Optimal convergence speed of Bergman metrics on symplectic manifolds

TL;DR

This paper proves that embeddings of symplectic manifolds via eigensections of the Bochner Laplacian converge rapidly to the symplectic form, generalizing Donaldson's bounds to almost-Kähler manifolds.

Contribution

It establishes a $1/p^2$ convergence rate of Fubini-Study forms to the symplectic form, extending Donaldson's bounds to the almost-Kähler setting.

Findings

Fubini-Study forms converge at rate $1/p^{2}$ to the symplectic form.

Generalization of Donaldson's bounds on the Calabi functional to almost-Kähler manifolds.

Provides a quantitative convergence result for symplectic embeddings.

Abstract

It is known that a compact symplectic manifold endowed with a prequantum line bundle can be embedded in the projective space generated by the eigensections of low energy of the Bochner Laplacian acting on high $p$-tensor powers of the prequantum line bundle. We show that the Fubini-Study forms induced by these embeddings converge at speed rate $1/p^{2}$ to the symplectic form. This result implies the generalization to the almost-K\"ahler case of the lower bounds on the Calabi functional given by Donaldson for K\"ahler manifolds, as shown by Lejmi and Keller.

Equations

$$\Big{|}\frac{1}{p}\Phi_{p}^{\ast}(\omega_{{}_{\rm FS}})-\omega\Big{|}_{\mathscr{C}^{2}}\leqslant\frac{C}{p^{1/2}}\,\cdot$$

$$\frac{\sqrt{-1}}{2\pi }{R}^L=\omega .$$

$$\big{\langle}s_{1},s_{2}\big{\rangle}=\int_{X}\big{\langle}s_{1},s_{2}\big{\rangle}(x)dv_{X}(x).$$

$$\Delta^{L^{p}}=-\sum_{k}\Big{[}\big{(}\nabla_{e_{k}}^{L^{p}}\big{)}^{2}-\nabla_{\nabla_{e_{k}}^{TX}e_{k}}^{L^{p}}\Big{]}.$$

$${\Delta }_{p,\Phi }={\Delta }^{L^p}-2\pi np+\Phi .$$

$$\text{Spec}({\Delta }_{p,\Phi })\subset [-C_L,C_L]\cup [4\pi p-C_L,+\infty ),$$

$$\dim {\mathcal{H}}_p={\int }_X\mathrm{Td}(T^{(1,0)}X)e^{p\omega },$$

$$\Phi_{p}:X\backslash\textup{Bl}(\mathcal{H}_{p})\rightarrow\mathbb{P}(\mathcal{H}^{\ast}_{p}),\quad\Phi_{p}(x)=\big{\{}s\in\mathcal{H}_{p}:s(x)=0\big{\}}\,,$$

$$\Big{|}\frac{1}{p}\Phi_{p}^{\ast}(\omega_{{}_{\rm FS}})-\omega\Big{|}_{\mathscr{C}^{\ell}}\leqslant\frac{C_{\ell}}{p^{2}}\,,$$

$$\displaystyle\|\boldsymbol{\cdot}\|_{L^{2}}=\Big{(}\int_{\mathbb{R}^{2n}}|\,\boldsymbol{\cdot}\,|^{2}dZ\Big{)}^{1/2}\ \textup{on}\ \mathbb{R}^{2n},$$

$$b_j=-2\frac{\partial }{\partial z_j}+\pi {\overline{z}}_j,b_j^{+}=2\frac{\partial }{\partial {\overline{z}}_j}+\pi z_j,b=(b_1,\dots ,b_n),\mathcal{L}=\sum _{j=1}^n{b}_j{b}_j^{+},$$

$$\displaystyle\textup{Spec}(\mathscr{L})=\big{\{}4\pi|\alpha|:\alpha\in\mathbb{N}^{n}\big{\}},$$

$$\displaystyle b^{\alpha}\big{(}z^{\beta}\exp\big{(}-\pi\sum_{j}|z_{j}|^{2}/2\big{)}\big{)},\ \ \textup{with}\ \beta\in\mathbb{N}^{n}.$$

$$\displaystyle\bigg{\{}\phi_{\beta}(z)=\Big{(}\frac{\pi^{|\beta|}}{\beta!}\Big{)}^{1/2}z^{\beta}e^{-\pi\sum_{j}|z_{j}|^{2}/2}:\beta\in\mathbb{N}^{n}\bigg{\}}.$$

$$\displaystyle\mathscr{P}(Z,Z^{\prime})=\exp\Big{(}-\frac{\pi}{2}\sum_{j=1}^{n}\big{(}|z_{j}|^{2}+|z^{\prime}_{j}|^{2}-2z_{j}\overline{z}^{\prime}_{j}\big{)}\Big{)}.$$

$$\displaystyle\big{(}b^{+}_{j}\mathscr{P}\big{)}(Z,Z^{\prime})=0,\ \ \big{(}b_{j}\mathscr{P}\big{)}(Z,Z^{\prime})=2\pi(\overline{z}_{j}-\overline{z}^{\prime}_{j})\mathscr{P}(Z,Z^{\prime}).$$

$$\begin{array}{rl}[b_j,b_k^{+}]& =b_j{b}_k^{+}-b_k^{+}{b}_j=-4\pi {\delta }_{jk},\\ [b_j,b_k]& =[b_j^{+},b_k^{+}]=0,\\ [g(z,\overline{z}),b_j]& =2\frac{\partial }{\partial z_j}g(z,\overline{z}),\\ [g(z,\overline{z}),b_j^{+}]& =-2\frac{\partial }{\partial {\overline{z}}_j}g(z,\overline{z}).\end{array}$$

$$\displaystyle\big{(}\mathscr{P}b^{\alpha}z^{\beta}\mathscr{P}\big{)}(Z,Z^{\prime})=\begin{cases}\big{(}z^{\beta}\mathscr{P}\big{)}(Z,Z^{\prime}),\ &\textup{if}\ |\alpha|=0,\\ 0,\ \ &\textup{if}\ |\alpha|>0.\end{cases}$$

