a The almost CR-structure on M is said to be integrable if the distribution H (1,0) (M) is involutive. An integrable almost CR-structure is called a CR-structure and a manifold equipped with a CR-structure a CR-manifold. In this paper we consider only CR-hypersurfaces, i.e. CR-manifolds of CR-codimension 1.
A real hypersurface M in a complex manifold (M, J ) has an induced CR-structure: Conversely, every analytic CRhypersurface is locally realizable as such real hypersurface of CR-dimension dim C M − 1. In smooth situation a realization is not always possible, but in this article we restrict to real analytic CR-structures and hence make no distinction between abstract and embedded CR-hypersurfaces.
The As shown in classical works [7][8][9]40,41], see also [5,6], the dimension of the symmetry algebra s(M) of a Levi-nondegenerate connected CR-hypersurface M of CR-dimension n does not exceed n 2 + 4n + 3. If dim s(M) attains this bound then M is spherical, i.e. locally CR-equivalent to an open subset of the hyperquadric for some 0 ≤ k ≤ n/2. The Levi form of Q k has signature (k, n − k) everywhere and dim s(Q k ) = n 2 + 4n + 3 for all k. Thus, for the class of Levi-nondegenerate connected CR-hypersurfaces of CR-dimension n one has D max = n 2 + 4n + 3. Further, D smax = n 2 + 3 in the strongly pseudoconvex (Levi-definite) case for n > 1 and D smax = n 2 + 4 in the Levi-indefinite case [30]. The situation n = 1 is exceptional with D smax = 3 [7,8,31].
In the absence of Levi-nondegeneracy, finding the maximal and submaximal dimensions of the symmetry algebra is much harder. As is customary, assume the CR-manifold M and the vector fields forming the symmetry algebra to be real-analytic. Then s(M) = hol(M) is finite-dimensional provided that M is holomorphically nondegenerate, see [2, §11.3, §12.5], [11,39]. Regarding the maximal symmetry dimension D max , the following is a variant of Beloshapka's conjecture, cf. [4, p. 38]. The authors of [28] argument that for n = 1 this is a version of Poincaré's problème local [35]. For n = 1 the above conjecture holds true since a 3-dimensional holomorphically nondegenerate CR-hypersurface always has points of Levi-nondegeneracy. For n = 2 the conjecture was established in [21] where the proof relied on a reduction of 5-dimensional uniformly Levi-degenerate 2-nondegenerate CR-structures to absolute parallelisms (see [2, §11.1] for the definition of k-nondegeneracy). Thus, for real-analytic connected holomorphically nondegenerate CR-hypersurfaces of CR-dimension 1 ≤ n ≤ 2 one has, just as in the Levi-nondegenerate case, D max = n 2 + 4n + 3.

Conjecture 1.1 For any real-analytic connected holomorphically nondegenerate CRhypersurface M of CR-dimension n one has
It was shown in [28] that for n = 1 the condition dim hol(M, x) > 5 for x ∈ M implies that M is spherical near x, where hol(M, x) is the Lie algebra of germs at x of real-analytic vector fields on M whose flows consist of CR-transformations. In [19] we gave a short proof of this fact, and, applying the argument of [19] to the symmetry algebra s(M) instead of hol(M, x), one also obtains D smax = 5. Notice that the result of [19,28] improves on the statement of Conjecture 1.1 for n = 1 by replacing the assertion of generic sphericity of M by that of sphericity everywhere.
Further, in the recent paper [20] we considered the case n = 2. It was shown that in this situation either dim s(M) = 15 and M is spherical, or dim s(M) ≤ 11 with the equality occurring only if on a dense open subset M is spherical with Levi form of signature (1,1). This result improves on the statement of Conjecture 1.1 for n = 2 as it yields sphericity near every point of M. In addition, we constructed a series of examples of pairwise nonequivalent CR-hypersurfaces with dim s(M) = 11 thus establishing D smax = 11. This fact also led to the following analogue of the result of [28] for n = 2: the condition dim hol(M, x) > 11 for x ∈ M implies that M is spherical near x, and this estimate is sharp.

Main results
In the present paper we assume that n is arbitrary and that the Levi-nondegeneracy locus is nonempty, which is a condition stronger than holomorphic nondegeneracy. Of course, in this case M is Levi-nondegenerate on a dense open subset of M, perhaps with different Levisignatures at different points, and the symmetry dimension is finite. One of our goals is to determine the maximal and submaximal dimensions in this situation.

Theorem 1.2
Assume that M is a real-analytic connected CR-hypersurface of CR-dimension n ≥ 1 having a point of Levi-nondegeneracy. Then for its symmetry algebra s = hol(M) exactly one of the two situations is possible: (i) dim s = n 2 + 4n + 3 and M is spherical everywhere, (ii) dim s ≤ n 2 + 2n + 2 + δ 2,n and in the case of equality M is spherical on its Levinondegeneracy locus with fixed signature of the Levi form.
Moreover, the upper bound in (ii) is realizable and so the submaximal dimension is D smax = n 2 + 2n + 2 + δ 2,n .
This result improves on the statement of Conjecture 1.1. Note that the result is global in M, even if one takes M = U to be a small fixed neighborhood of a point x ∈ M. The proof of the theorem also leads to the following local version of the result, generalizing theorems from [19,20,28] for arbitrary n.

