In mathematics, a CR manifold is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge.
The bundle L is called a CR structure on the manifold M.
- 1 Introduction and motivation
- 2 Embedded CR manifolds
- 3 Abstract CR structures
- 4 Examples
- 5 See also
- 6 Notes
- 7 References
Introduction and motivation
The notion of a CR structure attempts to describe intrinsically the property of being a hypersurface in complex space by studying the properties of holomorphic vector fields which are tangent to the hypersurface.
Suppose for instance that M is the hypersurface of C2 given by the equation
where z and w are the usual complex coordinates on C2. The holomorphic tangent bundle of C2 consists of all linear combinations of the vectors
The distribution L on M consists of all combinations of these vectors which are tangent to M. In detail, the tangent vectors must annihilate the defining equation for M, so L consists of complex scalar multiples of
Note that L gives a CR structure on M, for [L,L] = 0 (since L is one-dimensional) and since ∂/∂z and ∂/∂w are linearly independent of their complex conjugates.
More generally, suppose that M is a real hypersurface in Cn, with defining equation F(z1, ..., zn) = 0. Then the CR structure L consists of those linear combinations of the basic holomorphic vectors on Cn:
which annihilate the defining function. In this case, for the same reason as before. Moreover, [L,L] ⊂ L since the commutator of vector fields annihilating F is again a vector field annihilating F.
Embedded and abstract CR manifolds
There is a sharp contrast between the theories of embedded CR manifolds (hypersurface and edges of wedges in complex space) and abstract CR manifolds (those given by the Lagrangian distribution L). Many of the formal geometrical features are similar. These include:
- A notion of convexity (supplied by the Levi form)
- A differential operator, analogous to the Dolbeault operator, and an associated cohomology (the tangential Cauchy-Riemann complex).
This article first treats the geometry of embedded CR manifolds, shows how to defined these structures intrinsically, and then generalizes these to the abstract setting.
Embedded CR manifolds
Embedded CR manifolds are, first and foremost, submanifolds of Cn. Define a pair of subbundles of the complexified tangent bundle C ⊗ TC'n by:
- T(1,0)Cn consists of the complex vectors annihilating the antiholomorphic functions. In the holomorphic coordinates:
- T(0,1)Cn consists of the complex vectors annihilating the holomorphic functions. In coordinates:
Also relevant are the characteristic annihilators from the Dolbeault complex:
- Ω(1,0)Cn = (T(0,1)Cn)⊥. In coordinates,
- Ω(0,1)Cn = (T(1,0)Cn)⊥. In coordinates,
The exterior products of these are denoted by the self-evident notation Ω(p,q), and the Dolbeault operator and its complex conjugate map between these spaces via
Furthermore, there is a decomposition of the usual exterior derivative via .
Real submanifolds of complex space
Let M ⊂ Cn be a real submanifold, defined locally as the locus of a system of smooth real-valued functions
- F1 = 0, F2 = 0, ..., Fk = 0.
Suppose that this system has maximal rank, in the sense that the differentials satisfy the following independence condition:
Note that this condition is strictly stronger than needed to apply the implicit function theorem: in particular, M is a manifold of real dimension 2n − k. We say that M is an embedded CR manifold of CR codimension k. In most applications, k = 1, in which case the manifold is said to be of hypersurface type.
Let L ⊂ T(1,0)Cn|M be the subbundle of vectors annihilating all of the defining functions F1, ..., Fk. Note that, by the usual considerations for integrable distributions on hypersurfaces, L is involutive. Moreover, the independence condition implies that L is a bundle of constant rank n − k.
Henceforth, suppose that k = 1 (so that the CR manifold is of hypersurface type), unless otherwise noted.
The Levi form
This determines a metric on L. M is said to be strictly pseudoconvex if h is positive definite (or pseudoconvex in case h is positive semidefinite). Many of the analytic existence and uniqueness results in the theory of CR manifolds depend on the strict pseudoconvexity of the Levi form.
This nomenclature comes from the study of pseudoconvex domains: M is the boundary of a (strictly) pseudoconvex domain in Cn if and only if it is (strictly) pseudoconvex as a CR manifold. (See plurisubharmonic functions and Stein manifold.)
Abstract CR structures
An abstract CR structure on a manifold M of dimension n consists of a subbundle L of the complexified tangent bundle which is formally integrable, in the sense that [L,L] ⊂ L, which is linearly independent of its complex conjugate. The CR codimension of the CR structure is k = n - 2 dim L. In case k = 1, the CR structure is said to be of hypersurface type. Most examples of abstract CR structures are of hypersurface type, unless otherwise made explicit.
The Levi form and pseudoconvexity
h defines a sesquilinear form on L since it does not depend on how v and w are extended to sections of L, by the integrability condition. This form extends to a hermitian form on the bundle by the same expression. The extended form is also sometimes referred to as the Levi form.
The Levi form can alternatively be characterized in terms of duality. Consider the line subbundle of the complex cotangent bundle annihilating V
For each local section α∈Γ(H0M), let
The form hα is a complex-valued hermitian form associated to α.
Generalizations of the Levi form exist when the manifold is not of hypersurface type, in which case the form no longer assumes values in a line bundle, but rather in a vector bundle. One may then speak, not of a Levi form, but of a collection of Levi forms for the structure.
The tangential Cauchy–Riemann complex
The canonical example of a CR manifold is the real sphere as a submanifold of . The bundle described above is given by
where is the bundle of holomorphic vectors. The real form of this is given by , the bundle given at a point concretely in terms of the complex structure, , on by
and the almost complex structure on is just the restriction of .
- See (Levi 909, p. 207): the Levi form is the differential form associated to the differential operator C, according to Levi's notation.
- Capogna, Luca; Danielli, Donatella; Pauls, Scott; Tyson, Jeremy (2007). Applications of Heisenberg Geometry. "An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem". Progress in Mathematics 259 (Berlin: Birkhauser). pp. 45–48.
- Levi, Eugenio Elia (1910), "Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse", Annali di Matematica Pura e Applicata, s. III, (in Italian) XVII (1): 61–87, doi:10.1007/BF02419336, JFM 41.0487.01. An important paper in the theory of functions of several complex variables. An English translation of the title reads as:-"studies on essential singular points of analytic functions of two or more complex variables".
- Boggess, Albert (1991). CR Manifolds and the Tangential Cauchy Riemann Complex. CRC Press.
- Hill, D. and Nacinovich, M. (1995). "Duality and distribution cohomology of CR manifolds". Ann. Scuola Norm. Sup. Pisa 22 (2): 315–339.
- Chern S. S. and Moser, J.K. (1974). "Real hypersurfaces in complex manifolds". Acta Math. 133: 219–271. doi:10.1007/BF02392146.
- Harvey, F.R. and Lawson, H.B., Jr. (1978). "On boundaries of complex analytic varieties". Ann. Math. 102 (2): 223–290. doi:10.2307/1971032. JSTOR 1971032.