Berezin–Toeplitz quantization for lower energy forms

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_ℂM = 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.,

(1.1)

Let g^TM: = ω(⋅,J⋅) be the J-Riemannian metric on TM. The Riemannian volume form of g^TM is denoted by dv_M. On the space of smooth sections with compact support we introduce the L²-scalar product associated to the metrics h and the Riemannian volume form dv_M by

(1.2)

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

(1.3)

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

(1.4)

(1.5)

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 L² 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:

1. (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}.

2. 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⊗_ℝℂ induced by . The Riemannian volume form dv_M of (M,Θ) satisfies . For every q = 0,1,…,n, the Hermitian metric ⟨ ⋅ ,⋅ ⟩ on TM⊗_ℝℂ 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,

(1.6)

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

(1.7)

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 kth tensor power of the line bundle (L,h). Let ( ⋅ ,⋅ )_k and ( ⋅ ,⋅ ) denote the global L² 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

(1.8)

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

(1.9)

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

(1.10)

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:

1. D₀⋐M(j) and j≠q,

2. D₀⋐M(q) and .

Then, for every N>1, m∈ℕ, there exists C_N,m>0 independent of k such that

(1.11)

If D₀⋐M(q) there exists a symbol

(1.12)

(1.13)

(1.14)

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+2m+1). Then we deduce from (1.12) that

(1.15)

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,

(1.16)

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 x₀ and k₀>0, such that for all k≥k₀ we have

(1.17)

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:

1. D₀⋐M(j) and j≠q,

2. D₀⋐M(q) and .

Then, for every N>1, m∈ℕ, there exists C_N,m>0 independent of k such that

(1.18)

If D₀⋐M(q) there exists a symbol

(1.19)

(1.20)

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 3n and 2n 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 L² operator norm if for any , there exists C>0 independent of k such that , for k large, where _∥⋅∥op denotes the L² 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

(1.21)

(1.22)

(1.23)

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>2n. Then

(1.24)

(1.25)

(1.26)

Since , we can identify to an element of . Then

(1.27)

(1.28)

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>2n. If M is compact, then

(1.29)

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

(1.30)

In particular,

(1.31)

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, k₀∈ℕ, such that for all k≥k₀ 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

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:

1. D₀⋐M(j) and j≠q,

2. D₀⋐M(q), has an O(k^−N) small spectral gap on D₀ and .

Then

Assume that D₀⋐M(q) and has an O(k^−N) small spectral gap on D₀. 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

(1.32)

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 dv_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

(1.33)

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

(1.34)

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 D₀⊂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⊗_ℝℂ 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 π₁ and π₂ 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 L² inner product on . By using this L² 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^1,0M and T^0,1M 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^1,0M. In local holomorphic coordinates , if , then . We extend the Hermitian metric ⟨ ⋅ ,⋅ ⟩ to TM⊗_ℝℂ in a natural way. The Hermitian metric ⟨ ⋅ ,⋅ ⟩ on TM⊗_ℝℂ 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

(2.1)

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

Let L^k, k>0, be the kth 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 L²–Hermitian inner products on the spaces and are given by

(2.2)

Let be a k-dependent continuous operator with smooth kernel A_k(x,y). Let s, be local trivializing holomorphic sections of L on D₀⋐M, D₁⋐M, respectively, , , where D₀, D₁ are open sets. The localized operator of A_k on is given by

(2.3)

(2.4)

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

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

Consider a singular Hermitian metric h on a holomorphic line bundle L over M. If h₀ 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 h₀. 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

(2.10)

(2.11)

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

(2.12)

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”)

(2.13)

(2.14)

(2.15)

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 π₁ and π₂ 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

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 D₀⋐M, D₁⋐M, respectively, , , where D₀, D₁ are open sets. We denote by the localization given by (2.3).

Let and be orthonormal frames of on D₀ and D₁, respectively. Then,

(3.1)

(3.2)

The goal of this section is to prove the following.

Theorem 3.1.

With the notations used above, we assume that D₀⋐M(j), j≠q, j∈{0,1,…,n} or D₀⋐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 D₀⋐M(q). Then, for every N>1, m∈ℕ, there exists C_N,m>0 independent of k such that

Assume that D₀⋐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∈D₀, let be local holomorphic coordinates centered at p satisfying

(3.3)

(3.4)

Moreover, when q = 0, we have

(3.5)

Fix N>1. Let be an orthonormal frame for , where d_k∈ℕ∪{∞}. On D₀, D₁, we write

(3.6)

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

(3.7)

Lemma 3.4.

Assume that D₀⋐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

(3.8)

In view of Theorem 3.2, we see that

(3.9)

(3.10)

Lemma 3.4 provides the proof of Theorem 3.1 in the case D₀⋐M(j), j≠q. Now, we assume that D₀⋐M(q). Fix p∈D₀, I₀, J₀ strictly increasing with , and . Put

(3.11)

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

(3.12)

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

(3.13)

Lemma 3.6.

