Some Remarks on the Total CR $Q$ and $Q^\prime$-Curvatures

We prove that the total CR $Q$-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the $P^\prime$-operator and the CR invariance of the total $Q^\prime$-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold.


Introduction
The Q-curvature, which was introduced by T. Branson [3], is a fundamental curvature quantity on even dimensional conformal manifolds. It satisfies a simple conformal transformation formula and its integral is shown to be a global conformal invariant. The ambient metric construction of the Q-curvature [9] also works for a CR manifold M of dimension 2n + 1, and we can define the CR Q-curvature, which we denote by Q. The CR Q-curvature is a CR density of weight −n − 1 defined for a fixed contact form θ and is expressed in terms of the associated pseudo-hermitian structure. If we take another contact form θ = e Υ θ, Υ ∈ C ∞ (M ), it transforms as where P is a CR invariant linear differential operator, called the (critical) CR GJMS operator. Since P is formally self-adjoint and kills constant functions, the integral called the total CR Q-curvature, is invariant under rescaling of the contact form and gives a global CR invariant of M . However, it follows readily from the definition of the CR Qcurvature that Q vanishes identically for an important class of contact forms, namely the pseudo-Einstein contact forms. Since the boundary of a Stein manifold admits a pseudo-Einstein contact form [5], the CR invariant Q vanishes for such a CR manifold. Moreover, it has been shown that on a Sasakian manifold the CR Q-curvature is expressed as a divergence [1], and hence Q also vanishes in this case. Thus, it is reasonable to conjecture that the total CR Q-curvature vanishes for any CR manifold, and our first result is the confirmation of this conjecture: Theorem 1.1. Let M be a compact strictly pseudoconvex CR manifold. Then the total CR Q-curvature of M vanishes: Q = 0.
For three dimensional CR manifolds, Theorem 1.1 follows from the explicit formula of the CR Q-curvature; see [9]. In higher dimensions, we make use of the fact that a compact strictly pseudoconvex CR manifold M of dimension greater than three can be realized as the boundary arXiv:1711.01724v2 [math.DG] 14 Feb 2018 of a complex variety with at most isolated singularities [2,10,11]. By resolution of singularities, we can realize M as the boundary of a complex manifold X which may not be Stein. In this setting, the total CR Q-curvature is characterized as the logarithmic coefficient of the volume expansion of the asymptotically Kähler-Einstein metric on X [15]. By a simple argument using Stokes' theorem, we prove that there is no logarithmic term in the expansion.
Although the vanishing of Q is disappointing, there is an alternative Q-like object on a CR manifold which admits pseudo-Einstein contact forms. Generalizing the operator of Branson-Fontana-Morpurgo [4] on the CR sphere, Case-Yang [7] (in dimension three) and Hirachi [12] (in general dimensions) introduced the P -operator and the Q -curvature for pseudo-Einstein CR manifolds. Let us denote the set of pseudo-Einstein contact forms by PE and the space of CR pluriharmonic functions by P. Two pseudo-Einstein contact forms θ, θ ∈ PE are related by θ = e Υ θ for some Υ ∈ P. For a fixed θ ∈ PE, the P -operator is defined to be a linear differential operator on P which kills constant functions and satisfies the transformation formula under the rescaling θ = e Υ θ. The Q -curvature is a CR density of weight −n − 1 defined for θ ∈ PE, and satisfies Q = Q + 2P Υ + P Υ 2 for the rescaling. Thus, if P is formally self-adjoint on P, the total Q -curvature gives a CR invariant of M . In dimension three and five, the formal self-adjointness of P follows from the explicit formulas [6,7]. In higher dimensions, Hirachi [12,Theorem 4.5] proved the formal self-adjointness under the assumption that M is the boundary of a Stein manifold X; in the proof he used Green's formula for the asymptotically Kähler-Einstein metric g on X, and the global Kählerness of g was needed to assure that a pluriharmonic function is harmonic with respect to g. In this paper, we slightly modify his proof and prove the self-adjointness of P for general pseudo-Einstein manifolds: Theorem 1.2. Let M be a compact strictly pseudoconvex CR manifold. Then the P -operator for a pseudo-Einstein contact form satisfies Consequently, the CR invariance of Q holds for any CR manifold which admits a pseudo-Einstein contact form: Theorem 1.3. Let M be a compact strictly pseudoconvex CR manifold which admits a pseudo-Einstein contact form. Then the total Q -curvature is independent of the choice of θ ∈ PE.
We note that Q is a nontrivial CR invariant since it has a nontrivial variational formula; see [13]. We also give an alternative proof of Theorem 1.3 by using the characterization [12, Theorem 5.6] of Q as the logarithmic coefficient in the expansion of some integral over a complex manifold with boundary M .
2 Proof of Theorem 1.1 We briefly review the ambient metric construction of the CR Q-curvature; we refer the reader to [9,12,13] for detail.
Let X be an (n + 1)-dimensional complex manifold with strictly pseudoconvex CR boundary M , and let r ∈ C ∞ (X) be a boundary defining function which is positive in the interior X. The restriction of the canonical bundle K X to M is naturally isomorphic to the CR canonical bundle The density bundles over X and M are defined by We call E(w) the CR density bundle of weight w. The space of sections of E(w) and E(w) are also denoted by the same symbols. We define a C * -action on X by δ λ u = λ n+2 u for λ ∈ C * and u ∈ X. Then a section of E(w) can be identified with a function on X which is homogeneous with respect to this action: Similarly, sections of E(w) are identified with homogeneous functions on N . Let ρ ∈ E(1) be a density on X and (z 1 , . . . , z n+1 ) local holomorphic coordinates. We set Since J [ρ] is invariant under changes of holomorphic coordinates, J defines a global differential operator, called the Monge-Ampère operator. Fefferman [8] showed that there exists ρ ∈ E(1) unique modulo O(r n+3 ) which satisfies J [ρ] = 1 + O(r n+2 ) and is a defining function of N . We fix such a ρ and define the ambient metric g by the Lorentz-Kähler metric on a neighborhood of N in X which has the Kähler form −i∂∂ρ.
Recall that there exists a canonical weighted contact form θ ∈ Γ(T * M ⊗ E(1)) on M , and the choice of a contact form θ is equivalent to the choice of a positive section τ ∈ E(1), called a CR scale; they are related by the equation θ = τ θ. For a CR scale τ ∈ E(1), we define the CR Q-curvature by where ∆ = − ∇ I ∇ I is the Kähler Laplacian of g and τ ∈ E(1) is an arbitrary extension of τ . It can be shown that Q is independent of the choice of an extension of τ , and the total CR Q-curvature Q is invariant by rescaling of τ .
The total CR Q-curvature has a characterization in terms of a complete metric on X. We note that the (1, 1)-form −i∂∂ log ρ descends to a Kähler form on X near the boundary. We extend this Kähler metric to a hermitian metric g on X. The Kähler Laplacian ∆ = −g ij ∇ i ∇ j of g is related to ∆ by the equation near N in X \ N . In the right-hand side, we have regarded f as a function on X.
For any contact form θ on M , there exists a boundary defining function ρ such that With this formula, we prove the following characterization of Q.

