THE BIGRADED RUMIN COMPLEX. 1. Introduction

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1 THE BIGRADED RUMIN COMPLEX JEFFREY S. CASE Abstract. We give a detailed description of the bigraded Rumin complex in dimensions three and five, including formulae for all of its constituent operators. The de Rham complex. Introduction C (M d Ω (M d... d Ω n (M d 0 plays an important role in differential geometry, where M is an n-dimensional manifold, Ω k (M is the space of k-forms, and d is the exterior derivative. One reason for this is that the de Rham complex computes the real cohomology groups of M. Another reason is that the exterior derivative, being defined independently of the metric, is in particular a conformally invariant operator. As such, it plays an important role in the construction of many conformally covariant operators. In CR geometry, the corresponding complex is the bigraded Rumin complex. Suppose first that (M 2n+, H is a contact manifold and that θ is a contact form for H. The graded Rumin complex [3] is defined as follows: Let I be the C (M- module generated by θ and dθ and let J be the C (M-module of differential forms annihilated by θ and dθ; i.e. I = {θ β + dθ γ β, γ Ω(M}, J = {α Ω(M α θ = 0 = α dθ}, where Ω(M = 2n+ k=0 Ωk (M is the space of differential forms. It is a fact that Ω k (M/I = 0 for k n + and that J Ω k (M = 0 for k n. We thus define E k = Ω k (M/I for 0 k n and F k = J Ω k (M for n + k 2n +. Moreover, the exterior derivative gives rise to well defined maps d: E k E k+, if 0 k n, d: F k F k+, if n + k 2n. An important observation of Rumin [3] is that if ω Λ n H, then there is a unique β Λ n H such that d(ω + β θ F n+. It follows that the operator D : E n F n+ given by D(ω = d(ω + α is well-defined. Moreover, one obtains a complex (. E 0 d E d... d E n D F n+ d F n+2... d F 2n+ d 0, and this complex computes the real cohomology groups of M. Date: October 6, 205.

2 2 JEFFREY S. CASE In the case when (M 2n+, H is a CR manifold that is, when H admits an integrable almost complex structure J : H H we obtain even more. First, the spaces I and J are well-defined independently of the choice of contact form θ, and hence the operators appearing in the Rumin complex (. are all CR invariant operators. Second, the almost complex structure J induces a grading on the space of k-forms for every k, and hence a grading on the spaces E k and F k. Specifically, we have the splitting H C = T,0 T 0, for T,0 and T 0, the +i- and ieigenspaces of J. We then define Ω,0 to be the annihilator of T 0,, and Ω 0, to be the annihilator of T,0. Note that Ω,0 Ω 0, = θ. Then for each 0 k n we may decompose k E k = E j,k j, where [ω] E j,k j if and only if every representative ω [ω] is such that j=0 0 = i Z i Zj+ ω, 0 = i Z i Zk j+ ω for every Z,..., Z j+ T,0 and every Z,..., Z k j+ T 0,. Similarly, for each n + k 2n + we may decompose k F k = F j,k j, where ω F j,k j if and only if every representative ω [ω] is such that j= 0 = i Z i Zj ω, 0 = i Z i Zk j+ ω for every Z,..., Z j T,0 and every Z,..., Z k j+ T 0,. The purpose of this note is to describe explicitly the spaces and the maps in the (bigraded Rumin complex in dimensions three and five. Some remarks about the use of these operators in CR geometry are also included. To emphasize the CR invariance of the operators and spaces in the Rumin complex, we will give our final description in terms of spaces of sections of density bundles. Specifically, for w C, we will consider the space E(w of densities of weight w: Choose an orientiation of H and let G denote the R + -subbundle of T M of positively-oriented contact forms for θ. Then { } E(w := u: M G C u(x, e σ θ = e wσ(x u(x, θ for all σ C (M. Similarly, if V is a vector bundle over M, we will denote by V (w := V E(w the space of sections of V which transform as does an element of E(w; i.e. an element X V (w is a function X : M G V such that X(p, θ V p for all p M, X(p, e σ θ = e wσ(p X(p, θ for all σ C (M. Lastly, we will use Penrose notation to denote the spaces of sections of vector bundles; e.g. E α denotes the space of sections of E,0, E β denotes the space of sections of E 0,, and E (α β0 denotes the space of sections of E,, which can be identified with tracefree (with respect to the Levi form sections of E,0 E 0, (see