We have

Proof.

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

(3.14)

From (3.14), it is easy to see that

(3.15)

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

(3.16)

From (3.16), we have

(3.17)

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

(3.18)

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

Lemma 3.7.

With the notations and assumptions above, assume that for k large, , ∀y∈D₁, 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

(3.19)

(3.20)

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

(3.21)

(3.22)

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 D₀⋐M, D₁⋐M, respectively, , . Assume that D₀⋐M(j), j≠q. Then

Assume now that D₀⋐M(q) and has an O(k^−N) small spectral gap on D₀. 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 D₀⋐M, D₁⋐M, respectively, , . Let and be orthonormal frames of on D₀ and D₁, respectively. Then, {e^J; |J| = q, J strictly increasing}, {w^J; |J| = q, J strictly increasing} are orthonormal frames of on D₀ and D₁, respectively. As in (3.1), we write

(4.1)

Let and be orthonormal bases of , where d_k∈ℕ∪{∞}. On D₀, D₁, we write

(4.2)

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

(4.3)

Lemma 4.1.

Assume that D₀⋐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 , I₀, J₀ strictly increasing with , and . Take and so that

(4.4)

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

(4.5)

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

(4.6)

Now, we assume that D₀⋐M(q). Fix and take , τ = 1 on D₀.

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

Proof.

Fix , , I₀, J₀ are strictly increasing and . Take so that

(4.7)

Assume that , where a_I,J(x,y,k) is as in (3.11). From (3.12) and (3.10), we have

(4.8)

Now, we assume that . We define now u_k is as in (3.13) and g₁ as in (3.19). Since and if , where c>0 is a constant independent of k, we conclude that

(4.9)

From (3.21) and (3.10), we have

(4.10)

(4.11)

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

Proof.

Fix , , I₀, J₀ are strictly increasing and . Take so that

(4.12)

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

(4.13)

(4.14)

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 y₀ such that

(4.15)

From (3.11), we can check that

(4.16)

(4.17)

(4.18)

(4.19)

From (3.21) and (3.10), we have

(4.20)

From Lemmas 4.1–4.3 we deduce:

Theorem 4.4.

Let s, be local trivializing holomorphic sections of L on D₀⋐M and D₁⋐M, respectively. Assume that D₀⋐M(j), j≠q or D₀⋐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 D₀⋐M and D₁⋐M, respectively. Assume that D₀⋐M(j), j≠q. Then,

Assume that D₀⋐M(q) and has O(k^−N) small spectral gap on D₀. 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 D₀⋐M(q), then, for every N>1, m∈ℕ, there exists C_N,m>0 independent of k such that

(4.21)

(4.22)

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 D₀⋐M(q) and has O(k^−N) small spectral gap on D₀, then

(4.23)

Moreover, if D₀⋐M(j), j≠q, j∈{0,1,2,…,n}, then

(4.24)

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

(5.1)

(5.2)

Let be an orthonormal frame of on D. Then,

(5.3)

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

(5.4)

Take and be orthonormal frames for , where d_k∈ℕ∪{∞}. On D, we write

(5.5)

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,y₀)∈D×D, strictly increasing I₀, J₀, , 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

(5.6)

(5.7)

Take and so that

(5.8)

From (5.8) and (5.7), we get

(5.9)

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

(5.10)

It is known by [Citation18, Theorem 3.11] that

(5.11)

From (5.11) and (5.6), we obtain

(5.12)

(5.13)

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

(5.14)

(5.15)

(5.16)

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

(5.17)

Put

(5.18)

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

Proof.

From the stationary phase formula of Melin–Sjöstrand [Citation28], there is a complex phase function with Ψ₁(x,x) = 0, on D×D, for some c>0, such that for every bounded function f∈𝒞^∞(M), we have

(5.19)

(5.20)

(5.21)

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

(5.22)

(5.23)

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

(5.24)

We claim that

(5.25)

Note that Ψ(x,x) = Ψ₁(x,x) = 0. We assume that there are , and , such that

(5.26)

From (5.19) and (5.20), we have

(5.27)

(5.28)

It is obviously that

(5.29)

From (5.27) to (5.29), we get a contradiction. The claim follows. Since τ is arbitrary, Ψ and Ψ₁ are independent of τ, we conclude that Ψ₁(x,y)−Ψ(x,y) vanishes to infinite order on D×D. Thus, we can replace Ψ₁ 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 D₀⋐M. Assume that D₀⋐M(q). Then, for every N>1, m∈ℕ, there exists independent of k such that

(5.30)

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 D₀⋐M and D₁⋐M, respectively, , , where D₀ and D₁ are open sets. Assume that D₀⋐M(j), j≠q or D₀⋐M(q) and . Then, for every m∈ℕ, N>1, there exists C_N,m>0 independent of k such that

(6.1)

Proof.

