Renormalized characteristic forms of the Cheng–Yau metric and global CR invariants

Abstract

For each invariant polynomial Φ, we construct a global CR invariant via the renormalized characteristic form of the Cheng–Yau metric on a strictly pseudoconvex domain. When the degree of Φ is 0, the invariant agrees with the total $Q^{\prime }$-curvature. When the degree is equal to the CR dimension, we construct a primed pseudo-hermitian invariant ${\mathcal{I}}_{\mathrm{\Phi }}^{\prime }$ which integrates to the corresponding CR invariant. These are generalizations of the ${\mathcal{I}}^{\prime }$-curvature on CR five-manifolds, introduced by Case–Gover.

Introduction

The Cheng–Yau metric g is a complete Kähler–Einstein metric on a bounded strictly pseudoconvex domain $\mathrm{\Omega }\subset {\mathbb{C}}^{n+1}$, given by the Kähler form $-i\partial \bar{\partial }\log \rho$ with a defining function ρ which solves the complex Monge–Ampère equation [9]. Since g is biholomorphically invariant, one may try to construct biholomorphic invariants of Ω or CR invariants of the boundary M by using geometric quantities of this metric. However, due to the singularity of g at the boundary, we need some renormalization procedure to extract finite values from invariants of g.

Burns–Epstein [2] introduced such a renormalization for the Levi-Civita connection of g. Let $\psi _i{}^j$ be the connection 1-forms of g in a $(1,0)$-coframe $\{{\theta }^1,\dots ,{\theta }^{n+1}\}$. They defined the renormalized connection $\bar{\mathrm{\nabla }}$ by the connection 1-forms

$$\theta _i{}^j:=\psi _i{}^j+\frac{1}{\rho }({\rho }_k\delta _i{}^j+{\rho }_i\delta _k{}^j){\theta }^k(\partial \rho ={\rho }_i{\theta }^i),$$

and showed that it extends to a linear connection on $\bar{\mathrm{\Omega }}$. The transformation (1.1) is so-called a c-projective transformation [3], and this renormalization procedure is an example of c-projective compactifications, introduced by Čap–Gover [5].

Burns–Epstein also defined the renormalized curvature, which is continuous up to M, by setting

$$W_i{}^j:=\mathrm{\Psi }_i{}^j+(g_{k\bar{l}}\delta _i{}^j+g_{i\bar{l}}\delta _k{}^j){\theta }^k\wedge {\theta }^{\bar{l}},$$

where $\mathrm{\Psi }_i{}^j$ is the curvature form of g. Since g satisfies $\mathrm{Ric}(g)+(n+2)g=0$, this agrees with both the c-projective Weyl curvature and the Bochner curvature of the Cheng–Yau metric. Also, this coincides with the $(1,1)$-component of the curvature $\mathrm{\Theta }_i{}^j$ of $\bar{\mathrm{\nabla }}$. For an Ad-invariant polynomial Φ on $\mathfrak{gl}(n+1,\mathbb{C})$, they showed $\mathrm{\Phi }(W)=\mathrm{\Phi }(\mathrm{\Theta })$ and called it the renormalized characteristic form. When Φ has degree $n+1$, the integral of $\mathrm{\Phi }(W)$ over Ω converges to a biholomorphic invariant of Ω, which is called the characteristic number of the domain. By using the Chern–Simons transgression and the Lee–Melrose expansion of the Monge–Ampère solution, they proved that this number is determined by local geometric data of the boundary. However, since the transgression is based on the global coordinates of ${\mathbb{C}}^{n+1}$, they needed a complicated topological procedure to relate the invariant to intrinsic CR geometry of M, and it does not provide a global CR invariant which can be written in terms of Tanaka–Webster curvature quantities.

As an exceptional case, they derived a Chern–Gauss–Bonnet formula for $c_2(\mathrm{\Theta })$ whose boundary term gives a global invariant of CR three-manifolds, called the Burns–Epstein invariant [1]. The author [22] generalized the renormalized Chern–Gauss–Bonnet formula and the Burns–Epstein invariant to higher dimensional cases in a more general setting: We consider a strictly pseudoconvex domain Ω in a complex manifold X of dimension $n+1$. If the boundary M admits a pseudo-Einstein contact form θ, there exists a global approximate solution ρ to the Monge–Ampère equation such that $\theta =(i/2)(\partial \rho -\bar{\partial }\rho ){|}_{TM}$, and we can define the Cheng–Yau metric on Ω as a hermitian metric g which agrees with $-i\partial \bar{\partial }\log \rho$ near M; see §3.2 for detail. Let Θ be the curvature form of the renormalized connection defined via ρ. Then we have