$$\mathcal{R}=\sum _j{Z}_j\frac{\partial }{\partial Z_j}=z+\overline{z}=Z.$$

$$\big{\{}(Z,Z^{\prime})\in TX\times_{X}TX:|Z|,|Z^{\prime}|<\varepsilon\big{\}}\ni(Z,Z^{\prime})\longmapsto P_{\mathcal{H}_{p},x_{0}}(Z,Z^{\prime})\in\mathbb{C}\,,$$

$${\mathbb{R}}^{2n}\ni (Z_1,\dots ,Z_{2n})⟼\sum _{j=1}^{2n}{Z}_j{e}_j\in T_{x_0}X.$$

$$d{v}_X(Z)={\kappa }_{x_0}(Z)d{v}_{TX}(Z),{\kappa }_{x_0}(0)=1,$$

$$\displaystyle\big{(}S_{t}s\big{)}(Z):=s(Z/t),\ \ \ \mathscr{L}_{t}:=S^{-1}_{t}\kappa^{1/2}t^{2}\Delta_{p,\Phi}\kappa^{-1/2}S_{t}.$$

$${\nabla }_{0,\bullet }={\nabla }_{\bullet }+\frac{1}{2}{R}_{x_0}^L(Z,\cdot ),{\mathcal{L}}_0=-\sum _{j=1}^{2n}({\nabla }_{0,e_j}{)}^2-2n\pi =\sum _{j=1}^n{b}_j{b}_j^{+}=\mathcal{L}.$$

$${\mathcal{L}}_t={\mathcal{L}}_0+\sum _{r=1}^m{t}^r{\mathcal{O}}_r+\mathcal{O}(t^{m+1}).$$

$${\mathcal{O}}_1(Z)=$$

$$\begin{split}\mathcal{O}_{2}(Z)&=\frac{1}{3}\Big{\langle}R^{TX}_{x_{0}}(\mathcal{R},e_{i})\mathcal{R},e_{j}\Big{\rangle}_{x_{0}}\nabla_{0,e_{i}}\nabla_{0,e_{j}}\\ &\qquad+\Big{[}\frac{2}{3}\Big{\langle}R_{x_{0}}^{TX}(\mathcal{R},e_{j})e_{j},e_{i}\Big{\rangle}_{x_{0}}-\frac{1}{2}\sum_{|\alpha|=2}(\partial^{\alpha}R^{L})_{x_{0}}\frac{Z^{\alpha}}{\alpha!}(\mathcal{R},e_{i})\Big{]}\nabla_{0,e_{i}}\\ &\qquad-\frac{1}{4}\nabla_{e_{i}}\Big{(}\sum_{|\alpha|=2}(\partial^{\alpha}R^{L})_{x_{0}}(\mathcal{R},e_{i})\frac{Z^{\alpha}}{\alpha!}\Big{)}-\frac{1}{9}\sum_{i}\Big{[}\sum_{j}(\partial_{j}R^{L})_{x_{0}}(\mathcal{R},e_{i})Z_{j}\Big{]}^{2}\\ &\qquad-\frac{1}{12}\Big{[}\mathscr{L}_{0},\Big{\langle}R_{x_{0}}^{TX}(\mathcal{R},e_{i})\mathcal{R},e_{i}\Big{\rangle}_{x_{0}}\Big{]}+\Phi_{x_{0}}.\end{split}$$

$${\mathcal{F}}_r(Z,Z^{\prime })=J_r(Z,Z^{\prime })\mathcal{P}(Z,Z^{\prime }),J_0=1,$$

$$\sup_{|\alpha|+|\alpha^{\prime}|\leqslant m}\Big{|}\frac{\partial^{|\alpha|+|\alpha^{\prime}|}}{\partial Z^{\alpha}{\partial Z^{\prime}}^{\alpha^{\prime}}}\Big{(}\frac{1}{p^{n}}P_{\mathcal{H}_{p}}(Z,Z^{\prime})-\sum_{r=0}^{k}\mathscr{F}_{r}(\sqrt{p}Z,\sqrt{p}Z^{\prime})\kappa^{-\frac{1}{2}}(Z)\kappa^{-\frac{1}{2}}(Z^{\prime})p^{-\frac{r}{2}}\Big{)}\Big{|}_{\mathscr{C}^{\ell}(X)}\leqslant Cp^{-\frac{k-m+1}{2}},$$

$$\begin{array}{rl}{\mathcal{F}}_1& =-{\mathcal{P}}^{\perp }{\mathcal{L}}^{-1}{\mathcal{O}}_1\mathcal{P}-\mathcal{P}{\mathcal{O}}_1{\mathcal{L}}^{-1}{\mathcal{P}}^{\perp },\\ {\mathcal{F}}_2& ={\mathcal{L}}^{-1}{\mathcal{P}}^{\perp }{\mathcal{O}}_1{\mathcal{L}}^{-1}{\mathcal{P}}^{\perp }{\mathcal{O}}_1\mathcal{P}-{\mathcal{L}}^{-1}{\mathcal{P}}^{\perp }{\mathcal{O}}_2\mathcal{P}\\ & +\mathcal{P}{\mathcal{O}}_1{\mathcal{L}}^{-1}{\mathcal{P}}^{\perp }{\mathcal{O}}_1{\mathcal{L}}^{-1}{\mathcal{P}}^{\perp }-\mathcal{P}{\mathcal{O}}_2{\mathcal{L}}^{-1}{\mathcal{P}}^{\perp }\\ & +{\mathcal{P}}^{\perp }{\mathcal{L}}^{-1}{\mathcal{O}}_1\mathcal{P}{\mathcal{O}}_1{\mathcal{L}}^{-1}{\mathcal{P}}^{\perp }-\mathcal{P}{\mathcal{O}}_1{\mathcal{L}}^{-2}{\mathcal{P}}^{\perp }{\mathcal{O}}_1\mathcal{P}.\end{array}$$

Full text

W. L. supported by National Natural Science Foundation of China (Grant Nos. 11401232, 11871233)

X. M. partially supported by NNSFC No. 11829102 and funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative

G. M. partially supported by DFG funded project SFB TRR 191

0. Introduction

A very useful tool in the study of canonical Kähler metrics is the use of Bergman metrics to approximate arbitrary Kähler metrics in a given integral cohomology class, see e. g., [7, 11, 17].

