## ABSTRACT

Let *M* be an arbitrary complex manifold and let *L* be a Hermitian holomorphic line bundle over *M*. We introduce the Berezin–Toeplitz quantization of the open set of *M* where the curvature on *L* is nondegenerate. In particular, we quantize any manifold admitting a positive line bundle. The quantum spaces are the spectral spaces corresponding to [0,*k*^{−N}], where *N*>1 is fixed, of the Kodaira Laplace operator acting on forms with values in tensor powers *L*^{k}. We establish the asymptotic expansion of associated Toeplitz operators and their composition in the semiclassical limit *k*→∞ and we define the corresponding star-product. If the Kodaira Laplace operator has a certain spectral gap this method yields quantization by means of harmonic forms. As applications, we obtain the Berezin–Toeplitz quantization for semi-positive and big line bundles.

## 1. Introduction and statement of the main results

The aim of the geometric quantization theory of Kostant and Souriau is to relate the classical observables (smooth functions) on a phase space (a symplectic manifold) to the quantum observables (bounded linear operators) on the quantum space (sections of a line bundle). Berezin–Toeplitz quantization is a particularly eﬃcient version of the geometric quantization theory [Citation2, Citation3, Citation14, Citation21, Citation22, Citation31]. Toeplitz operators and more generally Toeplitz structures were introduced in geometric quantization by Berezin [Citation3] and Boutet de Monvel–Guillemin [Citation6]. We refer to [Citation22, Citation26, Citation30] for reviews of Berezin–Toeplitz quantization.

The setting of Berezin–Toeplitz quantization on Kähler manifolds is the following. Let (*M*,*ω*,*J*) be a Kähler manifold of dim_{ℂ}*M* = *n* with Kähler form *ω* and complex structure *J*. Let (*L*,*h*) be a holomorphic Hermitian line bundle on *X*, and let ∇^{L} be the holomorphic Hermitian connection on (*L*,*h*) with curvature . We assume that (*L*,*h*,∇^{L}) is a prequantum line bundle, i.e.,

Let *g*^{TM}: = *ω*(⋅,*J*⋅) be the *J*-Riemannian metric on *TM*. The Riemannian volume form of *g*^{TM} is denoted by *d**v*_{M}. On the space of smooth sections with compact support we introduce the *L*^{2}-scalar product associated to the metrics *h* and the Riemannian volume form *d**v*_{M} by

The completion of with respect to (1.2) is denoted as usual by . We denote by the closed subspace of consisting of holomorphic sections. The Bergman projection is the orthogonal projection . For a bounded function *f*∈*𝒞*^{∞}(*M*), set

*f*is the pointwise multiplication by

*f*. The map which associates to

*f*∈

*𝒞*

^{∞}(

*M*) the family of bounded operators {

*T*

_{f, k}} on is called the

*Berezin–Toeplitz quantization*. A

*Toeplitz operator*is a sequence of bounded linear endomorphisms of verifying , such that there exist a sequence such that for any

*p*≥0, there exists

*C*

_{p}>0 with for any

*k*∈

*ℕ*, where

_{∥ ⋅ ∥op}denotes the operator norm on the space of bounded operators.

Assume now that (*M*,*ω*,*J*) is a compact Kähler manifold. Then Bordemann et al. [Citation5] and Schlichenmaier [Citation29] (using the analysis of Toeplitz structures of Boutet de Monvel–Guillemin [Citation6]), Charles [Citation7] (inspired by semiclassical analysis of Boutet de Monvel–Guillemin [Citation6]) and Ma–Marinescu [Citation25] (using the expansion of the Bergman kernel [Citation9, Citation24]) showed that the composition of two Toeplitz operators is a Toeplitz operator, in the sense that for any *f*,*g*∈*𝒞*^{∞}(*M*), the product has an asymptotic expansion

*C*

_{p}are bidifferential operators of order ≤2

*r*, satisfying

*C*

_{0}(

*f*,

*g*) =

*fg*and . Here { ⋅ , ⋅ } is the Poisson bracket on (

*M*,2

*πω*). We deduce from (1.4),

In [Citation24, Citation25] Ma–Marinescu extended the Berezin–Toeplitz quantization to symplectic manifolds and orbifolds by using as quantum space the kernel of the Dirac operator acting on powers of the prequantum line bundle twisted with an arbitrary vector bundle with arbitrary metric on manifolds. Recently, Charles [Citation8] introduced a semiclassical approach for symplectic manifolds inspired from the Boutet de Monvel–Guillemin theory [Citation6].

In this paper, we extend the Berezin–Toeplitz quantization in several directions. Firstly, we consider an arbitrary Hermitian manifold (*M*,*Θ*,*J*) endowed with arbitrary Hermitian holomorphic line bundle (*L*,*h*) and we quantize the open set *M*(0) where the curvature of (*L*,*h*) is positive. Since there are no holomorphic *L*^{2} sections in general, we use as quantum spaces the spectral spaces of the Kodaira Laplacian on , corresponding to energy less than *k*^{−N}, *N*>1 fixed, decaying to 0 polynomially in *k*, as *k*→∞. Secondly, we consider the same construction for the Kodaira Laplacian acting on (0,*q*)-forms. In this case, we quantize the open set *M*(*q*) where the curvature of (*L*,*h*) is nondegenerate and has exactly *q* negative eigenvalues (and hence *n*−*q* positive ones). Quantization using (0,*q*)-forms was introduced in [Citation24, Section 8.2] for bundles with mixed curvature of signature (*q*,*n*−*q*) everywhere on a compact manifold. It was based on the asymptotic of Bergman kernel developed in Ma and Marinescu [Citation23].

The idea underlying the construction used in this paper comes from the local holomorphic Morse inequalities [Citation4, Citation11, Citation18, Citation24]. Roughly speaking, the harmonic (0,*q*)-forms with values in *L*^{k} tend to concentrate on *M*(*q*) as *k*→∞. More precisely, the semiclassical limit of the kernel of the spectral projectors considered above was determined in [Citation18, Theorem 1.1], see also [Citation18, Theorems 1.6 –1.10] for important particular cases. This is the main technical ingredient used in this paper, which is in turn based on techniques of microlocal and semiclassical analysis [Citation13, Citation28], especially the stationary phase method of Melin–Sjöstrand [Citation28].

We now formulate the main results. We refer to Section 2 for some standard notations and terminology used here. We are working in the following general setting:

(

*M*,*Θ*,*J*) is a Hermitian manifold of complex dimension*n*, where*Θ*is a smooth positive (1,1)-form and*J*is the complex structure. Moreover, (*L*,*h*) is a holomorphic Hermitian line bundle over*M*, where*h*is the Hermitian fiber metric on*L*, and*q*∈{0,1,…,*n*}.*f*,*g*∈*𝒞*^{∞}(*M*) are smooth bounded functions.

Let be the Riemannian metric on *TM* induced by *Θ* and *J* and let ⟨ ⋅ ,⋅ ⟩ be the Hermitian metric on *ℂTM*: = *TM*⊗_{ℝ}*ℂ* induced by . The Riemannian volume form *d**v*_{M} of (*M*,*Θ*) satisfies . For every *q* = 0,1,…,*n*, the Hermitian metric ⟨ ⋅ ,⋅ ⟩ on *TM*⊗_{ℝ}*ℂ* induces a Hermitian metric ⟨ ⋅ ,⋅ ⟩ on the bundle of (0,*q*) forms of *M*.

We will denote by *ϕ* the local weights of the Hermitian metric *h* on *L* (see (2.1)). Let ∇^{L} be the holomorphic Hermitian connection on (*L*,*h*) with curvature . We will identify the curvature form *R*^{L} with the Hermitian matrix satisfying for every , *x*∈*M*,

Let , where , are the eigenvalues of *Ṙ*^{L} with respect to ⟨ ⋅ ,⋅ ⟩. For *j*∈{0,1,…,*n*}, let

We denote by *W* the subbundle of rank *j* of generated by the eigenvectors corresponding to negative eigenvalues of *Ṙ*^{L}. Then is a rank one sub-bundle. Here is the dual bundle of the complex conjugate bundle of *W* and is the vector space of all finite sums of , . We denote by the orthogonal projection from onto .