Lemma 2.1 ([15, Proposition A.3]).
For an arbitrary defining function ρ, we have where lp denotes the coefficient of log in the asymptotic expansion in .
Proof . Since the coefficient of log in the volume expansion is independent of the choice of ρ [15, Proposition 4.1], we may assume that ρ satisfies (2.2) for a fixed contact θ on M . We take τ ∈ E(1) such that ρ = τ ρ. Then, θ is the contact form corresponding to the CR scale τ | N . By the same argument as in the proof of [12, Lemma 3.1], we can take F ∈ E(0), G ∈ E(−n − 1) which satisfy We set G := τ n+1 G ∈ E(0). By (2.1) and the equation ρ ∆ log ρ = n + 1, we have Then, by using (2.4), we compute as Thus we complete the proof.
Proof of Theorem 1.1. Let ρ be an arbitrary defining function of M , and τ ∈ E(1) the density on X defined by ρ = τ ρ. Then α := −i∂∂ log τ is a closed (1, 1)-form on X. The volume form of g is given by vol g = ω n+1 /(n + 1)! with the fundamental 2-form ω = ig jk θ j ∧ θ k . Near the boundary M in X, we have Since the logarithmic term in the volume expansion is determined by the behavior of vol g near the boundary, we compute as The first term in the last line is 0 since α is smooth up to the boundary. Using −i∂∂ log ρ = d(ϑ/ρ) and dα = 0, we also have Thus, by Lemma 2.1 we obtain Q = 0.