Let $(X,\omega)$ be a compact Kähler manifold endowed with a Hermitian holomorphic line bundle $(L,h^{L})$ such that $\frac{\sqrt{-1}}{2\pi}R^{L}=\omega$. Since the bundle $L$ is positive, Kodaira’s theorem shows that high powers $L^{p}$ give rise to holomorphic embeddings $\Phi_{p}:X\to\mathbb{P}(H^{0}(X,L^{p})^{*})$. The Bergman form $\omega_{p}$ at level $p$ is defined as the rescaled induced Fubini-Study form $\frac{1}{p}\Phi_{p}^{*}\omega_{{}_{\rm FS}}$, where $\omega_{{}_{\rm FS}}$ is the natural Fubini-Study form on $\mathbb{P}(H^{0}(X,L^{p})^{*})$. Tian [17] showed that $\omega_{p}$ converges to $\omega$ in the $\mathscr{C}^{2}$ topology with speed rate $p^{-1/2}$, as $p\to\infty$, that is, there exists $C>0$ such that for any $p\in\mathbb{N}^{*}$ we have

[TABLE]

This was improved by Ruan [15] to convergence in $\mathscr{C}^{\infty}$ with speed rate $p^{-1}$ (see also [13, Theorem 5.1.4]). Tian’s result was motivated by a problem of Yau [18].

The process described above can be seen in the general framework of quantization. The Bergman forms $\omega_{p}$ can be thought as quantization at level $p$ of the original Kähler form $\omega$. The number $1/p$ is to be thought of as analogous to Planck’s constant and in the semiclassical limit $p\to\infty$ the quantized objects $\omega_{p}$ converge to the original Kähler one.

The proof of the convergence in [15, 17] is based on the diagonal expansion of the Bergman kernel up to second order. A full diagonal asymptotic expansion of the Bergman kernel in powers of $p$ in the $\mathscr{C}^{\infty}$ topology was obtained by Catlin [5] and Zelditch [19] as an application of Boutet de Monvel and Sjöstrand’s work [4], see also [6, 14] for different approaches and generalizations. We refer to [13] for a comprehensive study of several analytic and geometric aspects of Bergman kernel. One advantage of the expansion in the $\mathscr{C}^{\infty}$ topology is that it easily implies the convergence of the Bergman forms $\omega_{p}$ to $\omega$ with speed rate $p^{-2}$, see [13, (5.1.23)]. This convergence speed is optimal. Note that the scalar curvature is up to a multiplicative constant the coefficient of the second term of the Bergman kernel expansion. The purpose of this paper is to extend this optimal result to the case of symplectic manifolds.

The Bergman kernel of a holomorphic line bundle $L$ on a complex manifold is the smooth kernel of the orthogonal projection from the space of square integrable sections on the space of holomorphic sections, or, equivalently, on the kernel of the Kodaira Laplacian $\Box^{L}=\overline{\partial}^{L}\,\overline{\partial}^{L*}+\overline{\partial}^{L*}\,\overline{\partial}^{L}$ on $L$. In order to find a suitable notion of “holomorphic section” of a prequantum line bundle on a compact symplectic manifold, Guillemin and Uribe [9] introduced a renormalized Bochner Laplacian $\Delta_{p,0}$ (cf. (0.5)) which reduces to $2\Box^{L}$ in the Kähler case.

We describe this construction in detail. Let $(X,\omega)$ be a compact symplectic manifold of real dimension $2n$. Let $(L,h^{L})$ be a Hermitian line bundle on $X$, and let $\nabla^{L}$ be a Hermitian connection on $(L,h^{L})$ with the curvature $R^{L}=(\nabla^{L})^{2}$. We will assume throughout the paper that $(L,h^{L},\nabla^{L})$ is a prequantum line bundle of $(X,\omega)$, i.e.,

[TABLE]

We choose an almost complex structure $J$ such that $\omega$ is $J$-invariant and $\omega(\cdot,J\cdot)>0$. The almost complex structure $J$ induces a splitting $TX\otimes_{\mathbb{R}}\mathbb{C}=T^{(1,0)}X\oplus T^{(0,1)}X$, where $T^{(1,0)}X$ and $T^{(0,1)}X$ are the eigenbundles of $J$ corresponding to the eigenvalues $\sqrt{-1}$ and $-\sqrt{-1}$, respectively.

Let $g^{TX}(\cdot,\cdot):=w(\cdot,J\cdot)$ be the Riemannian metric on $TX$ induced by $\omega$ and $J$. The Riemannian volume form $dv_{X}$ of $(X,g^{TX})$ has the form $dv_{X}=\omega^{n}/n!$. The $L^{2}$-Hermitian product on the space $\mathscr{C}^{\infty}(X,L^{p})$ of smooth sections of $L^{p}$ on $X$, with $L^{p}:=L^{\otimes p}$, is given by

[TABLE]

Let $\nabla^{TX}$ be the Levi-Civita connection on $(X,g^{TX})$ with curvature $R^{TX}$, and let $\nabla^{L^{p}}$ be the connection on $L^{p}$ induced by $\nabla^{L}$. Let $\{e_{k}\}$ be a local orthonormal frame of $(TX,g^{TX})$. The Bochner Laplacian acting on $\mathscr{C}^{\infty}(X,L^{p})$ is given by

[TABLE]

Given $\Phi\in\mathscr{C}^{\infty}(X,\mathbb{R})$, the renormalized Bochner Laplacian is defined by

[TABLE]

By [9], [12, Corollary 1.2], there exists $C_{L}>0$ independent of $p$ such that

[TABLE]

where $\textup{Spec}(A)$ denotes the spectrum of the operator $A$. Since $\Delta_{p,\Phi}$ is an elliptic operator on a compact manifold, it has discrete spectrum and its eigensections are smooth. Let $\mathcal{H}_{p}$ be the direct sum of eigenspaces of $\Delta_{p,\Phi}$ corresponding to the eigenvalues lying in $[-C_{L},C_{L}]$. In mathematical physics terms, the operator $\Delta_{p,\Phi}$ is a semiclassical Schrödinger operator and the space $\mathcal{H}_{p}$ is the space of its bound states as $p\to\infty$. The space $\mathcal{H}_{p}$ proves to be an appropriate replacement for the space of holomorphic sections $H^{0}(X,L^{p})$ from the Kähler case. In particular, we have for $p$ large enough (cf. [13, (8.3.3)]),

[TABLE]

where $\operatorname{Td}(T^{(1,0)}X)$ is the Todd class of $T^{(1,0)}X$, which corresponds to the Riemann-Roch-Hirzebruch formula from complex geometry.