$$\underset{\mathrm{\Omega }}{\int }{c}_{n+1}(\mathrm{\Theta })=\chi (\mathrm{\Omega })+\underset{M}{\int }\mathrm{\Pi }(R_{\alpha \bar{\beta }\gamma \bar{\mu }},A_{\alpha \beta })\theta \wedge {(d\theta )}^n,$$

where Π is a linear combination of the complete contractions of polynomials in the Tanaka–Webster curvature and torsion without covariant derivatives. In the case of the approximate Cheng–Yau metric, $c_{n+1}(\mathrm{\Theta })$ agrees with $c_{n+1}(W)$ only near the boundary, so it depends on the choice of ρ or θ; see Proposition 4.5. Nevertheless, it is shown that the boundary integral in (1.2) is independent of the choice of θ and gives a CR invariant of M.

In this paper, we show that the Burns–Epstein invariant given by (1.2) can be further decomposed into the sum of global CR invariants. We fix a Fefferman defining function ρ associated with a pseudo-Einstein contact form θ, and set $\omega :=-i\partial \bar{\partial }\log \rho$. Then we can expand $c_{n+1}(\mathrm{\Theta })$ as

$$c_{n+1}(\mathrm{\Theta })=c_{n+1}(\mathrm{\Psi })+\sum _{m=0}^n{\omega }^{n+1-m}\wedge {\mathrm{\Phi }}_m(\mathrm{\Theta }),$$

where ${\mathrm{\Phi }}_m$ is an invariant polynomial of degree m. Our main theorem states that the finite part of the integral of each term in this expansion gives a CR invariant:

Theorem 1.1

Let Φ be an Ad-invariant homogeneous polynomial of degree $m(0\le m\le n)$ on $\mathfrak{gl}(n+1,\mathbb{C})$. Let ρ be the Fefferman defining function associated with a pseudo-Einstein contact form θ. Then we have

$$\mathrm{fp}\underset{\rho >ϵ}{\int }{\omega }^{n+1-m}\wedge \mathit{\Phi }(\mathit{\Theta })=\underset{M}{\int }{\mathcal{F}}_{\mathit{\Phi }}(R_{\alpha \bar{\beta }\gamma \bar{\mu }},A_{\alpha \beta },{\mathrm{\nabla }}_{\alpha })\theta \wedge {(d\theta )}^n,$$

where ${\mathcal{F}}_{\mathit{\Phi }}$ is a linear combination of the complete contractions of polynomials in the Tanaka–Webster curvature, torsion and their covariant derivatives. Moreover, the integral is independent of the choice of θ and gives a CR invariant of M.

A usual characteristic form for g can be written by a combination of powers of ω and renormalized characteristic forms; see §4.2. Hence Theorem 1.1 implies that the finite part

$$\mathrm{fp}\underset{\rho >ϵ}{\int }{\omega }^{n+1-m}\wedge \mathrm{\Phi }(\mathrm{\Psi })$$

also gives a CR invariant of M. Moreover, it follows from (1.2) and (1.3) that the finite part of the usual Gauss–Bonnet integral

$$\mathrm{fp}\underset{\rho >ϵ}{\int }{c}_{n+1}(\mathrm{\Psi })$$

is the sum of $\chi (\mathrm{\Omega })$ and a CR invariant of M.

When $\mathrm{\Phi }=1(m=0)$, the integrand of the left-hand side of (1.4) is equal to the volume form of g up to a compactly supported form determined by the difference between ω and the Kähler form of g. The volume renormalization of the Cheng–Yau metric with respect to a Fefferman defining function is considered in [17], and it is shown that

$$\mathrm{fp}\underset{\rho >ϵ}{\int }{\omega }^{n+1}=k_n\underset{M}{\int }{Q}^{\prime },$$

where $Q^{\prime }$ is the $Q^{\prime }$-curvature, introduced by Case–Yang [8] for $n=1$ and generalized by Hirachi [16] to higher dimensions. Therefore, our global CR invariants in Theorem 1.1 can be considered as generalizations of the total $Q^{\prime }$-curvature.

