Invariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry

We study Kohn-Dirac operators $D_\theta$ on strictly pseudoconvex CR manifolds with ${\rm spin}^{\mathbb C}$ structure of weight $\ell\in{\mathbb Z}$. Certain components of $D_\theta$ are CR invariants. We also derive CR invariant twistor operators of weight $\ell$. Harmonic spinors correspond to cohomology classes of some twisted Kohn-Rossi complex. Applying a Schr\"odinger-Lichnerowicz-type formula, we prove vanishing theorems for harmonic spinors and (twisted) Kohn-Rossi groups. We also derive obstructions to positive Webster curvature.

We study in this paper the Kohn-Dirac operator D θ for spin C structures of weight ∈ Z on strictly pseudoconvex CR manifolds with adapted pseudo-Hermitian structure θ. Our construction 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-Hermitian structure. However, similar as in Kähler geometry, the spinor bundle Σ decomposes with respect to the CR structure into eigenbundles Σ µq of certain eigenvalues µ q . For µ q = − the restriction D of the Kohn-Dirac operator to Γ Σ µq acts CR-covariantly. This observation gives rise to 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 special solutions of the corresponding twistor equation, which realize some lower bound for the square of the first non-zero eigenvalue of the Kohn-Dirac operator D θ . For µ q = the corresponding twistor operator P is a CR invariant.
Analyzing the Clifford multiplication on the spinor bundle for spin C structures over strictly pseudoconvex 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 of the 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 Schrödinger-Lichnerowicz-type formula gives rise to vanishing theorems for twisted Kohn-Rossi groups.
For example, on a closed, strictly pseudoconvex CR manifold M of even CR dimension m ≥ 2 with spin structure given by a square root √ K of the canonical line bundle, we have for µ q = = 0 the Schrödinger-Lichnerowicz-type formula  [6] for the given CR structure is 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 CRcovariant components D and P are determined in Section 7. In Section 8 we recall the Schrödinger-Lichnerowicz-type formula and derive a basic vanishing theorem for harmonic spinors (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 we discuss CR circle bundles of Kähler manifolds and relate holomorphic cohomology groups to Kohn-Rossi groups. Finally, in Section 12 we construct closed, strictly pseudoconvex CR manifolds with H Throughout this paper we will deal with strictly pseudoconvex CR structures on M . For example, in the generic case when the Levi form is non-degenerate, the smooth boundary of a domain of holomorphy in C m+1 is strictly pseudoconvex. The complex structure J extends C-linearly to H(M ) ⊗ C, the complexification of the Levi distribution, and induces a decomposition H(M ) ⊗ C = T 10 ⊕ T 01 into ±i-eigenbundles. Then a complex-valued p-form η on M is said to be of type (p, 0) if ι Z η = 0 for all Z ∈ T 01 . This gives rise to the complex vector bundle Λ p,0 (M ) of (p, 0)-forms on M . For the (m + 1)st exterior power Λ m+1,0 (M ) of Λ 1,0 (M ) we write K = K(M ). This is the canonical line bundle of the CR manifold M with first Chern class c 1 (K) ∈ H 2 (M, Z). Its dual is the anticanonical bundle, denoted by K −1 .
When dealing with a strictly pseudoconvex CR manifold, we will often assume the existence and choice of some (m + 2)nd root E(1) of the anticanonical bundle K −1 , that is a complex line bundle over M with The dual bundle of this root is denoted by E(−1). Then, for any integer p ∈ Z, we have the pth power E(p) of E(1). We call p the weight of E(p). In particular, the canonical bundle K has weight −(m + 2), whereas the anticanonical bundle K −1 has weight m + 2.
In general, the existence of an (m + 2)nd root E(1) is restrictive to the global nature of the underlying CR structure on M . For the application of tractor calculus in CR geometry this assumption is basic. In fact, the standard homogeneous model of CR geometry on the sphere allows a natural choice for E(1) (see [2]). For our treatment of spin C structures in CR geometry the choice of some E(1) is useful as well.

