Blow-ups and infinitesimal automorphisms of CR-manifolds

For a real-analytic connected CR-hypersurface M of CR-dimension n ě 1 having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra s “ spMq: (i) either dim s “ n ` 4n ` 3 and M is spherical everywhere; (ii) or dim s ď n ` 2n` 2` δ2,n 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.

1 Introduction 1.1 Formulation of the problem hyperquadric Q k :" ! pz, wq P C nˆC : Im w " k ÿ j"1 |z j | 2´n ÿ j"k`1 |z j | 2 ) (1.1) for some 0 ď k ď n{2. The Levi form of Q k has signature pk, n´kq everywhere and dim spQ k q " n 2`4 n`3 for all k. Thus, for the class of Levi-nondegenerate connected CR-hypersurfaces of CR-dimension n one has D max " n 2`4 n`3. Further, D smax " n 2`3 in the strongly pseudoconvex (Levi-definite) case for n ą 1 and D smax " n 2`4 in the Levi-indefinite case [K2]. The situation n " 1 is exceptional with D smax " 3 [C, KT].
In the absence of Levi-nondegeneracy, finding the maximal and submaximal dimensions of the symmetry algebra is much harder. To simplify the setup, in this case one usually switches to the real-analytic category by assuming the manifolds and the vector fields forming the symmetry algebra to be real-analytic rather than just smooth. In order to guarantee the finite-dimensionality of spMq " holpMq it then suffices to require that M is holomorphically nondegenerate, see [BER,§11.3,§12.5], [E, St]. Regarding the maximal possible value for dim spMq in this situation, let us mention the following variant of a conjecture due to V. Beloshapka,cf. [B2,p. 38]. The authors of [KS2] argument that for n " 1 this is a version of Poincaré's problème local [Po].
Conjecture 1.1 For any real-analytic connected holomorphically nondegenerate CRhypersurface M of CR-dimension n one has dim spMq ď n 2`4 n`3, with the maximal value n 2`4 n`3 attained only if on a dense open set M is spherical.
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 [IZ] where the proof relied on a reduction of 5-dimensional uniformly Levi-degenerate 2-nondegenerate CR-structures to absolute parallelisms (see [BER,§11.1] for the definition of k-nondegeneracy). Thus, for realanalytic connected holomorphically nondegenerate CR-hypersurfaces of CR-dimension 1 ď n ď 2 one has, just as in the Levi-nondegenerate case, D max " n 2`4 n`3.
It was shown in [KS2] that for n " 1 the condition dim holpM, xq ą 5 for x P M implies that M is spherical near x, where holpM, xq is the Lie algebra of germs at x of real-analytic vector fields on M whose flows consist of CR-transformations. In [IK1] we gave a short proof of this fact, and, applying the argument of [IK1] to the symmetry algebra spMq instead of holpM, xq, one also obtains D smax " 5. Notice that the result of [KS2,IK1] 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 [IK2] we considered the case n " 2. It was shown that in this situation either dim spMq " 15 and M is spherical, or dim spMq ď 11 with the equality occurring only if on a dense open subset M is spherical with Levi form of signature p1, 1q. 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 spMq " 11 thus establishing D smax " 11. This fact also led to the following analogue of the result of [KS2] for n " 2: the condition dim holpM, xq ą 11 for x P 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 Levi-signatures 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 CRdimension n ě 1 having a point of Levi-nondegeneracy. Then for its symmetry algebra s " holpMq exactly one of the two situations is possible: (i) dim s " n 2`4 n`3 and M is spherical everywhere, (ii) dim s ď n 2`2 n`2`δ 2,n and in the case of equality M is spherical on its Levi-nondegeneracy 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`2 n`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 P M. The proof of the theorem also leads to the following local version of the result, generalizing theorems from [KS2,IK1,IK2] for arbitrary n. Corollary 1.3 With the assumptions of Theorem 1.2 in the case n ě 3 the condition dim holpM, xq ą n 2`2 n`2 for x P M implies that M is spherical in a neighborhood of the point x, and this estimate is sharp.
As in papers [IK1,IK2], our argument relies on the techniques from Lie theory, notably on the description of proper subalgebras of maximal dimension of supp, qq obtained in Theorem 3.3, where 1 ď p ď q, p`q ě 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.
Theorem 1.4 All maximal parabolic subalgebras of the pseudo-unitary algebra supp, qq are realizable as the symmetry of a certain blow up of the standard hyperquadric (1.1).
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 [IK2]. 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 [KS1,KL], 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 Section 2.1. The relation of such blow-up to symmetry is not straightforward. We discuss it in Sections 2.2-2.3. 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 [IK2] and we construct more examples in Section 4.1. Acually, Theorem 4.1 gives a series of examples of pairwise nonequivalent CR-hypersurfaces with the submaximal value dim spMq " n 2`2 n`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 Section 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.
Theorem 1.5 Under the assumptions of Theorem 1.2 the automorphism group G " HolpMq satisfies one of the alternatives: (i) dim G " n 2`4 n`3 and M is spherical everywhere, (ii) dim G ď n 2`2 n`2`δ 2,n and in the case of equality M is spherical on its Levi-nondegeneracy 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 Section 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 Section 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 Section 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.
Acknowledgements. My first and foremost thanks go to Aleksander 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.