For *k*>0, let be the *k*th tensor power of the line bundle (*L*,*h*). Let ( ⋅ ,⋅ )_{k} and ( ⋅ ,⋅ ) denote the global *L*^{2} inner products on and induced by ⟨ ⋅ ,⋅ ⟩ and *h*^{k}, respectively (see (2.2)). We denote by and the completions of and with respect to ( ⋅ ,⋅ )_{k} and ( ⋅ ,⋅ ), respectively.

Let be the Kodaira Laplacian acting on (0,*q*)–forms with values in *L*^{k}, cf. (2.6). We denote by the same symbol the Gaffney extension of the Kodaira Laplacian, cf. (2.9). It is well-known that is self-adjoint and the spectrum of is contained in (see [Citation24, Proposition 3.1.2]). For a Borel set *B*⊂ℝ let *E*(*B*) be the spectral projection of corresponding to the set *B*, where *E* is the spectral measure of (see Davies [Citation10, Section 2]) and for λ∈ℝ we set *E*_{λ} = *E*((−∞,λ]) and

If λ = 0, then is the space of global harmonic sections. The *spectral projection* of is the orthogonal projection

Fix *f*∈*𝒞*^{∞}(*M*) be a bounded function. Let λ≥0. The *Berezin–Toeplitz quantization for * is the operator

Let be the Schwartz kernel of , see (2.13), (2.14). Since is elliptic, we have .

Let be a *k*-dependent continuous operator with smooth kernel *A*_{k}(*x*,*y*) and let be open trivializations with trivializing sections *s* and , respectively. In this paper, we will identify *A*_{k} and *A*_{k}(*x*,*y*) on with the localized operators and , respectively (see (2.3)).

The first main result of this work is the following.

Theorem 1.1.

Under the assumptions (A) and (B) let *j*∈{0,1,…,*n*} and on which *L* is trivial. Suppose that one of the following conditions is fulfilled:

*D*_{0}⋐*M*(*j*) and*j*≠*q*,*D*_{0}⋐*M*(*q*) and .

Then, for every *N*>1, *m*∈*ℕ*, there exists *C*_{N,m}>0 independent of *k* such that

If *D*_{0}⋐*M*(*q*) there exists a symbol

*N*>1,

*m*∈

*ℕ*, there exists independent of

*k*such that

We collect more properties for the phase *Ψ* in Theorem 3.3. The results says that, roughly speaking, the Toeplitz kernel acting on (0,*q*)-forms, decays rapidly as *k*→∞ outside *M*(*q*) and off-diagonal, and admits an asymptotic expansion on the set *M*(*q*).

Let *ℓ*,*m*∈*ℕ* be fixed and choose *N*≥2(*n*+*ℓ*+2*m*+1). Then we deduce from (1.12) that

Note that if *M* is compact complex manifold endowed with a positive line bundle *L* (i.e., *M*(0) = *M*) we have by [Citation27, Theorem 0.1] for any *ℓ*,*m*∈*ℕ*,

Actually, in this case, due to the spectral gap of the Kodaira Laplacian [Citation24, Theorem 1.5.5] we have for *k* large enough, so (1.15) follows from (1.16). The expansion (1.15) bears resemblance to the expansion of the Toeplitz kernels for functions *f*∈*𝒞*^{p}(*M*) (see [Citation1, (3.19)]), for arbitrary *p*∈*ℕ*. In (1.15) the upper bound for the order of expansion *ℓ* is due to the size *k*^{−N} of the spectral parameter, while in case of symbols of class *𝒞*^{p}(*M*) is due to the order of differentiability *p*.

It is interesting to note that Theorem 1.1 and the following results provide a generalization of various expansions for Toeplitz operators in the case of an arbitrary complex manifold endowed with a positive line bundle. In this case, we have simply *M* = *M*(0). Of course, in such generality, the quantum spaces have to be spectral spaces .

The first three coeﬃcients of the kernel expansions of Toeplitz operators and of their composition for the quantization of a compact Kähler manifold with positive line bundle were calculated by Ma–Marinescu [Citation27] in the presence of a twisting vector bundle *E* and later by Hsiao [Citation17] for *E* = *ℂ*. Both [Citation17, Citation27] work with a general not necessarily Kähler base metric *Θ* which might not be polarized, that is, in general. We will calculate the top coeﬃcients *b*_{f,1}(*x*,*x*) and *b*_{f,2}(*x*,*x*) of the expansion (1.12) in Section 7. The coeﬃcients *b*_{f,0}(*x*,*x*) and *b*_{f,1}(*x*,*x*) were given in [Citation7] for *E* = *ℂ* and . It is a remarkable manifestation of universality, that the coeﬃcients for the quantization with holomorphic sections [Citation17, Citation27] and for the quantization with spectral spaces used in this paper are given by the same formulas. We refer to [Citation32] for an interpretation in graph-theoretic terms of the Toeplitz kernel expansion. The formulas from [Citation27] play an essential role in the quantization of the Mabuchi energy [Citation15] and Laplace operator [Citation20]. On the set where the curvature of *L* is degenerate we have the following behavior.

Theorem 1.2.

Under the general assumptions (A) and (B), set

Then for every , *𝜀*>0, *N*>1 and every *j*∈{0,1,…,*n*}, there exist a neighborhood *U* of *x*_{0} and *k*_{0}>0, such that for all *k*≥*k*_{0} we have

We consider next the composition of two Berezin–Toeplitz quantizations. We have first the following expansion of the kernels of Toeplitz operators.

Theorem 1.3.

Under the assumptions (A) and (B) let *j*∈{0,1,…,*n*} and on which *L* is trivial. Suppose that one of the following conditions is fulfilled:

*D*_{0}⋐*M*(*j*) and*j*≠*q*,*D*_{0}⋐*M*(*q*) and .

Then, for every *N*>1, *m*∈*ℕ*, there exists *C*_{N,m}>0 independent of *k* such that

If *D*_{0}⋐*M*(*q*) there exists a symbol

*N*>1,

*m*∈

*ℕ*, there exists independent of

*k*such that

It should be noticed that Theorem 1.3 holds for any Hermitian manifold *M*, not necessarily compact. Note that the estimates in Theorem 1.3 involve the power compared to in Theorem 1.1. We will explain why there are different exponents 3*n* and 2*n* in the proof of Theorem 6.1.

We will calculate the top coeﬃcients *b*_{f,1}(*x*,*x*), *b*_{f,2}(*x*,*x*) and *b*_{f,g,1}(*x*,*x*), *b*_{f,g,2}(*x*,*x*) of the expansions (1.12) and (1.19) in Section 7 (see Theorems 7.1 and 7.4).

We come now to the asymptotic expansion of the composition of two Toeplitz operators in the operator norm. Let be *k*-dependent continuous operator. We say that as *k*→∞, locally in the *L*^{2} operator norm if for any , there exists *C*>0 independent of *k* such that , for *k* large, where _{∥⋅∥op} denotes the *L*^{2} operator norm. We also denote by ⟨ ⋅ ,| ⋅ ⟩_{ω} the Hermitian metric on induced by .

Theorem 1.4.

Under the assumptions (A) and (B) suppose moreover that *f*,*g*∈*𝒞*^{∞}(*M*) have compact support in *M*(0). Then for every *N*>1, there exist functions , *p*∈*ℕ*, such that for any *ℓ*∈*ℕ* the product has the asymptotic expansion

*L*

^{2}operator norm. Moreover,

*f*,

*g*} is the Poisson bracket on (

*M*(0),2

*πω*).

We will give formulas for the coeﬃcients *C*_{j}(*f*,*g*), *j* = 0,1,3, in Corollary 7.5. They have the same form as those in the expansion of the Toeplitz operators acting on spaces of holomorphic sections, see [Citation17, (1.29)], [Citation27, (0.20)]. Formula (1.23) represents the semiclassical correspondence principle between classical and quantum observables. Theorem 1.4 allows us to introduce a star-product on the set where a line positive is positive, see Remark 6.5.

As an application of Theorems 1.1 and 1.2, we obtain:

Theorem 1.5.

Assume (A) and (B) are fulfilled and let *N*>2*n*. Then

*M*(

*q*), for every

*D*⋐

*M*, there exists

*C*

_{D}>0 independent of

*k*such that

_{M(q)}denotes the characteristic function of

*M*(

*q*), we have the pointwise convergence:

Since , we can identify to an element of . Then

*M*is compact, we define

For λ = 0, we set , , , .

From (1.24)–(1.26), we get Weyl’s formula for Berezin–Toeplitz quantization.

Theorem 1.6.

