Invariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2021), 011, 25 pages Invariant Dirac Operators, Harmonic Spinors,and Vanishing Theorems in CR Geometry
Felipe LEITNERUniversit¨at Greifswald, Institut f¨ur Mathematik und Informatik,Walter-Rathenau-Str. 47, D-17489 Greifswald, Germany
E-mail: [email protected]
Received July 23, 2020, in final form January 22, 2021; Published online February 04, 2021https://doi.org/10.3842/SIGMA.2021.011
Abstract.
We study Kohn–Dirac operators D θ on strictly pseudoconvex CR manifolds withspin C structure of weight (cid:96) ∈ Z . Certain components of D θ are CR invariants. We also de-rive CR invariant twistor operators of weight (cid:96) . Harmonic spinors correspond to cohomologyclasses of some twisted Kohn–Rossi complex. Applying a Schr¨odinger–Lichnerowicz-type for-mula, we prove vanishing theorems for harmonic spinors and (twisted) Kohn–Rossi groups.We also derive obstructions to positive Webster curvature. Key words:
CR geometry; spin geometry; Kohn–Dirac operator; harmonic spinors; Kohn–Rossi cohomology; vanishing theorems
The classical Schr¨odinger–Lichnerowicz formula D = ∆ + scal4of Riemannian geometry relates the square of the Dirac operator to the spinor Laplacian andscalar curvature. This Weitzenb¨ock formula can be used to prove vanishing theorems for har-monic spinors on closed manifolds. Via Hodge theory and Dolbeault’s theorem this give rise tovanishing theorems for holomorphic cohomology on K¨ahler manifolds (see [5]). Moreover, via the index theorem for elliptic differential operators, the ˆ A -genus is understood to be an obstructionto positive scalar curvature on spin manifolds (see [14]).Due to J.J. Kohn there is also an harmonic theory for the Kohn–Laplacian on strictly pseu-doconvex CR manifolds (see [4, 7]). Even though the Kohn–Laplacian is not elliptic, this theoryshows that classes in the cohomology groups of the tangential Cauchy–Riemann complex (or Kohn–Rossi complex ) are represented by harmonic forms. In particular, the (non-extremal)
Kohn–Rossi groups are finite dimensional over closed manifolds.In [21] Tanaka describes this harmonic theory for the Kohn–Laplacian on ( p, q )-forms withvalues in some CR vector bundle E over (abstract) strictly pseudoconvex CR manifolds. TheKohn–Laplacian is defined with respect to some pseudo-Hermitian structure θ and the corre-sponding canonical connection . In particular, Tanaka derives Weitzenb¨ock formulas and provesvanishing theorems for the Kohn–Rossi groups. On the other hand, in [18] R. Petit introduces spinor calculus and Dirac-type operators to strictly pseudoconvex CR manifolds with adaptedpseudo-Hermitian structure (cf. also [12, 20]). Deriving some Schr¨odinger–Lichnerowicz-type for-mula for the Kohn–Dirac operator , this approach gives rise to vanishing theorems for harmonicspinors over closed CR manifolds (cf. also [9]). a r X i v : . [ m a t h . DG ] F e b F. LeitnerWe study in this paper the Kohn–Dirac operator D θ for spin C structures of weight (cid:96) ∈ Z on strictly pseudoconvex CR manifolds with adapted pseudo-Hermitian structure θ . Our con-struction of D θ uses the Webster–Tanaka spinor derivative, only. The Kohn–Dirac operator D θ does not behave naturally with respect to conformal changes of the underlying pseudo-Hermitianstructure. However, similar as in K¨ahler geometry, the spinor bundle Σ decomposes with respectto the CR structure into eigenbundles Σ µ q of certain eigenvalues µ q . For µ q = − (cid:96) the restric-tion D (cid:96) of the Kohn–Dirac operator to Γ (cid:0) Σ µ q (cid:1) acts CR-covariantly. This observation gives riseto CR invariants for the underlying strictly pseudoconvex CR manifold.Complementary to D θ we also have twistor operators . In the spin case [12] we discuss specialsolutions of the corresponding twistor equation, which realize some lower bound for the squareof the first non-zero eigenvalue of the Kohn–Dirac operator D θ . For µ q = (cid:96) the correspondingtwistor operator P (cid:96) is a CR invariant.Analyzing the Clifford multiplication on the spinor bundle for spin C structures over strictlypseudoconvex CR manifolds shows that the Kohn–Dirac operator is a square root of the Kohn–Laplacian acting on (0 , q )-forms with values in some CR line bundle E . Thus, our discussion ofthe Kohn–Dirac operator fits well to Kohn’s harmonic theory, as described in [21]. In particular,harmonic spinors correspond to cohomology classes of certain twisted Kohn–Rossi complexes.Computing the curvature term of the corresponding Schr¨odinger–Lichnerowicz-type formulagives rise to vanishing theorems for twisted Kohn–Rossi groups.For example, on a closed, strictly pseudoconvex CR manifold M of even CR dimension m ≥ √K of the canonical line bundle, we have for µ q = (cid:96) = 0 the Schr¨odinger–Lichnerowicz-type formula D ∗ D = ∆ tr + scal W D of D θ , where ∆ tr denotes the spinor sub-Laplacian and scal W is the Webster scalar curvature. In this case harmonic spinors correspond to cohomology classesin the Kohn–Rossi group H m (cid:0) M, √K (cid:1) . Positive Webster scalar curvature scal W > M im-mediately implies that this Kohn–Rossi group is trivial. On the other hand, H m (cid:0) M, √K (cid:1) (cid:54) = { } poses an obstruction to the existence of any adapted pseudo-Hermitian structure θ on M of posi-tive Webster scalar curvature. In this case the Yamabe invariant in [6] for the given CR structureis non-positive.In Sections 2 to 5 we introduce CR manifolds and pseudo-Hermitian geometry with spin C structures. In Section 6 the Kohn–Dirac and twistor operators are constructed. The CR-covariant components D (cid:96) and P (cid:96) are determined in Section 7. In Section 8 we recall theSchr¨odinger–Lichnerowicz-type formula and derive a basic vanishing theorem for harmonic spi-nors (see Proposition 8.1). Section 9 briefly reviews the harmonic theory for the Kohn–Laplacian.In Section 10 we derive vanishing theorems for twisted Kohn–Rossi groups. In Section 11 wediscuss CR circle bundles of K¨ahler manifolds and relate holomorphic cohomology groups toKohn–Rossi groups. Finally, in Section 12 we construct closed, strictly pseudoconvex CR mani-folds with H m (cid:0) M, √K (cid:1) (cid:54) = 0. Let M n be a connected and orientable, real C ∞ -manifold of odd dimension n = 2 m + 1 ≥ H ( M ) , J ) of a corank 1 subbundle H ( M ) of the tangent bundle T ( M )and a bundle endomorphism J : H ( M ) → H ( M ) with J ( X ) = − X for any X ∈ H ( M ). Thenvariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry 3Lie bracket [ · , · ] of vector fields defines a bilinear skew pairing {· , ·} : H ( M ) × H ( M ) → T ( M ) /H ( M ) , ( X, Y ) (cid:55)→ − [ X, Y ] mod H ( M ) , with values in the real line bundle T ( M ) /H ( M ).We call the pair ( H ( M ) , J ) a strictly pseudoconvex CR structure on M (of hypersurface typeand CR dimension m ≥
1) if the following conditions are satisfied: • { J X, Y } + { X, J Y } = 0 mod H ( M ) for any X, Y ∈ H ( M ) and • the symmetric pairing {· , J ·} on H ( M ) is definite, i.e., { X, J X } (cid:54) = 0 for any X (cid:54) = 0, • the Nijenhuis tensor N J ( X, Y ) = [
X, Y ] − [ J X, J Y ] + J ([ J X, Y ] + [
X, J Y ]) vanishes iden-tically for any
X, Y ∈ H ( M ).Throughout this paper we will deal with strictly pseudoconvex CR structures on M . For exam-ple, in the generic case when the Levi form is non-degenerate, the smooth boundary of a domainof holomorphy in C m +1 is strictly pseudoconvex.The complex structure J extends C -linearly to H ( M ) ⊗ C , the complexification of the Levidistribution , and induces a decomposition H ( M ) ⊗ C = T ⊕ T into ± i-eigenbundles. Then a complex-valued p -form η on M is said to be of type ( p,
0) if ι Z η = 0for all Z ∈ T . This gives rise to the complex vector bundle Λ p, ( M ) of ( p, M . Forthe ( m + 1)st exterior power Λ m +1 , ( M ) of Λ , ( M ) we write K = K ( M ). This is the canonicalline bundle of the CR manifold M with first Chern class c ( K ) ∈ H ( M, Z ). Its dual is the anticanonical bundle , denoted by K − .When dealing with a strictly pseudoconvex CR manifold, we will often assume the existenceand choice of some ( m + 2)nd root E (1) of the anticanonical bundle K − , that is a complex linebundle over M with E (1) m +2 = K − . The dual bundle of this root is denoted by E ( − p ∈ Z , we have the p thpower E ( p ) of E (1). We call p the weight of E ( p ). In particular, the canonical bundle K hasweight − ( m + 2), whereas the anticanonical bundle K − has weight m + 2.In general, the existence of an ( m + 2)nd root E (1) is restrictive to the global nature of theunderlying CR structure on M . For the application of tractor calculus in CR geometry thisassumption is basic. In fact, the standard homogeneous model of CR geometry on the sphereallows a natural choice for E (1) (see [2]). For our treatment of spin C structures in CR geometrythe choice of some E (1) is useful as well. Let (cid:0) M m +1 , H ( M ) , J (cid:1) , m ≥