Corollary 1.3
With the assumptions of Theorem 1.2 in the case n ≥ 3 the condition dim hol(M, x) > n 2 + 2n + 2 for x ∈ M implies that M is spherical in a neighborhood of the point x, and this estimate is sharp.
As in papers [19,20], our argument relies on the techniques from Lie theory, notably on the description of proper subalgebras of maximal dimension of su( p, q) obtained in Theorem 3.3, where 1 ≤ p ≤ q, p + q ≥ 3. These pseudo-unitary algebras are precisely the maximal symmetry algebras of spherical models. We show that among proper maximal subalgebras of those the maximal dimension is attained on certain parabolic subalgebras. This raises the question if all parabolic subalgebras can be symmetries of CR-hypersurfaces. To this we answer affirmatively as follows. We suggest that other (non-maximal) parabolic subalgebras can be realized as symmetries of iterated blow-ups, and we demonstrate this in the first non-trivial case of CR-dimension n = 2. This contributes to the models with large symmetry algebras considered in [20]. The general problem is discussed in the conclusion of the paper.
Note that a blow-up construction in CR-geometry has been discussed so far only phenomenologically [24,27], and even a formal definition of this procedure was lacking in general (so rather a blow-down has been identified in loc.cit.). We approach the general problem in Sect. 2.1. The relation of such blow-up to symmetry is not straightforward. We discuss it in Sects. 2.2 and 2.3. For instance, we will show that an iterative blow-up (which can be considered as one blow-up from the naïve topological viewpoint) can reduce the symmetry beyond expectations.
It is not true that all sub-maximally symmetric models can be obtained by the proposed blow-up construction. This concerns the series of models in [20] and we construct more examples in Sect. 4.1. Actually, Theorem 4.1 gives a series of examples of pairwise nonequivalent CR-hypersurfaces with the submaximal value dim s(M) = n 2 + 2n + 2 for n = 2. However all examples we constructed and investigated can be shown (in many cases a-posteriori) to be obtained by a blow-up with an additional ramified covering that we describe in Sect. 4.3. This gives a new powerful tool for generating symmetric models in CR-geometry.
Finally, let us characterize Lie groups of automorphisms with large dimensions. (i) dim G = n 2 + 4n + 3 and M is spherical everywhere, (ii) dim G ≤ n 2 + 2n + 2 + δ 2,n and in the case of equality M is spherical on its Levinondegeneracy locus with fixed signature of the Levi form.
The upper bound in (ii) is realizable, implying that the submaximal dimension of the automorphism group is the same D smax as in the Lie algebra case.
The structure of the paper is as follows. In Sect. 2 we introduce the CR blow-up, as our main tool to create examples, and we construct some models with large symmetry algebra/group of automorphims. In Sect. 3, using the algebraic and analytic techniques, we derive a sharp upper bound on the symmetry dimension, thus proving the maximal and submaximal symmetry bounds; the reader interested in the gap phenomenon can proceed directly there. Then in Sect. 4 we provide further examples, containing an infinite sequence of submaximally symmetric and other models with large symmetry. Finally, in the Conclusion we formulate a more general conjecture on the symmetry dimension of CR-hypersurfaces and discuss other relevant problems.

The blow-up construction
Recall a construction from affine geometry. Let L be a subspace of a vector space V , codim(L, V ) = m. The blow-up of V along L (below is a subspace and x a point) is This works over any field, in particular for complex V , L, the blow-up is a complex algebraic manifold. The projection π L : The construction canonically extends to complex analytic geometry: if L is a complex submanifold of a complex manifold V , apply the above formula using local charts V ⊃ U α C n , straightening L ∩U α and patching the charts to obtain Bl L V , see e.g. [16]. The projection π L : Bl L V → V is holomorphic and satisfies the same properties: Everywhere below we will assume that m = codimL > 1, because otherwise Bl L V V for m = 1. Our aim is to extends this construction from complex geometry to CR-geometry. Though such a construction can be given on the abstract level, it is convenient to present a version for embedded CR-surfaces and we restrict to hypersurfaces. In this section we formulate only the standard blow-up; variations on it, like iterated blow-ups, weighted blow-ups and ramified coverings will be discussed in Sect. 4.3.

Blow-up in CR-geometry
Let ι : M → V be a real hypersurface in a complex manifold of dimension n + 1 and π L : Bl L V → V a blow-up along a complex submanifold L meeting M ≡ ι(M). In general, L does not belong to M and the germ of L along M ⊂ V is uniquely determined by This subset has singular points M ⊂ π −1 L (L ). For our purposes it is enough to describe singularities in an affine chart: V = C n+1 and L ⊂ V a subspace. Proof Let M be the zero set of a non-singular function f : Since π L is a diffeomorphism outside L, we can restrict to x ∈ L . With such x one readily verifies that the image of dx π L : Tx Bl L V → T x V atx = (x, ) coincides with , and so it belongs to the kernel of d x f if and only if ⊂ H (x).
Thusx is non-singular unless T x L = T x L ⊂ ⊂ H (x). A priori it could happen that M possesses another defining functionf near suchx that is not a pullback π * L f , yet a closer analysis shows that the singularity atx is conical and hence essential.
Removing singularities fromM we obtain what we call CR-blowup of M along L: Proof Note at first that the construction is defined because every real-analytic CR-surface admits a closed real-analytic CR-embedding as a hypersurface to a complex manifold V [1]. Next, by Theorem 1.12 of loc.cit. such an embedding is unique up to a biholomorphism of (the germ of) a neighborhood of ι(M) ⊂ V . Since biholomorphisms naturally induce maps of blow-ups the first claim follows. The second claim of the proposition follows from Corollary 2.2.