The blow-up construction
Recall a construction from affine geometry. Let L Ă V be a subspace of codimpL, V q " m. The blow-up of V along L (below Π is a subspace and x a point) is The projection π L : Bl L V Ñ V , px, Πq Þ Ñ x, is a biholomorphism when restricted onto V zL, and π´1 L pxq " PpV {Lq » CP m´1 for x P L. The projection p L : Bl L V Ñ PpV {Lq, px, Πq Þ Ñ Π, is the tautological line bundleˆL.
This construction canonically extends to complex geometry: if L is a submanifold of a manifold V , apply the above formula using local charts V Ą U α » C n , straightening L X U α and patching the charts, see e.g. [H]. 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 more convenient to present a version for embedded CR-surfaces (we restrict to hypersurfaces). In this section we present only the standard blow-up; variations on it, like iterated blow-ups, weighted blow-ups and ramified coverings will be discussed in Section 4.3.

Blow-up in CR-geometry
Let ι : M ãÑ V be a real hypersurface in a complex manifold of dimension n`1 and π L : Bl L V Ñ V a blow-up along a complex submanifold L meeting ιpMq. Definẽ This subset has singular points ΣM Ă π´1 L pM X Lq. For our purposes it is enough to assume V " C n`1 and L Ă V a subspace, so let us describe singularities in the affine version. These correspond to the points px, Πq with x P L X M and Π Ă Hpxq, where Hpxq is the CR-plane at the point x. Removing singularities we obtain Note that L does not belong to M in general. In this case we can encode L by L 1 " L X M: the germ of L along M Ă V is uniquely determined by L 1 . We continue however writing Bl L M (and not Bl L 1 M) using an embedding ι.
Note also that by the above description of singularities for x P L with m " codimpL, V q ą 1 the fiber π´1 L pxq is either CP m´1 or C m´1 " CP m´1 zCP m´2 . Hence, π´1 L pL 1 q is connected if L 1 " L X M is connected.
Proposition 2.1 For real-analytic CR-hypersurfaces M the CR-blowup construction is well-defined, i.e., a change of the embedding ι results in a CR-equivalence of Bl L M.
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 [AF]. Next, by Theorem 1.12 of loc.cit. such an embedding is unique up to a biholomorphism of (the germ of) a neighborhood of ιpMq Ă V . Since biholomorphisms naturally induce maps of blow-ups the claim follows. l Example 2.2 Let us blow-up the hyperquadric Q " tImpwq " }z} 2 u Ă C n pzqˆCpwq at the point o " p0, 0q, where }z} 2 " ř n j"1 σ j |z j | 2 , σ j "˘1, z " pz 1 , . . . , z n q. The blow-up contains the following open dense subset with π o pz, wq " pw¨z, wq. The model M for n " 1 appeared in [KS1].
The whole blow-up is obtained from !`p z 1 , . . . , z n , wq, rζ 1 :¨¨¨: ζ n : ̟s˘P C n`1ˆC P n , Impwq " }z} 2 , by removing singularities. In the chart ̟ ‰ 0 we get U 0 " M as above. For 1 ď k ď n in the chart ζ k ‰ 0 we get The singularities of U k are Σ k " tz k " 0, w " 0u, so U 1 k " U k zΣ k is the nonsingular part. The projections π k o : U 1 k Ñ Q and the gluing maps ϕ k : U 1 k Ñ U 0 (whenever defined) are given by the formulae: π k o pz 1 , . . . , z n , wq " pz 1 z k , . . . , z k´1 z k , z k , z k z k`1 , . . . , z k z n , z k wq, ϕ k pz 1 , . . . , z n , wq "´z . . , U n by gluing via ϕ 1 , . . . , ϕ n . In what follows we often change the blow-up Bl o Q to M.
Example 2.3 More generally, let C n pzq " C n´k pz 1 qˆC k pz 2 q be the direct product and let }z} 2 " }z 1 } 2`} z 2 } 2 be the quadric of signature pp,qq, where both quadrics }z 1 } 2 and }z 2 } 2 are nondegenerate. Let L " C k pz 2 q. The corresponding blow-up contains the following model with π L pz 1 , z 2 , wq " pw¨z 1 , z 2 , wq.