Let $\mathbb{P}(\mathcal{H}^{\ast}_{p})$ be the projective space associated to the dual space of $\mathcal{H}_{p}$; we identify $\mathbb{P}(\mathcal{H}^{\ast}_{p})$ with the Grassmannian of hyperplanes in $\mathcal{H}_{p}$. The base locus of $\mathcal{H}_{p}$ is the set \textup{Bl}(\mathcal{H}_{p})=\big{\{}x\in X:s(x)=0 for all s\in\mathcal{H}_{p}\big{\}}. We define the Kodaira map

[TABLE]

which sends $x\in X\backslash\textup{Bl}(\mathcal{H})$ to the hyperplane of sections vanishing at $x$. Note that $\mathcal{H}_{p}$ is endowed with the induced $L^{2}$ Hermitian product (0.3) so there is a well-defined Fubini-Study metric $g_{{}_{\rm FS}}$ on $\mathbb{P}(\mathcal{H}^{\ast}_{p})$ with the associated form $\omega_{{}_{\rm FS}}$.

The symplectic Kodaira embedding theorem [14, Theorem 3.6], [13, Theorem 8.3.12], states that for large $p$ the Kodaira maps $\Phi_{p}:X\rightarrow\mathbb{P}(\mathcal{H}^{\ast}_{p})$ are embeddings and the Bergman forms converge to the symplectic form with speed rate $p^{-1}$. We note that in this case the near-diagonal expansion of the Bergman kernel is essential for the proof, in contrast to the the Kähler case, where the diagonal expansion already implies the result. Let us also observe that [14, Theorem 3.6] and [13, Theorem 8.3.12] are valid in a more general context, namely when $g^{TX}$ is an arbitrary $J$-invariant Riemannian metric.

There exists in the literature another replacement of the notion of holomorphic section, see e. g., [2, 16]. It is based on a construction by Boutet de Monvel and Guillemin [3] of a first-order pseudodifferential operator $D_{b}$ on the circle bundle of $L^{*}$. The associated Szegő kernels are well defined modulo smooth operators on the associated circle bundle, even though $D_{b}$ is neither canonically defined nor unique. Indeed, Boutet de Monvel–Guillemin define the Szegő kernels first, and construct the operator $D_{b}$ from the Szegő kernels. For these spaces the Bergman forms converge to the symplectic form with speed rate $p^{-1}$, too.

The main result of this paper is as follows.

Theorem 0.1**.**

Let $(X,\omega)$ be a compact symplectic manifold and $(L,h^{L})$ be a Hermitian line bundle endowed with a Hermitian connection $\nabla^{L}$ such that $\frac{\sqrt{-1}}{2\pi}R^{L}=\omega$ holds. Let $J$ be an almost complex structure on $TX$ such that $g^{TX}(\cdot,\cdot):=\omega(\cdot,J\cdot)$ is a $J$-invariant Riemannian metric on $TX$. Then for any $\ell\in\mathbb{N}$, there exists $C_{\ell}>0$ such that

[TABLE]

where $\Phi_{p}$ is the Kodaira map (0.8) defined by the space $\mathcal{H}_{p}$ of bound states of the renormalized Bochner Laplacian $\Delta_{p,\Phi}$ associated with $g^{TX},\nabla^{L},\Phi$ in (0.5).

The proof is based on the near diagonal expansion of the Bergman kernel of $\mathcal{H}_{p}$ from [13, 14]. The sharp bound of $\mathscr{O}(p^{-2})$ is due to some remarkable cancellations of the coefficients in this expansions, reminiscent of the local properties of the curvature of Kähler metrics.

The main motivation for approximating Kähler metrics by Fubini-Study metrics arises from questions about the existence and uniqueness of Kähler metrics of constant scalar curvature, or more generally, Kähler-Einstein metrics, see [7, 8, 17, 18]. It is natural to study such questions also in the symplectic framework, for example, it is interesting to generalize to the almost-Kähler case the lower bounds on the Calabi functional given by Donaldson [8]. This is done by Lejmi and Keller [10]. Theorem 0.1 plays a crucial role in their proof in the symplectic case.

The organization of this paper is as follows. In Section 1, we recall the formal calculus on $\mathbb{C}^{n}$ for the model operator $\mathscr{L}$ (cf. (1.2)), which is the main ingredient of our approach. In Section 2, we review the asymptotic expansion of the generalized Bergman kernel. In Section 3, we reduce the proof of Theorem 0.1 to Theorem 3.3. In Section 4, we prove Theorem 3.3 and thus finish the proof of Theorem 0.1.

We shall use the following notations. For $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}$, $z\in\mathbb{C}^{n}$, we set $|\alpha|=\sum^{n}_{j=1}\alpha_{j}$, $\alpha!=\prod_{j}(\alpha_{j}!)$ and $z^{\alpha}:=z_{1}^{\alpha_{1}}\cdots z^{\alpha_{n}}_{n}$. Moreover, when an index variable appears twice in a single term, it means that we are summing over all its possible values.

Acknowledgments. We would like to thank Mehdi Lejmi, Julien Keller and Gábor Székelyhidi for motivating and helpful discussions which led to the writing of this paper.

1. Kernel calculus on $\mathbb{C}^{n}$

In this section, we recall the formal calculus on $\mathbb{C}^{n}$ for the model operator $\mathscr{L}$ introduced in [14, § 1.4], [13, § 4.1.6] (with $a_{j}=2\pi$ therein). This calculus is the main ingredient of our approach.

Let us consider the canonical coordinates $(Z_{1},\ldots,Z_{2n})$ on the real vector space $\mathbb{R}^{2n}$. On the complex vector space $\mathbb{C}^{n}$ we consider the complex coordinates $(z_{1},\ldots,z_{n})$. The two sets of coordinates are linked by the relation $z_{j}=Z_{2j-1}+\sqrt{-1}Z_{2j}$, $j=1,\ldots,n$.