Example 2.4 Let us blow-up the hyperquadric
The blow-up contains the following open dense subset The model M for n = 1 appeared in [27].
The whole blow-up is obtained from (z 1 , . . . , z n , w), [ζ 1 : · · · : ζ n : ] ∈ C n+1 × CP n , by removing singularities. In the chart = 0 we get U 0 = M as above. For 1 ≤ k ≤ n in the chart ζ k = 0 we get The projections π k o : U k → Q and the gluing maps ϕ k : U k \{w = 0} → U 0 are given by the formulae: Thus Bl o Q is obtained from the union of U 0 , U 1 , . . . , U n by gluing via ϕ 1 , . . . , ϕ n . Since ∪ n k=1 π k o (U k ) ∩ {w = 0} (this is empty only for the sign definite norm z 2 ) is the null-cone In what follows we often change the blow-up Bl o Q to M.
be the direct product and let z 2 = z 2 + z 2 be the quadric of signature (p,q), where both quadrics z 2 and z 2 are nondegenerate of signatures ( p , q ) and ( p , q ) with p + p =p, q + q =q. Let L = C k (z ). The corresponding blow-up contains the following model

Symmetry of a blow-up
Next we describe how the symmetry algebra of M changes upon the blow-up construction. Recall that the Levi-degeneracy locus in M is an analytic subset.

Theorem 2.6 Let M be a connected real analytic CR-hypersurface having Levi nondegenerate points. If L = L assume that either each component of L contains a Levi-nondegenerate point or that the Levi-degeneracy locus in M has
Thus in the case when m = codim(L) > 2 the rank of the Levi form is at most dim < 2n and so , correspond to the singularity stratum M that is removed, and the argument applies as well. Alternatively, we note that when L ⊂ M the blow-up contains the complex hypersurface π −1 L (L) C m−1 × L and so Bl L M is not minimal along it. Our further arguments can be applied component-wise, so we can assume L (and L) to be connected.
If a symmetry is not tangent to this complex submanifold, then a flow along it generates an open subset of Levi-degenerate points, which is impossible because Levi-nondegenerate points are dense in M. If L L then dim π −1 L (L ) = 2n − 1. Any symmetry must be therefore tangent to π −1 L (L ) if Levi degeneracy locus has codim > 1. Alternatively, if L contains a Levi-nondegenerate point x, then any point from a small neighborhood U x ⊂ M of x is Levi-nondegenerate. Thus a symmetry must be tangent to π −1 L (L ∩ U x ) and hence, by analyticity of vector fields from s(M), this symmetry is tangent to π −1 L (L ) everywhere. We conclude that in any case the symmetries of Bl L M must be tangent to π −1 L (L ). Thus they descend to the blow-down manifold M. Indeed, consider a symmetry s ∈ s(Bl L M) restricted to Bl L M\π −1 L (L ) M\L . Choose a neighborhood U ⊂ V of x ∈ L and adapted complex coordinates to the submanifold L ⊂ V . In these coordinates the components of s are holomorphic functions that analytically extend to L by the Hartogs principle applied to U (it is important here that m = codimL > 1). In other words, L is a removable singularity for the symmetry s on M.
Therefore we get a map q L : s(Bl L (M)) → s(M), which is clearly a homomorphism of Lie algebras. Since π L : Bl L M → M is a biholomorphism over M\L the map q L is injective. Indeed, q L (s) = 0 fors ∈ s(Bl L M) impliess| U = 0 for U ⊂ M\L and thereforẽ s = 0 by analyticity and connectedness of Bl L M.
It is clear that the vector fields s ∈ s(M) that lift to Bl L M must preserve L . Conversely, if s preserves L it lifts to the blow up of V along L. Since s is also a symmetry of M, it restrict toM and then to the non-singular part Bl L M. Thus q L has the required image: The proof in the case of germs of symmetries is completely analogous.

Remark 2.7
The Levi-nondegeneracy assumptions in Theorem 2.6 can be relaxed to knondegeneracy for k > 1, as was kindly communicated to us by a reviewer: A combination of [12, Theorem 1.1,Theorem 1.4] implies that every pointx of π −1 L (L ) is not of finite type in the sense of Kohn and Bloom-Graham, so not finitely-nondegenerate by [2, Remark 11.5.14]. On holomorphically nondegenerate M every point in the complement to a proper analytic set is finitely-nondegenerate [2, Theorem 11.5.1]. This example implies the following statement.

Corollary 2.9
The first parabolic subalgebra p 1,n+1 ⊂ su( p, q) is realizable as symmetry of an analytic CR-hypersurface of CRdim = n containing Levi nondegenerate points. As such one can take either the constructed blow-up or its submanifold M ⊂ Bl o Q.
Proof That the symmetry of Bl o Q is as indicated follows from Theorem 2.6. Let us also show that the symmetry does not grow upon restriction to the submanifold M. The subset π o (M) ⊂ Q is obtained from the quadric by removing the hyperplane {w = 0} punctured at o. The symmetry algebra of both π o (M) and Q is su( p, q). Now the same argument as in the above proof shows that q L : s(M) → s(π o (M)) is an injective map with the image consisting of symmetry fields vanishing at o.