Assume (A) and (B) are fulfilled and let *N*>2*n*. If *M* is compact, then

From Theorem 1.6 we deduce the following (see Section 8).

Theorem 1.7.

Under assumptions (A) and (B) suppose that *M* is compact and *M*(*q*−1) = *∅*, *M*(*q*+1) = *∅*. Then

In particular,

Let’s consider *q* = 0 and *f*≡1 in (1.31). If *M*(1) = *∅*, we obtain as *k*→∞. Therefore, as *k*→∞, provided *M*(0)≠*∅* and *M*(1) = *∅*. This is a form of Demailly’s criterion for a line bundle to be big, which answers the Grauert–Riemenschneider conjecture, see [Citation11], [Citation24, Theorem 2.2.27].

We wish now to link the quantization scheme, we proposed above by using spectral spaces to the more traditional quantization using holomorphic sections (or, more generally, harmonic forms). For this purpose we need the notion of *O*(*k*^{−N}) small spectral gap property introduced in [Citation18, Definition 1.5]:

Definition 1.8.

Let *D*⊂*M*. We say that has * O(k^{−N}) small spectral gap on D* if there exist constants

*C*

_{D}>0,

*N*∈

*ℕ*,

*k*

_{0}∈

*ℕ*, such that for all

*k*≥

*k*

_{0}and , we have

Let be open sets and be *k*-dependent continuous operators with smooth kernels . We write locally uniformly on or locally uniformly on if

*N*>1.

The following result describes the asymptotics of the kernels of Toeplitz operators corresponding to *harmonic forms* in the case of small spectral gap.

Theorem 1.9.

Under the assumptions (A) and (B) let *j*∈{0,1,…,*n*} and on which *L* is trivial. Suppose that one of the following conditions is fulfilled:

*D*_{0}⋐*M*(*j*) and*j*≠*q*,*D*_{0}⋐*M*(*q*), has an*O*(*k*^{−N}) small spectral gap on*D*_{0}and .

Then

Assume that *D*_{0}⋐*M*(*q*) and has an *O*(*k*^{−N}) small spectral gap on *D*_{0}. Then,

There are several geometric situations when there exists a spectral gap. For example, if *L* is a positive line bundle on a compact manifold *M*, or more generally, if *L* is uniformly positive on a complete manifold (*M*,*Θ*) with and *∂Θ* bounded below, then the Kodaira Laplacian has a “large" spectral gap on *M*, that is, there exists a constant *C*>0 such that for all *k* we have (see [Citation24, Theorem 1.5.5], [Citation24, Theorem 6.1.1, (6.1.8)]). Therefore, we can recover from Theorem 1.9 results about quantization of noncompact manifolds, such as [Citation24, Theorem 7.5.1], [Citation25, Theorem 5.3], [Citation26, Theorem 2.30].

In this paper, as applications of Theorem 1.9, we establish Berezin–Toeplitz quantization for semipositive and big line bundles. We assume now that (*M*,*Θ*) is compact and we set

Note that by Siu’s criterion [Citation24, Theorem 2.2.27], *L* is big under the hypotheses of Theorem 1.10 below. By [Citation24, Lemma 2.3.6], *ℳ*(*L*)≠*∅*. Set

Theorem 1.10.

Let (*M*,*Θ*) be a compact Hermitian manifold. Let (*L*,*h*)→*M* be a Hermitian holomorphic line bundle with smooth Hermitian metric *h* having semipositive curvature and with *M*(0)≠*∅*. Let *f*,*g*∈*𝒞*^{∞}(*M*) and let be an open set on which *L* is trivial. Then

Let us consider a singular Hermitian holomorphic line bundle (*L*,*h*)→*M* (see e.g., [Citation24, Definition 2.3.1]). We assume that *h* is smooth outside a proper analytic set *Σ* and the curvature current of *h* is strictly positive. The metric *h* induces singular Hermitian metrics *h*^{k} on *L*^{k}. We denote by *ℐ*(*h*^{k}) the Nadel multiplier ideal sheaf associated to *h*^{k} and by the space of global sections of the sheaf (see (2.12)), where . We denote by ( ⋅ ,⋅ )_{k} the natural inner products on induced by *h* and the volume form *d**v*_{M} on *M* (see (2.11) and see also (2.10) for the precise meaning of ). The (multiplier ideal) Bergman kernel of is the orthogonal projection

Let *f*∈*𝒞*^{∞}(*M*). The multiplier ideal Berezin–Toeplitz operator is

*f*the multiplication operator on by

*f*. Let be the distribution kernel of . Note that .

Theorem 1.11.

Let (*L*,*h*) be a singular Hermitian holomorphic line bundle over a compact Hermitian manifold (*M*,*Θ*). We assume that *h* is smooth outside a proper analytic set *Σ* and the curvature current of *h* is strictly positive. Let *f*,*g*∈*𝒞*^{∞}(*M*). Let *D*_{0}⊂*M*∖*Σ* be an open set on which *L* is trivial. Then

The paper is organized as follows. In Section 2, we collect terminology, definitions and statements we use throughout. In Sections 3 and 4 prove the off-diagonal decay for the kernels and . In Section 5, we establish the full asymptotic of the Berezin–Toeplitz kernels and prove Theorem 1.1. Section 6 is devoted to the expansion of the composition of two Toeplitz operators and contains the proof of Theorems 1.3, 1.4, and 1.9–1.11. In Section 7, we calculate the leading coeﬃcients of the various expansions we established. Finally, in Section 8, we prove Theorems 1.2 and 1.7.

## 2. Preliminaries

**Some standard notations.** We denote by *ℕ* = {0,1,2,…} the set of natural numbers and by ℝ the set of real numbers. We use the standard notations *w*^{α}, for multi-indices , *w*∈*ℂ*^{m}, .

Let Ω be a *𝒞*^{∞} paracompact manifold equipped with a smooth density of integration. We let *TΩ* and *T**Ω denote the tangent bundle of Ω and the cotangent bundle of Ω, respectively. The complexified tangent bundle of Ω and the complexified cotangent bundle of Ω will be denoted by *ℂTΩ*: = *TΩ*⊗_{ℝ}*ℂ* and , respectively. We write ⟨ ⋅ ,⋅ ⟩ to denote the pointwise duality between *TΩ* and *T**Ω. We extend ⟨ ⋅ ,⋅ ⟩ bilinearly to .

Let *E* be a *𝒞*^{∞} vector bundle over Ω. We write *E** to denote the dual bundle of *E*. The fiber of *E* at *x*∈Ω will be denoted by *E*_{x}. We denote by *End* (*E*) the vector bundle over Ω with fiber over *x*∈Ω.

Let *F* be a vector bundle over another *𝒞*^{∞} paracompact manifold Ω^{′}. We introduce the vector bundle over Ω^{′}×Ω, where *π*_{1} and *π*_{2} are the projections of Ω^{′}×Ω on the first and second factor (see [Citation24, p. 337]). The fiber of *F*⊠*E** over (*x*,*y*)∈Ω^{′}×Ω consists of the linear maps from *E*_{y} to *F*_{x}.

Let *Y*⊂Ω be an open set and take any *L*^{2} inner product on . By using this *L*^{2} inner product, in this paper, we will consider a distribution section of *E* over *Y* is a continuous linear form on . From now on, the spaces distribution sections of *E* over *Y* will be denoted by *𝒟*^{′}(*Y*,*E*). Let *ℰ*^{′}(*Y*,*E*) be the subspace of *𝒟*^{′}(*Y*,*E*) whose elements have compact support in *Y*. For *m*∈ℝ, we let *H*^{m}(*Y*,*E*) denote the Sobolev space of order *m* of sections of *E* over *Y*. Put

Let *M* be a complex manifold of dimension *n*. We always assume that *M* is paracompact. Let *T*^{1,0}*M* and *T*^{0,1}*M* denote the holomorphic tangent bundle of *M* and the antiholomorphic tangent bundle of *M*, respectively. Let be the holomorphic cotangent bundle of *M* and let be the antiholomorphic cotangent bundle of *M*. For *p*,*q*∈*ℕ*, let be the bundle of (*p*,*q*) forms of *M*.

For an open set *D*⊂*M* we let Ω^{p,q}(*D*) denote the space of smooth sections of over *D* and let be the subspace of Ω^{0,q}(*D*) whose elements have compact support in *D*. Similarly, if *E* is a vector bundle over *D*, then we let Ω^{p,q}(*D*,*E*) denote the space of smooth sections of over *D*. Let be the subspace of Ω^{p,q}(*D*,*E*) whose elements have compact support in *D*.