Pseudo-Hermitian geometry
Let M 2m+1 , H(M ), J , m ≥ 1, be strictly pseudoconvex. Since M is orientable, there exists some 1-form θ on M , whose kernel Ker(θ) defines the contact distribution H(M ). The differential dθ is a non-degenerate 2-form on H(M ), and the conditions ι T θ = θ(T ) = 1 and ι T dθ = 0 define 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 with 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 dθ = 0 for the Lie derivatives with respect to T θ . Furthermore, defines a non-degenerate, symmetric bilinear form, i.e., a metric on H(M ), which is either positive or negative definite. In case g θ is positive definite, we call θ ∈ Ω 1 (M ) an adapted pseudo-Hermitian structure for M 2m+1 , H(M ), J . Note that any two pseudo-Hermitian structures θ andθ differ only by some positive function or conformal scale, i.e.,θ = e 2f 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 for X ∈ Γ(H(M )). The torsion is given by some obligatory part with transverse X, Y in H(M ) and, furthermore, by 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. The composition 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 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 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 tensor on H(M ). On the other hand, the pseudo-Hermitian Ricci form is given by for any X, Y ∈ H(M ). Then we have for any X, Y ∈ H(M ), where ρ θ corresponds to the J-invariant and τ is the J-antiinvariant part of Ric W on H(M ). If m ≥ 2 and the pseudo-Hermitian Ricci form ρ θ is a multiple of dθ we call θ a pseudo-Einstein structure on the CR manifold M 2m+1 , H(M ), J (see [11]). For m = 1 this condition is vacuous. However, for m > 1 the pseudo-Einstein condition implies Ric W (T, JX) = 1 4m X scal W for any 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 4m dθ and the Webster scalar curvature of some pseudo-Einstein structure θ need not be constant. In fact, it is constant if and only if Ric W (T ) = tr θ ∇ W · τ (·) = 0.

Spin C structures
Recall that the group Spin C (2m) is a central extension of SO(2m) given by the exact sequence This gives a twisted product Spin C (2m) = Spin(2m) × Z 2 U(1) with the spin group. We have Spin C (2m)/U(1) ∼ = SO(2m), and a group homomorphism λ : Spin C (2m) → SO(2m) as well as Spin C (2m)/Spin(2m) ∼ = U(1). Note that there is also a canonical homomorphism which is the lift of ι × det : U(m) → SO(2m) × U(1). Now let θ be a pseudo-Hermitian form on the strictly pseudoconvex CR manifold M 2m+1 , H(M ), J , m ≥ 1. This gives rise to the metric g θ on the Levi distribution H(M ). We denote by SO(H(M )) the principal SO(2m)-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 is some principal Spin C (2m)-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 (2m).
Let (P, Λ) be some fixed spin C structure for (M, θ). Then P 1 := P/Spin(2m) → M is a principal U(1)-bundle, and we denote the associated complex line bundle by L → M . This is the determinant bundle of the spin C structure. The corresponding fiber bundle map Λ 1 : P → SO(H(M )) × P 1 over M is a twofold covering. On the other hand, let L(β) → M be a complex line bundle determined by some integral class β ∈ H 2 (M, Z). Then, if 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 (2m). The corresponding determinant bundle is K −1 . All other spin C structures differ from the canonical one by multiplication with a principal U(1)-bundle, related to some line bundle E(α), α ∈ H 2 (M, Z). The corresponding determinant bundle L(β) satisfies E(α) 2 = K ⊗ L(β). Spin C structures with the same determinant bundle L(β) are parametrized by the elements in H 1 (M, Z 2 ) (see [10,18]).
In particular, if c 1 (K) ≡ 0 mod 2, then θ on M admits some spin C structure with trivial determinant 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)nd root E(1) of K −1 . Then −c 1 (K) = (m + 2)c 1 (E(1)), and a spin C structure for (M, θ) with determinant bundle L = E(p) exists when We say that a spin C structure with determinant bundle L = E(p) has weight p. In the following we assume that spin C structures to θ on M exist for all necessary weights p ∈ Z.

Spinors and connections
Let M 2m+1 , H(M ), J be a strictly pseudoconvex CR manifold of hypersurface type and CR dimension m ≥ 1, and let (P, Λ) be a spin C structure of weight ∈ Z for some given pseudo-Hermitian form θ on M . The choice of (P, Λ) gives rise to an associated spinor bundle which satisfies X · ψ, φ = − ψ, X · φ 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 Cl(H(M )) of the Levi distribution.
The Webster-Tanaka connection ∇ W to θ stems from a principal fiber bundle connection on the unitary frame bundle, contained in SO(H(M )). This gives rise to a covariant derivative ∇ W on any root of K −1 and its powers, in particular, for E(1) and the determinant bundle L = E( ).