Symmetry of a Blow-up
Next we describe how symmetry of M changes upon a blow-up along L in the case of our current interest. By ι : M Ñ V we denote a CR-embedding (realization), as above.
Theorem 2.4 Let M be a real analytic CR-hypersurface having Levi-nondegenerate points and spMq be its symmetry algebra. Let L Ă V be a complex submanifold of codimension m ą 1. Assume that either L Ă M or m ą n 2`1 . Then the symmetry algebra of the blow-up spBl L Mq is the subalgebra in spMq consisting of symmetries preserving L, i.e., tangent to L along M.
The same is true for the germs of symmetries, i.e., holpBl L M,xq Ă holpM, π L pxqq is determined by the condition of preserving L for everyx P Bl L M, π L pxq P L 1 .
Proof. We can assume M to be connected: the claims for every connected component of M considered independently, yield the same result for the whole M.
By [AF,Theorem 1.12] and [BER,Proposition 12.4.22], the infinitesimal symmetries of M are bijective with holomorphic vector fields on the germ of M in V that along M are tangent to HpMq, the holomorphic tangent bundle of M. In other words, every real-analytic infinitesimal CR-automorphism defined on an open subset U 1 Ă M is the real part of a holomorphic vector field defined on an open subset U Ă V with ιpU 1 q Ă ιpMq X U. The condition in the theorem is easily verified to be independent of the choice of an embedding ι. Now we claim that Bl L M is Levi-degenerate along π´1 L pL 1 q. If L Ă M then actually Bl L M is non-minimal: π´1 L pL 1 q is a complex hypersurface in Bl L M, implying the claim. If m ą n 2`1 , then π´1 L pxq is a complex submanifold through x P L 1 of dimension ą n 2 . Therefore the Levi form L Bl L M restricts to zero on Txπ´1 L pxq, π L pxq " x, and L Bl L M pxq is degenerate for all such pointsx.
We conclude that the symmetries of Bl L M must be tangent to π´1 L pL 1 q. Thus they descend to the blow-down manifold M and we get a map q L : spBl L pMqq Ñ spMq. This map is clearly a homomorphism of Lie algebras.
Since π L : Bl L M Ñ M is a biholomorphism over MzL 1 the map q L is injective. Indeed, q L psq " 0 fors P spBl L Mq impliess| U " 0 for U Ă MzL 1 and therefores " 0 by analyticity and connectedness of Bl L M.
On the other hand, it is clear that the vector fields s P spMq that lift to Bl L M must preserve L. If s does so, 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 is surjective. l Example 2.5 The symmetry algebra of the hyperquadric Q is supp, qq, where pp,qq " pp´1, q´1q is the signature of the Levi form. The isotropy algebra of a point is the first parabolic subalgebra p 1,n`1 , and hence this is the symmetry of the blow-up model Bl o Q constructed in Example 2.2. In Section 4.2 we will give explicit formulae for the symmetry fields of the open dense submanifold M Ă Bl o Q.
This example implies the following statement.
Corollary 2.6 The first parabolic subalgebra p 1,n`1 Ă supp, qq 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 Ă Bl o Q.
Proof. That the symmetry of the Bl o Q is as indicated follows from Theorem 2.4. Let us also show that the symmetry does not grow upon restriction to the submanifold M. The subset π o pMq Ă Q is obtained from the quadric by removing the hyperplane tw " 0u punctured at o. The symmetry algebra of both π o pMq and Q is supp, qq. Now the same argument as in the above proof shows that q L : spMq Ñ spπ o pMqq is an injective map with the image consisting of symmetry fields vanishing at o. l Example 2.7 Considering the more general blow-up model from Example 2.3 with k " dim C L P p0, nq we conclude that its symmetry is smaller in dimension than the parabolic subalgebra fixing a subspace of dimension pk`1q in linear representation. For instance, forp "q " 1 the symmetry algebra spBl L Qq has dimension 8, while the corresponding parabolic algebra p 2 Ă sup2, 2q has dimension 11. This is in accordance with Theorem 2.4, if one verifies the action of spQq on L.