1, be strictly pseudoconvex. Since M is orientable, there existssome 1-form θ on M , whose kernel Ker( θ ) defines the contact distribution H ( M ). The differen-tial d θ is a non-degenerate 2-form on H ( M ), and the conditions ι T θ = θ ( T ) = 1 and ι T d θ = 0define a unique vector field T = T θ , which is the Reeb vector field of θ . We use to call T the characteristic vector . The tangent bundle T ( M ) splits into the direct sum T ( M ) = H ( M ) ⊕ R T F. Leitnerwith corresponding projection π θ : T ( M ) → H ( M ). We use to say that vectors in H ( M ) are transverse (to the characteristic direction of T ). Note that L T θ = L T d θ = 0 for the Liederivatives with respect to T θ .Furthermore, g θ ( X, Y ) := 12 d θ ( X, J Y ) , X, Y ∈ H ( M ) , defines a non-degenerate, symmetric bilinear form, i.e., a metric on H ( M ), which is eitherpositive or negative definite. In case g θ is positive definite, we call θ ∈ Ω ( M ) an adapted pseudo-Hermitian structure for (cid:0) M m +1 , H ( M ) , J (cid:1) . Note that any two pseudo-Hermitian structures θ and ˜ θ differ only by some positive function or conformal scale , i.e., ˜ θ = e f θ for some f ∈ C ∞ ( M ).Let us fix some adapted pseudo-Hermitian structure θ on M . To θ we have the Webster–Tanaka connection ∇ W on T ( M ) (see [21, 22]), for which by definition the characteristic vector T ,the metric g θ and the complex structure J on H ( M ) are parallel. Hence, the structure group of ∇ W is the unitary group U( m ). In characteristic direction, we have ∇ W T X = 12 ([ T, X ] − J [ T, J X ])for X ∈ Γ( H ( M )).The torsion is given by some obligatory part ∇ W X Y − ∇ W Y X − [ X, Y ] = d θ ( X, Y ) T with transverse X , Y in H ( M ) and, furthermore, byTor W ( T, X ) = −
12 ([
T, X ] + J [ T, J X ]) , X ∈ H ( M ) . We call the latter part τ ( X ) := Tor W ( T, X ) , X ∈ H ( M ) , Webster torsion tensor of θ on ( M, H ( M ) , J ). This is a symmetric and trace-free tensor. Thecomposition τ ◦ J = − J ◦ τ is symmetric and trace-free as well. We set τ ( X, Y ) = g θ ( τ X, Y ), X, Y ∈ H ( M ).As usual the curvature operator R W ( X, Y ) of ∇ W is defined by R W ( X, Y ) := ∇ W X ∇ W Y − ∇ W Y ∇ W X − ∇ W[ X,Y ] for any X, Y ∈ T ( M ). Since ∇ W is metric, R W ( X, Y ) is skew-symmetric with respect to g θ on H ( M ). The first Bianchi identity for X, Y, Z ∈ H ( M ) is given by the cyclic sum (cid:88) XY Z R W ( X, Y ) Z = (cid:88) XY Z d θ ( X, Y ) τ ( Z ) . (3.1)For any X ∈ T ( M ), the Webster–Ricci endomorphism
Ric W ( X ) is the g θ -trace of R ( X, · )( · ),and the Webster scalar curvature is the trace scal W = tr θ Ric W of the Webster–Ricci tensoron H ( M ). On the other hand, the pseudo-Hermitian Ricci form is given by ρ θ ( X, Y ) := 12 tr θ (cid:0) g θ (cid:0) R W ( X, Y, J · , · ) (cid:1)(cid:1) for any X, Y ∈ H ( M ). Then we haveRic W ( X, Y ) = ρ θ ( X, J Y ) + 2( m − τ ( X, J Y )nvariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry 5for any
X, Y ∈ H ( M ), where ρ θ corresponds to the J -invariant and τ is the J -antiinvariant partof Ric W on H ( M ).If m ≥ ρ θ is a multiple of d θ we call θ a pseudo-Einsteinstructure on the CR manifold (cid:0) M m +1 , H ( M ) , J (cid:1) (see [11]). For m = 1 this condition is vacuous.However, for m > W ( T, J X ) = m X (cid:0) scal W (cid:1) forany X ∈ H ( M ). This is a suitable replacement for the Einstein condition when m = 1 (see [3]).In any case we have ρ θ = scal W m d θ and the Webster scalar curvature of some pseudo-Einsteinstructure θ need not be constant. In fact, it is constant if and only ifRic W ( T ) = tr θ (cid:0) ∇ W · τ (cid:1) ( · ) = 0 . C structures Recall that the group Spin C (2 m ) is a central extension of SO(2 m ) given by the exact sequence1 → Z → Spin C (2 m ) → SO(2 m ) × U(1) → . This gives a twisted product Spin C (2 m ) = Spin(2 m ) × Z U(1) with the spin group. We haveSpin C (2 m ) / U(1) ∼ = SO(2 m ), and a group homomorphism λ : Spin C (2 m ) → SO(2 m ) as well asSpin C (2 m ) / Spin(2 m ) ∼ = U(1). Note that there is also a canonical homomorphism j : U( m ) → Spin C (2 m ) , which is the lift of ι × det : U( m ) → SO(2 m ) × U(1).Now let θ be a pseudo-Hermitian form on the strictly pseudoconvex CR manifold (cid:0) M m +1 ,H ( M ) , J (cid:1) , m ≥
1. This gives rise to the metric g θ on the Levi distribution H ( M ). We denoteby SO( H ( M )) the principal SO(2 m )-bundle of orthonormal frames in H ( M ). A spin C structure to θ on M is a reduction ( P, Λ) of the frame bundle SO( H ( M )). This means here, P → M issome principal Spin C (2 m )-bundle with fiber bundle map Λ : P → SO( H ( M )) such that Λ( p · s ) =Λ( p ) · λ ( s ) for all p ∈ P and s ∈ Spin C (2 m ).Let ( P, Λ) be some fixed spin C structure for ( M, θ ). Then P := P/ Spin(2 m ) → M isa principal U(1)-bundle, and we denote the associated complex line bundle by L → M . This isthe determinant bundle of the spin C structure. The corresponding fiber bundle map Λ : P → SO( H ( M )) × P over M is a twofold covering. On the other hand, let L ( β ) → M be a complexline bundle determined by some integral class β ∈ H ( M, Z ). Then, if β ≡ − c ( K ) mod 2 , there exists a spin C structure ( P, Λ) to θ on M with determinant bundle L ( β ).There exists always the canonical spin C structure to θ on M , which stems from the lift j : U( m ) → Spin C (2 m ). The corresponding determinant bundle is K − . All other spin C struc-tures differ from the canonical one by multiplication with a principal U(1)-bundle, related tosome line bundle E ( α ), α ∈ H ( M, Z ). The corresponding determinant bundle L ( β ) satisfies E ( α ) = K ⊗ L ( β ). Spin C structures with the same determinant bundle L ( β ) are parametrizedby the elements in H ( M, Z ) (see [10, 18]).In particular, if c ( K ) ≡ θ on M admits some spin C structure with trivialdeterminant bundle. This represents an ordinary spin structure for the Levi distribution H ( M )with metric g θ (cf. [12]). More generally, let us consider the powers E ( p ), p ∈ Z , of an ( m + 2)ndroot E (1) of K − . Then − c ( K ) = ( m + 2) c ( E (1)), and a spin C structure for ( M, θ ) withdeterminant bundle L = E ( p ) exists when( m + 2 − p ) c ( E (1)) ≡ . F. Leitner
Lemma 4.1.
Let E (1) be an ( m + 2) nd root of K − → M m +1 . Then θ on M admits a spin C structure with determinant bundle L = E ( p ) , p ∈ Z , if ( i ) E (1) itself admits some square root, or ( ii ) m and p ∈ Z are odd, or ( iii ) m and p are even. We say that a spin C structure with determinant bundle L = E ( p ) has weight p . In thefollowing we assume that spin C structures to θ on M exist for all necessary weights p ∈ Z . Let (cid:0) M m +1 , H ( M ) , J (cid:1) be a strictly pseudoconvex CR manifold of hypersurface type and CRdimension m ≥
1, and let ( P, Λ) be a spin C structure of weight (cid:96) ∈ Z for some given pseudo-Hermitian form θ on M . The choice of ( P, Λ) gives rise to an associated spinor bundleΣ( H ( M )) := P × ρ m Σover M , where ρ m denotes the representation of Spin C (2 m ) on the complex spinor module Σ.Note that the center U(1) acts by complex scalar multiplication on Σ. The spinor bundle hasrk C (Σ( H ( M )) = 2 m .The spinor bundle Σ( H ( M )) is equipped with a Hermitian inner product (cid:104)· , ·(cid:105) , and we havea Clifford multiplication c : H ( M ) ⊗ Σ( H ( M )) → Σ( H ( M )) , ( X, φ ) (cid:55)→ X · φ, which satisfies (cid:104) X · ψ, φ (cid:105) = −(cid:104) ψ, X · φ (cid:105) for any transverse X ∈ H ( M ) and φ ∈ Σ( H ( M )), given at some point of M . The multiplication c extends to the complex Clifford bundle C l( H ( M )) of the Levi distribution.The Webster–Tanaka connection ∇ W to θ stems from a principal fiber bundle connection onthe unitary frame bundle, contained in SO( H ( M )). This gives rise to a covariant derivative ∇ W on any root of K − and its powers, in particular, for E (1) and the determinant bundle L = E ( (cid:96) ).Recall that ( P, Λ) induces a twofold covering map P → SO( H ( M )) × P . Then the Webster–Tanaka connection lifts to P , which in turn gives rise to some covariant derivative on spinorfields: ∇ Σ : Γ( T ( M )) ⊗ Γ(Σ( H ( M )) → Γ(Σ( H ( M )) , ( X, φ ) (cid:55)→ ∇ Σ X φ. Note that this construction does not need an auxiliary connection on the determinant bundle L .We only use the Webster–Tanaka connection on L and call ∇ Σ the Webster–Tanaka spinorderivative to the given spin C structure of weight (cid:96) .The spinor derivative satisfies the rules ∇ Σ Y ( X · φ ) = (cid:0) ∇ W Y X (cid:1) · φ + X · ∇ Σ Y φ and Y (cid:104) φ, ψ (cid:105) = (cid:10) ∇ Σ Y φ, ψ (cid:11) + (cid:10) φ, ∇ Σ Y ψ (cid:11) nvariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry 7for any X ∈ Γ( H ( M )), Y ∈ Γ( T ( M )) and φ, ψ ∈ Γ(Σ( H ( M ))). Locally, with respect to someorthonormal frame s = ( s , . . . , s m ), the spinor derivative is given by the formula ∇ Σ φ = d φ + 12 m (cid:88) j
1, be strictly pseudoconvex. We have H ( M ) ⊗ C = T ⊕ T andany real transverse vector X ∈ H ( M ) can be written as X = X + X with X = X − i J X ∈ T and X = X + i J X ∈ T . If e = ( e , . . . , e m ) denotes a complex orthonormal basis of ( H, J, g θ ), i.e., s = ( e , J e , . . . , e m ,J e m ) is a real orthonormal basis of ( H, g θ ), we set E α := ( e α ) = e α − i J e α , α = 1 , . . . , m. F. LeitnerThe vectors ( E , . . . , E m ) form an orthogonal basis with respect to the Levi form on T . Aselements in the complexified Clifford algebra C l( H ( M )) we have E α E α = 0 and E α E β + E β E α = − δ αβ for any α, β = 1 , . . . , m . Moreover, m (cid:88) α =1 E α E α = −