Example 2.10
Considering the more general blow-up model from Example 2.5 with k = dim C L ∈ (0, n) we conclude that its symmetry is smaller in dimension than the parabolic subalgebra fixing a subspace of dimension (k + 1) in linear representation. For instance, forp =q = 1 the symmetry algebra s(Bl L Q) has dimension 8, while the corresponding parabolic algebra p 2 ⊂ su(2, 2) has dimension 11. This is in accordance with Theorem 2.6, if one verifies the action of s(Q) on L.

Automorphisms of a blow-up
The argument of the previous theorem extends to the Lie group case and we get: The second statement follows by dimension comparison of Levi degeneracy loci.
We give an application of this theorem. Let p + q = n + 2, 1 ≤ s ≤ p ≤ q. Recall that the parabolic subgroup P s,n−s+2 ⊂ SU ( p, q) is the stabilizer of a null s-plane (and thus also of the orthogonal co-isotropic (n − s + 2)-plane) in the standard representation of SU ( p, q) on C n+2 , its Lie algebra is the parabolic subalgebra p s,n−s+2 .

Example 2.12 Let
Clearly, L has dimension k and lies in Q p−1 . An open dense subset M of the blow-up Bl L Q p−1 belongs to the hypersurface S ⊂ C n (z) × C(w) given by with the projection π L : S → Q p−1 given by The hypersurface S contains the hyperplane {w = 0} = π −1 L (L), and for every x ∈ L the fiber π −1 L (x) is an (n − k)-dimensional vector subspace of C n+1 . The singular locus of S is given by The CR-hypersurface M is obtained by excluding S from S. Thus M ⊂ S is an open subset containing an open subset of the hyperplane {w = 0}. By Theorem 2.6 the symmetry algebra s(M) is p s,n−s+2 . We do not provide details of this derivation here because in Sect. 4.2 we present these symmetries explicitly. This will realize all maximal parabolic subalgebras of su( p, q).
Note that the automorphism group of the spherical surface Q p−1 is not SU ( p, q), but its parabolic subgroup P 1,n+1 due to incompleteness, and so the automorphism group of its blow-up is not P s,n−s+2 (even for s = 1).

Example 2.13
Let us consider the compact version, possessing the automorphism group of maximal size. For this embed the hyperquadric Q = Q p−1 into projective space and take the closure: where the hyperplane at infinity is CP n = {ξ = 0}. The Lie group G = PSU( p, q) acts transitively on Q. Moreover, it acts transitively on the manifold N of linear subspaces of CP n+1 of dimension k that lie in Q with dim N = (k + 1)(2n − 3k + 1). The stabilizer of a point L in N is the parabolic subgroup P s,n−s+2 ⊂ P SU ( p, q), and one can verify using Theorem 2.11 that this is indeed the automorphism group of Bl L Q.

The gap phenomenon
In this section we prove Theorem 1.2, Corollary 1.3, Theorem 1.5 and further results.

An algebraic dimension bound
Consider the simple Lie algebra su( p, q), 1 ≤ p ≤ q, p + q = n + 2 ≥ 3, where p counts the number of positive eigenvalues and q the number of negative ones in the signature of the corresponding Hermitian form. The case of sign-definite metric, i.e. the algebra su(n + 2), will be excluded from consideration.
The algebra has type A n+1 , and its parabolic subalgebra corresponding to the crossed nodes that form a subset I of the nodes of the Satake diagram is denoted by p I . In particular, the maximal parabolic subalgebras are p s,n−s+2 for 1 ≤ s ≤ p, where for n = 2m − 2 we identify p m,m with p m . Recall that a cross can be imposed only on a white node of the Satake diagram; any two white nodes related by an arrow shall be crossed simultaneously. Here are some examples: Proposition 3.1 Dimension of the maximal parabolic subalgebra p s,n−s+2 ⊂ su( p, q) is d n (s) = n 2 − 2sn + 3s 2 + 4n − 4s + 3.
Proof For g = su( p, q) the grading g = g −ν ⊕ · · · ⊕ g 0 ⊕ · · · ⊕ g ν corresponding to a parabolic subalgebra p g 0 ⊕ · · · ⊕ g ν of g has g 0 z(g 0 ) ⊕ g ss 0 , where the first summand is the center of dimension equal to the number of crosses in the Satake diagram of g and the second (semisimple) summand corresponds to the Satake diagram obtained by removing the crosses. Thus, a maximal parabolic subalgebra of g, independently of the coloring of the nodes, satisfies: . The case n = 2m, s = m + 1 is special yet subject to this formula.
Some initial values of d n (s) are as follows:  Proof By Mostow's theorem [34], a maximal subalgebra of a real simple Lie algebra is either parabolic, or the centralizer of a pseudotorus, or semisimple.
If h C falls in Case (i), we have n + 2 = st (1 < s ≤ t < n + 2, hence n ≥ 2) and h C = sl(s, C) ⊕ sl(t, C) is embedded in sl(n + 2, C) via the representation on C s ⊗ C t = C n+2 .
The maximum of the function s 2 + (n+2) 2 s 2 −2 on the interval 2 ≤ s ≤ n + 1 is attained at s = n + 1 and is clearly seen to be strictly less than dim p 1,n+1 .
In Case (ii) we first assume that h C is a classical Lie subalgebra of sl(n + 2, C). If h C has type A, then dim C h C is maximal if h C = sl(k + 1, C) ⊂ sl(n + 2, C), k ≤ n, which does not give the optimal dimension as dim C sl(k + 1, C) ≤ n 2 + 2n < dim p 1,n+1 .
If h C has type B or D, then dim C h C is maximal if h C = so(n+2, C) ⊂ sl(n+2, C), which does not give the optimal dimension since dim C so(n+2, C) = 1 2 (n 2 +3n+2) < dim p 1,n+1 . Suppose that h C has type C and write n = 2k + r , where r is either 0 or 1. Then dim C h C is maximal if h C = sp(2k + 2, C) ⊂ sl(n + 2, C), which again does not give the optimal dimension as dim C sp(2k + 2, C) = (k + 1)(2k + 3). Indeed, this number is strictly less than dim p 1,n+1 for n = 2 and is strictly less than 11 = dim p 2 for n = 2.
Consider now the exceptional Lie algebras. The representation V of minimal dimension of g 2 = Lie(G 2 ) has dimension 7 (V = R λ 1 ), so if h C = g 2 we have n ≥ 5. Hence g 2 does not give the optimal dimension since dim C g 2 = 14 < 5 2 + 2 · 5 + 2.
Similarly, the representation V of minimal dimension of the exceptional Lie algebra f 4 = Lie(F 4 ) has dimension 26 (V = R λ 4 ), so if h C = f 4 we have n ≥ 24. Hence f 4 does not give the optimal dimension since dim C f 4 = 52 < 24 2 + 2 · 24 + 2.
Thus, all semisimple subalgebras of su( p, q) have dimensions strictly smaller than the maximal possible dimension of a parabolic subalgebra.