Automorphisms of a Blow-up
The argument of the previous theorem extends to the group case and we get: Theorem 2.8 Under the assumptions of Theorem 2.4, if G is the automorphism group of M, then the automorphism group for Bl L pMq is the stabilizer of L in G. l We give an application of this theorem. Let p`q " n`2, 1 ď s ď p ď q. Recall that the parabolic subgroup P s,n´s`2 Ă SUpp, qq is the stabilizer of a null s-plane (and thus also of the orthogonal co-isotropic pn´s`2q-plane) in the standard representation of SUpp, qq on C n`2 , its Lie algebra is the parabolic subalgebra p s,n´s`2 .
Example 2.9 Let k " s´1. Consider the hyperquadric Q p´1 Ă C n pzqˆCpwq defined as Let L :" tpz, wq P C n`1 : z j " 0 p1 ď j ď kq, z ℓ " 0 p2k`1 ď ℓ ď nq, w " 0u. (2.2) 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 Ă C n pzqˆCpwq given by with the projection π L : S Ñ Q p´1 given by π L pz 1 , . . . , z n , wq " pz 1 w, . . . , z k w, z k`1 , . . . , z 2k , z 2k`1 w, . . . , z n w, wq. (2.4) The hypersurface S contains the hyperplane tw " 0u " π´1 L pLq, and for every x P L the fiber π´1 L pxq is an pn´kq-dimensional vector subspace of C n`1 . The singular locus of S is given by The CR-hypersurface M is obtained by excluding Σ S from S. Thus M Ă S is an open subset containing an open subset of the hyperplane tw " 0u. By Theorem 2.4 the symmetry algebra spMq is p s,n´s`2 . We do not provide details of this derivation here because in Section 4.2 we present these symmetries explicitly. This will realize all maximal parabolic subalgebras of supp, qq.
Note that the automorphism group of the spherical surface Q p´1 is not SUpp, qq, 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 2.10 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 " tξ " 0u. The Lie group G " PSUpp, qq 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 " pk`1qp2n´3k`1q. The stabilizer of a point L in N is the parabolic subgroup P s,n´s`2 Ă P SUpp, qq, and one can verify using Theorem 2.8 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 supp, qq, 1 ď p ď q, p`q " n`2 ě 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 supn`2q, 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 ď s ď 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 Ă supp, qq is d n psq :" n 2´2 sn`3s 2`4 n´4s`3.
Proof. For g :" supp, qq the grading g " g´ν '¨¨¨' g 0 '¨¨¨' g ν corresponding to a parabolic subalgebra p » g 0 '¨¨¨'g ν of g has g 0 » zpg 0 q'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: dim p s,n´s`2 " 1 2 pdim g`dim g 0 q " 1 2 pdim A n`1`2`2 dim A s´1`d im A n´2s`1 q " 1 2 ppn`2q 2`p n´2s`2q 2 q`s 2´1 " d n psq, where we set dim A s :" 0 for s ď 0. l Some initial values of d n psq are as follows: n å s 1 2 3 4 1 5 2 10 11 3 17 16 4 26 23 26 5 37 32 33 6 50 43 42 47 7 65 56 53 56 Corollary 3.2 The maximal dimension of a parabolic subalgebra of supp, qq is uniquely given by dim p 1,n`1 " n 2`2 n`2 except for n " 2 where the maximum is attained by dim p 2 " 11 ą dim p 1,3 and n " 4 where dim p 3 " dim p 1,5 " 26. Now we restrict the dimension of a proper subalgebra of the pseudounitary algebra, which simultaneously gives a bound for subgroups of the pseudounitary group.
Theorem 3.3 A proper subalgebra of supp, qq of maximal dimension is a parabolic subalgebra, as described in Corollary 3.2.
Proof. By Mostow's theorem [M], a maximal subalgebra of a real simple Lie algebra is either parabolic, or the centralizer of a pseudotorus, or semisimple.
The centralizers of pseudotoric subalgebras of supp, qq have the maximal possible dimension for either upp, q´1q or upp´1, qq, both of dimension pn`1q 2 ă dim p 1,n`1 .
Next, fix a semisimple subalgebra h Ă supp, qq; by complexifying it we obtain a subalgebra h C Ă supp, qq C " slpn`2, Cq. By Dynkin's theorem (see [D] and also [GOV,Chap. 6,Sect. 3.2]) a maximal semisimple subalgebra of the simple Lie algebra of type A n`1 is either (i) nonsimple irreducible, or (ii) simple irreducible.
If h C falls in Case (i), we have n`2 " st (1 ă s ď t ă n`2, hence n ě 2) and h C " slps, Cq ' slpt, Cq is embedded in slpn`2, Cq via the representation on C s b C t " C n`2 . Then dim C h C " s 2`t2´2 " s 2`p n`2q 2 s 2´2 . The maximum of the function s 2`p n`2q 2 s 2´2 on the interval 2 ď s ď 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 slpn`2, Cq. If h C has type A, then dim C h C is maximal if h C " slpk`1, Cq Ă slpn`2, Cq, k ď n, which does not give the optimal dimension as dim C slpk`1, Cq ď n 2`2 n ă dim p 1,n`1 .
If h C has type B or D, then dim C h C is maximal if h C " sopn`2, Cq Ă slpn`2, Cq, which does not give the optimal dimension since dim C sopn`2, Cq " 1 2 pn 2`3 n`2q ă 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 " spp2k`2, Cq Ă slpn`2, Cq, which again does not give the optimal dimension as dim C spp2k`2, Cq " pk`1qp2k`3q. 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 " LiepG 2 q has dimension 7 (V " R λ 1 ), so if h C " g 2 we have n ě 5. Hence g 2 does not give the optimal dimension since dim C g 2 " 14 ă 5 2`2¨5`2 .
Similarly, the representation V of minimal dimension of the exceptional Lie algebra f 4 " LiepF 4 q has dimension 26 (V " R λ 4 ), so if h C " f 4 we have n ě 24. Hence f 4 does not give the optimal dimension since dim C f 4 " 52 ă 24 2`2¨2 4`2.
Thus, all semisimple subalgebras of supp, qq have dimensions strictly smaller than the maximal possible dimension of a parabolic subalgebra. l Remark 3.4 By [IK2, Proposition 2.1; Remark 2.5], for n " 2 every proper subalgebra of supp, qq of dimension 10 " n 2`2 n`2 is also parabolic and conjugate to p 1,3 .