12 ( m + Θ) , m (cid:88) α =1 E α E α = −
12 ( m − Θ) . Now let θ be a pseudo-Hermitian form on M with spin C structure of weight (cid:96) ∈ Z . Thespinorial derivative ∇ Σ on Σ( H ( M )) is induced by the Webster–Tanaka connection. In thefollowing, we allow covariant derivatives with respect to Z ∈ H ( M ) ⊗ C . This is defined by C -linear extension and denoted by ∇ tr Z . This derivative in transverse direction decomposes into ∇ tr = ∇ ⊕ ∇ , i.e., for any spinor φ ∈ Γ(Σ( H ( M ))), we have locally ∇ φ = m (cid:88) α =1 E ∗ α ⊗ ∇ tr E α φ and ∇ φ = m (cid:88) α =1 E α ∗ ⊗ ∇ tr E α φ with respect to some frame ( E , . . . , E m ) of T .Recall that Clifford multiplication is denoted by c . Then we can define the first order differ-ential operators D − φ = c ( ∇ φ ) and D + φ = c ( ∇ φ )for spinors φ ∈ Γ(Σ( H ( M ))). Locally, the two operators are given by D − φ = 2 m (cid:88) α =1 E α · ∇ tr E α φ and D + φ = 2 m (cid:88) α =1 E α · ∇ tr E α φ. Note that Θ · X − X · Θ = − X for X ∈ T . This shows T · Σ µ q ⊆ Σ µ q +1 and T · Σ µ q ⊆ Σ µ q − for any q ∈ { , . . . , m } . Hence, the operator D + maps spinors from Γ (cid:0) Σ µ q (cid:1) to Γ (cid:0) Σ µ q +1 (cid:1) .Similarly, D − : Γ (cid:0) Σ µ q (cid:1) → Γ (cid:0) Σ µ q − (cid:1) . In fact, we have [Θ , D + ] = − D + and [Θ , D − ] = 2 D − .We compute the square of D + . Locally, around any p ∈ M , we can choose a synchronizedframe of the form ( e , . . . , e m ) with ∇ W e α e β ( p ) = 0 , α, β ∈ { , . . . , m } . Then ( D + ) φ = 4 m (cid:88) α,β =1 E α E β ∇ tr E α ∇ tr E β φ = 2 (cid:88) α,β E α E β · R Σ (cid:0) E α , E β (cid:1) φ = − (cid:18) (cid:88) α,β E α τ ( E α ) E β E β + E α E α E β τ (cid:0) E β (cid:1)(cid:19) φ = 0 , where we use (3.1), (5.1) and the fact that τ , τ ◦ J are trace-free. Similarly, we obtain D − = 0.Thus, we have constructed two chain complexes0 → Γ (cid:0) Σ µ (cid:1) D + −→ Γ (cid:0) Σ µ (cid:1) D + −→ · · · D + −→ Γ (cid:0) Σ µ m − (cid:1) D + −→ Γ (cid:0) Σ µ m (cid:1) → → Γ (cid:0) Σ µ m (cid:1) D − −→ Γ (cid:0) Σ µ m − (cid:1) D − −→ · · · D − −→ Γ (cid:0) Σ µ (cid:1) D − −→ Γ (cid:0) Σ µ (cid:1) → . From the discussions in Section 10 it will become clear that these complexes produce finitedimensional cohomology groups. This compares to the construction of spinorial cohomology onK¨ahler manifolds as described in [15].Next we define D θ φ = c (cid:0) ∇ tr · φ (cid:1) = ( D + + D − ) φ. This is a first order, subelliptic differential operator acting on spinor fields φ ∈ Γ(Σ( H ( M )). Wecall D θ the Kohn–Dirac operator to θ with spin C structure of weight (cid:96) on M (see [18]; cf. [12, 20]).Locally, with respect to an orthonormal frame ( s , . . . , s m ), the Kohn–Dirac operator is given by D θ φ = m (cid:88) i =1 s i · ∇ tr s i φ. Obviously, D θ does not preserve the decomposition of spinors with respect to Θ-eigenvalues.However, we have the identity D θ = D + D − + D − D + , which shows that the square of the Kohn–Dirac operator maps sections of Σ µ q ( H ( M )) to sectionsof Σ µ q ( H ( M )) again, i.e., D θ : Γ (cid:0) Σ µ q (cid:1) → Γ (cid:0) Σ µ q (cid:1) , q = 0 , . . . , m. On the spinor bundle, we have the L -inner product defined by( φ, ψ ) := (cid:90) M (cid:104) φ, ψ (cid:105) vol θ for compactly supported spinors φ, ψ ∈ Γ c (Σ), wherevol θ := θ ∧ (d θ ) m denotes the induced volume form of the pseudo-Hermitian structure θ on M . The Kohn–Diracoperator D θ is formally self-adjoint with respect to this L -inner product ( · , · ) on Γ c (Σ) (see [12]).Complementary to the Kohn–Dirac operator D θ , we have twistor operators P ( µ q ) acting onΓ (cid:0) Σ µ q ( H ( M )) (cid:1) for q = 0 , . . . , m . In fact, there are orthogonal decompositions T ∗ ⊗ Σ µ q ∼ = Ker( c ) ⊕ Σ µ q − and T ∗ ⊗ Σ µ q ∼ = Ker( c ) ⊕ Σ µ q +1 , where Ker( c ) denote the corresponding kernels of the Clifford multiplication. Then with a q := 12( q + 1) and b q := 12( m − q + 1)we have for the derivatives ∇ φ µ q and ∇ φ µ q of a spinor φ µ q ∈ Γ (cid:0) Σ µ q (cid:1) the decompositions ∇ φ µ q = P φ µ q − b q m (cid:88) α =1 E ∗ α ⊗ E α · D − φ µ q , ∇ φ µ q = P φ µ q − a q m (cid:88) α =1 E α ∗ ⊗ E α · D + φ µ q , c ) by P ( φ µ q ) = m (cid:88) α =1 E ∗ α ⊗ (cid:0) ∇ E α φ µ q + b q E α · D − φ µ q (cid:1) ,P ( φ µ q ) = m (cid:88) α =1 E α ∗ ⊗ (cid:0) ∇ E α φ µ q + a q E α · D + φ µ q (cid:1) , respectively. The sum P ( µ q ) = P + P is given with respect to a local orthonormal frame s by P ( µ q ) φ µ q = m (cid:88) i =1 s ∗ i ⊗ (cid:18) ∇ tr s i φ µ q + a q s i + i J s i D + φ µ q + b q s i − i J s i D − φ µ q (cid:19) . This is the projection of ∇ tr φ µ q to the kernel Ker( c ). In the previous section we have introduced Kohn–Dirac operators D θ and twistor operators P ( µ q ) for spin C structures of weight (cid:96) ∈ Z . We have only used the Webster–Tanaka connection fortheir construction. Now we compute the transformation law for D θ and P ( µ q ) under conformalchange of the pseudo-Hermitian structure. It turns out that certain components of D θ and P ( µ q ) are CR invariants.Let θ and ˜ θ = e f θ be two adapted pseudo-Hermitian structures on (cid:0) M m +1 , H ( M ) , J (cid:1) , m ≥
1. We denote by ∇ W and ∇ Σ derivatives with respect to θ . The derivatives with respectto ˜ θ are simply denote by (cid:101) ∇ . Note that the structure group of the Webster–Tanaka connectionis U( m ) for any pseudo-Hermitian form. We have the transformation rule (cid:101) ∇ X Y = ∇ W X Y + 2 X ( f ) Y + 2 Y ( f ) X − g θ ( X , Y ) grad ( f ) , (cid:101) ∇ X Y = ∇ W X Y + 2 X ( f ) Y + 2 Y ( f ) X − g θ ( X , Y ) grad ( f ) , (7.1)where X = X + X and Y = Y + Y are transverse vectors (see, e.g., [11]). The gradientgrad θ ( f ) ∈ Γ( H ( M )) with complex components grad ( f ) ∈ Γ( T ) and grad ( f ) ∈ Γ( T ) isdual via g θ to the restriction of the differential d f to H ( M ).Now let ( P, Λ) be some spin C structure of weight (cid:96) ∈ Z to θ on (cid:0) M m +1 , H ( M ) , J (cid:1) . Thecanonical bundle K and all line bundles E ( p ), p ∈ Z , are natural for the underlying CR structure.In particular, the determinant bundle L → M of weight (cid:96) is natural, and the correspondingprincipal U(1)-bundles P and ˜ P of frames in L with respect to θ and ˜ θ , respectively, arenaturally identified. The same is true for the orthonormal frames in H ( M ) to θ and ˜ θ . Thus,there exists a unique spin C structure (cid:0) ˜ P , ˜Λ (cid:1) with respect to ˜ θ on M , whose spinor frames are naturally identified with those of ( P, Λ). Of course, the determinant bundle to (cid:0) ˜ P , ˜Λ (cid:1) hasweight (cid:96) again, and there exists an unitary isomorphismΣ( H ( M )) ∼ = (cid:101) Σ( H ( M )) ,φ (cid:55)→ ˜ φ, between the two kinds of spinor bundles such that X · φ is sent to e − f X ˜ · ˜ φ for any transversevector X ∈ H ( M ) and spinor φ ∈ Σ( H ( M )). Also note that (cid:101) Θ ˜ φ = (Θ φ ) (cid:101) , i.e., the decompositionΣ( H ( M )) = ⊕ mq =0 Σ µ q ( H ( M )) into Θ-eigenspaces is CR-invariant.We compare now the spinor derivatives with respect to θ and ˜ θ , respectively. First, let σ = E ∧ · · · ∧ E m be a local section in K − → M and ˜ σ = (cid:102) E ∧ · · · ∧ (cid:103) E m the correspondingnvariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry 11section with respect to ˜ θ . Then ˜ σ = e − mf σ and with (7.1) we obtain the transformation rule (cid:101) ∇ X ˜ σ = ( A σ ( X ) + ( m + 2) X ( f )) ˜ σ = A ˜ σ ( X )˜ σ, (cid:101) ∇ X ˜ σ = ( A σ ( X ) − ( m + 2) X ( f )) ˜ σ = A ˜ σ ( X )˜ σ,X = X + X , for the local connections forms of K − . This gives A ˜ σ ( X ) − A σ ( X ) = − i( m + 2)( J X )( f ) , X ∈ H ( M ) . Accordingly, A ˜ σ ( X ) − A σ ( X ) = − i (cid:96) ( J X )( f ) , X ∈ H ( M ) , for the local connection forms on L = E ( (cid:96) ). With formulas in [12] we obtain for spinors φ thetransformation rule (cid:101) ∇ X ˜ φ = (cid:94) ∇ Σ X φ − ( X · grad ( f ) · φ ) (cid:101) + (cid:96) − X ( f ) ˜ φ − X ( f )(Θ φ ) (cid:101) , (cid:101) ∇ X ˜ φ = (cid:94) ∇ Σ X φ − ( X · grad ( f ) · φ ) (cid:101) − (cid:96) + 22 X ( f ) ˜ φ + 12 X ( f )(Θ φ ) (cid:101) . This gives for φ µ q ∈ Γ(Σ µ q ), q ∈ { , . . . , m } ,˜ D − ˜ φ µ q = e − f (cid:18) D − φ µ q + (cid:18) m + 1 + µ q + (cid:96) (cid:19) grad ( f ) φ µ q (cid:19) (cid:101) , ˜ D + ˜ φ µ q = e − f (cid:18) D + φ µ q + (cid:18) m + 1 − µ q + (cid:96) (cid:19) grad ( f ) φ µ q (cid:19) (cid:101) , and we obtain˜ D − (cid:0) e − v − f ˜ φ µ q (cid:1) = e − ( v − +1) f (cid:94) D − φ µ q for v − = m + 1 + µ q + (cid:96) , ˜ D + (cid:0) e − v + f ˜ φ µ q (cid:1) = e − ( v + +1) f (cid:94) D + φ µ q for v + = m + 1 − µ q + (cid:96) . Hence, for the Θ-eigenvalue µ q = − (cid:96) , we have D ˜ θ (cid:0) e − ( m +1) f ˜ φ − (cid:96) (cid:1) = e − ( m +2) f (cid:94) D θ φ − (cid:96) , (7.2)i.e., the restriction of the Kohn–Dirac operator D θ of weight (cid:96) to Γ (cid:0) Σ − (cid:96) (cid:1) acts CR-covariantly.Recall that the given spin C structure of weight (cid:96) ∈ Z on M is determined by some complexline bundle E ( α ), α ∈ H ( M, Z ). Definition 7.1.