Establishing the submaximal symmetry dimension
We assumed that M has a point of Levi-nondegeneracy, which implies that M is holomorphically nondegenerate, see [2, Theorem 11.5.1]. The condition of holomorphic nondegeneracy for a real-analytic hypersurface in complex space was introduced in [39] and requires that for every point of the hypersurface there exists no nontrivial holomorphic vector field tangent to the hypersurface near the point. Extensive discussions of this condition can be found in [2, §11.3], [11], but we only make a note of the fact, stated in [2, Corollary 12.5.5], that the holomorphic nondegeneracy of M is equivalent to the finite-dimensionality of all the algebras hol(M, x). Notice that together with [2, Proposition 12.5.1] this corollary implies that the finite-dimensionality of hol(M, x 0 ) for some x 0 ∈ M implies the finite-dimensionality of hol(M, x) for all x ∈ M.
Clearly, s(M) = hol(M) may be viewed as a subalgebra of hol(M, x) for any x. Therefore for a holomorphically nondegenerate M, and in particular for the case we consider, the symmetry algebra s(M) is finite-dimensional.

Proof of Theorem 1.2
For n = 1 the theorem was obtained in [19,28], for n = 2 its stronger variant was proven in [20], so we assume that n ≥ 3.
Consider Case (i) first. We will show that M is spherical everywhere. Since we already established sphericity on U , fix a point x 0 ∈ S M . Consider the isotropy subalgebra hol 0 (M) of hol(M) at x 0 . Clearly, dim hol 0 (M) ≥ (n + 2) 2 − 1 − (2n + 1) = n 2 + 2n + 2. Hence, appealing to Theorem 3.3 once again, we see that one of the following holds: In Case (ia), the orbit of x 0 under the corresponding local action of the group SU ( p, q) is open, so it contains a spherical point x ∈ U , and hence M is spherical near the point x 0 as well.
In Case (ib), by the Guillemin-Sternberg theorem [15, pp. 113-115], the action of the simple Lie algebra su( p, q) is linearizable near x 0 , and we obtain a nontrivial (2n + 1)-dimensional representation of su( p, q). But the lowest-dimensional representation of su( p, q) is the standard C p,q of real dimension 2n + 4, which is a contradiction.
Consider now Case (ii) and assume that dim hol(M) = n 2 + 2n + 2. Then by Theorem 3.3 the algebra hol(M) is isomorphic either to the parabolic subalgebra p 1,n+1 of su( p, q), or, if n = 4 and p = q = 3, to the parabolic subalgebra p 3 of su (3,3). As all such parabolic subalgebras are pairwise nonisomorphic, we see that p and q are determined uniquely. Therefore, the Levi form of M has fixed signature on U .
Finally, the obtained upper bound for the symmetry dimension is realizable due to Corollary 2.9. This finished the proof.

Some results on spherical points
Let us further discuss the result of Theorem 1.2. First note that the exceptional case n = 2 can be included into part of the statement as follows.

Proposition 3.5 Assume that M is a real-analytic connected CR-hypersurface of CRdimension n ≥ 1 having a Levi-nondegenerate point. If dim s(M) ≥ n 2 + 2n + 2, then M is spherical on its Levi-nondegeneracy locus with fixed signature of the Levi form.
Proof Only the case n = 2 is special in regard to the proof of Sect. 3.2. In this case dim s = 10 = n 2 + 2n + 2, and the statement follows by Remark 3.4 since the parabolic subalgebras p 1,3 ⊂ su(1, 3) and p 1,3 ⊂ su(2, 2) derived in [20] are not isomorphic.
Next, set We found that its symmetry algebra is spanned by the vector fields This algebra is isomorphic to , where R is the grading element, S is the center and J is the complex structure on the contact subspace in