We consider the $L^{2}$-norm

[TABLE]

where $dZ=dZ_{1}\ldots dZ_{2n}$ is the Lebesgue measure. We define the differential operators:

[TABLE]

which extend to closed densely defined operators on \big{(}L^{2}(\mathbb{R}^{2n}),\|\boldsymbol{\cdot}\|_{L^{2}}\big{)}. As such, $b^{+}_{j}$ is the adjoint of $b_{j}$ and $\mathscr{L}$ defines as a densely defined self-adjoint operator on \big{(}L^{2}(\mathbb{R}^{2n}),\|\boldsymbol{\cdot}\|_{L^{2}}\big{)}. The following result was established in [14, Theorem 1.15] (cf. also [13, Theorem 4.1.20]).

Theorem 1.1**.**

The spectrum of $\mathscr{L}$ on $L^{2}(\mathbb{R}^{2n})$ is given by

[TABLE]

and an orthogonal basis of the eigenspace of $4\pi|\alpha|$ is given by

[TABLE]

In particular, an orthonormal basis of $\operatorname{Ker}(\mathscr{L})$ is

[TABLE]

Let $\mathscr{P}(Z,Z^{\prime})$ denote the kernel of the orthogonal projection $\mathscr{P}:L^{2}(\mathbb{R}^{2n})\rightarrow\operatorname{Ker}(\mathscr{L})$ with respect to $dZ^{\prime}$. Set $\mathscr{P}^{\bot}=\textup{Id}-\mathscr{P}$.

Obviously $\mathscr{P}(Z,Z^{\prime})=\sum_{\beta}\phi_{\beta}(z)\overline{\phi_{\beta}(z^{\prime})}$, so we infer from (1.5) that

[TABLE]

By (1.2) and (1.6), we obtain

[TABLE]

The following commutation relations are very useful in the computations. Namely, for any polynomial $g(z,\overline{z})$ in $z$ and $\overline{z}$, we have

[TABLE]

For a polynomial $F$ in $Z,Z^{\prime}$, we denote by $F\mathscr{P}$ the operator on $L^{2}(\mathbb{R}^{2n})$ defined by the kernel $F(Z,Z^{\prime})\mathscr{P}(Z,Z^{\prime})$ and the volume form $dZ$.

In the calculations involving the kernel $\mathscr{P}(\boldsymbol{\cdot},\boldsymbol{\cdot})$, we prefer however to use the orthogonal decomposition of $L^{2}(\mathbb{R}^{2n})$ given in Theorem 1.1 and the fact that $\mathscr{P}$ is an orthogonal projection, rather than integrating against the expression (1.6) of $\mathscr{P}(\boldsymbol{\cdot}\,,\boldsymbol{\cdot})$. This point of view leads to streamlined computations and to a better understanding of the operators involved. As an example, Theorem 1.1 implies that

[TABLE]

We will also identify $z$ to $\sum_{j}z_{j}\frac{\partial}{\partial z_{j}}$ and $\overline{z}$ to $\sum_{j}\overline{z}_{j}\frac{\partial}{\partial\overline{z}_{j}}$ when we consider $z$ and $\overline{z}$ as vector fields, and

[TABLE]

2. Asymptotic expansion of the

generalized Bergman kernel

Let $a^{X}$ be the injectivity radius of $(X,g^{TX})$. We denote by $B^{X}(x,\varepsilon)$ and $B^{T_{x}X}(0,\varepsilon)$ the open balls in $X$ and $T_{x}X$ with center $x$ and radius $\varepsilon$, respectively. Then the exponential map $T_{x}X\ni Z\rightarrow\textup{exp}^{X}_{x}(Z)\in X$ is a diffeomorphism from $B^{T_{x}X}(0,\varepsilon)$ onto $B^{X}(x,\varepsilon)$ for $\varepsilon\leqslant a^{X}$. From now on, we identify $B^{T_{x}X}(0,\varepsilon)$ with $B^{X}(x,\varepsilon)$ via the exponential map for $\varepsilon\leqslant a^{X}$.

We fix $x_{0}\in X$. For $Z\in B^{T_{x_{0}}X}$ we identify $(L_{Z},h_{Z}^{L})$ to $(L_{x_{0}},h_{x_{0}}^{L})$ by parallel transport with respect to the connection $\nabla^{L}$ along the curve $\gamma_{Z}:[0,1]\ni u\rightarrow\textup{exp}^{X}_{x_{0}}(uZ)$.

In general, for functions in normal coordinates, we will add a subscript $x_{0}$ to indicate the base point $x_{0}\in X$. Similarly, $P_{\mathcal{H}_{p}}(x,y)$ induces in terms of the above trivialization (note that $\operatorname{End}({L^{p}_{x_{0}}})=\mathbb{C}$) a smooth function

[TABLE]

which also depends smoothly on the parameter $x_{0}$.

Let us choose an orthonormal basis $\{w_{j}\}^{n}_{j=1}$ of $T_{x_{0}}^{(1,0)}X$. Then $e_{2j-1}=\frac{1}{\sqrt{2}}(w_{j}+\overline{w}_{j})$ and $e_{2j}=\frac{\sqrt{-1}}{\sqrt{2}}(w_{j}-\overline{w}_{j})$, $j=1,\ldots,n$, forms an orthonormal basis of $T_{x_{0}}X$. We use coordinates on $T_{x_{0}}X\simeq\mathbb{R}^{2n}$ given by the identification

[TABLE]

In the sequel we also use complex coordinates $z=(z_{1},\ldots,z_{n})$ on $\mathbb{C}^{n}\simeq\mathbb{R}^{2n}$.

Let $dv_{TX}$ be the Riemannian volume form on $(T_{x_{0}}X,g^{T_{x_{0}}X})$. Let $\kappa_{x_{0}}:T_{x_{0}}X\rightarrow\mathbb{R}$, $Z\mapsto\kappa_{x_{0}}(Z)$ be a smooth positive function defined by

[TABLE]

where the subscript $x_{0}$ of $\kappa_{x_{0}}(Z)$ indicates the base point $x_{0}\in X$.