3 THE BIGRADED RUMIN COMPLEX 3 Section 3 for further discussion. Our use of Penrose notation is consistent with [], and the space E(w is the same as the space E(w, w in that reference. 2. Three-dimensional manifolds Let (M 3, H, J be a three-dimensional CR manifold and fix a contact form θ. E 0 = E 0,0 is the space E(0 of densities of weight zero. E = Ω (M/ θ is generated by θ α and θ β modulo θ. Hence can we identify E,0 = E α (0 and E 0, = E β(0. F 2 consists of all two-forms which vanish upon wedging with θ. Thus F 2 is generated by θ θ α and θ θ β. Since θ has weight, we may identify E α ( ω α = i ωα θ θ α F 2,0, E β( ω β = i ω β θ θ α F,. The factors of ±i are included to ensure that the operator d: F 2 F 3 is the divergence operator under this identification. F 3 is Ω 3 (M. Since θ dθ is nowhere vanishing and has weight 2, we may identify F 2, = E( 2 via E( 2 u = u θ dθ F 2,. The Rumin complex is the complex (2. E α (0 E α ( E(0 E( 2. E β(0 E β( The operators in the complex are defined as follows: Operators defined on zero-forms: (2.2a E(0 u b α u E α (0, (2.2b (2.3a (2.3b (2.3c (2.3d Operators defined on one-forms: E(0 u b βu E β(0. E α (0 ω α D γ γ ω α i 0 ω α E α (, E α (0 ω α D β γ ω γ + ia βγ ω γ E β(, E β(0 ω β E β(0 ω β Operators defined on two-forms: D + α ρ ω ρ ia α ρ ω ρ E α (, D ρ ρ ω β + i 0 ω β E β(. (2.4a (2.4b E α ( ω α E β( ω β b γ ω γ E( 2, b ρ ω ρ E( 2.

4 4 JEFFREY S. CASE The formulae (2.2 come from writing the formulae for the exterior derivative and splitting according to type. The formulae (2.3 follow from the observation that if ω = ω α θ α + ω βθ β, then f = i( σ ω σ γ ω γ is such that (2.5 d (ω + fθ = i ( γ γ ω α i 0 ω α α ρ ω ρ ia α ρ ω ρ θ θ α i ( ρ ρ ω β + i 0 ω β β γ ω γ + ia βγ ω γ θ θ β. Since d(ω + fθ has no component of type θ α θ β, (2.5 yields the formula for Rumin s D operator. Splitting according to types yields (2.3. The formulae (2.4 come from the formula ( d iω α θ θ α iω β θ θ β = ( γ ω γ + ρ ω ρ θ dθ and splitting according to type. Besides conformal invariance, the key property of the Rumin complex (2. is that it is a bigraded complex; i.e. if A and B are any two bundles in the diagram which are separated by a column, then the sum of all paths from A to B through the complex equals zero. For example, D b + D + b : E(0 E α ( is the zero map. This follows immediately from the identity d 2 = 0 and the definitions of the Rumin operators. This observation is also the one used by Lee [2] to prove commutator formulae. 3. Five-dimensional manifolds Let (M 5, H, J be a three-dimensional CR manifold and fix a contact form θ. As in the three-dimensional case, E 0,0 = E(0 and we may identify E,0 = E α (0 and E 0, = E β(0. E 2 = Ω 2 (M/ θ, dθ is generated by θ α θ γ, θ β θ σ, and θ α θ β mod dθ, θ θ α, θ θ β. Furthermore, since dθ = ih α βθ α θ β, elements of θ α θ β in E 2 are determined by their tracefree part. In summary, we may identify E αγ (0 ω αγ = 2 ω αγ θ α θ γ E 2,0, E (α β0 (0 ω α β = iωα β θ α θ β E,, E β σ (0 ω β σ = 2 ω β σ θ β θ σ E 0,2 with the conventions that elements of E αγ and E β σ are skew-symmetric and elements of E (α β0 are tracefree; i.e. ω γ γ = 0 for all ω α β E (α β0. F 3 consists of all three-forms which are annihilated by θ and dθ. Clearly elements of θ θ α θ γ, θ θ β θ σ have this property. Finally, a three-form ω α β θ θ α θ β has this property if and only if ω γ γ = 0. In summary, we may identify E αγ ( ω αγ = 2 ω αγ θ θ α θ γ F 3,0, E (α β0 ( ω α β = iωα β θ θ α θ β F 2,, E β σ ( ω β σ = 2 ω β σ θ θ β θ σ F,2.