Group version of the main results
Let us first note that the situations in (ii) can correspond to the existence of Levi-degenerate points as in Theorem 1.2 (the models are in Example 2.13), but the dimension can also drop by purely topological reasons, reducing the pseudo-unitary group to its subgroup. For instance, removing from hyperquadric (1.1) a subspace L s−1 of dimension (s − 1) reduces P SU ( p, q) to its maximal parabolic subgroup P s,n−s+2 .
The global infinitesimal automorphisms are un-altered by this removal of L s−1 , but some of the vector fields from s(M) become incomplete resulting in reduction of G. This is the only global effect and it is manifested in a remarkably short proof of Theorem 1.5 given below. In fact, it is a simpler statement than that for the global infinitesimal automorphisms since realization, indicated in the previous paragraph, follows from the very definition of the parabolic subgroup as the stabilizer of a linear subspace in the projective version of the flat model and does not appeal to blow-ups.
Proof of Theorem 1.5 Let s be the infinitesimal automorphism algebra of M and g = Lie(G) the Lie algebra of G. Because g ⊂ s the assumption of case (i) in Theorem 1.5 implies the assumption of case (i) in Theorem 1.2 and consequently the implications align.
Consider now case (ii) in Theorem 1.5. If g = s then the implications align again and we are done. Otherwise dim s > dim g and this implies, by Theorem 3.3, that dim s = n 2 + 4n + 3, so we are under the assumption of case (i) in Theorem 1.2, which yields sphericity of M everywhere.

Models with large symmetry
We will now elaborate on CR-hypersurfaces with submaximal symmetry dimension. First we exhibit a countably many non-equivalent models with the symmetry algebra being the first parabolic subalgebra p 1,n+1 .
Then we realize in two non-equivalent ways all maximal parabolic subalgebras proving Theorem 1.4. In particular, for n = 2 we obtain an example of a CR-hypersurface with dim s(M) = n 2 + 2n + 3 = 11 that is more elementary compared to those discussed in [20]. For n = 4 we get an example of a CR-hypersurface with dim s(M) = n 2 + 2n + 2 = 26; its algebra dim s(M) = p 3 ⊂ su(3, 3) yields yet another model with symmetry of the same dimension as p 1,5 .
Finally we show other means to produce models with large symmetry: iterated blow-ups and ramified coverings. In fact, both the series of examples in Sect. 4.1 and those from [20] can be seen as a combination of a blow-up and a ramified covering.
We now observe that every point (z, w) ∈ M m,ε satisfies the equation In fact, for every value of ε, Eq. (4.6) describes 2m pairwise CR-equivalent smooth hypersurfaces, with (4.5) being one of them. The other hypersurfaces are obtained from (4.5) by multiplying w by a root of order 2m of either 1 or −1. One obtains m hypersurfaces from the roots of 1 and the other m ones from the roots of −1 (notice that two opposite roots lead to the same equation). All these hypersurfaces intersect along {w = 0}. For example, when m = 1 the set described by Eq. (4.6) is the union of the following two smooth hypersurfaces: v = εu tan 1 2 arcsin( z 2 ) and u = −εv tan 1 2 arcsin( z 2 ) , z < 1.
Each of the 2m hypersurfaces given by (4.6) is spherical away from S m,ε . Indeed, fix a point (z 0 , w 0 ) satisfying (4.6) with w 0 = 0. Then Re(w 2m 0 ) = 0, and setting σ = ε sgn Re(w 2m 0 ), we see that the map (z, w) → (zw m , σ w 2m ) transforms a neighborhood of (z 0 , w 0 ) on the relevant hypersurface to an open subset of the hyperquadric (1.1) that we rewrite so Proof It is straightforward to check that the following vector fields span the algebra s(M m,ε ), where 1 ≤ j ≤ n, j < ≤ n and summation over repeated indices is not assumed: Re(w∂ w ), Re w 2m (mξ + w∂ w ) , Re z j w m (mξ + w∂ w ) + imεσ j w m ∂ z j , Re i z j w m (mξ + w∂ w ) + mεσ j w m ∂ z j , (4.8) with ξ = n j=1 z j ∂ z j . Furthermore, one can check that vector fields (4.8) define a faithful representation of the parabolic subalgebra p 1,n+1 ⊂ su( p, q).
Since the surface M m,ε is not everywhere spherical, Theorem 1.2 yields the upper bound dim s(M m,ε ) ≤ n 2 + 2n + 2 + δ 2,n . Moreover in case n = 2 the equality is only attained for the parabolic subalgebra p 2 ⊂ su(2, 2), and since the other parabolic p 1,3 does not embed into p 2 , we conclude that in fact dim s(M m,ε ) ≤ n 2 + 2n + 2. But since vector fields (4.8) in totality n 2 + 2n + 2 constitute the symmetries of the model, we conclude the opposite inequality and hence s(M m,ε ) = p 1,n+1 .
Similarly, for any connected neighborhood U of a point x ∈ S m,ε in M m,ε we have hol(U ) = p 1,n+1 , while if U ∩ S m,ε = ∅ we get hol(U ) = su( p, q).
Next, formulas (4.8) show that all elements of s(M m,ε ) vanish precisely at the origin. Hence, if a real-analytic CR-diffeomorphism F establishes equivalence between M m,ε and M k,δ , we have F(0) = 0. Observe now that the highest order of the vanishing of a vector field in the algebra s(M m,ε ) at the origin is 2m + 1, and this number must be preserved by F. This shows that m = k. The same argument yields the nonequivalence of the germs of M m,ε and M k,δ at the origin by means of a real-analytic CR-diffeomorphism unless m = k.
Further, if a real-analytic CR-diffeomorphism F establishes equivalence between M m,−1 and M m,+1 , we have again F(0) = 0. Since F holomorphically extends to a neighbourhood of the origin, let us write it as (z, w) → ( f (z, w), g(z, w)), with f (z, w) = Az + Bw + · · · , g(z, w) = Cw + · · · , where dots denote higher-order terms and A, B, C are complex matrices of sizes n × n, n × 1, 1 × 1, respectively (note that g contains no linear terms in z because F must preserve S m,ε ). The condition that F maps M m,−1 to M m,+1 is written as the identity =0.
The terms linear in u yield ImC = 0. Then ReC = 0 and the next terms in decomposition of the above identity imply Since signature is an invariant of the quadric, for p = q this condition is impossible. The same argument yields the nonequivalence of the germs of M m,−1 and M m,+1 at the origin by means of a real-analytic CR-diffeomorphism.