Rescaling $\Delta_{p,\Phi}$ and Taylor expansion. For $s\in\mathscr{C}^{\infty}(\mathbb{R}^{2n},\mathbb{C})$, $Z\in\mathbb{R}^{2n}$, $|Z|\leqslant\varepsilon$, and for $t=\frac{1}{\sqrt{p}}$, set

[TABLE]

For $U\in T_{x_{0}}X$, we denote $\nabla_{U}$ the ordinary differential in direction $U$. Set

[TABLE]

By [14, Theorem 1.4], there exist second order differential operators $\mathcal{O}_{r}$ such that we have an asymptotic expansion in $t$ when $t\rightarrow 0$,

[TABLE]

Moreover,

[TABLE]

and

[TABLE]

From (2.1) and (2.3), as in [13, Remark 4.1.8], $\mathscr{L}_{t}$ is a formally self-adjoint elliptic operator with respect to $\|\boldsymbol{\cdot}\|_{L^{2}}$ on $\mathbb{R}^{2n}$ and is a smooth family of operators with respect to the parameter $x_{0}\in X$. Thus $\mathscr{L},\mathscr{L}_{0}$ and $\mathcal{O}_{r}$ in (2.5) are formally self-adjoint with respect to $\|\boldsymbol{\cdot}\|_{L^{2}}$.

By [13, Theorem 8.3.8], the following asymptotic expansion of the generalized Bergman kernel holds.

Theorem 2.1**.**

There exist polynomials $J_{r}(Z,Z^{\prime})$ in $Z,Z^{\prime}$ with the same parity as $r$ and of degree $\deg J_{r}(Z,Z^{\prime})\leqslant 3r$, such that if we define

[TABLE]

then for any $k,\ell,m\in\mathbb{N}$, $q>0$, there exists $C>0$ such that if $p\geqslant 1$, $Z,Z^{\prime}\in T_{x_{0}}X$ and $|Z|,|Z^{\prime}|\leqslant\frac{q}{\sqrt{p}}$, we have

[TABLE]

*where $\mathscr{C}^{\ell}(X)$ is $\mathscr{C}^{\ell}$-norm for the parameter $x_{0}\in X$. *

Moreover, by [13, (4.1.93), (8.3.45)], $\mathscr{F}_{1}$ and $\mathscr{F}_{2}$ are given by (cf. [13, (8.3.65)], [14, (1.111)])

[TABLE]

From Theorem 2.1, we get in particular [13, Theorem 8.3.3]: there exist $\boldsymbol{b}_{r}\in\mathscr{C}^{\infty}(X,\mathbb{R})$ such that for any $k,\ell\in\mathbb{N}$, there exists $C_{k,\ell}>0$ such that

[TABLE]

and

[TABLE]

3. Proof of Theorem 0.1

In this section we reduce Theorem 0.1 to Theorem 3.3. Let us fix $x_{0}\in X$. As in section 2, we identify a small geodesic ball $B^{X}(x_{0},\varepsilon)$ to $B^{T_{x_{0}}X}$ by means of the exponential map and we trivialize $L$ by using a unit frame $e_{L}(Z)$ which is parallel with respect to $\nabla^{L}$ along the curve $[0,1]\ni u\rightarrow uZ$ for $Z\in B^{T_{x_{0}}X}(0,\varepsilon)$.

Set $d_{p}:=\dim\mathcal{H}_{p}$ and for $v=(v_{1},\ldots,v_{d_{p}})\in\mathbb{C}^{d_{p}}$, set $\|v\|^{2}=\sum^{d_{p}}_{j=1}|v_{j}|^{2}$. We can now express the Fubini-Study form in the homogeneous coordinate $[v]=[v_{1},\ldots,v_{d_{p}}]\in\mathbb{P}(\mathcal{H}^{\ast}_{p})$ as

[TABLE]

Let $\{s_{j}\}$ be an orthonormal basis of $\mathcal{H}_{p}$, and let $\{s^{j}\}$ be its dual basis. We write locally $s_{j}=f_{j}e^{\otimes p}_{L}$, then by (0.8), as in [13, (5.1.17)], we have

[TABLE]

Set

[TABLE]

Then

[TABLE]

By (3.1), (3.2) and (3.3), we get

[TABLE]

where \big{|}_{x\,=\,y\,=\,x_{0}} means the pull-back by the diagonal map $\jmath:X\rightarrow X\times X$, $x_{0}\mapsto(x_{0},x_{0})$.

By (3.4), $P_{\mathcal{H}_{p}}(x,y)$ is represented by $f^{p}(x,y)$ under our trivialization of $L$. Since we work with normal coordinates, we get from (2.2) (cf. [13, (4.1.101)])

[TABLE]

By (2.9), (2.12), (3.5) and (3.6), we get

[TABLE]

From (1.6) and (2.8), we obtain

[TABLE]

As $J_{r}$ is a polynomial in $Z,Z^{\prime}$ with the same parity as $r$, we know from (1.6) and (2.8) that for $\alpha,\alpha^{\prime}\in\mathbb{N}^{2n}$, there exists a polynomial $J_{r,\alpha,\alpha^{\prime}}$ in $Z,Z^{\prime}$ with the same parity as $r-|\alpha|-|\alpha^{\prime}|$ such that

[TABLE]

In particular, (3.9) yields

[TABLE]

By (1.6) and (2.8), we get

[TABLE]

Substituting (2.12), (3.8), (3.10) and (3.11) into (3.7) yields

[TABLE]

Recall that for a tensor $\psi$, $\nabla^{X}\psi$ is the covariant derivative of $\psi$ induced by the Levi-Civita connection $\nabla^{TX}$. We will denote by $\left\langle\,\cdot,\,\cdot\right\rangle$ the $\mathbb{C}$-bilinear form on $TX\otimes_{\mathbb{R}}\mathbb{C}$ induced by $g^{TX}$.

The following observation [13, (8.3.54)] is very useful.

Lemma 3.1**.**

For $U\in T_{x_{0}}X$, $\nabla_{U}^{X}J$ is skew-adjoint and the tensor \big{\langle}(\nabla_{\boldsymbol{\cdot}}^{X}J)\boldsymbol{\cdot},\boldsymbol{\cdot}\big{\rangle} is of type \big{(}T^{\ast(1,0)}X\big{)}^{\otimes 3}\oplus\big{(}T^{\ast(0,1)}X\big{)}^{\otimes 3}.