Recall that (P, Λ) induces a twofold covering map P → SO(H(M )) × P 1 . Then the Webster-Tanaka connection lifts to P , which in turn gives rise to some covariant derivative on spinor fields: 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 spinor derivative to the given spin C structure of weight . The spinor derivative satisfies the rules for any X ∈ Γ(H(M )), Y ∈ Γ(T (M )) and φ, ψ ∈ Γ(Σ(H(M ))). Locally, with respect to some orthonormal frame s = (s 1 , . . . , s 2m ), the spinor derivative is given by the formula where A W,s denotes the local Webster-Tanaka connection form on P 1 with values in iR. The curvature R Σ of the spinor derivative ∇ Σ is then given by for any X, Y ∈ T M and spinor φ ∈ Γ(Σ(H(M ))). Note that is a multiple of the pseudo-Hermitian Ricci form ρ θ . The underlying pseudo-Hermitian form θ gives rise to further structure on the spinor bundle Σ(H(M )). In fact, recall that dθ is ∇ W -parallel and basic, i.e., ι T dθ = 0. We set Θ := i dθ 2 ∈ Cl(H(M )) in the complex Clifford bundle. Then Θ acts by real eigenvalues µ q = m − 2q, q ∈ {0, . . . , m} on Σ(H(M )). We obtain the decomposition into Θ-eigenspaces Σ µq (H(M )) of rank ( m q ) to the eigenvalue µ q (see [18]). We call the bundles to the Θ-eigenvalues µ q = ±m extremal. (We also define Σ a = {0} to be trivial for any a > m and a < −m.) Accordingly, we can decompose any spinor φ on M into where Θφ µq = (m − 2q)φ µq . This decomposition is compatible with the spinor derivative ∇ Σ .

Kohn-Dirac and twistor operators
Let M 2m+1 , H(M ), J , m ≥ 1, be strictly pseudoconvex. We have H(M ) ⊗ C = T 10 ⊕ T 01 and any real transverse vector X ∈ H(M ) can be written as X = X 10 + X 01 with If e = (e 1 , . . . , e m ) denotes a complex orthonormal basis of (H, J, g θ ), i.e., s = (e 1 , Je 1 , . . . , e m , Je m ) is a real orthonormal basis of (H, g θ ), we set The vectors (E 1 , . . . , E m ) form an orthogonal basis with respect to the Levi form on T 10 . As elements in the complexified Clifford algebra Cl(H(M )) we have Now let θ be a pseudo-Hermitian form on M with spin C structure of weight ∈ Z. The spinorial derivative ∇ Σ on Σ(H(M )) is induced by the Webster-Tanaka connection. In the following, 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 i.e., for any spinor φ ∈ Γ(Σ(H(M ))), we have locally Recall that Clifford multiplication is denoted by c. Then we can define the first order differential operators and for spinors φ ∈ Γ(Σ(H(M ))). Locally, the two operators are given by Note that Θ · X 10 − X 10 · Θ = −2X 10 for X 10 ∈ T 10 . This shows for any q ∈ {0, . . . , m}. Hence, the operator D + maps spinors from Γ Σ µq to Γ Σ µ q+1 . Similarly, We compute the square of D + . Locally, around any p ∈ M , we can choose a synchronized frame of the form (e 1 , . . . , e m ) with Then where we use (3.1), (5.1) and the fact that τ , τ • J are trace-free. Similarly, we obtain D 2 − = 0. Thus, we have constructed two chain complexes From the discussions in Section 10 it will become clear that these complexes produce finite dimensional cohomology groups. This compares to the construction of spinorial cohomology on Kähler manifolds as described in [15]. Next we define This is a first order, subelliptic differential operator acting on spinor fields φ ∈ Γ(Σ(H(M )). We call D θ the Kohn-Dirac operator to θ with spin C structure of weight on M (see [18]; cf. [12,20]). Locally, with respect to an orthonormal frame (s 1 , . . . , s m ), the Kohn-Dirac operator is given by Obviously, D θ does not preserve the decomposition of spinors with respect to Θ-eigenvalues. However, we have the identity which shows that the square of the Kohn-Dirac operator maps sections of Σ µq (H(M )) to sections of Σ µq (H(M )) again, i.e., On the spinor bundle, we have the L 2 -inner product defined by denotes the induced volume form of the pseudo-Hermitian structure θ on M . The Kohn-Dirac operator D θ is formally self-adjoint with respect to this L 2 -inner product (·, ·) on Γ c (Σ) (see [12]).