The $Q^{\prime }$-curvature is a pseudo-hermitian invariant defined for each pseudo-Einstein contact form, and the CR invariance of its integral follows from the transformation formula under rescaling, which is described in terms of the CR GJMS operator and the $P^{\prime }$-operator. Thus, it is natural to expect that the CR invariant given by Theorem 1.1 is also the integral of a “$Q^{\prime }$-like” curvature on M whose transformation formula explains the CR invariance. In this paper, we construct such a curvature quantity ${\mathcal{I}}_{\mathrm{\Phi }}^{\prime }$ for each invariant polynomial Φ of degree $m=n$, which gives a generalization of the ${\mathcal{I}}^{\prime }$-curvature of Case–Gover [6].

In [6], they consider a local CR invariant $\mathcal{I}\in \mathcal{E}(-3,-3)$ which is an analogue of the Fefferman–Graham invariant in conformal geometry. When the CR manifold M is 5-dimensional, it integrates to a global CR invariant, which is though equal to 0. They introduced an alternative “primed” pseudo-hermitian invariant ${\mathcal{I}}^{\prime }$ and showed that it integrates to a non-trivial CR invariant which equals the difference between the Burns–Epstein invariant and the total $Q^{\prime }$-curvature. The transformation formula of ${\mathcal{I}}^{\prime }$ is described by a CR invariant tensor $X_{\alpha }$ such that $(X_{\alpha }{\theta }^{\alpha }+X_{\bar{\alpha }}{\theta }^{\bar{\alpha }})\wedge \theta \wedge d\theta$ gives a representative of the second Chern class $c_2(T^{1,0}M)\in H^4(M;\mathbb{R})$ when θ is pseudo-Einstein. It then follows from the vanishing of this cohomology class that the integral of ${\mathcal{I}}^{\prime }$ is independent of the choice of pseudo-Einstein contact form.

By using CR tractor calculus, we generalize ${\mathcal{I}}^{\prime }$ to ${\mathcal{I}}_{\mathrm{\Phi }}^{\prime }$ on $(2n+1)$-dimensional CR manifolds for any invariant polynomial Φ of degree n on $\mathfrak{gl}(n,\mathbb{C})$. This can be considered as a primed analogue of a local CR invariant ${\mathcal{I}}_{\mathrm{\Phi }}\in \mathcal{E}(-n-1,-n-1)$. For a rescaling $\hat{\theta }=e^{\mathrm{\Upsilon }}\theta$ of a pseudo-Einstein contact form, it satisfies

$${\hat{\mathcal{I}}}_{\mathrm{\Phi }}^{\prime }={\mathcal{I}}_{\mathrm{\Phi }}^{\prime }-X_{\alpha }^{\mathrm{\Phi }}{\mathrm{\Upsilon }}^{\alpha }-X_{\bar{\alpha }}^{\mathrm{\Phi }}{\mathrm{\Upsilon }}^{\bar{\alpha }}$$

with a CR invariant tensor $X_{\alpha }^{\mathrm{\Phi }}$ such that $n^2(X_{\alpha }^{\mathrm{\Phi }}{\theta }^{\alpha }+X_{\bar{\alpha }}^{\mathrm{\Phi }}{\theta }^{\bar{\alpha }})\wedge \theta \wedge {(d\theta )}^{n-1}$ is a representative of $\mathrm{\Phi }(T^{1,0}M)\in H^{2n}(M;\mathbb{R})$. Using the vanishing of this characteristic class, due to Takeuchi [25], we can prove that the total ${\mathcal{I}}_{\mathrm{\Phi }}^{\prime }$-curvature gives a CR invariant. Then we prove that two global CR invariants associated with Φ of degree n coincide up to a constant multiple (Theorem 6.6):

$$\mathrm{fp}\underset{\rho >ϵ}{\int }\omega \wedge \mathrm{\Phi }(\mathrm{\Theta })=-n\underset{M}{\int }{\mathcal{I}}_{\mathrm{\Phi }}^{\prime }.$$

Here, we regard Φ as an Ad-invariant polynomial on both $\mathfrak{gl}(n+1,\mathbb{C})$ and $\mathfrak{gl}(n,\mathbb{C})$ by representing it as a polynomial in $T_p(A):=\mathrm{tr}{(iA)}^p$. To prove this equality, we relate the renormalized characteristic form to the ambient metric and establish an explicit correspondence between the ambient metric and the CR tractor calculus in the Graham–Lee setting.

