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Berezin–Toeplitz quantization for lower energy forms
Chin-Yu Hsiao & George Marinescu
To cite this article: Chin-Yu Hsiao & George Marinescu (2017) Berezin–Toeplitz quantization for lower energy forms, Communications in Partial Differential Equations, 42:6, 895-942, DOI:
10.1080/03605302.2017.1330340
To link to this article: https://doi.org/10.1080/03605302.2017.1330340
Accepted author version posted online: 05 Jun 2017.
Published online: 21 Jul 2017.
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2017, VOL. 42, NO. 6, 895–942
https://doi.org/10.1080/03605302.2017.1330340
Berezin–Toeplitz quantization for lower energy forms
Chin-Yu Hsiaoaand George Marinescub,c
aInstitute of Mathematics, Academia Sinica and National Center for Theoretical Sciences, Taipei, Taiwan;
bMathematisches Institut, Universität zu Köln, Germany;cInstitute of Mathematics ‘Simion Stoilow’, Romanian Academy, Bucharest, Romania
ABSTRACT
LetMbe an arbitrary complex manifold and letLbe a Hermitian holo- morphic line bundle overM. We introduce the Berezin–Toeplitz quanti- zation of the open set ofMwhere the curvature onLis nondegenerate.
In particular, we quantize any manifold admitting a positive line bundle.
The quantum spaces are the spectral spaces corresponding toh 0,k−Ni
, whereN > 1 is fixed, of the Kodaira Laplace operator acting on forms with values in tensor powersLk. We establish the asymptotic expansion of associated Toeplitz operators and their composition in the semiclas- sical limitk→ ∞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.
ARTICLE HISTORY Received 10 March 2015 Accepted 28 February 2017
KEYWORDS Berezin–Toeplitz quantization; Kodaira Laplace operator; local holomorphic Morse inequalities; Melin–Sjöstrand stationary phase method;
positive line bundle; spectral asymptotics
MATHEMATICS SUBJECT CLASSIFICATION 53D50; 32A25; 47B35; 81S10
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 [2,3,14,21,22,31]. Toeplitz operators and more generally Toeplitz structures were introduced in geometric quantization by Berezin [3] and Boutet de Monvel–Guillemin [6].
We refer to [22,26,30] 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 dimCM = nwith Kähler formωand complex structure J. Let(L,h)be a holomorphic Hermitian line bundle onX, and let∇Lbe the holomorphic Hermitian connection on(L,h)with curvatureRL = (∇L)2. We assume that(L,h,∇L)is a prequantum line bundle, i.e.,
ω=
√−1
2π RL. (1.1)
LetgTM := ω(·,J·)be theJ-Riemannian metric onTM. The Riemannian volume form of gTMis denoted bydvM. On the space of smooth sections with compact supportC∞
0 (M,Lk) we introduce theL2-scalar product associated to the metricshand the Riemannian volume
CONTACTGeorge Marinescu [email protected] Mathematisches Institut, Universität zu Köln, Weyertal 86-90, Köln 50931, Germany; Institute of Mathematics ‘Simion Stoilow’, Romanian Academy, Bucharest, Romania.
© 2017 Taylor & Francis
formdvMby
s1,s2
= Z
M
s1(x),s2(x)
hkdvM(x). (1.2)
The completion ofC∞
0 (M,Lk)with respect to (1.2) is denoted as usual byL2(M,Lk). We denote byH0(2)(M,Lk)the closed subspace ofL2(M,Lk)consisting of holomorphic sections.
The Bergman projection is the orthogonal projectionPk : L2(M,Lk)→ H0(2)(M,Lk). For a bounded functionf ∈C∞(M), set
Tf,k:L2(M,Lk)−→L2(M,Lk), Tf,k=Pkf Pk, (1.3) where the action off is the pointwise multiplication byf. The map which associates tof ∈ C∞(M)the family of bounded operators{Tf,k}onL2(M,Lk)is called theBerezin–Toeplitz quantization. AToeplitz operatoris a sequence{Tk}k∈Nof bounded linear endomorphisms of L2(M,Lk)verifyingTk =PkTkPk, such that there exist a sequencegℓ∈ C∞(M)such that for anyp>0, there existsCp>0 withkTk−Pp
ℓ=0Tgℓ,kk−ℓkop6Cp k−p−1for anyk∈N, wherek · kopdenotes the operator norm on the space of bounded operators.