Lemma 3.2**.**

We have

[TABLE]

Proof.

By (1.2) and (2.6), we have (cf. [13, (8.3.51)])

[TABLE]

From Theorem 1.1, (1.7), (3.14) and Lemma 3.1, we get (cf. [13, (8.3.67)])

[TABLE]

Note that if $K$ is an operator on \big{(}\mathbb{R}^{2n},\|\boldsymbol{\cdot}\|_{L^{2}}\big{)} with smooth kernel $K(Z,Z^{\prime})$ with respect to $dZ^{\prime}$, then the kernel $K^{\ast}(Z,Z^{\prime})$ of the adjoint $K^{\ast}$ of $K$, with respect to $dZ^{\prime}$, is given by

[TABLE]

As $\mathscr{L}$, $\mathcal{O}_{1}$ are formally self-adjoint with respect to $\|\boldsymbol{\cdot}\|_{L^{2}}$, thus $\mathscr{P}\mathcal{O}_{1}\mathscr{L}^{-1}\mathscr{P}^{\bot}$ is the adjoint of $\mathscr{L}^{-1}\mathscr{P}^{\bot}\mathcal{O}_{1}\mathscr{P}$. From Lemma 3.1, (3.15) and (3.16), we get

[TABLE]

As the coefficients of $\mathscr{P}(Z,Z^{\prime})$ in (3.15) and (3.17) are polynomials of degree $3$, from (2.10), (3.15) and (3.17), we get (3.13). The proof of Lemma 3.2 is completed. ∎

Theorem 3.3**.**

The following identity holds,

[TABLE]

Lemma 3.2, Theorem 3.3 and (3.12) yield Theorem 0.1.

4. Proof of Theorem 3.3

This section is devoted to the proof of Theorem 3.3. We will compute the contribution of each term in (2.10) to $\mathscr{F}_{2}$. Set

[TABLE]

For $j\in\{1,\ldots,6\}$, let $I_{j}(Z,Z^{\prime})$ be the smooth kernel of the operator $I_{j}$ with respect to $dZ^{\prime}$. By (2.10),

[TABLE]

In the context of (3.16), by denoting b_{jk}=\dfrac{\partial^{2}K}{\partial Z_{j}\partial Z^{\prime}_{k}}(Z,Z^{\prime})\Big{|}_{Z=Z^{\prime}=0} we have

[TABLE]

Since the operators $\mathcal{O}_{r}$ from (2.5) are formally self-adjoint with respect to $\|\boldsymbol{\cdot}\|_{L^{2}}$, (4) implies that $I_{1}$ and $I_{2}$ are the adjoints of $I_{3}$ and $I_{4}$, respectively, as operators acting on \big{(}\mathbb{R}^{2n},\|\boldsymbol{\cdot}\|_{L^{2}}\big{)}. Hence by (4.3),

[TABLE]

4.1. Evaluation of $(d_{x}d_{y}I_{j})(0,0)$

for $j=1,3,5,6$

To simplify the notation, for polynomials $Q_{1},Q_{2}$ in $Z,Z^{\prime}$, we will denote

[TABLE]

if the constant coefficient and the coefficient of $Z^{\prime}_{j}$ for all $j$ in $Q_{1}-Q_{2}$ as a polynomial in $Z$ are zero; we denote

[TABLE]

if the constant coefficient and the coefficients of $Z_{j},Z^{\prime}_{k},Z_{j}Z^{\prime}_{k}$ for all $j,k$ in $Q_{1}-Q_{2}$ are zero.

Set

[TABLE]

From Lemma 3.1, (1.7), (3.14) and (3.15), we get

[TABLE]

where in the last relation $\sim$ of (4.8) we used b_{i}\Big{\langle}\big{(}\nabla_{\overline{z}}^{X}J\big{)}\overline{z},\frac{\partial}{\partial\overline{z}_{i}}\Big{\rangle}\mathscr{P}=\Big{\langle}\big{(}\nabla_{\overline{z}}^{X}J\big{)}\overline{z},-2\pi\overline{z}^{\prime}\Big{\rangle}\mathscr{P}.

By Theorem 1.1, (1.7) and (1.8), we get b^{+}_{i}b_{j}b_{k}=b_{j}b_{k}b^{+}_{i}+4\pi\big{(}\delta_{ij}b_{k}+\delta_{ik}b_{j}\big{)} and

[TABLE]

By (1.7), (1.8), (4), (4.8) and (4.9) we obtain

[TABLE]

Recall that $\mathscr{J}_{sti}$ is anti-symmetric on $t$ and $i$, thus the contribution of $-2\delta_{ij}\delta_{kt}z_{s}-2\delta_{ik}\delta_{jt}z_{s}$ in (4.10) is zero. Thus (4.10) yields

[TABLE]

By Lemma 3.1, (1.6) and (4.11), we get

[TABLE]

From (4.4), (4.7) and (4.12), we get

[TABLE]

By (1.6), (3.15), (3.17) and (4), we get

[TABLE]

where in the last equation we use $\mathscr{P}(0,0)=1$, since we need to compute the constant coefficient of $\mathscr{P}$ in \mathscr{P}\circ\big{(}\overline{z}^{\prime\prime}_{s}\overline{z}^{\prime\prime}_{t}z^{\prime\prime}_{j}z^{\prime\prime}_{k}\mathscr{P}\big{)}.