Let (cid:0) M m +1 , H ( M ) , J (cid:1) , m ≥
1, be strictly pseudoconvex with pseudo-Hermi-tian form θ and spin C structure of weight (cid:96) ∈ Z .(a) A spinor φ ∈ Γ(Σ( H ( M )) in the kernel of the Kohn–Dirac operator, i.e., D θ φ = 0, iscalled harmonic . We denote by H q ( α ) the space of harmonic spinors with Θ-eigenvalue µ q , q ∈ { , . . . , m } . Its dimension is denoted h q ( α ).(b) For weight (cid:96) ∈ {− m, − m + 2 , . . . , m − , m } the differential operator D (cid:96) : Γ (cid:0) Σ − (cid:96) (cid:1) → Γ (cid:0) Σ − (cid:96) +2 (cid:1) ⊕ Γ (cid:0) Σ − (cid:96) − (cid:1) denotes the restriction of D θ (of weight (cid:96) ) to spinors of Θ-eigenvalue − (cid:96) . We call D (cid:96) the (cid:96) th (CR-covariant) component of the Kohn–Dirac operator.(c) A spinor φ in the kernel of D (cid:96) is called harmonic of weight (cid:96) .Since D (cid:96) acts by (7.2) CR-covariantly, harmonic spinors of weight (cid:96) are CR invariants of( H ( M ) , J ) on M . The dimension h m + (cid:96) ( α ) is a CR invariant as well. In Section 10 we will seethat in fact all dimensions h q ( α ), 0 < q < m , are CR-invariant numbers.2 F. LeitnerLet us consider the twistor operators P and P . Calculating as above we find (cid:101) ∇ X (cid:0) e − w − f ˜ φ µ q (cid:1) + b q X ˜ D − (cid:0) e − w − f ˜ φ µ q (cid:1) = e − w − f · (cid:0) ∇ X φ µ q + b q X D − φ µ q (cid:1) (cid:101) exactly for w − = (cid:96) − µ q −
1, and (cid:101) ∇ X (cid:0) e − w + f ˜ φ µ q (cid:1) + a q X ˜ D + (cid:0) e − w + f ˜ φ µ q (cid:1) = e − w + f · (cid:0) ∇ X φ µ q + a q X D − φ µ q (cid:1) (cid:101) exactly for w + = µ q − (cid:96) −
1. Hence, the twistor operator P ( µ q ) = P + P is CR-covariant forthe Θ-eigenvalue µ q = (cid:96) . Definition 7.2.
Let (cid:0) M m +1 , H ( M ) , J (cid:1) , m ≥
1, be strictly pseudoconvex with pseudo-Hermi-tian form θ and spin C structure of weight (cid:96) ∈ Z .(a) For weight (cid:96) ∈ {− m, − m + 2 , . . . , m − , m } the differential operator P (cid:96) : Γ (cid:0) Σ (cid:96) (cid:1) → Γ(Ker( c )) ⊂ Γ (cid:0) H ( M ) ⊗ Σ (cid:96) (cid:1) denotes the (cid:96) th component of the twistor operator to θ .(b) A (non-trivial) element in the kernel of P (cid:96) is called CR twistor spinor of weight (cid:96) . Thedimension p (cid:96) of Ker( P (cid:96) ) denotes a CR invariant.In the non-extremal cases, i.e., for (cid:96) (cid:54) = ± m , the twistor equation is overdetermined. In fact,similar as in [19], we suppose the existence of a twistor connection such that CR twistor spinorscorrespond to parallel sections in certain twistor bundles . This would imply that for (cid:96) (cid:54) = ± m the CR invariants p (cid:96) are numbers. Example 7.3.
Parallel spinors of weight (cid:96) ∈ {− m, − m +2 , . . . , m − , m } are CR twistor spinors.For the spin case ( (cid:96) = 0) we discuss parallel spinors to any eigenvalue µ q in [13]. They occuron pseudo-Einstein spin manifolds. In Section 12 we demonstrate the construction of closed CRmanifolds admitting parallel spinors with (cid:96) = µ q = 0, i.e., CR twistor spinors. Example 7.4.
In [12] we describe pseudo-Hermitian Killing spinors for the case of a spinstructure on M . These spinors are in the kernel of the twistor operator and realize a certainlower bound for the non-zero eigenvalues of the Kohn–Dirac operator D θ . In particular, we findCR twistor spinors of weight (cid:96) = 0 on the standard spheres S m +1 of even CR dimension m ≥ θ is related to some 3 -Sasakian structure on M .However, in this situation the CR dimension m is odd and (cid:96) = 0 is impossible. Such Killingspinors are not CR twistor spinors. Let (cid:0) M m +1 , H ( M ) , J (cid:1) , m ≥
1, be strictly pseudoconvex with pseudo-Hermitian structure θ andKohn–Dirac operator D θ to some spin C structure of weight (cid:96) ∈ Z . The operator D θ is formallyself-adjoint and there exists a Schr¨odinger–Lichnerowicz-type formula (see [18]; cf. [12]). We usethis formula to derive vanishing theorems for harmonic spinors.Let ∇ tr denote the transversal part of the Webster–Tanaka spinorial derivative to a chosenspin C structure of weight (cid:96) ∈ Z . Then ∆ tr = − tr θ (cid:0) ∇ tr ◦ ∇ tr (cid:1) denotes the spinor sub-Laplacian ,and we have∆ tr = ∇ ∗ ∇ + ∇ ∗ ∇ , nvariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry 13where ∇ ∗ and ∇ ∗ are the formal adjoint to ∇ and ∇ , respectively. As in [12] we obtainwith (5.1) the equation D θ φ = ∆ tr φ − i (cid:96) m + 2) ρ θ φ + 14 scal W · φ − d θ ∇ Σ T φ for the square of the Kohn–Dirac operator. For the spinorial derivative in characteristic directionwe compute ∇ Σ T φ = i4 m (cid:18) ∇ ∗ ∇ − ∇ ∗ ∇ + i ρ θ − (cid:96) scal W m + 2) (cid:19) ( φ ) . This results in the Schr¨odinger–Lichnerowicz-type formula (cf. [12, 18]) D θ = (cid:18)(cid:18) − Θ m (cid:19) ∇ ∗ ∇ + (cid:18) m (cid:19) ∇ ∗ ∇ (cid:19) − i2 (cid:18) (cid:96)m + 2 + Θ m (cid:19) ρ θ + (cid:18) (cid:96) Θ m ( m + 2) (cid:19) scal W . (8.1)The curvature part in (8.1) acts by Clifford multiplication as a self-adjoint operator on spinors.In case that this operator is positive definite, at each point of some closed manifold M , weimmediately obtain vanishing results for harmonic spinors in the non-extremal bundles Σ µ q ( µ q (cid:54) = ± m ). We aim to specify the situation. Let us call the pseudo-Hermitian Ricci form ρ θ positive (resp. negative) semidefinite if all eigenvalues are nonnegative (resp. nonpositive) on M .We can state our basic vanishing result as follows. Proposition 8.1.
Let θ be some pseudo-Hermitian structure on a closed CR manifold M m +1 , m ≥ , with spin C structure of weight (cid:96) ∈ Z . ( a ) A non-extremal bundle Σ µ q allows no harmonic spinors under the following conditions: (1) µ q = − m(cid:96)m +2 and scal W ≥ on M with scal W ( p ) > at some point p ∈ M . (2) µ q > − m(cid:96)m +2 , ρ θ semidefinite on M and ( m + 2 − (cid:96) )scal W > at some point p ∈ M . (3) µ q < − m(cid:96)m +2 , ρ θ semidefinite on M and ( m + 2 + (cid:96) )scal W > at some point p ∈ M . ( b ) If | (cid:96) | > m + 2 and ρ θ (cid:54)≡ is negative semidefinite then any harmonic spinor is a sectionof the extremal bundles Σ m ⊕ Σ − m . ( c ) If | (cid:96) | < m + 2 and ρ θ (cid:54)≡ is positive semidefinite then any harmonic spinor is a section ofthe extremal bundles Σ m ⊕ Σ − m . Proof .