Remark 4.2
The last statement of Theorem 4.1 does not hold for p = q (just interchange the groups of variables z 1 , . . . , z n/2 and z n/2+1 , . . . , z n ). The invariance of the pair (m, ε) for n = 1 was claimed in [4].

Realization of maximal parabolics
We will now construct realizations of all the maximal parabolic subalgebras p s,n−s+2 , 1 ≤ s ≤ p.
Finding such realizations is interesting in its own right, as this adds up to the study of symmetry of polynomial CR models, cf. [3,25,26].
The first model has been already introduced in Example 2.12, see Eq. (2.3) applicable to all 1 ≤ s ≤ n 2 + 1. It is a blow-up (2.4) of the hyperquadric Qp along the subspace L given by (2.2). Let us denote this model M s I . Its locus of Levi degeneracy is a complex submanifold S s I of real dimension 2n. Indeed, S s I is an open subset of the hyperplane {w = 0} (coincides with it for s = 1). The second model is applicable for 1 < s < n 2 + 1, i.e. k = s − 1 ∈ (0, n 2 ). It is a blow up along the following subspace in C n (z) × C(w) of dimension n − k: An important difference between (2.2) and (4.9) is that the latter L is not contained in Qp, so the blow-up happen along the real-analytic subvariety L = L ∩ Qp. An open subset of Bl L Qp embeds into the hypersurface S ⊂ C n (z) × C(w) given by Im(w) = k j=1 z j wz k+ j + z k+ jwz j + z 2 (4.10) ( z 2 has the same meaning as in Example 2.12) with the projection given by π L (z 1 , . . . , z n , w) = (z 1 w, . . . , z k w, z k+1 , . . . , z 2k , z 2k+1 , . . . , z n , w). (4.11) The hypersurface S contains the real-analytic subvariety S = { z = 0, w = 0} = π −1 L (L ), and for every x ∈ L the fiber π −1 L (x) is a k-dimensional vector subspace of C n+1 . This subvariety has real dimension Excluding the singular locus Of course, this assertion implies Theorem 1.4. Note that complementarity of dimensions of L in both cases (s − 1 and n − s + 1) reflects certain duality and it gives light to the fact that both surfaces have the same symmetry algebra.
Proof The symmetries of both models are obtained by straightforward but very demanding computations (involving many Maple experiments). For (2.3), denoting ζ = n j=2k+1 z j ∂ z j − w∂ w , ξ = k a=1 z a ∂ z a + ζ, η = k a=1 z a+k ∂ z a+k + w∂ w , the following are the generators of s(M s I ) with indices in the range 1 ≤ a, b ≤ k, a < c ≤ k, 2k + 1 ≤ j ≤ n, j < ≤ n: Re(∂ z a + 2i z a+k η), Re(i∂ z a + 2z a+k η), Re(∂ z a+k − 2i z a ξ), Re(i∂ z a+k − 2z a ξ), Re w(∂ z a+k + 2i z a η) , Re w(i∂ z a+k + 2z a η) , Re(wη), Re(ζ − w∂ w ), Re(∂ z j + 2iσ j z j wη), Re(i∂ z j + 2σ j z j wη). (4.12) Similarly, for (4.10), if we denote and use the same range for indices, then the following are the generators of s(M s II ): Re(∂ z a + 2i z a+k ξ), Re(i∂ z a + 2z a+k ξ), Re(∂ z a+k − 2i z a η), Re(i∂ z a+k − 2z a η), Re(w ∂ z a+k + 2i z a ξ) , Re(w i∂ z a+k + 2z a ξ) , Re(wξ ), Re(ζ + w∂ w ), Clearly, Qp is singular if r > 1. We will now blow up Qp at the origin by a weighted analogue of map π L with L = o, namely, by the map where m ∈ N. The result of the blow-up is the hypersurface R r ,m = π −1 o,m ( Qp), which is described by the equation (4.14) Fix m ∈ N and set r = 2m. One can rewrite Eq. (4.14) of R 2m,m as Taking squares, we obtain Im(w 2m The pair of equations in (4.15) for σ = ±1 describes the same set of points as the pair of equations in (4.6) for ε = ±1. This set is formed by 4m smooth hypersurfaces all intersecting along w = 0. Otherwise said, the models of Theorem 4.1 can be considered as a ramified covering of a weighted blow-up.