Establishing the submaximal symmetry dimension
We assumed that M has a point of Levi-nondegeneracy, which implies that M is holomorphically nondegenerate, see [BER,Theorem 11.5.1]. The condition of holomorphic nondegeneracy for a real-analytic hypersurface in complex space was introduced in [St] 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 [BER,§11.3], [E], but we only make a note of the fact, stated in [BER,Corollary 12.5.5], that the holomorphic nondegeneracy of M is equivalent to the finite-dimensionality of all the algebras holpM, xq. Notice that together with [BER,Proposition 12.5.1] this corollary implies that the finite-dimensionality of holpM, x 0 q for some x 0 P M implies the finite-dimensionality of holpM, xq for all x P M.
Clearly, spMq " holpMq may be viewed as a subalgebra of holpM, xq for any x. Therefore for a holomorphically nondegenerate M, and in particular for the case we consider, the symmetry algebra spMq is finite-dimensional.
Proof of Theorem 1.2. For n " 1 the theorem was obtained in [KS2,IK1], for n " 2 its stronger variant was proven in [IK2], so we assume that n ě 3.
Let Choose a point x P U. The natural map holpMq Ñ holpUq Ñ holpM, xq is injective. If x is not spherical, [K2] implies dim holpMq ď n 2`4 that is less than n 2`2 n`2.
Thus every point of U is spherical. Then holpMq is a subalgebra of supp, qq for some 1 ď p ď q, p`q " n`2, and by Theorem 3.3 there is an alternative: (i) holpMq " supp, qq; (ii) dim holpMq ď n 2`2 n`2.
Consider Case (i) first. We will show that M is spherical everywhere. Since we already established sphericity on U, fix a point x 0 P S M . Consider the isotropy subalgebra hol 0 pMq of holpMq at x 0 . Clearly, dim hol 0 pMq ě pn`2q 2´1´p 2n`1q " n 2`2 n`2. Hence, appealing to Theorem 3.3 once again, we see that one of the following holds: (ia) dim hol 0 pMq " n 2`2 n`2; (ib) hol 0 pMq " holpMq " supp, qq.
In Case (ia), the orbit of x 0 under the corresponding local action of the group SUpp, qq is open, so it contains a spherical point x P U, and hence M is spherical near the point x 0 as well.
In Case (ib), by the Guillemin-Sternberg theorem [GS,, the action of the simple Lie algebra supp, qq is linearizable near x 0 , and we obtain a nontrivial p2n`1qdimensional representation of supp, qq. But the lowest-dimensional representation of supp, qq is the standard C p,q of real dimension 2n`4, which is a contradiction.
Consider now Case (ii) and assume that dim holpMq " n 2`2 n`2. Then by Theorem 3.3 the algebra holpMq is isomorphic either to the parabolic subalgebra p 1,n`1 of supp, qq, or, if n " 4 and p " q " 3, to the parabolic subalgebra p 3 of sup3, 3q. 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 2.6. This finished the proof. l Proof of Corollary 1.3. If M is holomorphically nondegenerate, then for every x P M there exists a connected neighborhood U of x in M for which the natural map holpUq Ñ holpM, xq is surjective [BER,Proposition 12.5.1]; for any such U we have holpM, xq " holpU, xq " holpUq. Taking U instead of M in Theorem 1.2, the statement of the corollary follows. l

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 ě 1 having a Levi-nondegenerate point. If dim spMq ě n 2`2 n`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 Section 3.2. In this case dim s " 10 " n 2`2 n`2, and the statement follows by Remark 3.4 since the parabolic subalgebras p 1,3 Ă sup1, 3q and p 1,3 Ă sup2, 2q derived in [IK2] are not isomorphic. l Next, set d 0 :" Proposition 3.6 Under the assumption of Proposition 3.5 the inequality dim s ą d 0 implies that M is spherical on its Levi-nondegeneracy locus, possibly with different signatures of the Levi form at different points.
Proof. Let S be the Levi-nondegeneracy locus of M. If there exists a point of S near which M is not spherical, then, since the natural map holpMq Ñ holpM, xq is injective for every x P M, by [C], [K2] we have dim holpMq ď d 0 . l Remark 3.7 Concerning Proposition 3.6, [KS1, Example 6.2] actually shows that it is possible for the Levi-nondegeneracy locus S of a real-analytic CR-hypersurface M to be disconnected, for the signature of the Levi form of M to be different on different connected components of S, and for M to be locally CR-equivalent to different hyperquadrics near different points. By Proposition 3.5 such an effect is impossible if the algebra holpMq has large dimension. The hypersurface M Ă C 3 from [KS1, Example 6.2] is given by the equation We found that its symmetry algebra is spanned by the vector fields R :" 2 Rep´z 2 B z 2`2 wB w q, S :" Repiz 1 B z 1 q, J :"´2 Repiz 2 B z 2 q, X :" Repiz 1 z 2 wB z 1`B z 2`2 iz 2 w 2 B w q, Y :" Repz 1 z 2 wB z 1`i B z 2`2 z 2 w 2 B w q, Z :" Repz 1 wB z 1`2 w 2 B w q.
This algebra is isomorphic to R 3 iheis 3 , where R is the grading element, S is the center and J is the complex structure on the contact subspace in heis 3 " xX, Y, Z : rX, Y s " Zy.

Group version of the main results
Let us first note that the situations in (ii) can correspond to the existence of Levidegenerate points as in Theorem 1.2 (the models are in Example 2.10), 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 ps´1q reduces P SUpp, qq 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 spMq 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 " LiepGq the Lie algebra of G. Because g Ă 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 ą dim g and this implies, by Theorem 3.3, that dim s " n 2`4 n`3, so we are under the assumption of case (i) in Theorem 1.2, which yields sphericity of M everywhere. l

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 CRhypersurface with dim spMq " n 2`2 n`3 " 11 that is more elementary compared to those discussed in [IK2]. For n " 4 we get an example of a CR-hypersurface with dim spMq " n 2`2 n`2 " 26; its algebra dim spMq " p 3 Ă sup3, 3q 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 Section 4.1 and those from [IK2] can be seen as a combination of a blow-up and a ramified covering.