Assume now that(M,ω,J)is a compact Kähler manifold. Then Bordemann et al. [5] and Schlichenmaier [29] (using the analysis of Toeplitz structures of Boutet de Monvel–Guillemin [6]), Charles [7] (inspired by semiclassical analysis of Boutet de Monvel–Guillemin [6]) and Ma–Marinescu [25] (using the expansion of the Bergman kernel [9,24]) showed that the composition of two Toeplitz operators is a Toeplitz operator, in the sense that for anyf,g ∈ C∞(M), the productTf,kTg,khas an asymptotic expansion
Tf,kTg,k= X∞ p=0
TCp(f,g),kk−p+O(k−∞) (1.4) whereCpare bidifferential operators of order6 2r, satisfyingC0(f,g) = fgandC1(f,g)− C1(g,f)=√
−1{f,g}. Here{ ·,· }is the Poisson bracket on(M, 2π ω). We deduce from (1.4), [Tf,k,Tg,k] =
√−1
k T{f,g},k+O(k−2). (1.5) In [24,25] Ma–Marinescu extended the Berezin–Toeplitz quantization to symplectic man- ifolds 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 [8] introduced a semiclassical approach for symplectic manifolds inspired from the Boutet de Monvel–Guillemin theory [6].
In this paper, we extend the Berezin–Toeplitz quantization in several directions. Firstly, we consider an arbitrary Hermitian manifold(M,2,J)endowed with arbitrary Hermitian holomorphic line bundle(L,h)and we quantize the open setM(0)where the curvature of (L,h)is positive. Since there are no holomorphicL2sections in general, we use as quantum spaces the spectral spaces of the Kodaira Laplacian2(0)k on L2(M,Lk), corresponding to energy less thank−N,N > 1 fixed, decaying to 0 polynomially ink, ask → ∞. Secondly, we consider the same construction for the Kodaira Laplacian2(q)k acting on (0,q)-forms.
In this case, we quantize the open setM(q)where the curvature of(L,h)is nondegenerate and has exactlyqnegative eigenvalues (and hencen−qpositive ones). Quantization using (0,q)-forms was introduced in [24, 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 [23].
The idea underlying the construction used in this paper comes from the local holomorphic Morse inequalities [4,11,18,24]. Roughly speaking, the harmonic(0,q)-forms with values in Lktend to concentrate onM(q)ask→ ∞. More precisely, the semiclassical limit of the kernel of the spectral projectors considered above was determined in [18, Theorem 1.1], see also [18, 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 [13,28], especially the stationary phase method of Melin–Sjöstrand [28].
We now formulate the main results. We refer to Section2for some standard notations and terminology used here. We are working in the following general setting:
(A) (M,2,J)is a Hermitian manifold of complex dimensionn, where2is a smooth positive (1, 1)-form andJis the complex structure. Moreover,(L,h)is a holomorphic Hermitian line bundle overM, wherehis the Hermitian fiber metric onL, andq∈ {0, 1,. . .,n}. (B) f,g∈C∞(M)are smooth bounded functions.
Letg2TM(·,·)=2(·,J·)be the Riemannian metric onTMinduced by2andJand leth ·,· i be the Hermitian metric onCTM := TM⊗RCinduced bygTM2 . The Riemannian volume formdvM of(M,2)satisfiesdvM =2n/n!. For everyq=0, 1,. . .,n, the Hermitian metric h ·,· ionTM⊗RCinduces a Hermitian metrich ·,· ion30,q(T∗M)the bundle of(0,q)forms ofM.
We will denote byφthe local weights of the Hermitian metrichonL(see (2.1)). Let∇Lbe the holomorphic Hermitian connection on(L,h)with curvatureRL=(∇L)2. We will identify the curvature formRLwith the Hermitian matrixR˙L ∈ C∞(M, End(T1,0M))satisfying for everyU,V∈Tx1,0M,x∈M,
hRL(x),U∧Vi = h ˙RL(x)U,Vi. (1.6) Let detR˙L(x):=µ1(x) . . . µn(x), where{µj(x)}nj=1, are the eigenvalues ofR˙Lwith respect to h ·,· i. Forj∈ {0, 1,. . .,n}, let
M(j)=
x∈M;R˙L(x)is nondegenerate and has exactlyjnegative eigenvalues . (1.7) We denote by W the subbundle of rank j of T1,0M|M(j) generated by the eigenvectors corresponding to negative eigenvalues ofR˙L. Then detW∗ := 3jW∗ ⊂ 30,j(T∗M)|M(j)
is a rank one sub-bundle. HereW∗is the dual bundle of the complex conjugate bundle ofW and3jW∗is the vector space of all finite sums ofv1∧ · · · ∧vj,v1,. . .,vj ∈W∗. We denote byIdetW∗ ∈End(30,j(T∗M))the orthogonal projection from30,j(T∗M)onto detW∗.