Let φ = φ µ q (cid:54)≡ φ = ( m − q ) φ . We put Q q = − i2 (cid:18) (cid:96)m + 2 + µ q m (cid:19) ρ θ + (cid:18) (cid:96)µ q m ( m + 2) (cid:19) scal W A ( φ ) = (cid:90) M (cid:10) Q q φ, φ (cid:11) vol θ . The Schr¨odinger–Lichnerowicz-type formula (8.1) gives (cid:107) D θ φ (cid:107) = 2 qm (cid:107)∇ φ (cid:107) + 2( m − q ) m (cid:107)∇ φ (cid:107) + A ( φ ) . (8.3)4 F. LeitnerIf the eigenvalues of ρ θ have no different signs, we have by Cauchy–Schwarz inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) i ρ θ · φ, φ (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ scal W | φ | . Hence, A ( φ ) ≥ ( m − µ q )( m + 2 − (cid:96) ) m ( m + 2) (cid:90) M scal W | φ | vol θ for m(cid:96) + ( m + 2) µ q ≥
0, and A ( φ ) ≥ ( m + µ q )( m + 2 + (cid:96) ) m ( m + 2) (cid:90) M scal W | φ | vol θ for m(cid:96) + ( m + 2) µ q ≤ µ q (cid:54) = ± m is non-extremal. Then A ( φ ) is obviously positive for the threecases of part (a) of the proposition. In particular, the right hand side of (8.3) is always positive.This shows that no harmonic spinors exist in these three cases.If | (cid:96) | > m + 2 and ρ θ (cid:54)≡ µ q (cid:54) = ± m . If | (cid:96) | < m + 2 and ρ θ (cid:54)≡ µ q (cid:54) = ± m . (cid:4) Note that we have no vanishing results for harmonic spinors in the extremal bundles Σ − m or Σ m . Such spinors are simply holomorphic or antiholomorphic , respectively. So far we also haveno vanishing results for the canonical and anticanonical spin C structures when (cid:96) = ± ( m + 2).However, there are vanishing results in these cases (see [21] and Section 10). Example 8.2.
Any pseudo-Einstein spin manifold M (i.e., (cid:96) = 0) admits parallel spinors in theextremal bundles, no matter of the sign of the Webster scalar curvature (see [13]). For scal W > M .Let us consider the CR-covariant components D (cid:96) of the Kohn–Dirac operator. The formaladjoint of D (cid:96) , (cid:96) ∈ {− m, − m + 2 , . . . , m − , m } , is the restriction of D θ to the image of D (cid:96) , i.e., D ∗ (cid:96) = D θ : Im( D (cid:96) ) → Γ (cid:0) Σ − (cid:96) (cid:1) . Then the Schr¨odinger–Lichnerowicz-type formula (8.1) for D (cid:96) is expressed by D ∗ (cid:96) D (cid:96) = m + (cid:96)m ∇ ∗ ∇ + m − (cid:96)m ∇ ∗ ∇ + i (cid:96)ρ θ m ( m + 2) + (cid:18) − (cid:96) m ( m + 2) (cid:19) scal W . (8.4)Especially, for (cid:96) = 0, we have D ∗ D = ∆ tr + scal W . The latter formula looks like the classical Schr¨odinger–Lichnerowicz formula of Riemanniangeometry. This immediately shows that in the spin case h m ( α ) > (cid:96) = 0 exists) poses an obstruction to the existence of any adapted pseudo-Hermitianstructure on the CR manifold ( M, H ( M ) , J ) with positive Webster scalar scal W >
0. We givea more general version of this statement in terms of Kohn–Rossi cohomology in Corollary 10.4.Similarly, (8.4) implies that the CR invariants h m + (cid:96) ( α ) > | (cid:96) | < m , are obstructions to thepositivity of the Ricci form ρ θ > Example 8.3.
In Section 12 we construct closed CR manifolds over hyperK¨ahler manifolds which admit harmonic spinors of weight (cid:96) = 0. Such CR manifolds admit no adapted pseudo-Hermitian structure θ of positive Webster scalar curvature. Example 8.4.
There exist compact quotients of the
Heisenberg group , which are strictly pseu-doconvex and spin with harmonic spinors of weight (cid:96) = 0 (see [20]).
We briefly review here the
Kohn–Rossi complex [8] over CR manifolds, twisted with some CRvector bundle E . With respect to a pseudo-Hermitian form we construct the Kohn Laplacian (cid:3) E .Even though (cid:3) E is not an elliptic operator, there is a well behaving harmonic theory , similar to Hodge theory . In particular,
Kohn–Rossi cohomology groups are finite and cohomology classesadmit unique harmonic representatives over closed manifolds. This theory is due to J.J. Kohn(see [4, 7]). Our exposition of the topic follows [21] by N. Tanaka.Let (cid:0) M m +1 , H ( M ) , J (cid:1) be a closed manifold equipped with a strictly pseudoconvex CR struc-ture of hypersurface type and CR dimension m ≥
1. With respect to the complex structure J we have the decomposition H ( M ) ⊗ C = T ⊕ T of the Levi distribution. We define complexdifferential forms of degree ( p, q ) on H ( M ) byΛ p,q ( H ( M )) := Λ p T ∗ ⊗ Λ q T ∗ . Then Λ r ( H ( M )) ⊗ C = (cid:77) p + q = r Λ p,q ( H ( M )) . (Note that ( p, q )-forms on H ( M ) are not complex differentials form on M .)We are interested in the bundles Λ ,q ( H ( M )) = Λ q T ∗ of (0 , q )-forms. The correspondingspaces of smooth sections over M are denoted by C q ( M ), q = 0 , . . . , m . There exist tangentialCauchy–Riemann operators ¯ ∂ b : C q ( M ) → C q +1 ( M ) , q ∈ { , . . . , m } . These differential operators are by construction CR invariants and the sequence0 −→ C ( M ) ¯ ∂ b −→ C ( M ) ¯ ∂ b −→ · · · ¯ ∂ b −→ C m − ( M ) ¯ ∂ b −→ C m ( M ) −→ Kohn–Rossi complex . Its cohomology groups are denoted by H ,q ( M ), q ≥ E over M . We assume that E isequipped with some Cauchy–Riemann operator¯ ∂ E : Γ( E ) → Γ( E ⊗ T ∗ ) , i.e., ¯ ∂ E satisfies¯ ∂ E ( f u )( X ) = X ( f ) · u + f · (cid:0) ¯ ∂ E u (cid:1) ( X ) , ( ¯ ∂ E u )([ X , Y ]) = ¯ ∂ E (cid:0) ¯ ∂ E u ( X ) (cid:1) ( Y ) − ¯ ∂ E (cid:0) ¯ ∂ E u ( Y ) (cid:1) ( X )for any smooth C -valued function f and sections X , Y in T . We call (cid:0) E, ¯ ∂ E (cid:1) a CR vectorbundle over M . Smooth sections u of E with ¯ ∂ E u = 0 are holomorphic sections (see [21]).6 F. LeitnerFurthermore, for some CR vector bundle E over M , we set C q ( M, E ) = Λ q T ∗ ⊗ E and C q ( M, E ) = Γ( C q ( M, E )) for smooth sections. The holomorphic structure ¯ ∂ E on E extends toCauchy–Riemann operators¯ ∂ E : C q ( M, E ) → C q +1 ( M, E )for any q ∈ { , . . . , m } . This is by construction a twisted complex (cid:0) C q ( M, E ) , ¯ ∂ E (cid:1) , and wedenote the corresponding cohomology groups by H q ( M, E ), q ∈ { , . . . , m } . For E the trivialline bundle over M , these are the Kohn–Rossi cohomology groups H ,q ( M ) (see [21]).Let us assume now that a pseudo-Hermitian form θ is given on M and that the CR vectorbundle E → M is equipped with a Hermitian inner product (cid:104)· , ·(cid:105) E . In this setting we havea direct sum decomposition T ( M ) ⊗ C = T ⊕ T ⊕ R T θ , which gives rise to a unique identification of C q ( M, E ) with a subbundle of Λ q ( T ∗ ( M )) ⊗ E .And there exists a canonical connection D : Γ( E ) → Γ( T ∗ ( M ) ⊗ E ) compatible with (cid:104)· , ·(cid:105) E andrelated to the Cauchy–Riemann operator by D X u = ¯ ∂ E u ( X ), X ∈ T , for any u ∈ Γ( E ).Together with the Webster–Tanaka connection ∇ W we obtain covariant derivatives D : Γ (cid:0) Λ q ( M ) ⊗ E (cid:1) → Γ (cid:0) Λ q +1 ( M ) ⊗ E (cid:1) , q ∈ { , . . . , m } , and with respect to a local frame ( E , . . . , E m ) of T the Cauchy–Riemann operators are given by¯ ∂ E u = m (cid:88) α =1 E α ∗ ∧ D E α u for u ∈ C q ( M, E ).Moreover, for any q ∈ { , . . . , m } , the vector bundle C q ( M, E ) is equipped with a Hermitianinner product, which gives rise via vol θ to an L -inner product on C q ( M, E ). This allows theconstruction of a formally adjoint differential operator¯ ∂ ∗ E : C q +1 ( M, E ) → C q ( M, E )to ¯ ∂ E . With respect to a local frame ( E , . . . , E m ) the operator ¯ ∂ ∗ E is given by¯ ∂ ∗ E u = − m (cid:88) α =1 ι E α D E α u for u ∈ C q +1 ( M, E ).Finally, we can construct the
Kohn Laplacian (cid:3) E := ¯ ∂ ∗ E ¯ ∂ E + ¯ ∂ E ¯ ∂ ∗ E : C q ( M, E ) → C q ( M, E ) , q ∈ { , . . . , m } , with respect to θ on M . This is a 2nd order differential operator, which is formally self-adjointwith respect to ( · , · ) L on C q ( M, E ). Due to results of Kohn the operator (cid:3) E is sub- andhypoelliptic. We put H q ( M, E ) := (cid:8) u ∈ C q ( M, E ) | (cid:3) E u = 0 (cid:9) for the space of harmonic (0 , q )-forms. Since (cid:3) E is formally self-adjoint the harmonic equation (cid:3) E u = 0 is equivalent to ¯ ∂ E u = ¯ ∂ ∗ E u = 0 on M . It is known that H q ( M, E ) is finite dimensionalnvariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry 17for any q ∈ { , . . . , m − } . Moreover, every class in the Kohn–Rossi cohomology group H q ( M, E )admits a unique harmonic representative, i.e., H q ( M, E ) ∼ = H q ( M, E ) . In particular, the Kohn–Rossi cohomology groups H q ( M, E ) are finite dimensional for q ∈{ , . . . , m − } . The groups H ( M, E ) and H m ( M, E ) are infinite dimensional, in general.However, we still have H ( M, E ) ∼ = H ( M, E ) and H m ( M, E ) ∼ = H m ( M, E ) for any m ≥ m = 1 and M ⊆ C is embedded as CR manifold H ( M, E ) ∼ = H ( M, E ) and H ( M, E ) ∼ = H ( M, E ) are certainly true as well.