A series of different realizations of p 1,n`1
Fix n ě 1 and 1 ď p ď q with p`q " n`2, and set pp,qq :" pp´1, q´1q. The parabolic subalgebra g :" p 1,n`1 Ă supp, qq, which has a 2-grading g " g 0 ' g 1 ' g 2 , is abstractly isomorphic to g 0 i g`, where g 0 " supp,qq ' R 2 and g`" g 1 ' g 2 " Cp ,q i R is the Heisenberg algebra of dimension 2n`1.
For every m P N and ε "˘1 consider the real-analytic hypersurface M m,ε given in coordinates z 1 , . . . , z n , w " u`iv in C n`1 by v " εu tanˆ1 2m arcsinp}z} 2 q˙, }z} ă 1, (4.5) where }z} 2 :" is the standard Hermitian form of signature pp,qq. Here σ j :"`1 for 1 ď j ďp and σ j :"´1 for p ď j ď n (notice that σ j "´1 for all j in the Levi-definite case). For n " 1 this hypersurface was introduced in [B2] and also appeared in [KL]. Clearly, M m,ε contains the complex hypersurface Σ :" t}z} ă 1, w " 0u " M m,ε X tu " 0u and is Levi-nondegenerate with signature pp,qq away from Σ. The complement M m,ε zΣ has exactly two connected components; they are defined by the sign of u. The hypersurface M m,ε is not minimal, hence not of finite type (in the sense of Kohn and Bloom-Graham) at any point of Σ (see [BER,§1.5

]).
We now observe that every point pz, wq P M m,ε satisfies the equation Impw 2m q a 1´}z} 4 " ε Repw 2m q }z} 2 . (4.6) In fact, for every value of ε, equation (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 tw " 0u. For example, when m " 1 the set described by equation (4.6) is the union of the following two smooth hypersurfaces: v " εu tanˆ1 2 arcsinp}z} 2 q˙and u "´εv tanˆ1 2 arcsinp}z} 2 q˙, }z} ă 1.
Each of the 2m hypersurfaces given by (4.6) is spherical away from Σ. Indeed, fix a point pz 0 , w 0 q satisfying (4.6) with w 0 ‰ 0. Then Repw 2m 0 q ‰ 0, and setting σ :" ε sgn Repw 2m 0 q, we see that the map pz, wq Þ Ñ pzw m , σw 2m q transforms a neighborhood of pz 0 , w 0 q on the relevant hypersurface to an open subset of the hyperquadric (1.1) that we rewrite so Qp :" tpz, wq P C nˆC : v " }z} 2 u. (4.7) Theorem 4.1 For every m P N and ε "˘1 the symmetry algebra of M m,ε has dimension n 2`2 n`2; in fact one has s " p 1,n`1 . Furthermore, for m ‰ k and any ε, δ P t´1, 1u neither the hypersurfaces M m,ε and M k,δ nor their germs at the origin are equivalent by means of a real-analytic CR-diffeomorphism. In addition, for p ‰ q neither the hypersurfaces M m,´1 and M m,`1 nor their germs at the origin are equivalent by means of a real-analytic CR-diffeomorphism.
Proof. It is straightforward to check that the following vector fields span the algebra spM m,ε q, where 1 ď j ď n, j ă ℓ ď n and summation over repeated indices is not assumed: RepwB w q, Re`w 2m pmξ`wB w q˘, Re`z j w m pmξ`wB w q`imεσ j w m B z j˘, Re`iz j w m pmξ`wB w q`mεσ j w m B z j˘, with ξ :" ř n j"1 z j B z j . Furthermore, one can check that vector fields (4.8) define a faithful representation of the parabolic subalgebra p 1,n`1 Ă supp, qq.
Since the surface M m,ε is not everywhere spherical, Theorem 1.2 yields the upper bound dim spM m,ε q ď n 2`2 n`2`δ 2,n . Moreover in case n " 2 the equality is only attained for the parabolic subalgebra p 2 Ă sup2, 2q, and since the other parabolic p 1,3 does not embed into p 2 , we conclude that in fact dim spM m,ε q ď n 2`2 n`2. But since vector fields (4.8) in totality n 2`2 n`2 constitute the symmetries of the model, we conclude the opposite inequality and hence spM m,ε q " p 1,n`1 .
Similarly, for any connected neighborhood U of a point x P Σ in M m,ε we have holpUq " p 1,n`1 , while if U X Σ " H we get holpUq " supp, qq.
Next, formulas (4.8) show that all elements of spM m,ε q 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 p0q " 0. Observe now that the highest order of the vanishing of a vector field in the algebra spM m,ε q 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 p0q " 0. Since F holomorphically extends to a neighbourhood of the origin, let us write it as pz, wq Þ Ñ pf pz, wq, gpz, wqq, with f pz, wq " Az`Bw`¨¨¨, gpz, wq " Cw`¨¨¨, where dots denote higher-order terms and A, B, C are complex matrices of sizes nˆn, nˆ1, 1ˆ1, respectively (note that g contains no linear terms in z because F must preserve Σ). The condition that F maps M m,´1 to M m,`1 is written as the identity " Im`gpz, wq˘´Re`gpz, wq˘¨tanˆ1 2m arcsinp}f pz, wq} 2 q˙ˇˇˇˇw "u´iu tanp 1 2m arcsinp}z} 2 qq " 0.
The terms linear in u yield Im C " 0. Then Re C ‰ 0 and the next terms in decomposition of the above identity imply }z} 2 "´}Az} 2 .
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. l 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 pm, εq for n " 1 was claimed in [B2].