Complementary to the Kohn-Dirac operator D θ , we have twistor operators P (µq) acting on Γ Σ µq (H(M )) for q = 0, . . . , m. In fact, there are orthogonal decompositions where Ker(c) denote the corresponding kernels of the Clifford multiplication. Then with a q := 1 2(q + 1) and b q := 1 2(m − q + 1) we have for the derivatives ∇ 10 φ µq and ∇ 01 φ µq of a spinor φ µq ∈ Γ Σ µq the decompositions where the twistor operators map to Ker(c) by respectively. The sum P (µq) = P 10 + P 01 is given with respect to a local orthonormal frame s by This is the projection of ∇ tr φ µq to the kernel Ker(c).

Covariant components and spinorial CR invariants
In the previous section we have introduced Kohn-Dirac operators D θ and twistor operators P (µq) for spin C structures of weight ∈ Z. We have only used the Webster-Tanaka connection for their construction. Now we compute the transformation law for D θ and P (µq) under conformal change of the pseudo-Hermitian structure. It turns out that certain components of D θ and P (µq) are CR invariants. Let θ andθ = e 2f θ be two adapted pseudo-Hermitian structures on M 2m+1 , H(M ), J , m ≥ 1. We denote by ∇ W and ∇ Σ derivatives with respect to θ. The derivatives with respect toθ are simply denote by ∇. Note that the structure group of the Webster-Tanaka connection is U(m) for any pseudo-Hermitian form. We have the transformation rule where X = X 10 + X 01 and Y = Y 10 + Y 01 are transverse vectors (see, e.g., [11]). The gradient grad θ (f ) ∈ Γ(H(M )) with complex components grad 10 (f ) ∈ Γ(T 10 ) and grad 01 (f ) ∈ Γ(T 01 ) is dual via g θ to the restriction of the differential df to H(M ). Now let (P, Λ) be some spin C structure of weight ∈ Z to θ on M 2m+1 , H(M ), J . The canonical 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 is natural, and the corresponding principal U(1)-bundles P 1 andP 1 of frames in L with respect to θ andθ, respectively, are naturally identified. The same is true for the orthonormal frames in H(M ) to θ andθ. Thus, there exists a unique spin C structure P ,Λ with respect toθ on M , whose spinor frames are naturally identified with those of (P, Λ). Of course, the determinant bundle to P ,Λ has weight again, and there exists an unitary isomorphism between the two kinds of spinor bundles such that X · φ is sent to e −f X·φ for any transverse vector X ∈ H(M ) and spinor φ ∈ Σ(H(M )). Also note that Θφ = (Θφ) , i.e., the decomposition Σ(H(M )) = ⊕ m q=0 Σ µq (H(M )) into Θ-eigenspaces is CR-invariant. We compare now the spinor derivatives with respect to θ andθ, respectively. First, let the corresponding section with respect toθ. Thenσ = e −mf σ and with (7.1) we obtain the transformation rule X = X 10 + X 01 , for the local connections forms of K −1 . This gives Accordingly, for the local connection forms on L = E( ). With formulas in [12] we obtain for spinors φ the transformation rule This gives for φ µq ∈ Γ(Σ µq ), q ∈ {0, . . . , m}, and we obtaiñ Hence, for the Θ-eigenvalue µ q = − , we have i.e., the restriction of the Kohn-Dirac operator D θ of weight to Γ Σ − acts CR-covariantly. Recall that the given spin C structure of weight ∈ Z on M is determined by some complex line bundle E(α), α ∈ H 2 (M, Z). (a) A spinor φ ∈ Γ(Σ(H(M )) in the kernel of the Kohn-Dirac operator, i.e., D θ φ = 0, is called harmonic. We denote by H q (α) the space of harmonic spinors with Θ-eigenvalue µ q , q ∈ {0, . . . , m}. Its dimension is denoted h q (α).
denotes the restriction of D θ (of weight ) to spinors of Θ-eigenvalue − . We call D the th (CR-covariant) component of the Kohn-Dirac operator.
(c) A spinor φ in the kernel of D is called harmonic of weight .
Since D acts by (7.2) CR-covariantly, harmonic spinors of weight are CR invariants of (H(M ), J) on M . The dimension h m+ 2 (α) is a CR invariant as well. In Section 10 we will see that in fact all dimensions h q (α), 0 < q < m, are CR-invariant numbers.