For a multi-index we set |*J*| = *q*. We say that *J* is strictly increasing if . Let be a local frame for on an open set *D*⊂*M*. For a multi-index , we put . Let *E* be a vector bundle over *D* and let *f*∈Ω^{0,q}(*D*,*E*). *f* has the local representation

**Metric data.** Let (*M*,*Θ*) be a complex manifold of dimension *n*, where *Θ* is a smooth positive (1,1) form, which induces a Hermitian metric ⟨ ⋅ ,⋅ ⟩ on the holomorphic tangent bundle *T*^{1,0}*M*. In local holomorphic coordinates , if , then . We extend the Hermitian metric ⟨ ⋅ ,⋅ ⟩ to *TM*⊗_{ℝ}*ℂ* in a natural way. The Hermitian metric ⟨ ⋅ ,⋅ ⟩ on *TM*⊗_{ℝ}*ℂ* induces a Hermitian metric on also denoted by ⟨ ⋅ ,⋅ ⟩.

Let (*L*,*h*) be a Hermitian holomorphic line bundle over *M*, where the Hermitian metric on *L* is denoted by *h*. Until further notice, we assume that *h* is smooth. Given a local holomorphic frame *s* of *L* on an open subset *D*⊂*M* we define the associated local weight of *h* by

Let be the Chern curvature of *L*, where ∇^{L} is the Hermitian holomorphic connection. Then .

Let *L*^{k}, *k*>0, be the *k*th tensor power of the line bundle *L*. The Hermitian fiber metric on *L* induces a Hermitian fiber metric on *L*^{k} that we shall denote by *h*^{k}. If *s* is a local trivializing holomorphic section of *L* then *s*^{k} is a local trivializing holomorphic section of *L*^{k}. For *p*,*q*∈*ℕ*, the Hermitian metric ⟨ ⋅ ,⋅ ⟩ on and *h*^{k} induce a Hermitian metric on , denoted by . For , we denote the pointwise norm . We take as the induced volume form on *M*. The *L*^{2}–Hermitian inner products on the spaces and are given by

Let be a *k*-dependent continuous operator with smooth kernel *A*_{k}(*x*,*y*). Let *s*, be local trivializing holomorphic sections of *L* on *D*_{0}⋐*M*, *D*_{1}⋐*M*, respectively, , , where *D*_{0}, *D*_{1} are open sets. The localized operator of *A*_{k} on is given by

**A self-adjoint extension of the Kodaira Laplacian.** We denote by

*L*

^{k}and its formal adjoint with respect to ( ⋅ | ⋅)

_{k}, respectively. Let

*q*)–forms with values in

*L*

^{k}. We extend to by

*L*

^{2}space with respect to ( ⋅ ,⋅ )

_{k}. Let denote the Gaffney extension of the Kodaira Laplacian given by

*M*is complete, the Kodaira Laplacian is essentially self-adjoint [Citation24, Corollary 3.3.4] and the Gaffney extension coincides with the Friedrichs extension of .

Consider a singular Hermitian metric *h* on a holomorphic line bundle *L* over *M*. If *h*_{0} is a smooth Hermitian metric on *L* then for some function . The *Nadel multiplier ideal sheaf* of *h* is defined by *ℐ*(*h*) = *ℐ*(*φ*); the definition does not depend on the choice of *h*_{0}. Recall that the *Nadel multiplier ideal sheaf ℐ*(*φ*)⊂*𝒪*_{M} is the ideal subsheaf of germs of holomorphic functions *f*∈*𝒪*_{M,x} such that is integrable with respect to the Lebesgue measure in local coordinates near *x* for all *x*∈*M*. Put

_{| ⋅ |h}and denote the pointwise norms for sections induced by

*h*and

*h*

_{0}, respectively. With the help of

*h*and the volume form

*d*

*v*

_{M}we define an

*L*

^{2}inner product on

*𝒞*

^{∞}(

*M*,

*L*⊗

*ℐ*(

*h*)):

The singular Hermitian metric *h* induces a singular Hermitian metric on *L*^{k}, *k*>0. We denote by ( ⋅ ,⋅ )_{k} the natural inner products on defined as in (2.11) and by the completion of with respect to ( ⋅ ,⋅ )_{k}. The space of global sections in the sheaf is given by

**Schwartz kernel theorem and semiclassical Hörmander symbol spaces.** We recall here the Schwartz kernel theorem [Citation16, Theorems 5.2.1, 5.2.6], [Citation24, Thorem B.2.7]. Let Ω be a *𝒞*^{∞} paracompact manifold equipped with a smooth density of integration. Let *E* and *F* be smooth vector bundles over Ω. Any distribution (“kernel”)

*A*. Moreover, the following two statements are equivalent

If *A* satisfies (a) or (b), we say that *A* is a *smoothing operator*. Furthermore, *A* is smoothing if and only if is continuous, for all *N*≥0, *s*∈ℝ. Let be continuous operators. We write *A*≡*B* or *A*(*x*,*y*)≡*B*(*x*,*y*) if *A*−*B* is a smoothing operator.

We say that *A* is properly supported if the restrictions to *Supp A*(⋅,⋅) of the projections *π*_{1} and *π*_{2} from Ω×Ω to the first and second factor are proper.

We say that *A* is smoothing away the diagonal if is smoothing for all with .

We recall the definition of semiclassical Hörmander symbol spaces [Citation13, Chapter 8]:

Definition 2.1.

Let *U* be an open set in ℝ^{N}. Let

For *m*∈ℝ let

Hence (*a*(⋅,*k*))∈*S*^{m}(1;*U*) if for every , there exists *C*_{α}>0, such that on *W*. Consider a sequence , *j*∈*ℕ*, where *m*_{j}↘−∞, and let . We say that

*ℓ*∈

*ℕ*we have . For a given sequence

*a*

_{j}as above, we can always find such an asymptotic sum

*a*, which is unique up to an element in . We define

*S*

^{m}(1;

*Y*,

*E*) in the natural way, where

*Y*is a smooth paracompact manifold and

*E*is a vector bundle over

*Y*.

## 3. Spectral kernel estimates away the diagonal

The goal of this section is to prove the off-diagonal decay for the kernel of the spectral projection . For this purpose, we introduce a localization of the projection. Let *s*, be local trivializing holomorphic sections of *L* on *D*_{0}⋐*M*, *D*_{1}⋐*M*, respectively, , , where *D*_{0}, *D*_{1} are open sets. We denote by the localization given by (2.3).

Let and be orthonormal frames of on *D*_{0} and *D*_{1}, respectively. Then,

*D*

_{0}and

*D*

_{1}, respectively. We write

The goal of this section is to prove the following.

Theorem 3.1.

With the notations used above, we assume that *D*_{0}⋐*M*(*j*), *j*≠*q*, *j*∈{0,1,…,*n*} or *D*_{0}⋐*M*(*q*) and . Then, for every *N*>1, *m*∈*ℕ*, there exists *C*_{N,m}>0 independent of *k*, such that for all strictly increasing *I*,*J* with |*I*| = |*J*| = *q*,

As preparation, we recall the next result, established in [Citation18, Theorems 4.11 and 4.12]. The localization is defined as in (2.4).

Theorem 3.2.

With the notations used above, assume that *D*_{0}⋐*M*(*q*). Then, for every *N*>1, *m*∈*ℕ*, there exists *C*_{N,m}>0 independent of *k* such that

*b*(

*x*,

*y*,

*k*) is properly supported and is as in Theorem 1.1.

Assume that *D*_{0}⋐*M*(*j*), *j*≠*q*, *j*∈{0,1,2,…,*n*}. Then, for every *N*>1, *m*∈*ℕ*, there exists independent of *k* such that

The following properties of the phase function *Ψ* follow also from [Citation18, Theorem 3.8].

Theorem 3.3.

With the assumptions and notations used in Theorem 1.1, for a given point *p*∈*D*_{0}, let be local holomorphic coordinates centered at *p* satisfying

Moreover, when *q* = 0, we have

*N*∈

*ℕ*.

Fix *N*>1. Let be an orthonormal frame for , where *d*_{k}∈*ℕ*∪{∞}. On *D*_{0}, *D*_{1}, we write

It is not diﬃcult to check that for every strictly increasing *I*, *J*, with |*I*| = |*J*| = *q*, we have

Lemma 3.4.