Realization of maximal parabolics
We will now construct realizations of all the maximal parabolic subalgebras p s,n´s`2 , 1 ď s ď p. Finding such realizations is interesting in its own right, as this adds up to the study of symmetry of polynomial CR models, cf. [B1,KMZ,KM]. The first model has been already introduced in Example 2.9, see equation (2.3) applicable to all 1 ď s ď 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 tw " 0u (coincides with it for s " 1). The second model is applicable for 1 ă s ă n 2`1 , i.e. k :" s´1 P p0, n 2 q. It is a blow up along the following subspace in C n pzqˆCpwq of dimension n´k: L :" tpz, wq P C n`1 : z j " 0 p1 ď j ď kq, w " 0u. (4.9) An important difference between (2.2) and (4.9) is that the latter L is not contained in Qp, so in a sense we are blowing up the hyperquadric along the real-analytic subvariety L 1 :" LXQp. An open subset of Bl L Qp embeds into the hypersurface S Ă C n pzqˆCpwq given by Impwq " k ÿ j"1 pz j wz k`j`zk`jwzj q`}z 1 } 2 (4.10) (}z 1 } 2 has the same meaning as in Example 2.9) with the projection given by π L pz 1 , . . . , z n , wq " pz 1 w, . . . , z k w, z k`1 , . . . , z 2k , z 2k`1 , . . . , z n , wq. (4.11) The hypersurface S contains the real-analytic subvariety S 1 :" t}z 1 } " 0, w " 0u " π´1 L pL 1 q, and for every x P L 1 the fiber π´1 L pxq is a k-dimensional vector subspace of C n`1 . This subvariety has real dimension Excluding the singular locus from S, we obtain our second model denoted M s II . Its Levi-degeneracy locus is an open subset S s II of S 1 .
Theorem 4.3 Both models M s I (1 ď s ď n 2`1 ) and M s II (1 ă s ă n 2`1 ) have symmetry algebra s " p s,n´s`2 . They are neither globally CR equivalent nor locally equivalent near the Levi degeneracy 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 the following are the generators of spM s I q with indices ranging so: 1 ď a, b ď k, a ă c ď k, 2k`1 ď j ď n, j ă ℓ ď n.
Similarly, for (4.10), if we denote and use the same range for indices, then the following are the generators of spM s II q: RepB za`2 iz a`k ξq, RepiB za`2 z a`k ξq, RepB z a`k´2 iz a ηq, RepiB z a`k´2 z a ηq, Repw`B z a`k`2 iz a ξq˘, Repw`iB z a`k`2 z a ξq˘, Repwξq, Repζ`wB w q, Repz j B z a`k´σ j z a wB z j q, Repiz j B z a`k`i σ j z a wB z j q, RepwB z j`2 iσ j z j ξq, RepiwB z j`2 σ j z j ξq.
Finally note that the CR-manifolds M s I and M s II for 1 ă s ă n 2`1 are not equivalent even by means of a smooth CR-diffeomorphism. Indeed, any such diffeomorphism must preserve the points of Levi-degeneracy and therefore map S s

Clearly, r
Qp is singular if r ą 1. We will now blow up r Qp at the origin by a weighted analogue of map π L with L " o, namely, by the map π o,m : pz, wq Þ Ñ pzw m , wq, where m P N. The result of the blow-up is the hypersurface R r,m :" π´1 o,m p r Qpq, which is described by the equation Impw r q " σ|w| 2m }z} 2 . (4.14) Fix m P N and set r " 2m. One can rewrite equation (4.14) of R 2m,m as Impw 2m q " σ a Repw 2m q 2`I mpw 2m q 2 }z} 2 .
Taking squares, we obtain Impw 2m q 2 p1´}z} 4 q " Repw 2m q 2 }z} 4 , i.e. Impw 2m q a p1´}z} 4 q " σ|Repw 2m q|¨}z} 2 . (4.15) 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 [KL], see, e.g., Lemmas 22, 26 therein.
Let us note that surface (4.14) for r " 1 is an iterated blow-up of the hyperquadric (1.1). In fact, o " p0, 0q P C n pzqˆCpwq is the regular point of the surface Q m " tImpwq " |w| 2m }z} 2 u.
Note that Q 0 " Qp, wherep " p´1 and pp,qq is the signature of the quadric }z} 2 (p`q " n), and that Q j`1 " Bl o Q j for all j ě 0.
By Theorem 2.4 the symmetry algebra of Q 1 is obtained from that of Q 0 as stabilizer of o in supp, qq, 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 2.4 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 spQ m q " n 2`2 .
Actually, this dimension persists for the symmetry algebra of the ramified equation.
Proposition 4.8 The symmetry algebra of (4.14) for m ą 1 is upp,qq ' solp2q, where solp2q is the two-dimensional solvable Lie algebra.
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 ď ℓ ď n, ℓ ă j ď n.
The abstract Lie algebra structure is straightforward. l Finally, let us show an example of iterated blow-up giving a surface with a nonmaximal parabolic symmetry algebra. We do it in the simplest case n " 2 with parabolic p 1,2,3 in sup2, 2q being the Borel subalgebra.
The explanation is as follows. The surface (4.18) is only an open dense part of the blow-up of (4.16) along L 2 . The projection π L 2 is not epimorphic on M, the subset tpz 1 , z 2 , 0q : z 2 ‰ 0u is not in the image of π L 2 restricted to (4.18). This implies that the symmetry of M fixing this subset and L 1 (as per definition) also fixes their intersection, i.e., the point o " p0, 0, 0q leading to the symmetry algebra p 1,3 .