Let us consider the twistor operators P 10 and P 01 . Calculating as above we find Hence, the twistor operator P (µq) = P 10 + P 01 is CR-covariant for the Θ-eigenvalue µ q = .   In the non-extremal cases, i.e., for = ±m, the twistor equation is overdetermined. In fact, similar as in [19], we suppose the existence of a twistor connection such that CR twistor spinors correspond to parallel sections in certain twistor bundles. This would imply that for = ±m the CR invariants p are numbers. For the spin case ( = 0) we discuss parallel spinors to any eigenvalue µ q in [13]. They occur on pseudo-Einstein spin manifolds. In Section 12 we demonstrate the construction of closed CR manifolds admitting parallel spinors with = µ q = 0, i.e., CR twistor spinors.
Example 7.4. In [12] we describe pseudo-Hermitian Killing spinors for the case of a spin structure on M . These spinors are in the kernel of the twistor operator and realize a certain lower bound for the non-zero eigenvalues of the Kohn-Dirac operator D θ . In particular, we find CR twistor spinors of weight = 0 on the standard spheres S 2m+1 of even CR dimension m ≥ 2.
We also find Killing spinors in case that θ is related to some 3-Sasakian structure on M . However, in this situation the CR dimension m is odd and = 0 is impossible. Such Killing spinors are not CR twistor spinors.

Vanishing theorems for harmonic spinors
Let M 2m+1 , H(M ), J , m ≥ 1, be strictly pseudoconvex with pseudo-Hermitian structure θ and Kohn-Dirac operator D θ to some spin C structure of weight ∈ Z. The operator D θ is formally self-adjoint and there exists a Schrödinger-Lichnerowicz-type formula (see [18]; cf. [12]). We use this formula to derive vanishing theorems for harmonic spinors.
Let ∇ tr denote the transversal part of the Webster-Tanaka spinorial derivative to a chosen spin C structure of weight ∈ Z. Then ∆ tr = − tr θ ∇ tr • ∇ tr denotes the spinor sub-Laplacian, and we have where ∇ * 10 and ∇ * 01 are the formal adjoint to ∇ 10 and ∇ 01 , respectively. As in [12] we obtain with (5.1) the equation for the square of the Kohn-Dirac operator. For the spinorial derivative in characteristic direction we compute This results in the Schrödinger-Lichnerowicz-type formula (cf. [12,18]) 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 , we immediately obtain vanishing results for harmonic spinors in the non-extremal bundles Σ µq (µ q = ±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.  Proof . Let φ = φ µq ≡ 0 be a spinor with Θφ = (m − 2q)φ. We put and The Schrödinger-Lichnerowicz-type formula (8.1) gives If the eigenvalues of ρ θ have no different signs, we have by Cauchy-Schwarz inequality Hence, for m + (m + 2)µ q ≥ 0, and We assume that µ q = ±m is non-extremal. Then A(φ) is obviously positive for the three cases 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.
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 have no vanishing results for the canonical and anticanonical spin C structures when = ±(m + 2). However, there are vanishing results in these cases (see [21] and Section 10). Then the Schrödinger-Lichnerowicz-type formula (8.1) for D is expressed by Especially, for = 0, we have  8.3. In Section 12 we construct closed CR manifolds over hyperKähler manifolds which admit harmonic spinors of weight = 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 pseudoconvex and spin with harmonic spinors of weight = 0 (see [20]).

Harmonic theory for the Kohn-Rossi complex
We briefly review here the Kohn-Rossi complex [8] over CR manifolds, twisted with some CR vector bundle E. With respect to a pseudo-Hermitian form we construct the Kohn Laplacian E . Even though 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 classes admit 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 These differential operators are by construction CR invariants and the sequence is called Kohn-Rossi complex. Its cohomology groups are denoted by H 0,q (M ), q ≥ 0. More generally, let us consider a complex vector bundle E over M . We assume that E is equipped with some Cauchy-Riemann operator i.e.,∂ E satisfies for any smooth C-valued function f and sections X 01 , Y 01 in T 01 . We call E,∂ E a CR vector bundle over M . Smooth sections u of E with∂ E u = 0 are holomorphic sections (see [21]).