Remark 4.7
For n = 1 the weighted blow-up R 2m,m of a ramified cover over the hyperquadric was considered in [24], see, e.g., Lemmas 22, 26 therein.
Note that Q 0 = Qp, wherep = p − 1 and (p,q) is the signature of the quadric z 2 (p +q = n), and that Q j+1 = Bl o Q j for all j ≥ 0.
By Theorem 2.6 the symmetry algebra of Q 1 is obtained from that of Q 0 as stabilizer of o in su( p, q), the resulting algebra of vector fields has generators (4.12) for k = 0. This is the reduction to the parabolic subalgebra p 1,n+1 .
Again, using Theorem 2.6 the symmetry algebra of Q 2 is obtained from that of Q 1 as stabilizer of o in the algebra p 1,n+1 . This is obtained by removing the vector fields in the last line of (4.12) for k = 0. Further blow-ups do not change the dimension (only some coefficients are being modified), and we conclude dim s(Q m ) = n 2 + 2.
Actually, this dimension persists for the symmetry algebra of the ramified equation.
Proof Since we already restricted the dimension from above, it is enough to indicate the generators. They are given below with the index range 1 ≤ ≤ n, < j ≤ n: The abstract Lie algebra structure is straightforward.
Finally, let us show an example of iterated blow-up giving a surface with a non-maximal parabolic symmetry algebra. We do it in the simplest case n = 2 with parabolic p 1,2,3 in su(2, 2) being the Borel subalgebra.

Proof
The symmetry algebra has generators: They satisfy the structure equations of the Borel subalgebra in su (2,2). In fact, the stabilizer of L 1 in 11-dimensional symmetry algebra of (4.16) is the indicated 9-dimensional subalgebra p 1,2,3 ⊂ p 2 , in accordance with Theorem 2.6.
The blow-up Bl

Conclusion
Let us outline a possible generalization and formulate some open problems.

On generalization of the main result
Motivated by results in the present paper and a series of preceding works in complex analysis, we formulate the following claim, generalizing Conjecture 1.1. Let us support this claim. It holds for 1 ≤ n ≤ 2, and also for larger n, provided M is Levi nondegenerate somewhere. For the case n = 2 we utilized in [20] the following fact: a realanalytic connected holomorphically nondegenerate CR-hypersurface of dimension 5 with everywhere degenerate Levi form is generically 2-nondegenerate. By appealing to the main result of [21], this allowed to estimate the dimension of the symmetry algebra in everywhere Levi-degenerate case by 10, see also [32,33].
For n ≥ 3 some partial results generalizing this have been obtained in the literature. CR hypersurfaces that are 1-degenerate and 2-nondegenerate in the sense of Freeman with a certain additional condition were investigated in [37] for n = 3 and in [36] for general n. The upper bound on symmetry achieved in those references confirms our conjecture. Also in [38] all Levi degenerate homogeneous 7-dimensional CR hypersurfaces (n = 3) were classified. Again, the results align with Conjecture 5.1.
We expect that elaboration upon Cartan and Tanaka theories in the spirit of [29] can provide effective bounds on local symmetry important for this claim. Global topological behavior of M results in passing from a local algebra to a subalgebra and, by the results of Section 3.1, this cannot change the submaximal dimension bound.

On models with large symmetry
Realizations of many symmetry algebras remain beyond the scope of this paper. For instance, we conjecture that non-maximal parabolic subalgebras of the pseudo-unitary algebras can also be realized as symmetries of polynomial CR models. Realization of other maximal subalgebras, discussed in the proof of Theorem 3.3, is important too.
Large symmetry algebras can also be obtained via intersection of maximal subalgebras. For instance, blow-up of the hyperquadric Qp at two different points reduces the symmetry algebra su( p, q) to the intersection of two conjugated parabolics p 1,n+1 that vary in dimension depending on position of the points.
The same problem is interesting for the automorphism group. For n = 2 consider the lens space L m = S 5 /Z m , m > 1, where Z m ⊂ U (1) acts on the unit sphere S 5 ⊂ C 3 by complex multiplication. By [17, p. 37] the Lie group Hol(L m ) is U (3)/Z m , again of dimension 9. Note that L m is everywhere spherical falling into part (ii) of Theorem 1.5.
Finally, provided the Levi nondegeneracy locus is nonempty for n > 2, the models of symmetry dimension D max are spherical. Classification of CR-hypersurfaces with symme-try dimension D smax is not fully solved even for n = 1. The real difficulties show in the construction of the models in [20]. The approach taken in this paper suggests more tractable problems: Which weighted blow-ups and ramified coverings of the hyperquadric lead to the models with submaximal symmetry dimension? Can these be classified? We hope these directions show fruitful in the future.
Acknowledgements Open Access funding provided by UiT The Arctic University of Norway. My first and foremost thanks go to Alexander Isaev, who influenced several results in this paper. Our correspondence was of invaluable help. He brought to my attention a discussion, where Stefan Nemirovski suggested a method for blowing up hyperquadrics in order to construct CR-manifolds with submaximal symmetry dimension. I am grateful for this idea, inspiring the following progress. The suggestion that blow-ups can be useful for submaximal symmetry models was also independently communicated to the author by Ilya Kossovskiy. The DifferentialGeometry package of Maple was used for extensive experiments that underly symmetry computations for all models in this paper.
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