Assume that *D*_{0}⋐*M*(*j*), *j*≠*q*. Then, for every *N*>1, *m*∈*ℕ*, there exists *C*_{N,m}>0 independent of *k* such that for every strictly increasing *I*, *J*, with |*I*| = |*J*| = *q*,

Proof.

Fix *I*, *J* are strictly increasing, |*I*| = |*J*| = *q*, and let . By (3.7), we have

In view of Theorem 3.2, we see that

*C*

_{α}>0 is a constant independent of

*k*. Moreover, it is known (see [Citation18, Theorem 4.3]) that

*C*

_{β}>0 is a constant independent of

*k*. From (3.8) to (3.10), the lemma follows.

Lemma 3.4 provides the proof of Theorem 3.1 in the case *D*_{0}⋐*M*(*j*), *j*≠*q*. Now, we assume that *D*_{0}⋐*M*(*q*). Fix *p*∈*D*_{0}, *I*_{0}, *J*_{0} strictly increasing with , and . Put

*Ψ*(

*x*,

*y*) and

*b*(

*x*,

*y*,

*k*) are as in Theorem 3.2 and , for any

*I*,

*J*strictly increasing with |

*I*| = |

*J*| =

*q*.

Lemma 3.5.

Assume that . Then, there exists *C*_{α,β}>0 independent of *k* and the point *p* such that

Proof.

In view of Theorems 3.2 and (3.7), we see that

*C*

_{α}>0 is a constant independent of

*k*and the point

*p*. From (3.12), (3.8), and (3.10), the lemma follows.

Now, we assume that . Take so that *χ* = 1 if |*x*|≤1, *χ* = 0 if |*x*|>2. Put

*𝜀*>0 is a small constant and is as in (3.11). We need

Lemma 3.6.

We have

*s*

_{1}of

*L*on an open set

*W*⋐

*M*, , we have

Proof.

It is known from [Citation18, Theorems 3.11 and 3.12] that

From (3.14), it is easy to see that

Moreover, we have by [Citation18, Theorem 3.11],

From (3.16), we have

From (3.16) and semiclassical Gårding inequalities (see [Citation18, Lemma 4.1]), we obtain

From (3.18), (3.15), the lemma follows.

Lemma 3.7.

With the notations and assumptions above, assume that for *k* large, , ∀*y*∈*D*_{1}, where *c*>0 is a constant independent of *k*. Then, there exists *C*_{α,β}>0 independent of *K* and the point *p* such that

Proof.

Let us choose

From (3.10), (3.11), and Theorem 3.2, we conclude that

*C*>0 is a constant independent of

*k*and the point

*p*. From (3.7), (3.10), and (3.21), we have

*C*

_{1}>0 is a constant independent of

*k*and the point

*p*. From Lemma 3.6 and noting that if , where

*c*>0 is a constant independent of

*k*, we conclude that

From this observation and (3.22), the lemma follows.

From Lemmas 3.4, 3.5, 3.7, Theorem 3.1 follows.

We can repeat the proof of Theorem 3.1 and conclude:

Theorem 3.8.

Let *s* and be local trivializing holomorphic sections of *L* on open sets *D*_{0}⋐*M*, *D*_{1}⋐*M*, respectively, , . Assume that *D*_{0}⋐*M*(*j*), *j*≠*q*. Then

Assume now that *D*_{0}⋐*M*(*q*) and has an *O*(*k*^{−N}) small spectral gap on *D*_{0}. Suppose that . Then,

## 4. Berezin–Toeplitz kernel estimates away the diagonal

In this section, we prove the off-diagonal decay of the kernel of the Berezin–Toeplitz quantization (cf. (1.10)), where *f*∈*𝒞*^{∞}(*M*) is as usual a bounded function and *N*>1. This yields one half of Theorem 1.1, i.e., (1.11).

We consider as before the localization of as follows. Let *s*, be local trivializing holomorphic sections of *L* on open sets *D*_{0}⋐*M*, *D*_{1}⋐*M*, respectively, , . Let and be orthonormal frames of on *D*_{0} and *D*_{1}, respectively. Then, {*e*^{J}; |*J*| = *q*, *J* strictly increasing}, {*w*^{J}; |*J*| = *q*, *J* strictly increasing} are orthonormal frames of on *D*_{0} and *D*_{1}, respectively. As in (3.1), we write

Let and be orthonormal bases of , where *d*_{k}∈*ℕ*∪{∞}. On *D*_{0}, *D*_{1}, we write

It is not diﬃcult to check that for every strictly increasing *I*, *J*, |*I*| = |*J*| = *q*, we have

Lemma 4.1.

Assume that *D*_{0}⋐*M*(*j*), *j*≠*q*. Then, for every *N*>1 and *m*∈*ℕ*, there exists *C*_{N,m}>0 independent of *k* such that for every *I*, *J* strictly increasing, |*I*| = |*J*| = *q*, we have

Proof.

Fix , *I*_{0}, *J*_{0} strictly increasing with , and . Take and so that

This is always possible, see [Citation18, Proposition 4.5]. From (4.3) and (4.4), we see that

In view of Theorems 3.2 and (3.10), we see that

*C*

_{α}>0,

*C*

_{β}>0 are constants independent of

*k*and the points

*x*

_{0}and

*y*

_{0}. From (4.5) and (4.6), the lemma follows.

Now, we assume that *D*_{0}⋐*M*(*q*). Fix and take , *τ* = 1 on *D*_{0}.

Lemma 4.2.

With the assumptions and notations above, for every *N*>1 and *m*∈*ℕ*, there exists *C*_{N,m}>0 independent of *k* such that

*I*,

*J*, |

*I*| = |

*J*| =

*q*.

Proof.

Fix , , *I*_{0}, *J*_{0} are strictly increasing and . Take so that

Assume that , where *a*_{I,J}(*x*,*y*,*k*) is as in (3.11). From (3.12) and (3.10), we have

*C*

_{α,β}>0,

*C*>0 are constants independent of

*k*and the points

*p*,

*y*

_{0}.

Now, we assume that . We define now *u*_{k} is as in (3.13) and *g*_{1} as in (3.19). Since and if , where *c*>0 is a constant independent of *k*, we conclude that

From (3.21) and (3.10), we have

*C*

_{1}>0, are constants independent of

*k*and the points

*p*,

*y*

_{0}. From (4.9) and (4.10), we obtain

*k*and the points

*p*,

*y*

_{0}.

From (4.8) and (4.11), the lemma follows.

Lemma 4.3.

With the assumptions and notations above, assume that . Then, for every *N*>1 and *m*∈*ℕ*, there exists *C*_{N,m}>0 independent of *k* such that

*I*,

*J*strictly increasing, |

*I*| = |

*J*| =

*q*.

Proof.

Fix , , *I*_{0}, *J*_{0} are strictly increasing and . Take so that

Assume that , where *a*_{I,J}(*x*,*y*,*k*) is as in (3.11). We can repeat the procedure in the proof of Lemma 4.2 and conclude that

*C*

_{α,β}>0 is a constant independent of

*k*and the points

*p*and

*y*

_{0}. Now, we assume that . Let

*g*

_{1}be as in (3.19), where

*u*

_{k}is as in (3.13). From Lemma 3.6 and (3.13), we have

From (4.14), it is straightforward to see that for every *N*∈*ℕ*, there exists *C*_{N}>0 independent of *k* and the points *p* and *y*_{0} such that

From (3.11), we can check that

*C*

_{α}>0 is a constant independent of

*k*and the point

*p*. F rom (4.16) and (3.4), it is not-diﬃcult to check that

*C*

_{0}>0 is a constant independent of

*k*and the point

*p*. Moreover, from Theorem 3.1, we see that

*C*

_{β}>0 is a constant independent of

*k*and the points

*p*,

*y*

_{0}. From (4.15), (4.17), and (4.18), we conclude that

*C*

_{α,β}>0 is a constant independent of

*k*and the points

*p*,

*y*

_{0}.

From (3.21) and (3.10), we have

*C*

_{2}>0, are constants independent of

*k*and the point

*p*. From (4.19) and (4.20), the lemma follows.

From Lemmas 4.1–4.3 we deduce:

Theorem 4.4.