By (1.7) and (1.8), we get

[TABLE]

From Theorem 1.1 and (4.15), we get

[TABLE]

From (4.7), (4.14) and (4.16), we obtain

[TABLE]

By (3.15), (3.17) and (4), we get

[TABLE]

Thus by (4.7), (4.16) and (4.18), we get

[TABLE]

4.2. Evaluation of $(d_{x}d_{y}I_{2})(0,0)$: part I

Recall that by [14, Lemma 2.1] we have

[TABLE]

Set

[TABLE]

By Theorem 1.1, \mathscr{P}^{\bot}\Big{\langle}R^{TX}\Big{(}\frac{\partial}{\partial z_{i}},\frac{\partial}{\partial z_{j}}\Big{)}\frac{\partial}{\partial\overline{z}_{i}},\frac{\partial}{\partial\overline{z}_{j}}\Big{\rangle}\mathscr{P}=\mathscr{P}^{\bot}\Phi_{x_{0}}\mathscr{P}=0. Thus (4), (4.20) and (4.21) yield

[TABLE]

We evaluate first the contribution of $I_{2j}$, $j=1,3,5,6$, in $(d_{x}d_{y}I_{2})(0,0)$. We recall the following well-known symmetry properties of the curvature $R^{TX}$: for $U,V,W,Y\in TX$, we have

[TABLE]

Using (1.8) and (4.23), we have

[TABLE]

By (1.7) and (1.8), we get

[TABLE]

By Theorem 1.1, (4.21), (4.24) and (4.25), we get

[TABLE]

where

[TABLE]

Note that by Theorem 1.1, (1.7) and (1.8),

[TABLE]

Thus, from (1.7), (4.6) and (4.28), we get

[TABLE]

From (1.7) and (1.8), we get

[TABLE]

and

[TABLE]

By (1.7), (4.30) and (4.31), we get

[TABLE]

Substituting (3.11), (4.23), (4.29)–(4.32) into (4.26), we obtain

[TABLE]

By (4.21),

[TABLE]

By Theorem 1.1, (1.7) and (1.8),

[TABLE]

Note that by (1.7), \overline{z}_{s}\mathscr{P}=\Big{(}\dfrac{b_{s}}{2\pi}+\overline{z}^{\prime}_{s}\Big{)}\mathscr{P}. Thus from Theorem 1.1, we get

[TABLE]

As in (4.32), we get

[TABLE]

From (3.11), (4.23), (4.34), (4.35) and (4.37), we get

[TABLE]

Clearly, by Theorem 1.1 and (4.21),

[TABLE]

From Theorem 1.1, (1.7) and (4.25), we get

[TABLE]

and

[TABLE]

As in [13, (8.3.56), (8.3.63)], we have

[TABLE]

By (3.11), (4.23), (4.39), (4.40), (4.41) and (4.42), we get

[TABLE]

Then by Lemma 3.1 and (4.21), we get

[TABLE]

By Theorem 1.1, (1.7), (1.8) and (4.15), we get

[TABLE]

[TABLE]

By (1.7), (4.45) and (4.46), we get

[TABLE]

By (3.11), (4.44) and (4.47), we get

[TABLE]

By (4.48),

[TABLE]

Note that for $U,V,W\in TX$, \big{\langle}JU,V\big{\rangle}=\omega(U,V), thus (cf. [13, (8.3.48)]),

[TABLE]

From Lemma 3.1, (4.50) and \big{|}\frac{\partial}{\partial z_{j}}\big{|}^{2}=\frac{1}{2}, we have

[TABLE]

When we sum (4.51) over $r=q$, we get by (4.42) (cf. [13, (8.3.58)]),

[TABLE]

By (4.7) and (4.51), we get

[TABLE]

By Lemma 3.1, (4.50) and \big{|}\frac{\partial}{\partial z_{j}}\big{|}^{2}=\frac{1}{2}, we obtain

[TABLE]

By taking the conjugation of (4.51), we get

[TABLE]

Substituting (4.42), (4.52), (4.53), (4.54) and (4.55) into (4.49) yields

[TABLE]

4.3. Evaluation of $(d_{x}d_{y}I_{2})(0,0)$:

part II

We evaluate now the contribution of $I_{22}$, $I_{24}$ in $(d_{x}d_{y}I_{2})(0,0)$. The definitions of $\nabla^{X}\nabla^{X}J$ and $R^{TX}$ imply that for $U,V,W,Y\in TX$ (cf. [13, (8.3.59)]),

[TABLE]

Recall that [13, (8.3.71)],

[TABLE]

By (4.50) and (4.57), we get (cf. [13, (8.3.61)]): for $u_{1},u_{2},u_{3}\in T^{(1,0)}X$, $\overline{v}_{1},\overline{v}_{2}\in T^{(0,1)}X$,

[TABLE]

In particular, we have

[TABLE]

By (4.57) and (4.60), we get

[TABLE]

By (4.57) and (4.59) we get (cf. [13, (8.3.62)]),

[TABLE]

By (4.59), (4.60), (4.61) and (4.62), we get

[TABLE]

By (4.21), (4.58) and (4.63), we get

[TABLE]

By (1.7) and (1.8), we get

[TABLE]

Thus, by Theorem 1.1, (1.7), (1.8) and (4.65), as in (4.47), we get

[TABLE]

By (4.6) and (4.66), note that the last line of (4.66) has also the term $z\overline{z}^{\prime}$, we get

[TABLE]

Again by (1.7) and (1.8),

[TABLE]

By Theorem 1.1 and (4.68), we get

[TABLE]

By (1.7), (1.8) and (4.6), we get

[TABLE]

By (4.32), (4.69) and (4.70), we obtain

[TABLE]

Finally, by Theorem 1.1, (1.7) and (1.8), as in (4.29), we get

[TABLE]

From (3.11), (4.64), (4.67), (4.71) and (4.72), we obtain

[TABLE]

Note that by (4.23),

[TABLE]

By (4.23), (4.42), (4.52), (4.53), (4.54), (4.55), (4.73) and (4.74), we get

[TABLE]

By (4.57), (4.59) and (4.60), we get (cf. [14, (2.28)]),

[TABLE]

By (4.25) and (4.35), we get

[TABLE]

By (3.11), (4.42), (4.54), (4.76) and (4.77), we obtain

[TABLE]

Combining (4.22), (4.23), (4.33), (4.38), (4.42), (4.43), (4.56), (4.74), (4.75) and (4.78), we obtain

[TABLE]

By Lemma 3.1, (4.50), (4.51), (4.54), (4.55), (4.57) and (4.62), we get (cf. [13, (8.3.63)])

[TABLE]

Substituting (4.80) into (4.79) yields

[TABLE]

4.4. Proof of Theorem 3.3

By (4.4) and (4.81), we get

[TABLE]

Substituting (4.12), (4.13), (4.17), (4.19), (4.81) and (4.82) into (4.2), we finally obtain

[TABLE]

By [13, Theorem 8.3.4, Lemma 8.3.10],

[TABLE]

The identities (4.83) and (4.84) yield Theorem 3.3. This concludes the proof of Theorem 0.1.