Blow-ups and infinitesimal automorphisms of CR-manifolds
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Mathematische Zeitschrift
Blow-ups and infinitesimal automorphisms of CR-manifolds Boris Kruglikov1 Received: 18 December 2018 / Accepted: 26 January 2020 © The Author(s) 2020
Abstract For a real-analytic connected CR-hypersurface M of CR-dimension n 1 having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra s = s(M): (i) either dim s = n 2 + 4n + 3 and M is spherical everywhere; (ii) or dim s n 2 + 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. Keywords Real hypersurface in complex space · CR-automorphism · Holomorphic vector field · Submaximal symmetry dimension · Parabolic subalgebra · Gap phenomenon Mathematics Subject Classification 32V40 · 32C05 · 32M12 · 53C15
1 Introduction 1.1 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 Dmax of the symmetry dimension dim s(M) over all M ∈ C . Often the values immediately below Dmax 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 Dsmax , thus obtaining the interval (Dsmax , Dmax ) 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
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Boris Kruglikov [email protected] Department of Mathematics and Statistics, UiT the Arctic University of Norway, 90-37 Tromsö, Norway
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B. Kruglikov
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.
1.2 The status of knowledge Recall that an almost CR-structure on a smooth manifold M is a subbundle H (M) ⊂ T (M) of the tangent bundle, called the CR-distribution, endowed with a field of operators Jx : Hx (M) → Hx (M), Jx2 = −id, smoothly depending on x ∈ M. CR-dimension of M is CRdim M = 21 rank H (M), CR-codimension of M is dim M −rank H (M). The complexified CR-distribution spli
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