10 Vanishing theorems for twisted Kohn–Rossi cohomology
The harmonic theory of the previous section fits well to our discussion of Kohn–Dirac operatorsand harmonic spinors. In fact, the square D θ of the Kohn–Dirac operator has a natural interpre-tation as Kohn Laplacian (cid:3) E if we only make the appropriate choice for the CR line bundle E (see [18]). This justifies the name for D θ and gives rise via the Schr¨odinger–Lichnerowicz-typeformula to vanishing results for twisted Kohn–Rossi cohomology (see in [21, Section II, § (cid:0) M m +1 , H ( M ) , J (cid:1) be a closed manifold equipped with strictly pseudoconvex CR struc-ture of hypersurface type and CR dimension m ≥
1. We fix a pseudo-Hermitian form θ on M with spin C structure of weight (cid:96) ∈ Z . The corresponding spinor bundle Σ( H ( M )) → M decom-poses intoΣ( H ( M )) = m (cid:77) q =0 Σ µ q ( H ( M )) . The Kohn–Dirac operator is given by D θ = D + + D − . In particular, we have the spinorialcomplex (6.1) (cid:0) Γ (cid:0) Σ µ q (cid:1) , D + (cid:1) with cohomology groups, which we denote by S q ( M ), q = 0 , . . . , m (cf. the notion of spinorial cohomology in [15]).Recall that the chosen spin C structure on ( M, θ ) is uniquely determined by some complex linebundle E ( α ) → M , α ∈ H ( M, Z ), which is a square root of K ⊗ L , L = E ( (cid:96) ) the determinantbundle. Note that we can use the Webster–Tanaka connection ∇ W to define a holomorphicstructure on E ( α ) through ¯ ∂ E ( α ) η ( X ) := ∇ W X η , X ∈ T , for η ∈ Γ( E ( α )).Studying the spinor module Σ with Clifford multiplication c shows that the spinor bundleΣ( H ( M )) → M is isomorphic to m (cid:77) q =0 Λ q T ∗ ⊗ Σ µ ( H ( M )) . Moreover, the factor Σ µ ( H ( M )) is isomorphic to the line bundle E ( α ). In fact, we haveΣ µ q ( H ( M )) ∼ = Λ q T ∗ ⊗ E ( α ) of rank rk C = ( mq ). Hence, the identificationsΓ(Σ µ q ) ∼ = C q ( M, E ( α )) , q = 0 , . . . , m, (10.1)with the chain groups of the Kohn–Rossi complex, twisted by E ( α ).Examining the Clifford multiplication shows that the operator √ D + corresponds via (10.1)to the Cauchy–Riemann operator ¯ ∂ E ( α ) . In particular, we have S q ( M ) ∼ = H q ( M, E ( α )) , q ∈ { , . . . , m } , for the cohomology groups of the spinorial complex.8 F. LeitnerMoreover, the formal adjoint √ D − corresponds to ¯ ∂ ∗ E ( α ) . Hence, via (10.1) we have D θ = √ · (cid:0) ¯ ∂ E ( α ) + ¯ ∂ ∗ E ( α ) (cid:1) for the Kohn–Dirac operator and D θ = 2 (cid:3) E ( α ) for the square (cf. [18]). This shows that har-monic spinors with Θ-eigenvalue µ q = m − q , q = 0 , . . . , m , are in 1-to-1-correspondence withharmonic (0 , q )-forms with values in E ( α ). In particular, with results of Section 9 we can inter-pret harmonic spinors on closed M as representatives of twisted Kohn–Rossi cohomology classes. Theorem 10.1.
Let (cid:0) M m +1 , H ( M ) , J (cid:1) , m ≥ , be a strictly pseudoconvex and closed CRmanifold with pseudo-Hermitian form θ and spin C structure, determined by α ∈ H ( M, Z ) . . For the space of harmonic spinors to the eigenvalue µ q , q = 0 , . . . , m , we have H q ( α ) ∼ = H q ( M, E ( α )) ∼ = H q ( M, E ( α )) ∼ = S q ( M ) . . The space H q ( α ) of harmonic spinors on M is finite dimensional for any q ∈ { , . . . , m − } . Remark 10.2.
The cohomology groups H q ( M, E ( α )), q ∈ { , . . . , m } , of the twisted Kohn–Rossi complex are invariant objects of the underlying CR structure on M , whereas the construc-tion of the spaces of harmonic spinors H q ( α ) and harmonic (0 , q )-forms H q ( M, E ( α )) dependson the pseudo-Hermitian structure θ . It is only for µ q = − (cid:96) that we have seen in Section 7 thatharmonic spinors in H m + (cid:96) ( α ) are solutions of a CR invariant equation.Theorem 10.1 shows now that elements in H q ( α ), q ∈ { , . . . , m } , can be identified fordifferent pseudo-Hermitian forms θ and ˜ θ via the corresponding Kohn–Rossi cohomology groups.In particular, all the dimensions h q ( α ) of the spaces H q ( α ), q ∈ { , . . . , m − } , are CR invariantnumbers.Recall that a spin structure for the Levi distribution H ( M ) on M is given by some squareroot E ( α ) of the canonical bundle K → M . We denote the chosen square root by √K . On theother hand, the canonical spin C structure on M is given by the trivial line bundle E ( α ) = M × C .In this case (cid:96) = m + 2. We have the following vanishing results for Kohn–Rossi cohomology. Theorem 10.3.
Let (cid:0) M m +1 , H ( M ) , J (cid:1) , m ≥ , be a strictly pseudoconvex and closed CRmanifold with pseudo-Hermitian form θ and spin C structure of weight (cid:96) determined by α ∈ H ( M, Z ) . . If ρ θ (cid:54)≡ is negative semidefinite, | (cid:96) | > m + 2 and q ∈ { , . . . , m − } , then H q ( M, E ( α )) = { } for the q th ( α -twisted ) Kohn–Rossi cohomology group. . If ρ θ (cid:54)≡ is positive semidefinite, | (cid:96) | < m + 2 and q ∈ { , . . . , m − } , then H q ( M, E ( α )) = { } . . If ρ θ > is positive definite, E ( α ) = M × C and q ∈ { , . . . , m − } , then H ,q ( M ) = { } for the q th Kohn–Rossi group. . If scal W (cid:54)≡ is non-negative on a closed CR spin manifold M of even CR dimension m ,then H m (cid:0) M, √K (cid:1) = { } . nvariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry 19 Proof .
Part (1) and (2) of Theorem 10.3 follow immediately from Proposition 8.1 via theidentifications in Theorem 10.1. Part (3) is the statement of Proposition 7.4 on p. 62 in [21] forKohn–Rossi cohomology. We reprove this result here.The curvature term (8.2) decomposes into two summands as follows: Q q = − i2 (cid:18) (cid:96)m + 2 + µ q m (cid:19) ρ θ + (cid:18) (cid:96)µ q m ( m + 2) (cid:19) scal W (cid:16) µ q m (cid:17) (cid:18) − i2 ρ θ + scal W (cid:19) − i( (cid:96) − m − m + 2) (cid:18) ρ θ − scal W · d θ m (cid:19) =: 2( m − q ) m R ∗ + K. The second summand K vanishes for (cid:96) = m + 2. In any case we have tr θ K = 0.Via (10.1) Clifford multiplication on (0 , q )-forms u ∈ Λ q T ∗ is given by X · u = √ · (cid:0) X ∗ ∧ u − ι X u (cid:1) for any X ∈ H ( M ) (see, e.g., [16] and [18]). Then a short computations shows that R ∗ = − i2 ρ θ + scal W acts on u ∈ C q ( M, E ( α )) by( R ∗ u ) (cid:0) X , . . . , X q (cid:1) = q (cid:88) α =1 u (cid:0) X , . . . , ρ θ (cid:0) X α (cid:1) , . . . , X q (cid:1) for any X , . . . , X q ∈ Λ q T ∗ , i.e., R ∗ is the Ricci operator on C q ( M, E ( α )).On the other hand, the action of K is induced by the curvature of the canonical connec-tion on E ( α ) (which differs from the Webster–Tanaka connection, in general). Thus, Q q = m − q ) m R ∗ + K is exactly the curvature term of the Weitzenb¨ock formula in Proposition 5.1 onp. 47 of [21].In particular, for the case of the canonical spin C structure α = 0, we have (cid:96) = m + 2 and K = 0. If ρ θ > M , R ∗ is positive definite as well. Application of theWeitzenb¨ock formula for C q ( M, E ( α )) shows that there are no harmonic forms.Finally, in the spin case (cid:96) = 0 with even m , we have Q m = scal W and (8.1) shows that thereare no harmonic spinors. Hence, no harmonic forms in C m (cid:0) M, √K (cid:1) . (cid:4) On the other hand, non-trivial Kohn–Rossi groups pose obstructions to positive Webstercurvature on the underlying CR manifold. We putˆ q = ˆ q ( m, (cid:96) ) := m ( m + (cid:96) + 2)2( m + 2)and highlight the following result, which resembles the classical obstruction for positive scalarcurvature on K¨ahler manifolds (cf. [5, 14]). Corollary 10.4.
Let (cid:0) M m +1 , H ( M ) , J (cid:1) , m ≥ , be a strictly pseudoconvex and closed CRmanifold with spin C structure of weight (cid:96) ∈ Z to α ∈ H ( M, Z ) . If | (cid:96) | < m + 2 , ˆ q ∈ Z and H ˆ q ( M, E ( α )) (cid:54) = { } , then M admits no adapted pseudo-Hermitian structure θ of positive Webster scalar curvature scal W > . Proof .
