Logarithmic and nonlocal terms in the boundary asymptotic expansion of the harmonic projection kernels of the Bergman metric on strictly pseudoconvex domains

For a smoothly bounded strictly pseudoconvex domain in $\mathbb{C}^n$ equipped with its Bergman metric, Donnelly and Fefferman (1983) proved that the $L^2$ cohomology is infinite-dimensional exactly in middle degree $p+q=n$ and vanishes otherwise, and showed that the diagonal trace of the Schwartz kernel of the projection onto $L^2$-harmonic $(p,q)$-forms admits, in middle degree, a boundary asymptotic expansion that is a finite sum of Weyl invariants of the ambient Lorentz–Kähler metric, parallel to Fefferman's invariant expansion for the Bergman kernel. They noted that this expansion terminates after finitely many invariant terms and that logarithmic and nonlocal terms are expected to enter beyond that order, a phenomenon already present for the Bergman kernel itself. The model case was subsequently resolved: the logarithmic coefficient of the Bergman kernel was given a complete invariant-theoretic description by Hirachi (2000), the integral of the corresponding coefficient of the Szegő kernel was identified as a global invariant by Hirachi (2006), and the surrounding Q-prime-curvature and scattering-theory program was developed by several authors. However, no later work addresses the harmonic projection kernels of the Hodge Laplacian in middle degree, the actual new object of the 1983 note. The surviving open problem is to characterize and compute the logarithmic and nonlocal contributions in the full boundary asymptotic expansion of these middle-degree harmonic projection kernels and to place them in a geometric invariant framework analogous to the finite Weyl-invariant terms.

Background

A strictly pseudoconvex domain is a bounded domain $\Omega\subset\mathbb{C}^n$ with smooth boundary whose Levi form, the Hermitian form obtained by restricting the complex Hessian of a boundary defining function $\rho$ to the maximal complex subspace of the boundary tangent space, is positive definite; these are the domains on which complex analysis behaves most like the classical theory of the unit disc, and their boundary geometry is encoded in the Chern–Moser invariants, scalar curvature combinations that are unchanged by biholomorphic maps. The Bergman kernel $K(z,w)$ is the reproducing kernel of the Hilbert space of square-integrable holomorphic functions on $\Omega$; differentiating it on the diagonal yields the Bergman metric, a canonical complete Kähler metric (a Riemannian metric compatible with the complex structure) attached intrinsically to the domain.

On a complete Kähler manifold one can study square-integrable cohomology through harmonic forms. A $(p,q)$-form is a differential form with $p$ holomorphic and $q$ antiholomorphic indices; the Dolbeault complex of such forms computes holomorphic-type cohomology, and its Hodge Laplacian $\Delta$, the analogue for this complex of the classical Laplace operator, is essentially self-adjoint on square-integrable forms. The space $H_2^{p,q}$ of $L^2$ harmonic $(p,q)$-forms, the forms in the kernel of $\Delta$, then realizes the $L^2$ Dolbeault cohomology. Donnelly and Fefferman (1983) proved that for the Bergman metric of a strictly pseudoconvex domain this space is infinite-dimensional exactly in the middle degree $p+q=n$ and zero in all other degrees. The phenomenon is tied to the essential spectrum, the part of the operator's spectrum coming from behavior at infinity, here the boundary: zero lies at the bottom of the essential spectrum of the middle-degree Laplacian, while away from middle degree there is a spectral gap.

The same paper studied the Schwartz kernels, the integral kernels $X^{p,q}(x,y)$ realizing orthogonal projection onto $H_2^{p,q}$, and the pointwise trace of the restriction to the diagonal, $\mathrm{Tr}\,X^{p,q}(x,x)$. For the unit ball, the index theorem for homogeneous spaces identifies the alternating even/odd trace exactly with a curvature polynomial, and Donnelly and Fefferman proved the analogous boundary limit for general strictly pseudoconvex domains, together with the convergence of the middle-degree trace to its ball value $c_{p,q}$. Their sharper Theorem 3 announces, for $p+q=n$, an asymptotic expansion as $x$ approaches the boundary, of the form $\mathrm{Tr}\,X^{p,q}(x,x)\sim c_{p,q}(1+\sum_j\lambda_j W_j U^{\sigma(W_j)/2})$: here $u$ is the approximate solution of the complex Monge–Ampère equation, the nonlinear determinant equation of Kähler potential theory solved asymptotically by Cheng and Yau (1980); $U=|z_0|^2u$ is the Kähler potential of an ambient Lorentzian, that is indefinite, Kähler metric on $\mathbb{C}^*\times\Omega$; the $W_j$ are its Weyl invariants, curvature contractions expressible through the Chern–Moser boundary invariants; $\sigma(W_j)$ is the weight of the invariant, its degree under the natural dilations of the ambient space; and the coefficients $\lambda_j$ depend only on $(p,q)$. The formalism is that of Fefferman's parabolic invariant theory (Fefferman 1979), the program in which the strongest singularity of the Bergman kernel itself, $K\sim\varphi\rho^{-n-1}+\psi\log\rho$ in terms of a defining function $\rho$, is described invariant-theoretically (Fefferman 1974; Boutet de Monvel and Sjöstrand 1976).