Furthermore, for some CR vector bundle E over M , we set C q (M, E) = Λ q T * 01 ⊗ E and C q (M, E) = Γ(C q (M, E)) for smooth sections. The holomorphic structure∂ E on E extends to Cauchy-Riemann operators for any q ∈ {0, . . . , m}. This is by construction a twisted complex C q (M, E),∂ E , and we denote the corresponding cohomology groups by H q (M, E), q ∈ {0, . . . , m}. For E the trivial line bundle over M , these are the Kohn-Rossi cohomology groups H 0,q (M ) (see [21]).
Let us assume now that a pseudo-Hermitian form θ is given on M and that the CR vector bundle E → M is equipped with a Hermitian inner product ·, · E . In this setting we have a direct sum decomposition 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 ·, · E and related to the Cauchy-Riemann operator by D X 01 u =∂ E u(X 01 ), X 01 ∈ T 01 , for any u ∈ Γ(E). Together with the Webster-Tanaka connection ∇ W we obtain covariant derivatives and with respect to a local frame (E 1 , . . . , E m ) of T 10 the Cauchy-Riemann operators are given bȳ Moreover, for any q ∈ {0, . . . , m}, the vector bundle C q (M, E) is equipped with a Hermitian inner product, which gives rise via vol θ to an L 2 -inner product on C q (M, E). This allows the construction of a formally adjoint differential operator to∂ E . With respect to a local frame (E 1 , . . . , E m ) the operator∂ * E is given bȳ Finally, we can construct the Kohn Laplacian with respect to θ on M . This is a 2nd order differential operator, which is formally self-adjoint with respect to (·, ·) L 2 on C q (M, E). Due to results of Kohn the operator E is sub-and hypoelliptic. We put for the space of harmonic (0, q)-forms. Since E is formally self-adjoint the harmonic equation

Vanishing theorems for twisted Kohn-Rossi cohomology
The harmonic theory of the previous section fits well to our discussion of Kohn-Dirac operators and harmonic spinors. In fact, the square D 2 θ of the Kohn-Dirac operator has a natural interpretation as Kohn Laplacian 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 Schrödinger-Lichnerowicz-type formula to vanishing results for twisted Kohn-Rossi cohomology (see in [21, Section II, § 7]; cf. [11]).
Recall that the chosen spin C structure on (M, θ) is uniquely determined by some complex line bundle E(α) → M , α ∈ H 2 (M, Z), which is a square root of K ⊗ L, L = E( ) the determinant bundle. Note that we can use the Webster-Tanaka connection ∇ W to define a holomorphic structure on E(α) through∂ E(α) η(X 01 ) := ∇ W X 01 η, X 01 ∈ T 01 , for η ∈ Γ(E(α)). Studying the spinor module Σ with Clifford multiplication c shows that the spinor bundle Moreover, the factor Σ µ 0 (H(M )) is isomorphic to the line bundle E(α). In fact, we have Σ µq (H(M )) ∼ = Λ q T * 01 ⊗ E(α) of rank rk C = ( m q ). Hence, the identifications with the chain groups of the Kohn-Rossi complex, twisted by E(α).
Examining the Clifford multiplication shows that the operator 1 √ 2 D + corresponds via (10.1) to the Cauchy-Riemann operator∂ E(α) . In particular, we have for the cohomology groups of the spinorial complex.
In particular, for the case of the canonical spin C structure α = 0, we have = m + 2 and K = 0. If ρ θ > 0 is positive definite on M , R * is positive definite as well. Application of the Weitzenböck formula for C q (M, E(α)) shows that there are no harmonic forms.
Finally, in the spin case = 0 with even m, we have Q and highlight the following result, which resembles the classical obstruction for positive scalar curvature on Kähler manifolds (cf. [5,14]). Then scal W is constant on M . E.g., this happens when the Webster torsion τ is parallel, or when 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 and ρ θ ≡ 0. Then, either ρ θ > 0 and, for | | ≤ m + 2 and 1 < q < m, we have H q (M, E(α)) = {0} (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-Sasakian manifolds from Kähler geometry.

Cohomology of regular, torsion-free CR manifolds
Let M 2m+1 , H(M ), J , m ≥ 2, be strictly pseudoconvex with pseudo-Hermitian form θ. We call the characteristic vector T θ regular if all its integral curves are 1-dimensional submanifolds of M and the corresponding leaf space N is a smooth manifold of dimension 2m with smooth projection π : M → N . If, in addition, θ has vanishing Webster torsion τ = 0 then T θ is an infinitesimal automorphism of the underlying CR structure and a Killing vector for the Webster metric g W = g θ + θ • θ on M . Such a vector T θ is called transverse symmetry. (Here we understand transverse to H(M ).) It is straightforward to see that in this case the pseudo-Hermitian structure (H(M ), J, θ) projects to a Kähler structure on the leaf space N . Note that if 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 ≥ 2.