For ˆ q ∈ Z , we have µ ˆ q = − m(cid:96)m +2 ∈ Z and Q ˆ q = (cid:0) − (cid:96) ( m +2) (cid:1) scal W . If | (cid:96) | < m + 2, thenˆ q ∈ { , . . . , m − } and the functions Q ˆ q and scal W have the same sign. The non-vanishing of H ˆ q ( M, E ( α )) implies the existence of a harmonic spinor in Γ(Σ µ ˆ q ). This is impossible by (8.1)when scal W > (cid:4) Example 10.5. If M is a closed and strictly pseudoconvex CR spin manifold of even CRdimension m ≥
2, then the cohomology group H m (cid:0) M, √K (cid:1) poses an obstruction to scal W >
0. We give some concrete example of this case in Section 12(cf. also Corollary 11.2).
Example 10.6. If m = 4 and (cid:96) = −
3, then ˆ q = 1 and H ( M, E ( α )) poses an obstruction toscal W > Example 10.7.
Let (cid:0) M m +1 , θ (cid:1) , m ≥
2, be some pseudo-Einstein space with Ric W ( T ) = 0.Then scal W is constant on M . E.g., this happens when the Webster torsion τ is parallel, orwhen the characteristic vector T θ is a transverse symmetry of the underlying CR structure (cf.Section 11).We assume a spin C structure of weight (cid:96) and ρ θ (cid:54)≡
0. Then, either ρ θ > | (cid:96) | ≤ m + 2and 1 < q < m , we have H q ( M, E ( α )) = { } (by Theorem 10.3). E.g., Einstein–Sasakian manifolds of Riemannian signature give rise to such pseudo-Hermitian structures θ (see, e.g., [1]for the notion of Sasakian structures). In Section 12 we construct regular Einstein–Sasakianmanifolds from K¨ahler geometry.In the other case ρ θ < H q ( M, E ( α )) = { } for | (cid:96) | > m + 2 and any 1 < q < m . E.g., Einstein–Sasakian manifolds of Lorentzian signature fit to this situation.
11 Cohomology of regular, torsion-free CR manifolds
Let (cid:0) M m +1 , H ( M ) , J (cid:1) , m ≥
2, be strictly pseudoconvex with pseudo-Hermitian form θ . Wecall the characteristic vector T θ regular if all its integral curves are 1-dimensional submanifoldsof M and the corresponding leaf space N is a smooth manifold of dimension 2 m with smoothprojection π : M → N . If, in addition, θ has vanishing Webster torsion τ = 0 then T θ is aninfinitesimal automorphism of the underlying CR structure and a Killing vector for the
Webstermetric g W = g θ + θ ◦ θ on M . Such a vector T θ is called transverse symmetry . (Here weunderstand transverse to H ( M ).) It is straightforward to see that in this case the pseudo-Hermitian structure ( H ( M ) , J, θ ) projects to a K¨ahler structure on the leaf space N . Note thatif M is closed then π : M → N is a circle fiber bundle. In this case we call π : ( M, θ ) → N a (regular, torsion-free) CR circle bundle of complex dimension m ≥ Let us consider the underlying leaf space N with K¨ahler metric h , complex structure J andfundamental form ω . As for any complex manifold, we have the ( p, q )-forms Λ p,q ( N ) and theCauchy–Riemann operators¯ ∂ : Γ (cid:0) Λ p,q ( N ) (cid:1) → Γ (cid:0) Λ p,q +1 ( N ) (cid:1) , which in turn give rise to the Dolbeault cohomology groups H p,q ( N ), p, q ≥
0. In the following weare interested in the cohomology groups H ,q ( N ), q ≥
0. In fact, more generally, let E (cid:48) → N besome holomorphic vector bundle and O ( E (cid:48) ) the sheaf of local holomorphic sections of E (cid:48) . Thenwe have the q th cohomology group H q ( N, O ( E (cid:48) )) of the sheaf O ( E (cid:48) ), which is, by Dolbeault’stheorem , isomorphic to H ,q ( N, E (cid:48) ), q ≥ E (cid:48) be some complex line bundle over the K¨ahler manifold N . We assume that E (cid:48) is a root of some power of the anticanonical line bundle K (cid:48)− → N . This ensures that E (cid:48) isequipped with a holomorphic structure and Hermitian inner product, both compatible with thenvariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry 21Levi-Civita connection of the K¨ahler metric. The pullback E = π ∗ E (cid:48) is a line bundle over M with Hermitian inner product and the Webster–Tanaka connection induces some holomorphicstructure on E as well. Then any smooth section u (cid:48) ∈ Γ( E (cid:48) ) lifts to some smooth section u = π ∗ u (cid:48) of E → M . By construction, the Lie derivative L T u of the lift in characteristic direction vanishesidentically on M . In fact, any smooth section u ∈ Γ( E ) with L T u = 0 is the pullback of someunique section u (cid:48) in E (cid:48) → N . We call such sections in E → M projectable . More generally, for q ≥
0, we have the subspaces C q (0) ( M, E ) ⊆ C q ( M, E )of projectable (0 , q )-forms in the chain groups of the Kohn–Rossi complex with values in theline bundle E . These subgroups are naturally identified with the (0 , q )-forms Γ (cid:0) Λ ,q ⊗ E (cid:48) (cid:1) withvalues in E (cid:48) over the K¨ahler manifold N .Since the Webster torsion τ θ vanishes on M , the holomorphic structures on E (cid:48) and on itspullback E are compatible: ¯ ∂ E π ∗ v (cid:48) = π ∗ ¯ ∂ E (cid:48) v (cid:48) for any (0 , q )-form v (cid:48) on N . Also ¯ ∂ ∗ E π ∗ v (cid:48) = π ∗ ¯ ∂ ∗ E (cid:48) v (cid:48) is true. Now let γ ∈ H ,q ( N, E (cid:48) ) be a class in the q th Dolbeault group. By classical Hodgetheory γ is uniquely represented by some harmonic (0 , q )-form u (cid:48) with values in E (cid:48) . In general,the lift u = π ∗ u (cid:48) of some non-trivial harmonic u (cid:48) is a non-trivial element of C q (0) ( M, E ), which isharmonic with respect to the Kohn Laplacian (cid:3) E . Thus, we have an inclusion π ∗ : H q ( N, E (cid:48) ) (cid:44) → H q ( M, E )of spaces of harmonic forms. This again gives rise to a natural inclusion π ∗ : H q ( N, O ( E (cid:48) )) (cid:44) → H q ( M, E )of the holomorphic cohomology group H q ( N, O ( E (cid:48) )) into the Kohn–Rossi group H q ( M, E ) forany q ≥
0. We denote the image of this inclusion by H q (0) ( M, E ). By construction, classesin H q (0) ( M, E ) are represented by projectable harmonic (0 , q )-forms on M , i.e., by elementsof H q (0) ( M, E ).Let us assume now that the given holomorphic line bundle E (cid:48) → N is a square root of K (cid:48) ⊗ L (cid:48) ,where L (cid:48) → N is a line bundle of weight (cid:96) ∈ Z , i.e., L (cid:48) m +2 is the (cid:96) th power of the anticanonicalbundle K (cid:48)− over N . Then E (cid:48) determines a spin C structure on N with determinant bundle L (cid:48) of weight (cid:96) . This lifts to a spin C structure on ( M, θ ) with determinant bundle L = π ∗ L (cid:48) ofweight (cid:96) . Of course, the corresponding spinor bundle Σ (cid:48) → N pulls back to the spinor bundle Σover ( M, θ ) and harmonic spinors in Σ (cid:48) lift to harmonic spinors in Σ → M . Theorem 11.1.
Let π : ( M, θ ) → N be some CR circle bundle of complex dimension m ≥ with spin C structure of weight | (cid:96) | < m + 2 ( determined by some line bundle E = π ∗ E (cid:48) → M ) and assume ˆ q = m ( m + (cid:96) +2)2( m +2) ∈ Z . ( a ) If the scalar curvature scal h (cid:54)≡ is non-negative on the K¨ahler manifold N then thereexist no harmonic spinors on M to the Θ -eigenvalue µ ˆ q = − m(cid:96)m +2 and the Kohn–Rossicohomology group H ˆ q ( M, E ) = { } is trivial. ( b ) If the holomorphic cohomology group H ˆ q ( N, O ( E (cid:48) )) (cid:54) = { } is non-trivial then the strictlypseudoconvex CR manifold M admits no adapted pseudo-Hermitian structure of positiveWebster scalar curvature. Proof .
Since T θ is a transverse symmetry, we have ι T θ R W = 0 for the Webster curvatureoperator. Hence, the pseudo-Hermitian Ricci form ρ θ is the lift of the Ricci 2-form on N , andthe Webster scalar curvature on M is the lift of the Riemannian scalar curvature on N .2 F. LeitnerIf scal h (cid:54)≡ N , the curvature term Q ˆ q = (cid:0) − (cid:96) ( m +2) (cid:1) scal W (cid:54)≡ H ˆ q ( M, E ) = { } . Onthe other hand, we have an inclusion of H ˆ q ( N, O ( E (cid:48) )) into H ˆ q ( M, E ). Hence, if H ˆ q ( N, O ( E (cid:48) ))is non-trivial then H ˆ q ( M, E ) as well. By Corollary 10.4, this obstructs the existence of somepseudo-Hermitian form with positive Webster scalar curvature. (cid:4)
In case of spin structures we have the following special result.
Corollary 11.2.