As an example, we compute the total ${\mathcal{I}}_{\mathrm{\Phi }}^{\prime }$-curvatures for a circle bundle over a complete intersection Y in the complex projective space. It is given as a polynomial in the degrees of the defining polynomials of Y, and it shows that the invariants are non-trivial and independent of each other if we consider Φ modulo the first Chern form $c_1$. As for the other CR invariants given by Theorem 1.1, Takeuchi [26] computed them for Sasakian η-Einstein CR manifolds. His computation assures that the CR invariants in Theorem 1.1 are non-trivial if Φ is not 0 modulo $c_1$; see Remark 5.12.

After this work was completed, the author was informed by Jeffrey Case and Yuya Takeuchi that they had also constructed generalizations of the ${\mathcal{I}}^{\prime }$-curvature by a different method; they first generalize the CR invariant tensor $X_{\alpha }$ and construct the corresponding ${\mathcal{I}}^{\prime }$-type curvature. The curvatures thus obtained are exactly same as ours. The detail of the construction and the relation to Hirachi's conjecture will appear in their paper [7].

This paper is organized as follows: In §2, we review some materials in CR geometry such as the Tanaka–Webster connection and the CR tractor calculus. In §3, we define the Cheng–Yau metric and the ambient metric via Fefferman's approximate solution to the Monge–Ampère equation, and recall the Graham–Lee connection, which plays an important role in relating the Cheng–Yau metric to CR geometry on the boundary. In §4, we introduce the renormalized connection for the Cheng–Yau metric and the renormalized characteristic forms by following Burns–Epstein. Its relation to the ambient metric and the Graham–Lee connection will be discussed. Then we construct the CR invariants associated with each invariant polynomial Φ and prove Theorem 1.1. We also deal with the case of the exact Cheng–Yau metric. The ${\mathcal{I}}_{\mathrm{\Phi }}^{\prime }$-curvatures are defined in §5. We clarify its relation to characteristic classes on the CR manifold and prove the CR invariance of the integral. Then, in §6, we show that the total ${\mathcal{I}}_{\mathrm{\Phi }}^{\prime }$-curvature agrees with the CR invariant constructed in Theorem 1.1; the most part will be devoted to the preparation of the proof, where we establish the correspondence between the ambient construction and the CR tractor calculus in the Graham–Lee setting. Finally, we compute the total ${\mathcal{I}}_{\mathrm{\Phi }}^{\prime }$-curvatures for a circle bundle over a compact Kähler–Einstein manifold.

Notations: Throughout the paper, we denote the space of sections of a vector bundle by the same symbol as the bundle itself, and we adopt Einstein's summation convention. The symmetrization and the skew symmetrization of a tensor are represented by $(,)$ and $[,]$ respectively, and they are performed over barred indices and unbarred indices separately, e.g.,

$$B_{[{\alpha }_1{\bar{\beta }}_1{\alpha }_2{\bar{\beta }}_2]}:=\frac{1}{2!2!}(B_{{\alpha }_1{\bar{\beta }}_1{\alpha }_2{\bar{\beta }}_2}-B_{{\alpha }_2{\bar{\beta }}_1{\alpha }_1{\bar{\beta }}_2}-B_{{\alpha }_1{\bar{\beta }}_2{\alpha }_2{\bar{\beta }}_1}+B_{{\alpha }_2{\bar{\beta }}_2{\alpha }_1{\bar{\beta }}_1}).$$

The trace-free part of $B_{\alpha \bar{\beta }}$ is denoted by $B_{{(\alpha \bar{\beta })}_0}$.

When a function $I(ϵ)$ admits an asymptotic expansion

$$I(ϵ)=\sum _{m=0}^k\sum _{j=0}^{l_m}{a}_{m,j}{ϵ}^{-m}{(\log ϵ)}^j+o(1)$$

as $ϵ\to 0$ with some constants $a_{m,j}$, we define the logarithmic part and the finite part by

$$\mathrm{lp}I(ϵ):=a_{0,1},\mathrm{fp}I(ϵ):=a_{0,0}.$$

Acknowledgment The author is grateful to Jih-Hsin Cheng and Kengo Hirachi for invaluable comments and discussions. He thanks Yuya Takeuchi for informing the author of his computation of the CR invariants for Sasakian η-Einstein manifolds. He also thanks Jeffrey Case and Yuya Takeuchi for the information of their independent construction of generalizations of the ${\mathcal{I}}^{\prime }$-curvature and sharing their notes.