Let *s*, be local trivializing holomorphic sections of *L* on *D*_{0}⋐*M* and *D*_{1}⋐*M*, respectively. Assume that *D*_{0}⋐*M*(*j*), *j*≠*q* or *D*_{0}⋐*M*(*q*) and . Then, for every *m*∈*ℕ*, *N*>1, there exists *C*_{N,m}>0 independent of *k* such that

Theorem 4.4 implies immediately one half of Theorem 1.1, more precisely (1.11).

We can repeat the proof of Theorem 4.4 and deduce:

Theorem 4.5.

Let *s*, be local trivializing holomorphic sections of *L* on *D*_{0}⋐*M* and *D*_{1}⋐*M*, respectively. Assume that *D*_{0}⋐*M*(*j*), *j*≠*q*. Then,

Assume that *D*_{0}⋐*M*(*q*) and has *O*(*k*^{−N}) small spectral gap on *D*_{0}. Suppose that . Then,

Let’s explain why in Theorem 4.5, we have “≡0 mod *O*(*k*^{−∞})”. Recall that Theorem 4.4 is based on Theorem 3.2 which says that if *D*_{0}⋐*M*(*q*), then, for every *N*>1, *m*∈*ℕ*, there exists *C*_{N,m}>0 independent of *k* such that

*D*

_{0}⋐

*M*(

*j*),

*j*≠

*q*,

*j*∈{0,1,2,…,

*n*}, then, for every

*N*>1,

*m*∈

*ℕ*, there exists independent of

*k*such that

The estimates ≲*k*^{3n−N+2m} in (4.21) and (4.22) imply that we have the estimate in Theorem 4.4. Now, we consider the Bergman kernel. As in Theorem 3.2, assume that *D*_{0}⋐*M*(*q*) and has *O*(*k*^{−N}) small spectral gap on *D*_{0}, then

Moreover, if *D*_{0}⋐*M*(*j*), *j*≠*q*, *j*∈{0,1,2,…,*n*}, then

*O*(

*k*

^{−∞})”.

## 5. Asymptotic expansion of Berezin–Toeplitz quantization

In this section, we will establish the full asymptotic expansion for the kernel of the Toeplitz kernel corresponding to lower energy forms. This leads to the proof of Theorem 1.1.

Let *s* be a local trivializing holomorphic section of *L* on an open set *D*⋐*M*, . Fix *N*>1. We assume that *D*⋐*M*(*q*). Put

*Ψ*(

*x*,

*y*) and

*b*(

*x*,

*y*,

*k*) are as in Theorem 3.2. Fix an open set

*D*

_{0}⋐

*D*and with

*τ*= 1 on

*D*

_{0}. Put

Let be an orthonormal frame of on *D*. Then,

*D*. As in (3.1), we write

It is easy to see that for every |*I*| = |*J*| = *q*, *I*, *J* are strictly increasing, we have

Take and be orthonormal frames for , where *d*_{k}∈*ℕ*∪{∞}. On *D*, we write

Lemma 5.1.

With the assumptions and notations above, for every *N*>1 and *m*∈*ℕ*, there exists *C*_{N,m}>0 independent of *k* such that

Proof.

Fix (*p*,*y*_{0})∈*D*×*D*, strictly increasing *I*_{0}, *J*_{0}, , and . Assume that , where *a*_{I,J}(*x*,*y*,*k*) is as in (3.11). In view of the proof of Lemma 3.5, we see that

*C*

_{α}>0 is a constant independent of

*k*and the point

*p*. It is not diﬃcult to see that for every |

*I*| = |

*J*| =

*q*,

*I*,

*J*are strictly increasing, we have

Take and so that

From (5.8) and (5.7), we get

From (5.6), (5.8), and (3.10), we have

*C*>0,

*C*

_{α,β}>0 are constants independent of

*k*and the points

*p*and

*y*

_{0}.

It is known by [Citation18, Theorem 3.11] that

From (5.11) and (5.6), we obtain

*k*and the points

*p*and

*y*

_{0}. From (5.12), (5.10), and (5.9), we deduce that

*k*and the points

*p*and

*y*

_{0}.

Now, we assume that . Take , where *u*_{k} is as in (3.13). From Theorem 3.2 and Lemma 3.6, we can check that for every *N*>0, there is *C*_{N}>0 independent of *k* and the point *p* such that

*C*

_{α}>0 is a constant independent of

*k*and the point

*p*. Take so that

From (5.7), (5.14), (5.15), (5.16), and (3.10), we have

*N*>0, where are independent of

*k*and the points

*p*and

*y*

_{0}. From (5.17) and (5.13), the lemma follows.

Put

We can repeat the proof of Lemma 5.1 and conclude:

Lemma 5.2.

With the assumptions and notations above, for every *N*>1 and *m*∈*ℕ*, there exists independent of *k* such that

Lemma 5.3.

We have

*D*×

*D*, where in , , for any

*x*∈

*D*

_{0}.

Proof.

From the stationary phase formula of Melin–Sjöstrand [Citation28], there is a complex phase function with *Ψ*_{1}(*x*,*x*) = 0, on *D*×*D*, for some *c*>0, such that for every bounded function *f*∈*𝒞*^{∞}(*M*), we have

*D*×

*D*, where

*j*∈

*ℕ*. Moreover, for all

*x*∈

*D*

_{0}we have . Basically, here we used the fact that composition of complex Fourier integral operators is still a complex Fourier integral operator. Take

*f*= 1. Fix

*D*

^{′}⋐{

*τ*= 1}. We claim that

*j*∈

*ℕ*. The

*S*

_{k}constructed in [Citation18] is called approximated Szegö kernel.

*S*

_{k}satisfies

(see [Citation18, Theorems 3.11 and 3.12]). Relation (5.21) says that

*D*×

*D*. Since

*S*

_{k}is properly supported (see the discussion after (2.15) for the meaning of properly supported), the integral (5.22) is well-defined. Now,

Note that if |*x*−*y*|≥*c*, for some *c*>0. From this observation, we conclude that for (recall that *D*^{′}⋐{*τ* = 1})

We claim that

Note that *Ψ*(*x*,*x*) = *Ψ*_{1}(*x*,*x*) = 0. We assume that there are , and , such that

From (5.19) and (5.20), we have

It is obviously that

From (5.27) to (5.29), we get a contradiction. The claim follows. Since *τ* is arbitrary, *Ψ* and *Ψ*_{1} are independent of *τ*, we conclude that *Ψ*_{1}(*x*,*y*)−*Ψ*(*x*,*y*) vanishes to infinite order on *D*×*D*. Thus, we can replace *Ψ*_{1} by *Ψ* in (5.19). The lemma follows.

From Lemmas 5.1–5.3 and Lemma 4.2, we obtain the following.

Theorem 5.4.

With the notations above, let *s* be local trivializing holomorphic section of *L* on *D*_{0}⋐*M*. Assume that *D*_{0}⋐*M*(*q*). Then, for every *N*>1, *m*∈*ℕ*, there exists independent of *k* such that

*Ψ*is as in Theorem 3.2.

Proof (Proof of Theorem 1.1).

From Theorems 4.4 and 5.4, Theorem 1.1 follows.

## 6. Asymptotics of the composition of Toeplitz operators

In this section, we establish the expansion of the composition of two Toeplitz operators and prove Theorems 1.3, 1.4, 1.9, 1.10, and 1.11.

Let *f*,*g*∈*𝒞*^{∞}(*M*) be bounded. For λ≥0, put

Theorem 6.1.

Let *s*, be local trivializing holomorphic sections of *L* on *D*_{0}⋐*M* and *D*_{1}⋐*M*, respectively, , , where *D*_{0} and *D*_{1} are open sets. Assume that *D*_{0}⋐*M*(*j*), *j*≠*q* or *D*_{0}⋐*M*(*q*) and . Then, for every *m*∈*ℕ*, *N*>1, there exists *C*_{N,m}>0 independent of *k* such that

Proof.