Fork > 0, let(Lk,hk)be thekth tensor power of the line bundle(L,h). Let(·,·)kand (·,·)denote the globalL2 inner products on0,q0 (M,Lk) and0,q0 (M)induced byh ·,· i andhk, respectively (see (2.2)). We denote byL2(0,q)(M,Lk)andL2(0,q)(M)the completions of
0,q0 (M,Lk)and0,q0 (M)with respect to(·,·)kand(·,·), respectively.
Let2(q)k be the Kodaira Laplacian acting on(0,q)–forms with values inLk, cf. (2.6). We denote by the same symbol2(q)k the Gaffney extension of the Kodaira Laplacian, cf. (2.9).
It is well-known that2(q)k is self-adjoint and the spectrum of2(q)k is contained inR
+ (see [24, Proposition 3.1.2]). For a Borel setB ⊂ Rlet E(B)be the spectral projection of2(q)k corresponding to the setB, whereEis the spectral measure of2(q)k (see Davies [10, Section 2])
and forλ∈Rwe setEλ=E (−∞,λ] and Eq
λ(M,Lk)=RangeEλ⊂L2(0,q)(M,Lk). (1.8) Ifλ=0, thenEq
0(M,Lk)=Ker2(q)k =:Hq(M,Lk)is the space of global harmonic sections.
Thespectral projectionof2(q)k is the orthogonal projection P(q)k,λ:L2(0,q)(M,Lk)→Eq
λ(M,Lk). (1.9)
Fixf ∈ C∞(M)be a bounded function. Letλ ≥ 0. TheBerezin–Toeplitz quantization for Eq
λ(M,Lk)is the operator
Tk,λ(q),f :=P(q)k,λ◦f ◦Pk,λ(q):L2(0,q)(M,Lk)→Eq
λ(M,Lk). (1.10) LetTk,λ(q),f(·,·)be the Schwartz kernel ofTk,λ(q),f, see (2.13), (2.14). Since2(q)k is elliptic, we haveTk,λ(q),f(·,·)∈C∞ M×M,(Lk⊗30,q(T∗M))⊠(Lk⊗30,q(T∗M))∗
.
LetAk:L2(0,q)(M,Lk)→L2(0,q)(M,Lk)be ak-dependent continuous operator with smooth kernelAk(x,y)and letD0,D1 ⋐ Mbe open trivializations with trivializing sectionssand bs, respectively. In this paper, we will identifyAkandAk(x,y)onD0×D1with the localized
operatorsAk,s,bsandAk,s,bs(x,y), 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 D0,D1 ⋐ M on which L is trivial. Suppose that one of the following conditions is fulfilled:
(i) D0⋐M(j)and j6=q, (ii) D0⋐M(q)and D0T
D1= ∅.
Then, for every N>1, m∈N, there exists CN,m>0independent of k such that
Tk,k(q),f−N(x,y) Cm
(D0×D1)≤CN,mk2n−N2+2m. (1.11) If D0⋐M(q)there exists a symbol
bf(x,y,k)∈Sn(1;D0×D0,30,q(T∗M)⊠(30,q(T∗M))∗)
and a phase function9 ∈ C∞(D0×D0)such that for every N > 1, m ∈ N, there exists eCN,m >0independent of k such that
Tk,k(q),f−N(x,y)−eik9(x,y)bf(x,y,k) Cm
(D0×D0) ≤eCN,mk2n−N2+2m, (1.12) where bf(x,y,k)∼P∞
j=0bf,j(x,y)kn−jin the sense of Definition2.1and
bf,0(x,x)=(2π )−nf(x)detR˙L(x)IdetW∗(x), x∈D0, (1.13) and
9(x,y)∈C∞(D0×D0), 9(x,y)= −9(y,x),
(1.14)
∃c>0 : Im9≥cx−y2 ,9(x,y)=0⇔x=y.
We collect more properties for the phase9in Theorem3.3. The results says that, roughly speaking, the Toeplitz kernelT(q),f
k,k−N(·,·)acting on(0,q)-forms, decays rapidly ask → ∞ outsideM(q)and off-diagonal, and admits an asymptotic expansion on the setM(q).