Holomorphic cohomology and vanishing theorems
Let us consider the underlying leaf space N with Kähler metric h, complex structure J and fundamental form ω. As for any complex manifold, we have the (p, q)-forms Λ p,q (N ) and the Cauchy-Riemann operators which in turn give rise to the Dolbeault cohomology groups H p,q (N ), p, q ≥ 0. In the following we are interested in the cohomology groups H 0,q (N ), q ≥ 0. In fact, more generally, let E → N be some holomorphic vector bundle and O(E ) the sheaf of local holomorphic sections of E . Then we have the qth cohomology group H q (N, O(E )) of the sheaf O(E ), which is, by Dolbeault's theorem, isomorphic to H 0,q (N, E ), q ≥ 0. Now let E be some complex line bundle over the Kähler manifold N . We assume that E is a root of some power of the anticanonical line bundle K −1 → N . This ensures that E is equipped with a holomorphic structure and Hermitian inner product, both compatible with the Levi-Civita connection of the Kähler metric. The pullback E = π * E is a line bundle over M with Hermitian inner product and the Webster-Tanaka connection induces some holomorphic structure on E as well. Then any smooth section u ∈ Γ(E ) lifts to some smooth section u = π * u of E → M . By construction, the Lie derivative L T u of the lift in characteristic direction vanishes identically on M . In fact, any smooth section u ∈ Γ(E) with L T u = 0 is the pullback of some unique section u in E → N . We call such sections in E → M projectable. More generally, for q ≥ 0, we have the subspaces of projectable (0, q)-forms in the chain groups of the Kohn-Rossi complex with values in the line bundle E. These subgroups are naturally identified with the (0, q)-forms Γ Λ 0,q ⊗ E with values in E over the Kähler manifold N . Since the Webster torsion τ θ vanishes on M , the holomorphic structures on E and on its pullback E are compatible: . Let us assume now that the given holomorphic line bundle E → N is a square root of K ⊗L , where L → N is a line bundle of weight ∈ Z, i.e., L m+2 is the th power of the anticanonical bundle K −1 over N . Then E determines a spin C structure on N with determinant bundle L of weight . This lifts to a spin C structure on (M, θ) with determinant bundle L = π * L of weight . Of course, the corresponding spinor bundle Σ → N pulls back to the spinor bundle Σ over (M, θ) and harmonic spinors in Σ lift to harmonic spinors in Σ → M .
Theorem 11.1. Let π : (M, θ) → N be some CR circle bundle of complex dimension m ≥ 2 with spin C structure of weight | | < m + 2 (determined by some line bundle E = π * E → M ) and assumeq = m(m+ +2) 2(m+2) ∈ Z.  Proof . Since T θ is a transverse symmetry, we have ι T θ R W = 0 for the Webster curvature operator. Hence, the pseudo-Hermitian Ricci form ρ θ is the lift of the Ricci 2-form on N , and the Webster scalar curvature on M is the lift of the Riemannian scalar curvature on N .
If scal h ≡ 0 is non-negative on N , the curvature term Qq = 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 ≥ 2 with spin structure √ K → N .

Shifting cohomology
Let M 2m+1 , θ , m ≥ 2, be some closed pseudo-Hermitian manifold with vanishing Webster torsion τ θ = 0. We assume now that the characteristic vector T of θ is induced from some free U(1)-action on M . This implies that T is regular with underlying Kähler manifold N . In fact, here π : (M, θ) → N is a smooth principal U(1)-bundle and θ is a connection form. We call π : (M, θ) → N a CR principal U(1)-bundle. The corresponding holomorphic line bundle is denoted by F → N and F = π * F is its pullback to M .
Let C q (M ) denote the chain groups of the Kohn-Rossi complex on M . In [21] the differential operator In [21] it is shown that for any integers q, s ∈ Z the cohomology groups where the classes in H q (0) (M, F s ) are uniquely represented by projectable harmonic (0, q)-forms with values in F s . We call (11.1) a shift in Kohn-Rossi cohomology over the CR principal U(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 → M of the line bundle F → N associated to π : M → N has weight f ∈ Z. Then E := F s = E(sf ) determines a spin C structure of weight = 2sf + m + 2 on (M, θ).  Proof .