The proof is similar to the proof of Theorem 4.4, so we will insist here on the appearance of the exponent 3*n* in the power of *k* in (6.1), compared to 2*n* in the previous estimates. The argument holds for any complex manifold, *not necessarily compact*. For simplicity, we only consider *q* = 0. Let *s*, be local trivializing holomorphic sections of *L* on and , respectively, , , where and are open sets. Fix , , *D*_{0} and *D*_{1} are open sets. Let and *τ* = 1 on *D*_{0}. We will first show that how to estimate the kernel of on . Take

*d*

_{k}∈

*ℕ*∪{∞}. On , we write ,

*j*= 1,…,

*d*

_{k}. On , we write ,

*j*= 1,…,

*d*

_{k}. For every , put

Since and converge locally uniformly in *C*^{∞} topology, for fixed *y*, is a smooth section of *L*^{k}. It is easy to see that for every ,

When *M* is compact, *d*_{k} is finite and and it is easy to estimate (6.3). When *M* is *noncompact*, *d*_{k} could be infinite, so to estimate (6.3) we need a more detailed analysis. Now, we fix *p*∈*D*_{0}. From Theorem 3.2 and Lemma 3.6, we can find with ,

We take and obtain from (6.5) that

Now,

We claim that

Fix . We take so that , , *j* = 2,3,…,*d*_{k}. Then,

From (6.9), we can check that

Since *g* is a bounded function, , for some constant *C*>0 independent of *k*. Moreover, locally uniformly on *D*_{1}. From this observation and (6.10), the estimate (6.8) follows. Relation (6.4) yields

From (6.7), (6.8) , (6.11) and since , we conclude that

From (6.6) and (6.8), we have

Note that here we still get the exponent 2*n*−*N*. From (6.3), (6.12), and (6.13), we get

Thus, to estimate , we only need to estimate . Let and on *D*_{1}. We can repeat the procedure above and conclude that

Thus, to estimate , we only need to estimate . We now explain how to estimate . Take , *τ*_{1} = 1 on *Supp τ*. We have

The estimate of is as compact case since

*M*. We only need to show how to estimate . Note that

From (6.17), it is easy to see that

Now, fix and . We take and so that , , *j* = 2,3,…,*d*_{k}, , , *j* = 2,3,…,*d*_{k}. Thus,

From Lemma 4.2, we see that locally uniformly on . From this observation and since , we deduce that

Here we get the power 3*n*.

We have moreover:

Theorem 6.2.

With the notations above, let *s* be local trivializing holomorphic section of *L* on *D*_{0}⋐*M*. Assume that *D*_{0}⋐*M*(*q*). Then, for every *N*>1, *m*∈*ℕ*, there exists independent of *k* such that

*Ψ*is as in Theorem 3.2.

Proof.

The proof of this theorem is similar to the proof of Theorem 5.4. We only give the outline of the proof and for simplicity we consider only *q* = 0. Fix and take , *τ* = 1 on *D*_{0}. We may assume that the section *s* defined on . We can repeat the proof of Lemma 4.2 with minor changes and conclude that for every *N*>1 and *m*∈*ℕ*, there is *C*_{N,m}>0 independent of *k* such that

From (6.21), we only need to consider . Take , *τ*_{1} = 1 on *Supp τ*. We have

Let be the distribution kernels of and , respectively. We have

We first consider . Take

*d*

_{k}∈

*ℕ*∪{∞}. On , we write

It is straightforward to check that

From Lemmas 4.2, (3.10), and (6.24), it is not diﬃcult to see that for every *N*>1 and *m*∈*ℕ*, there exists *C*_{N,m}>0 independent of *k* such that

We now consider . We have

Put

*Ψ*(

*x*,

*y*) is as in Theorem 3.2 and are as in Theorem 5.4. Put

From Theorem 5.4, it is straightforward to see that for every *N*>1 and *m*∈*ℕ*, there exists *C*_{N,m}>0 independent of *k* such that

We claim that

We use now the theory of complex Fourier integral operator, in particular the fact that composition of complex Fourier integral operators is still a complex Fourier integral operator. Indeed, the complex stationary phase formula of Melin–Sjöstrand [Citation28] tells us that there is a complex phase with , such that for any *A*(*x*,*y*) = *e*^{ikΨ(x,y)}*a*(*x*,*y*,*k*), *C*(*x*,*y*) = *e*^{ikΨ(x,y)}*c*(*x*,*y*,*k*), where , and every , we have

*a*

_{0}and

*c*

_{0}denote the leading terms of

*a*(

*x*,

*y*,

*k*) and

*c*(

*x*,

*y*,

*k*), respectively. In the proof of Lemma 5.3, we proved that

*Ψ*(

*x*,

*y*)−

*Ψ*

_{1}(

*x*,

*y*) vanishes to infinite order on

*x*=

*y*(see (5.25)). Thus, we can replace

*Ψ*

_{1}in (6.29) by

*Ψ*and we get (6.28).

From (6.28), (6.27), (6.25), (6.23), and (6.21), the theorem follows.

Proof (Proof of Theorem 1.3).

Theorems 6.1 and 6.2 yield immediately Theorem 1.3.

Proof (Proof of Theorem 1.9).

This follows by using the asymptotics of the Bergman kernel proved in [Citation18, Theorem 1.6] in the case of an *O*(*k*^{−N}) small spectral gap and adapting the proofs of Theorems 1.1 and 1.3 to the current situation.

Proof (Proof of Theorem 1.10).

By [Citation18, Theorem 8.2], we know that has an *O*(*k*^{−N}) small spectral gap on every *D*⋐*M*^{′}∩*M*(0). This observation and Theorem 1.9 yield Theorem 1.10.

Proof (Proof of Theorem 1.11).

*M*∖*Σ* is a noncompact complex manifold. Let be the Gaffney extension of Kodaira Laplacian on *M*∖*Σ* and let be the associated Bergman projection. By a result of Skoda (see [Citation18, Lemma 7.2]), we know that

Moreover, we know that has *O*(*k*^{−N}) small spectral gap on every *D*⋐*M*∖*Σ* (see [Citation18, Theorem 9.1]). This observation, (6.30) and Theorem 1.9 imply Theorem 1.11.

In the following, we will prove Theorem 1.4. Fix *N*>1. Let , *D*⋐*M*(0). For simplicity, we may assume that *L*|_{D} is trivial and let *s* be a local trivializing holomorphic section of *L* on *D*, . Take with *τ* = 1 on *Supp f*∪*Supp g*. Put

We can repeat the proof of Lemma 4.2 with minor changes and obtain:

Lemma 6.3.

Let *s*_{1}, *s*_{2} be local trivializing holomorphic sections of *L* on *D*_{1}⋐*M* and *D*_{2}⋐*M*, respectively, where *D*_{1} and *D*_{2} are open sets. Then, for every *m*∈*ℕ*, there exists *C*_{m}>0 independent of *k* such that

In particular, locally in the *L*^{2} operator norm.

Let be as in Theorem 1.3. Then

Since , we can take , *j*∈*ℕ*. Note that *b*_{f,g}(*x*,*y*,*k*) and *b*_{f,g,j}(*x*,*y*) have uniquely determined Taylor expansion at *x* = *y*. Consider

In view of Theorem 1.3 and Lemma 6.3, we see that

Lemma 6.4.

For any *p*∈*ℕ* there exist such that

*p*∈

*ℕ*.

Proof.

Set

From (1.20) and (1.13), we see that

Note that *b*_{f,g,0}(*x*,*y*) and are holomorphic with respect to *x* and

From this observation and (6.34), it is easy to see that vanishes to infinite order on *x* = *y*. Thus, we can take so that and hence . Consider the expansion

*j*∈

*ℕ*. Set

From (1.13), we have and as in the discussion above, we can take so that and hence

Continuing inductively, the lemma follows.

Proof (Proof of Theorem 1.4).

Let , *j*_{0}∈*ℕ*. Consider the operator

By [Citation18, Theorem 3.11], we have

For every *p*∈*ℕ* put

As in (6.32), we can check that for *p* = 0,1,2,… ,

Moreover, from (6.37) and Lemma 6.4, we have

From (6.32), (6.38), and (6.39), we conclude that

*L*

^{2}operator norm. Moreover, we have by (6.33). We also have by (7.36), so as in [Citation27, (0.23)] we obtain

*f*,

*g*} is the Poisson bracket of the functions

*f*,

*g*with respect to the symplectic form 2

*πω*on

*M*(0) (see also [Citation25, (4.89)], [Citation24, (7.4.3)]). Therefore (1.23) follows.

Recall that the Poisson bracket { ⋅ , ⋅ } on (*M*,2*πω*) is defined as follows. For *f*,*g*∈*𝒞*^{∞}(*M*), let *ξ*_{f} be the Hamiltonian vector field generated by *f*, which is defined by 2*πω*(*ξ*_{f},⋅) = *df*. Then

Remark 6.5.