Proof of Theorem 1.2
We will recall the definitions of the P -operator and the Q -curvature. A CR scale τ ∈ E(1) is called pseudo-Einstein if it has an extension τ ∈ E(1) such that ∂∂ log τ = 0 near N in X.
The corresponding contact form θ is called a pseudo-Einstein contact form and characterized in terms of associated pseudo-hermitian structure; see [12,13,14]. If τ is a pseudo-Einstein CR scale, another τ is pseudo-Einstein if and only if τ = e −Υ τ for a CR pluriharmonic function Υ ∈ P. For any f ∈ P, we take an extension f ∈ E(0) such that ∂∂ f = 0 near M in X and define We note that the germs of τ and f along N is unique, and P f is assured to be a density by ∆ f | N = 0. The Q -curvature is defined by Here, the homogeneity of Q follows from the fact ∆ log τ | N = 0.
To prove the formal self-adjointness of P , we use its characterization in terms of the metric g. We define a differential operator ∆ by ∆ f = −g ij ∂ i ∂ j f . Since g is Kähler near the boundary, ∆ agrees with ∆ near M in X. . Let τ ∈ E(1) be a pseudo-Einstein CR scale and τ ∈ E(1) its extension such that ∂∂ log τ = 0 near N in X. Let ρ = ρ/ τ be the corresponding defining function. Then, for any f ∈ C ∞ (X) which is pluriharmonic in a neighborhood of M in X, there exist F, G ∈ C ∞ (X) such that F = O(ρ) and In the statement of [12,Lemma 4.4], the Laplacian ∆ is used, but we may replace it by ∆ since they agree near the boundary in X.
Proof of Theorem 1.2. We extend f j to a function on X such that ∂∂f j = 0 in a neighborhood of M in X. Let τ be a pseudo-Einstein CR scale and ρ = ρ/ τ the corresponding defining function. Then we have ω = −i∂∂ log ρ near M in X. We take F j , G j as in Lemma 3.1 so that u j := f j log ρ − F j − G j ρ n+1 log ρ satisfies ∆ u j = (n + 1)f j + O(ρ ∞ ). We consider the coefficient of log in the expansion of the integral which is symmetric in the indices 1 and 2. Since dω = 0, ∂∂f 2 = 0 near M in X, we have where (cpt supp) stands for a compactly supported form on X. Thus, The first and the third terms contain no log terms. Since ω = d(ϑ/ρ) near M in X, the second term is computed as The logarithmic term in the right-hand side is The coefficient of log in the first term is The second term is equal to 2 −n log Re ρ> i∂f 1 ∧ ∂f 2 ∧ (dϑ) n + −n log ρ> (cpt supp).
The first term in this formula is symmetric in the indices 1 and 2 while the second term gives no log term. Therefore, (3.1) should also be symmetric in 1 and 2, which implies the formal self-adjointness of P .

Proof of Theorem 1.3
The formal self-adjointness of the P -operator implies the CR invariance of the total Q -curvature. When n ≥ 2, the CR invariance can also be proved by the following characterization of Q in terms of the hermitian metric g on X whose fundamental 2-form ω = ig jk θ j ∧ θ k agrees with −i∂∂ log ρ near M in X: Theorem 4.1 ([12, Theorem 5.6]). Let τ ∈ E(1) be a pseudo-Einstein CR scale and τ ∈ E(1) its extension such that ∂∂ log τ = 0 near N in X. Let ρ = ρ/ τ be the corresponding defining function. Then we have lp r> i∂ log ρ ∧ ∂ log ρ ∧ ω n = (−1) n 2(n!) 2 Q (4.1) for any defining function r.
In [12,Theorem 5.6], it is assumed that X is Stein and ω = −i∂∂ log ρ globally on X, but as the logarithmic term is determined by the boundary behavior, it is sufficient to assume ω = −i∂∂ log ρ near M in X as above.
Proof of Theorem 1.3. Let τ , ρ be as in Theorem 4.1 and let ρ be the defining function corresponding to another pseudo-Einstein CR scale τ . Then we can write as ρ = e Υ ρ with Υ ∈ C ∞ (X) such that ∂∂Υ = 0 near M in X.
Using the defining function ρ for r in the formula (4.1), we compute as which implies that the third term is also 0. Thus, Q is independent of the choice of a pseudo-Einstein CR scale τ .