(2) This result follows directly from Theorem 11.1 and the shift (11.1).
Example 11.4. Let us assume that the CR dimension m ≥ 2 is even and the weight f of F → N is a positive factor of m 2 + 1, i.e., s := − m+2 2f ∈ Z. Then E 2 = F 2s = K, i.e., F s → M defines a spin structure on the CR manifold M . Hence, if the scalar curvature scal h of the underlying Kähler manifold is positive, the Kohn-Rossi subgroup H

Examples of CR principal U(1)-bundles
We discuss some examples of CR principal U(1)-bundles with applications of vanishing theorems for harmonic spinors and Kohn-Rossi cohomology. The construction here is based on the underlying Kähler manifold.
For our construction, let N 2m , ω, J , m ≥ 2, be a closed Kähler manifold such that a nontrivial multiple α := c[ω], c ∈ R \ {0}, of the Kähler class [ω] = 0 is integral, i.e., α ∈ H 2 (N, Z). By Lefschetz's theorem on (1, 1)-classes, there exists some Hermitian line bundle L → N with first Chern class c 1 (L ) = α. Let π : M → N be the corresponding principal U(1)-bundle to which L → N is associated. By the dd c -lemma, this bundle admits a connection form A ω with curvature dA ω = π * ω, the lift of the Kähler form. The lift of the complex structure J to the horizontal 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 := 2A ω . This is an adapted pseudo-Hermitian form, which has by construction vanishing torsion τ = 0, the characteristic vector T is regular and g θ This shows that ρ θ is positive definite in this case.
Let M be given some spin C structure of weight ∈ Z, determined by some line bundle E → M. For | | < m + 2, Proposition 8.1 shows that any harmonic spinor on M is a section of the extremal bundles. Accordingly, Theorem 10.3 shows that, for | | ≤ m + 2 and any 1 ≤ q ≤ m − 1, the qth Kohn-Rossi group H q (M, E) = {0} is trivial. In particular, if N 2m (m even) is spin, so is M and H m 2 M, √ K = {0}. For example, the Hopf fibration π : S 2m+1 → CP m over the complex projective space is a CR principal U(1)-bundle of this kind. In fact, the associated line bundle to π is K −1 , and S 2m+1 is equipped with the standard CR structure. Hence, for any spin C structure of weight | | ≤ m + 2 and 0 < q < m, we have H q S 2m+1 , E = {0} (cf. [21]). (Note that CP m is spin only for m odd. However, the standard CR manifold S 2m+1 is spin for any m.) Example 12.2. Let N 2m , m ≥ 2 and even, be some closed manifold with hyperKähler metric h, i.e., h is Ricci-flat and its holonomy group is contained in Sp m 2 . In particular, N is spin and admits some parallel spinor φ 0 in Σ 0 (N ), i.e., H m 2 N, O √ K = {0}. We assume now that h is a Hodge metric, i.e., the corresponding Kähler class [ω] ∈ H 2 (N, Z) is integral. E.g., any projective K3 surface admits some integral Kähler class [ω]. Then the CR [ω]-bundle π : M → N exists. Furthermore, the pullback of the spin structure on N gives rise to some spin structure √ K → M and the lift ψ 0 := π * φ 0 is a parallel section of Σ 0 (M(H)) with respect to the spinorial Webster-Tanaka connection ∇ Σ . In fact, the basic holonomy group of the adapted pseudo-Hermitian structure θ on M is contained in Sp m 2 (cf. [13]). In particular, ψ 0 is some harmonic spinor in Σ 0 (M(H)) and the Kohn-Rossi group H Observe that ψ 0 is in the kernel of the twistor operator P 0 as well, i.e., ψ 0 is an example for a CR twistor spinor of weight = 0 on some closed, strictly pseudoconvex CR manifold.
Finally, note that in the spin case there is a theory of Kählerian twistor spinors on the underlying N (see [17,19]). The lift of such twistor spinors gives rise to projectable spinors in the kernel of twistor operators on (M, θ). For the case µ q = = 0, this gives rise to CR twistor spinors. However, Kählerian 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 from the standard CR sphere, we do not know further examples (of non-extremal weight) on closed CR manifolds.