Berezin introduced in his ground-breaking work [Citation3] a star-product by using Toeplitz operators. Formal star-products are known to exist on symplectic manifolds by De Wilde and Lecomte [Citation12] and Fedosov [Citation14]. The Berezin–Toeplitz star-product gives a concrete geometric realization of such product. For compact Kähler manifold the Berezin–Toeplitz star product was introduced in Karabegov and Schlichenmaier [Citation19] and Schlichenmaier [Citation29]. For general compact symplectic manifolds this was realized in Ma and Marinescu [Citation24, Citation25] by using Toeplitz operators obtained by projecting on the kernel of the Dirac operator. Due to Theorem 1.4, we can also define an associative star-product on the set *M*(0) where a holomorphic line bundle *L*→*M* is positive, namely by setting for any ,

## 7. Calculation of the leading coeﬃcients

In this section, we will give formulas for the top coeﬃcients of the expansion (1.13) in the case *q* = 0, cf. Theorem 7.1. We introduce the geometric objects used in Theorem 7.1 below. Consider the (1,1)-form on *M*,

On *M*(0) the (1,1)-form *ω* is positive and induces a Riemannian metric . In local holomorphic coordinates , put

We notice that , , *j*,*k* = 1,…,*n*. Put

*h*

^{−1}is the inverse matrix of

*h*. The complex Laplacian with respect to

*ω*is given by

We notice that , *j*,*k* = 1,…,*n*. Put

*r* is called the scalar curvature with respect to *ω*. Let be the curvature of the canonical line bundle with respect to the real two form *Θ*. We recall that

Let be the Levi–Civita connection on , its curvature. Let *h* be as in (7.3). Put , , *j*,*k* = 1,…,*n*. *𝜃* is the Chern connection matrix with respect to *ω*. Then,

We denote by ⟨ ⋅ ,⋅ ⟩_{ω} the pointwise Hermitian metrics induced by on , *p*,*q*,*r*,*s*∈{0,1,…,*n*}, and by _{| ⋅ |ω} the corresponding norms.

Set

*T*

^{1,0}

*M*with respect to ⟨ ⋅ ,⋅ ⟩

_{ω}. It is straightforward to see that the definition of is independent of the choices of orthonormal frames. Thus, is globally defined. The Ricci curvature with respect to

*ω*is given by

*T*

^{1,0}

*M*with respect to ⟨ ⋅ ,⋅ ⟩

_{ω}. That is,

*Ric* _{ω} is a global (1,1) form.

Let

_{ω}. That is, in local coordinates , put

Then, for , we have

Theorem 7.1.

With the assumptions and notations used in Theorem 1.1, the coeﬃcients *b*_{f,1}(*x*,*x*) and *b*_{f,2}(*x*,*x*) in the expansion (1.12) for *q* = 0 have the following form: for every *x*∈*D*_{0},

The formulas given in Theorem 7.1 simplify if we assume that *ω* = *Θ*. In this case, we have and . See also [Citation26, Section 2.7], [Citation27, Remark 0.5], concerning the calculation of the coeﬃcients for an arbitrary underlying Hermitian metric *Θ*.

Let *q* = 0 and let

*j*∈

*ℕ*, and . We have

*j*∈

*ℕ*, and . In this section, we will calculate

*b*

_{1,f}(

*x*,

*x*) and

*b*

_{2,f}(

*x*,

*x*),

*x*∈

*D*

_{0}. Fix

*p*∈

*D*

_{0}. In a small neighborhood of the point

*p*there exist local coordinates , ,

*j*= 1,…,

*n*, and a local frame

*s*of

*L*, so that

Until further notice, we work with this local coordinates *x* and we identify *p* with the point *x* = *z* = 0. It is well-known (see [Citation18, Section 4.5]) that for every *N*∈*ℕ*, we have

We have

We notice that since *b*(*z*,*w*,*k*) is properly supported, we have

We recall the stationary phase formula of Hörmander (see [Citation16, Theorem 7.7.5]).

Theorem 7.2.

Let *K*⊂*D* be a compact set and *N* a positive integer. If , *F*∈*𝒞*^{∞}(*D*) and *Im F*≥0 in *D*, *Im F*(0) = 0, *F*^{′}(0) = 0, *det F*^{′′}(0)≠0, *F*^{′}≠0 in *K*∖{0} then

*C*is uniform when

*F*runs in a relatively compact set of

*𝒞*

^{∞}(

*D*), has a uniform bound and

Here

We now apply (7.21) to the integral in (7.19). Put

From (7.17) and (7.18), we see that

*h*is given by (7.23). Moreover, we can check that

From (7.24), (7.26) and using that *h* = *O*(^{|z|4}), it is not diﬃcult to see that

*L*

_{j}is given by (7.22). We notice that

From this observation, (7.27) becomes:

*N*≥0. From (7.28), (7.25), (7.21), (7.19) and (7.15), we get

Combining (7.29) with (7.16), we obtain

Theorem 7.3.

The coeﬃcients *b*_{f,j} of the expansion (7.16) of *b*_{f}(*x*,*y*,*k*), are given by

*j*= 0,1,… . In particular,

In [Citation18, Section 4.5], we determined all the derivatives of *b*_{0}(*x*,*y*), *b*_{1}(*x*,*y*), *b*_{2}(*x*,*y*) at (0,0). From this observation and Theorem 7.3, we can repeat the procedure in [Citation17, Section 4] and obtain Theorem 7.1. Since the calculation is the same, we omit the details.

Let be the (1,0) component of the Chern connection on induced by ⟨ ⋅ ,⋅ ⟩_{ω} (see the discussion after (7.11)). From Theorem 7.1 and the proof of Theorem 1.3, we can repeat the proof of [Citation17, Theorem 1.5] and get the following (see also Ma–Marinescu [Citation24] for another method).

Theorem 7.4.

With the notations as in Theorem 1.3, let *q* = 0. Then, for *b*_{f,g,1}, *b*_{f,g,2} in (1.19), we have

Corollary 7.5.

The coeﬃcients *C*_{1}(*f*,*g*) and *C*_{2}(*f*,*g*) of the expansion (1.21) of the composition of two Toeplitz operators are given by

Proof.

Formula (7.36) follows from (7.34) and

## 8. Behavior on the degenerate set and the Weyl law

In this section, we will prove Theorems 1.2 and 1.7. We recall first the following.

Theorem 8.1 ([Citation18, Theorem 1.3]).

Set

Then for every , *𝜀*>0, *N*>1 and every *m*∈{0,1,…,*n*}, there exist a neighborhood *U* of *x*_{0} and *k*_{0}>0, such that for all *k*≥*k*_{0} we have

Proof (Proof of Theorem 1.2).

Fix , *𝜀*>0 and *m*∈{0,1,…,*n*}. Let *U* be a small neighborhood of *x*_{0} as in Theorem 8.1. Let *p* be any point of *U* and let *s* be a local section of *L* defined in a small open set *D*⋐*U* of *p*, . Fix , *I*_{0}, *J*_{0} are strictly increasing. Take and be orthonormal frames for so that

*d*

_{k}∈

*ℕ*∪{∞} and on

*D*, we write

We have

From (8.2), it is not diﬃcult to see that

From (8.3) and (8.1), the theorem follows.

We now prove Theorem 1.7. We introduce some notations. For λ≥0, put

Recall that *E* denotes the spectral measure of . Let be the Schwartz kernel of . The trace of is given by

*M*is compact. We need the following.

Lemma 8.2.

There exists *C*>0 independent of *k* such that

Proof.

Let *p* be any point of *M* and let *s* be a local trivializing holomorphic section of *L* defined in a small open set *D*⋐*U* of *p*, . Fix , *I*_{0}, *J*_{0} are strictly increasing. Take , to be orthonormal frames for and

*d*

_{k}∈

*ℕ*∪{∞} and on

*D*, we write

We have

From (8.5), it is easy to see that

*C*

_{1}>0 is a constant independent of

*k*and the point

*p*. The lemma follows.

Proof (Proof of Theorem 1.7).

Since *M*(*q*−1) = *∅* and *M*(*q*+1) = *∅*, it is known [Citation18, Corollary 1.4], that for every *N*>1,

Moreover, it is easy to see that

From (8.6) and (8.7), we have

*C*

_{0}>0 is a constant independent of

*k*. From (8.8) and (8.4), we conclude that

In view of Theorem 1.5, we see that

### Acknowledgment

We are grateful to the referee for several suggestions which led to the improvement of the presentation.

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