Let π : ( M, θ ) → N be some CR circle bundle of even complex dimension m ≥ with spin structure √K (cid:48) → N . ( a ) If scal h (cid:54)≡ is non-negative on N , the Kohn–Rossi cohomology group H m (cid:0) M, √K (cid:1) = { } is trivial. ( b ) If the holomorphic cohomology group H m (cid:0) N, O (cid:0) √K (cid:48) (cid:1)(cid:1) (cid:54) = { } is non-trivial then thestrictly pseudoconvex CR manifold M admits no adapted pseudo-Hermitian structure ofpositive Webster scalar curvature. Let (cid:0) M m +1 , θ (cid:1) , m ≥
2, be some closed pseudo-Hermitian manifold with vanishing Webstertorsion τ θ = 0. We assume now that the characteristic vector T of θ is induced from somefree U(1)-action on M . This implies that T is regular with underlying K¨ahler manifold N . Infact, here π : ( M, θ ) → N is a smooth principal U(1)-bundle and θ is a connection form. Wecall π : ( M, θ ) → N a CR principal
U(1) -bundle . The corresponding holomorphic line bundle isdenoted by F (cid:48) → N and F = π ∗ F (cid:48) is its pullback to M .Let C q ( M ) denote the chain groups of the Kohn–Rossi complex on M . In [21] the differentialoperator N : C q ( M ) → C q ( M ) , q ≥ ,u (cid:55)→ i ∇ W T u, is introduced. Let us denote by C q ( λ ) ( M ) := (cid:8) u ∈ C q ( M ) : N u = λu (cid:9) the corresponding λ -eigenspace. Since the Webster torsion τ θ = 0 vanishes, we have in general ∇ W T u = L T u for the Lie derivative of any chain u ∈ C q ( M ). This shows that C q (0) ( M ) are exactlythe projectable (0 , q )-forms on M .The operator N is self-adjoint and commutes with ¯ ∂ and ¯ ∂ ∗ . Hence, N acts on the har-monic spaces H ,q ( M ), q ≥
0, as well. All eigenvalues λ are real. In fact, since H ,q ( M ) isfinite dimensional for 0 < q < m , we have finitely many real eigenvalues λ and a direct sumdecomposition H ,q ( M ) = (cid:77) λ H ,q ( λ ) ( M ) , which implies for the Kohn–Rossi groups: H ,q ( M ) = (cid:77) λ H ,q ( λ ) ( M ) , ≤ q ≤ m − . nvariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry 23In [21] it is shown that for any integers q, s ∈ Z the cohomology groups H ,q ( s ) ( M ) ∼ = H q (cid:0) N, O ( F (cid:48) s ) (cid:1) are naturally identified. Here F (cid:48) s denotes the s th power of the line bundle F (cid:48) → N , which isassociated to the given CR principal U(1)-bundle π : ( M, θ ) → N .On the other hand, let F = π ∗ F (cid:48) → M denote the pullback of F (cid:48) . Via the projection π : M → N the holomorphic cohomology group H q ( N, O ( F (cid:48) s )) is identified with the subgroup H q (0) ( M, F s ) ⊆ H q ( M, F s ) of F s -twisted Kohn–Rossi cohomology. This gives rise to a naturalidentification H ,q ( s ) ( M ) ∼ = H q (0) ( M, F s ) , (11.1)where the classes in H q (0) ( M, F s ) are uniquely represented by projectable harmonic (0 , q )-formswith values in F s . We call (11.1) a shift in Kohn–Rossi cohomology over the CR principalU(1)-bundle π : ( M, θ ) → N .We apply (11.1) to our situation of spin C structures. For this we assume that the pullback F = π ∗ F (cid:48) → M of the line bundle F (cid:48) → N associated to π : M → N has weight f ∈ Z . Then E := F s = E ( sf ) determines a spin C structure of weight (cid:96) = 2 sf + m + 2 on ( M, θ ). Theorem 11.3.
Let π : (cid:0) M m +1 , θ (cid:1) → N , m ≥ , be some CR principal U(1) -bundle of weight f ∈ Z and E := E ( sf ) , s ∈ Z . Then . H q (0) ( M, E ) = { } for q < m and s > . . If msfm +2 ∈ { − m, . . . , − } and scal h > then H , ˆ q ( s ) ( M ) = { } for ˆ q = m (cid:0) sfm +2 (cid:1) . Proof . (1) Comparing the Kohn Laplacian (cid:3) = ¯ ∂ ∗ ¯ ∂ + ¯ ∂ ¯ ∂ ∗ with (cid:3) = ∂ ∗ ∂ + ∂∂ ∗ gives (cid:3) + ( m − q ) N = (cid:3) (see [21]). Hence, the subgroups H q (0) ( M, E ) ∼ = H ,q ( s ) ( M ), q < m , are all trivial foreigenvalues s > (cid:4) Example 11.4.
Let us assume that the CR dimension m ≥ f of F → N is a positive factor of m + 1, i.e., s := − m +22 f ∈ Z . Then E = F s = K , i.e., F s → M definesa spin structure on the CR manifold M . Hence, if the scalar curvature scal h of the underlyingK¨ahler manifold is positive, the Kohn–Rossi subgroup H , m ( s ) ( M ) ∼ = H m (0) ( M, F s ) is trivial.In other words, in this situation the (untwisted) Kohn–Rossi subgroup H , m ( s ) ( M ) (cid:54) = { } to the negative N -eigenvalue s = − m +22 f ∈ Z is an obstruction to positive Webster scalarcurvature on M .
12 Examples of CR principal U(1)-bundles
We discuss some examples of CR principal U(1)-bundles with applications of vanishing theo-rems for harmonic spinors and Kohn–Rossi cohomology. The construction here is based on theunderlying K¨ahler manifold.For our construction, let (cid:0) N m , ω, J (cid:1) , m ≥
2, be a closed K¨ahler manifold such that a non-trivial multiple α := c [ ω ], c ∈ R \ { } , of the K¨ahler class [ ω ] (cid:54) = 0 is integral, i.e., α ∈ H ( N, Z ).By Lefschetz’s theorem on (1 , -classes , there exists some Hermitian line bundle L (cid:48) → N withfirst Chern class c ( L (cid:48) ) = α . Let π : M → N be the corresponding principal U(1)-bundle to4 F. Leitnerwhich L (cid:48) → N is associated. By the dd c -lemma , this bundle admits a connection form A ω withcurvature d A ω = π ∗ ω , the lift of the K¨ahler form. The lift of the complex structure J to the hor-izontal distribution H ( M ) of the connection A ω gives rise to a CR structure ( H ( M ) , J ) on M ,which is by construction strictly pseudoconvex of hypersurface type with CR dimension m .In addition, we set θ := 2 A ω . This is an adapted pseudo-Hermitian form, which has byconstruction vanishing torsion τ = 0, the characteristic vector T is regular and g θ = d θ ( · , J · )is the lift of the K¨ahler metric to H ( M ). Thus,( M , H ( M ) , J )is a CR principal U(1)-bundle with pseudo-Hermitian form θ over the K¨ahler manifold N . Letus call π : M → N the CR α -bundle . Example 12.1.
Let N m be a closed K¨ahler–Einstein manifold of positive scalar curvaturescal h > π : M → N be the principal U(1)-bundle to which the anticanonical bundle K (cid:48)− → N is associated. The Ricci 2-form is given by ρ = scal h m ω and the first Chern classis c ( N ) = − c ( K (cid:48) ) = (cid:2) π ρ (cid:3) ∈ H ( N, Z ). Hence, π : M → N is the CR c ( N )-bundle. Theadapted pseudo-Hermitian form θ is some multiple of the Levi-Civita connection on M , and thelift of the Ricci 2-form ρ along π : M → N is the pseudo-Hermitian Ricci form ρ θ of θ on M .This shows that ρ θ is positive definite in this case.Let M be given some spin C structure of weight (cid:96) ∈ Z , determined by some line bundle E → M . For | (cid:96) | < m + 2, Proposition 8.1 shows that any harmonic spinor on M is a sectionof the extremal bundles. Accordingly, Theorem 10.3 shows that, for | (cid:96) | ≤ m + 2 and any1 ≤ q ≤ m −
1, the q th Kohn–Rossi group H q ( M , E ) = { } is trivial. In particular, if N m ( m even) is spin, so is M and H m (cid:0) M , √K (cid:1) = { } .For example, the Hopf fibration π : S m +1 → C P m over the complex projective space is a CRprincipal U(1)-bundle of this kind. In fact, the associated line bundle to π is K (cid:48)− , and S m +1 isequipped with the standard CR structure . Hence, for any spin C structure of weight | (cid:96) | ≤ m + 2and 0 < q < m , we have H q (cid:0) S m +1 , E (cid:1) = { } (cf. [21]). (Note that C P m is spin only for m odd. However, the standard CR manifold S m +1 is spin for any m .) Example 12.2.
Let N m , m ≥ hyperK¨ahler metric h ,i.e., h is Ricci-flat and its holonomy group is contained in Sp (cid:0) m (cid:1) . In particular, N is spin andadmits some parallel spinor φ in Σ ( N ), i.e., H m (cid:0) N, O (cid:0) √K (cid:48) (cid:1)(cid:1) (cid:54) = { } .We assume now that h is a Hodge metric , i.e., the corresponding K¨ahler class [ ω ] ∈ H ( N, Z )is integral. E.g., any projective K3 surface admits some integral K¨ahler class [ ω ]. Then the CR[ ω ]-bundle π : M → N exists. Furthermore, the pullback of the spin structure on N gives riseto some spin structure √K → M and the lift ψ := π ∗ φ is a parallel section of Σ ( M ( H )) withrespect to the spinorial Webster–Tanaka connection ∇ Σ . In fact, the basic holonomy group ofthe adapted pseudo-Hermitian structure θ on M is contained in Sp (cid:0) m (cid:1) (cf. [13]). In particular, ψ is some harmonic spinor in Σ ( M ( H )) and the Kohn–Rossi group H m (cid:0) M , √K (cid:1) (cid:54) = { } is non-trivial. Accordingly, Corollary 10.4 states that the CR manifold M admits no pseudo-Hermitian structure of positive Webster scalar curvature. Similarly, we can apply Corollary 11.2,since H m (cid:0) N, O (cid:0) √K (cid:48) (cid:1)(cid:1) (cid:54) = { } .Observe that ψ is in the kernel of the twistor operator P as well, i.e., ψ is an example fora CR twistor spinor of weight (cid:96) = 0 on some closed, strictly pseudoconvex CR manifold.Finally, note that in the spin case there is a theory of K¨ahlerian twistor spinors on theunderlying N (see [17, 19]). The lift of such twistor spinors gives rise to projectable spinors inthe kernel of twistor operators on ( M, θ ). For the case µ q = (cid:96) = 0, this gives rise to CR twistorspinors. However, K¨ahlerian twistor spinors on N with µ q = 0 are necessarily parallel. Hence,as in our example, the corresponding CR twistor spinors on M are parallel as well. Apart fromnvariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry 25the standard CR sphere, we do not know further examples (of non-extremal weight) on closedCR manifolds. Acknowledgements
I would like to thank the referees for their helpful comments.
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