Crucially, the Theorem 3 expansion contains only a finite number of invariant terms, and the authors emphasized that this is a limitation of the method rather than of the geometry: even for the Bergman kernel, logarithmic terms (a smooth coefficient multiplied by $\log\rho$) and nonlocal terms (terms whose boundary value depends on the domain globally rather than on the germ of the boundary, hence not expressible through the local Weyl invariants) enter after a finite number of iterations of the underlying construction. For the Bergman and Szegő kernels, the Szegő kernel being the boundary-value analogue of the Bergman kernel, this full story was worked out later: the log term is governed by the obstruction function $\eta_1$ of Lee and Melrose (1982), the coefficient that first obstructs the formal solvability of the complex Monge–Ampère equation beyond finite order; the log coefficient was identified explicitly in dimension two by Graham (1987) and received a complete invariant-theoretic description in all dimensions by Hirachi (2000), with the strong coefficient settled by Bailey, Eastwood and Graham (1994); and on the global side, Hirachi (2006) proved that the integral of the Szegő log coefficient is a biholomorphic invariant of the domain, a starting point of the subsequent Q-prime-curvature and scattering program (the Q-prime curvature being a secondary curvature quantity naturally attached to that logarithmic coefficient). No corresponding theory exists for the harmonic projection kernels of the Hodge Laplacian in middle degree, which is the open problem recorded here.

Problem Statement

Let $\Omega$ be a smoothly bounded strictly pseudoconvex domain in $\mathbb{C}^n$ with its Bergman metric, $H_2^{p,q}$ the space of square-integrable harmonic $(p,q)$-forms for the Hodge Laplacian, and $X^{p,q}(x,y)$ the Schwartz kernel of the orthogonal projection onto $H_2^{p,q}$. Donnelly and Fefferman (1983) proved that $H_2^{p,q}$ is infinite-dimensional exactly in middle degree $p+q=n$ and zero otherwise, and announced that in middle degree the diagonal trace has a boundary asymptotic expansion consisting of a finite sum of Weyl invariants: as $x\to\partial\Omega$,

$$\mathrm{Tr}\,X^{p,q}(x,x)\sim c_{p,q}\Bigl(1+\sum_j\lambda_j W_j U^{\sigma(W_j)/2}\Bigr),$$

where $c_{p,q}$ is the ball value, $u$ is the approximate solution of the complex Monge-Ampère equation, $U=|z_0|^2u$ is the Kähler potential of the ambient Lorentz–Kähler metric on $\mathbb{C}^*\times\Omega$, the $W_j$ are its Weyl invariants (expressible through Chern–Moser boundary invariants) with weights $\sigma(W_j)$, and the coefficients $\lambda_j$ depend only on $(p,q)$; all in the formalism of Fefferman (1979). They stressed that this expansion contains only a finite number of terms and that, even for the Bergman kernel itself, logarithmic and nonlocal terms enter after a finite number of iterations. The problem is to characterize and compute these subsequent logarithmic and nonlocal contributions in the full boundary asymptotic expansion of the harmonic projection kernels for general strictly pseudoconvex domains: that is, to determine the complete asymptotic structure of $\mathrm{Tr}\,X^{p,q}(x,x)$ for $p+q=n$ beyond the finite Weyl-invariant sum, identifying which terms are local invariants, which are logarithmic, and which are nonlocal, and to give the logarithmic coefficients a geometric expression in the invariant-theoretic framework (ambient Weyl invariants, Chern–Moser theory) that describes the finite part of the expansion. The problem concerns every smoothly bounded strictly pseudoconvex domain in every dimension, and asks for the complete asymptotic structure of $\mathrm{Tr}\,X^{p,q}(x,x)$ for $p+q=n$, the classification of each term of that expansion as a local invariant, a logarithmic term, or a nonlocal term, and the explicit geometric expression for the logarithmic coefficients; here a nonlocal term is one whose boundary value depends on the domain as a whole rather than on the germ of the boundary. The object of the problem is the middle-degree harmonic projection kernels of the Hodge Laplacian, for which the already-settled logarithmic theory of the scalar Bergman and Szegő kernels is the model case.

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