5 THE BIGRADED RUMIN COMPLEX 5 The choice of identifications E αγ ( = F 3,0 and E β σ ( = F,2 ensure that d corresponds with the divergence operator when acting on these spaces. The choice of identification E (α β0 ( = F 2, ensures that Hermitian forms in E (α β0 ( are identified with real forms in F 2,. F 4 is generated by θ dθ θ α and θ dθ θ β. To ensure that d corresponds with the divergence operator, we make the identifications E α ( 2 ω α = iωα θ dθ θ α F 3,, E β( 2 ω β = iω β θ dθ θ β F 2,2. F 5 is Ω 5 (M. Since θ dθ dθ is nowhere vanishing, we make the identification E( 3 u = u θ dθ dθ F 3,2. The Rumin complex is the complex (3. E αγ (0 E αγ ( E α (0 E α ( 2 E(0 E (α β0 (0 E (α β0 ( E( 3. E β(0 E β( 2 E β σ (0 E β σ ( The operators in the complex are defined as follows: Operators defined on zero-forms: (3.2a (3.2b E(0 u α u E α (0, E(0 u βu E β(0. Operators defined on one-forms: (3.3a (3.3b (3.3c (3.3d E α (0 ω α α ω γ γ ω α E αγ (0, E α (0 ω α i ( βω α 2 h α β γ ω γ E (α β0 (0, E β(0 ω β i ( α ω β 2 h α β ρ ω ρ E (α β0 (0, E β(0 ω β βω σ σ ω β E β σ (0.

6 6 JEFFREY S. CASE (3.4a (3.4b (3.4c (3.4d (3.4e (3.4f (3.4g (3.5a (3.5b (3.5c (3.5d (3.6a (3.6b Operators defined on two-forms: E αγ (0 ω αγ i δ δ ω αγ 0 ω αγ E αγ (, E αγ (0 ω αγ β δ ω δα ia βδ ω δα E (α β0 (, E (α β0 (0 ω αγ ( α ρ + ia α ρ ω γ ρ ( γ ρ + ia γ ρ ω α ρ E αγ (, E (α β0 (0 ω αγ i β ρ ω α ρ i α δ ω δ β 0 ω α β E (α β0 (, E (α β0 (0 ω αγ ( β δ ia βδ ω δ σ ( σ δ ia σ δ ω δ β E β σ (, E β σ (0 ω αγ α ρ ω ρ β + ia α ρ ω ρ β E (α β0 (, E β σ (0 ω αγ i ρ ρ ω β σ 0 ω β σ E β σ (. Operators defined on three-forms: E αγ ( ω αγ δ ω δα E α ( 2, E (α β0 ( ω α β i ρ ω α ρ E α ( 2, E (α β0 ( ω α β i δ ω δ β E β( 2, Operators defined on four-forms: E β σ ω β σ ρ ω ρ β E β( 2. E α ( 2 ω α γ ω γ E( 3, E β( 2 ω β σ ω σ E( 3. The formulae for the operators (3.2 and (3.3 are obtained by writing the formulae for the exterior derivative and splitting according to type. The formulae for the operators (3.4 are obtained from the following two observations: First, if ω = 2 ω αγθ α θ γ for ω αγ E αγ (0 and if we define ξ = 2i δ ω δα θ α, then d (ω + ξ θ = ( i α δ ω δγ i γ δ ω δα + 0 ω αγ θ α θ γ θ (3.7 2 i ( β δ ω δα ia βδ ω δα θ θ α θ β. This yields (3.4a, (3.4b and their conjugates upon observing that δ δ ω αγ = α δ ω δγ γ δ ω δα. Second, if ω = iω α β θ α θ β for ω α β E (α β0 (0 and if we define ξ = ρ ω α ρ θ α + δ ω δ β θ β, then (3.8 d (ω + θ ξ = ( ρ γ ω α ρ + ia ρ α ω γ ρ θ θ α θ γ i ( i β ρ ω α ρ i α δ ω δ β 0 ω α β θ θ α θ β ( β δ ω δ σ ia σ δ ω δ β θ θ β θ σ. This yields (3.4c, (3.4d and (3.4e. The formulae for the operators (3.5 follow from the observations 2 d (ω αγ θ θ α θ γ = i δ ω δα θ dθ θ α, ( id ω α β θ θ α θ β = ρ ω α ρ θ dθ θ α δ ω δ β θ dθ θ β.

7 THE BIGRADED RUMIN COMPLEX 7 The formulae for the operators (3.6 follows from the observation d (iω α θ dθ θ α = δ ω δ θ (dθ 2. The Rumin complex (3. is a bigraded complex. References [] A. R. Gover and C. R. Graham. CR invariant powers of the sub-laplacian. J. Reine Angew. Math., 583: 27, [2] J. M. Lee. Pseudo-Einstein structures on CR manifolds. Amer. J. Math., 0(:57 78, 988. [3] M. Rumin. Formes différentielles sur les variétés de contact. J. Differential Geom., 39(2:28 330, McAllister Building, Penn State University, University Park, PA 680 address: jscase@psu.edu