The proof is similar to the proof of Theorem 4.4, so we will insist here on the appearance of the exponent 3n in the power of k in (6.1), compared to 2n 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 , , D₀ and D₁ are open sets. Let and τ = 1 on D₀. We will first show that how to estimate the kernel of on . Take

(6.2)

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 ,

(6.3)

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∈D₀. From Theorem 3.2 and Lemma 3.6, we can find with ,

(6.4)

(6.5)

We take and obtain from (6.5) that

(6.6)

Now,

(6.7)

We claim that

(6.8)

Fix . We take so that , , j = 2,3,…,d_k. Then,

(6.9)

From (6.9), we can check that

(6.10)

Since g is a bounded function, , for some constant C>0 independent of k. Moreover, locally uniformly on D₁. From this observation and (6.10), the estimate (6.8) follows. Relation (6.4) yields

(6.11)

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

(6.12)

From (6.6) and (6.8), we have

(6.13)

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

(6.14)

Thus, to estimate , we only need to estimate . Let and on D₁. We can repeat the procedure above and conclude that

(6.15)

Thus, to estimate , we only need to estimate . We now explain how to estimate . Take , τ₁ = 1 on Supp τ. We have

(6.16)

The estimate of is as compact case since

(6.17)

From (6.17), it is easy to see that

(6.18)

Now, fix and . We take and so that , , j = 2,3,…,d_k, , , j = 2,3,…,d_k. Thus,

(6.19)

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

Here we get the power 3n.

We have moreover:

Theorem 6.2.

With the notations above, let s be local trivializing holomorphic section of L on D₀⋐M. Assume that D₀⋐M(q). Then, for every N>1, m∈ℕ, there exists independent of k such that

(6.20)

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 D₀. 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

(6.21)

From (6.21), we only need to consider . Take , τ₁ = 1 on Supp τ. We have

(6.22)

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

(6.23)

We first consider . Take

It is straightforward to check that

(6.24)

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

(6.25)

We now consider . We have

(6.26)

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

(6.27)

We claim that

(6.28)

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

(6.29)

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

(6.30)

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

(6.31)

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

Lemma 6.3.

Let s₁, s₂ be local trivializing holomorphic sections of L on D₁⋐M and D₂⋐M, respectively, where D₁ and D₂ are open sets. Then, for every m∈ℕ, there exists C_m>0 independent of k such that

In particular, locally in the L² 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

(6.32)

Lemma 6.4.

For any p∈ℕ there exist such that

Proof.

Set

(6.33)

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

(6.34)

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

(6.35)

(6.36)

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 , j₀∈ℕ. Consider the operator

By [Citation18, Theorem 3.11], we have

(6.37)

For every p∈ℕ put

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

(6.38)

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

(6.39)

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

(6.40)

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

(6.41)

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 ,

(6.42)

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,

(7.1)

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

(7.2)

We notice that , , j,k = 1,…,n. Put

(7.3)

(7.4)

We notice that , j,k = 1,…,n. Put

(7.5)

(7.6)

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

(7.7)

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,

(7.8)

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

(7.9)

(7.10)

Ric _ω is a global (1,1) form.

Let

(7.11)

(7.12)

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∈D₀,

(7.13)

(7.14)

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

(7.15)

(7.16)

(7.17)

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

(7.18)

We have

(7.19)

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

(7.20)

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

(7.21)

(7.22)

Here

(7.23)

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

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

(7.24)

(7.25)

(7.26)

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

(7.27)

From this observation, (7.27) becomes:

(7.28)

(7.29)

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

(7.30)

(7.31)

(7.32)

(7.33)

In [Citation18, Section 4.5], we determined all the derivatives of b₀(x,y), b₁(x,y), b₂(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

(7.34)

(7.35)

Corollary 7.5.

The coefficients C₁(f,g) and C₂(f,g) of the expansion (1.21) of the composition of two Toeplitz operators are given by

(7.36)

(7.37)

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 x₀ and k₀>0, such that for all k≥k₀ we have

(8.1)

Proof (Proof of Theorem 1.2).

Fix , 𝜀>0 and m∈{0,1,…,n}. Let U be a small neighborhood of x₀ 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 , I₀, J₀ are strictly increasing. Take and be orthonormal frames for so that

We have

(8.2)

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

(8.3)

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

Lemma 8.2.

There exists C>0 independent of k such that

(8.4)

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 , I₀, J₀ are strictly increasing. Take , to be orthonormal frames for and

We have

(8.5)

From (8.5), it is easy to see that

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,

(8.6)

Moreover, it is easy to see that

(8.7)

From (8.6) and (8.7), we have

(8.8)

(8.9)

In view of Theorem 1.5, we see that

(8.10)