Conclusion
Let us outline a generalizations of our results and formulate some open problems.

On generalization of the main result
Motivated by the results in present paper and a series of previous results in complex analysis, we formulate the following claim, generalizing Conjecture 1.1.
Conjecture 5.1 Symmetry of any real-analytic connected holomorphically nondegenerate CR-hypersurface M of CR-dimension n satisfies dim spMq ď n 2`4 n`3, with the maximal value attained only if M is everywhere spherical. Otherwise dim spMq ď n 2`2 n`2`δ 2,n , with the maximal value attained only if on a dense open set M is spherical and of fixed signature of the Levi form.
Let us support this claim with some evidence. As shown it holds for 1 ď n ď 2, and also for larger n, provided M is Levi nondegenerate somewhere. It is instructive to recall that the case n " 2 in [IK2] utilized the following fact: a real-analytic connected holomorphically nondegenerate CR-hypersurface of dimension 5 with everywhere degenerate Levi form is generically 2-nondegenerate. By appealing to the main result of [IZ], this allowed to estimate the dimension of the symmetry algebra in everywhere Levi-degenerate case by 10, see also [MS, MP] for alternative approaches.
For n ě 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 [P] for n " 3 and in [PZ] for general n. The upper bound on symmetry achieved in those references confirms our conjecture. Also in [Sa] 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 (like in [K1]) for this problem can provide effective bounds on local symmetry important for this claim.
Global topological properties cannot change the submaximal bound, as it effects in passing from a local algebra to a subalgebra, and we have already investigated maximal subalgebras of the (conjecturally) maximal symmetry algebras supp, qq.

More models with large symmetry
While we computed many examples in this paper, some other interesting problems remain un-answered. For instance, realizations of other (non-maximal) parabolic subalgebras of the pseudo-unitary algebras. As we indicated in Example 4.9 an iterated blow-up yields such in the simplest case, and we expect this to work in general.
What about other maximal subalgebras, discussed in the proof of Theorem 3.3, namely reductive (stabilizers of pseudotoric subalgebras) and semi-simple algebrascan all of them be realized as the symmetry of some CR hypersurface?
Further large symmetry algebras can be obtained as intersection of those discussed. For instance, making blow-ups at two different points of the hyperquadric Qp reduces the symmetry algebra supp, qq to the intersection of two different parabolic subalgebras p 1,n`1 that are not aligned to the same choice of roots, and this intersection can vary in dimension depending on position of the points.
In [IK2] we presented a series of examples of 5-dimensional CR manifolds with symmetry dimension ď 11. In particular, for n " 9 we borrowed the following example from [KM]: M 5 " tImpwq " |z 1 | 2`| z 2 | 4 u. We determined generators of the symmetry algebra, but not its Lie algebra structure. It is remarkable, spM 5 q " up1, 2q, i.e., this is one of the reductive maximal subalgebras in sup2, 2q. Notice that it differs from the symmetry algebra p 1,2,3 , also of dimension 9, constructed in Example 4.9. Thus these CR models with 9-dimensional symmetry are non-equivalent.
The same problem is interesting for the automorphism group. Again, returning to symmetry dimension 9 for n " 2, consider the lens space L m :" S 5 {Z m , m ą 1, where Z m Ă Up1q acts on the unit sphere S 5 Ă C 3 by complex multiplication. By [I1, p. 37] the Lie group HolpL m q is naturally isomorphic to Up3q{Z m and thus has dimension 9. Note that L m is everywhere spherical falling into part (ii) of Theorem 1.5.
Finally, we know that (provided there are Levi nondegenerate points for n ą 2) the models of maximal symmetry dimension D max are spherical. However the classification of CR-hypersurfaces with the symmetry algebra of dimension D smax is a difficult task. It is not fully solved even for n " 1, but the real difficulties show up in the construction of the models in [IK2]. However the approach taken in this paper gives more hope. The following problems are open but appeal to be solved: Which weighted blow-ups of the hyperquadric lead to the models with symmetry of submaximal dimension? Which ramified coverings preserve this property? Is it possible to classify all such cases up to CR-automorphism? We hope these direction show fruitful in the future.