Letℓ,m ∈Nbe fixed and chooseN ≥2(n+ℓ+2m+1). Then we deduce from (1.12) that
Tk,k(q),f−N(x,x)= Xℓ
r=0
bf,r(x,x)kn−r+O(kn−ℓ−1) inCm(D0),D0⋐M(q). (1.15) Note that ifMis compact complex manifold endowed with a positive line bundle L (i.e., M(0)=M) we have by [27, Theorem 0.1] for anyℓ,m∈N,
Tk,0(0),f(x,x)= Xℓ
r=0
bf,r(x,x)kn−r+O(kn−ℓ−1) inCm(M). (1.16) Actually, in this case, due to the spectral gap of the Kodaira Laplacian [24, Theorem 1.5.5] we haveT(0)
f,k,k−N =Tf(0),k,0forklarge enough, so (1.15) follows from (1.16). The expansion (1.15) bears resemblance to the expansion of the Toeplitz kernels for functionsf ∈Cp(M)(see [1, (3.19)]), for arbitraryp∈N. In (1.15) the upper bound for the order of expansionℓis due to the sizek−N of the spectral parameter, while in case of symbols of classCp(M)is due to the order of differentiabilityp.
It is interesting to note that Theorem1.1and 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 simplyM = M(0). Of course, in such generality, the quantum spaces have to be spectral spacesEq
k−N(M,Lk).
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 [27] in the presence of a twisting vector bundleEand later by Hsiao [17] forE= C. Both [17,27] work with a general not necessarily Kähler base metric2which might not be polarized, that is,26= √2π−1RLin general. We will calculate the top coefficientsbf,1(x,x)andbf,2(x,x)of the expansion (1.12) in Section7. The coefficients bf,0(x,x)andbf,1(x,x)were given in [7] forE = Cand2 = √2π−1RL. It is a remarkable manifestation of universality, that the coefficients for the quantization with holomorphic sections [17,27] and for the quantization with spectral spaces used in this paper are given by the same formulas. We refer to [32] for an interpretation in graph-theoretic terms of the Toeplitz kernel expansion. The formulas from [27] play an essential role in the quantization of the Mabuchi energy [15] and Laplace operator [20]. On the set where the curvature ofLis degenerate we have the following behavior.
Theorem 1.2. Under the general assumptions (A) and (B), set Mdeg=n
x∈M;R˙Lis degenerate at x∈Mo .
Then for every x0∈Mdeg,ε >0, N>1and every j∈ {0, 1,. . .,n}, there exist a neighborhood U of x0and k0>0, such that for all k≥k0we have
Tk,k(j),f−N(x,x)
≤εkn, x∈U. (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 D0,D1 ⋐ M on which L is trivial. Suppose that one of the following conditions is fulfilled:
(i) D0⋐M(j)and j6=q, (ii) D0⋐M(q)and D0T
D1= ∅.
Then, for every N>1, m∈N, there exists CN,m>0independent of k such that
T(q),f
k,k−N◦T(q),g
k,k−N
(x,y) Cm
(D0×D1) ≤CN,mk3n−N2+2m. (1.18) If D0⋐M(q)there exists a symbol
bf,g(x,y,k)∈Sn(1;D0×D0,30,q(T∗M)⊠(30,q(T∗M))∗) such that for every N>1, m∈N, there existseCN,m>0independent of k such that
T(q),f
k,k−N◦T(q),g
k,k−N
(x,y)−eik9(x,y)bf,g(x,y,k)Cm
(D0×D0)≤eCN,mk3n−N2+2m, (1.19) where bf,g(x,y,k)∼P∞
j=0bf,g,j(x,y)kn−jin the sense of Definition2.1and
bf,g,0(x,x)=(2π )−nf(x)g(x)detR˙L(x)IdetW∗(x), x∈D0, (1.20) and9(x,y)∈C∞(D0×D0)is as in Theorem1.1.
It should be noticed that Theorem1.3holds for any Hermitian manifoldM, not necessarily compact. Note that the estimates in Theorem1.3involve the powerk3n−N2+2mcompared to k2n−N2+2m in Theorem1.1. We will explain why there are different exponents 3nand 2nin the proof of Theorem6.1.
We will calculate the top coefficientsbf,1(x,x),bf,2(x,x)andbf,g,1(x,x),bf,g,2(x,x)of the expansions (1.12) and (1.19) in Section7(see Theorems7.1and7.4).
We come now to the asymptotic expansion of the composition of two Toeplitz operators in the operator norm. LetAk:L2(M,Lk)→L2(M,Lk)bek-dependent continuous operator.
We say thatAk=O(km+km1)ask→ ∞, locally in theL2operator norm if for anyχ,χ1∈ C∞
0 (M), there existsC > 0 independent ofksuch thatkχAkχ1kop ≤ C(km+km1), fork large, wherek·kopdenotes theL2operator norm. We also denote byh ·,| · iωthe Hermitian metric onT∗M⊗RCinduced byω:= √2π−1RL.
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 Cp(f,g)∈C∞
0 (M(0)), p∈N, such that for anyℓ∈Nthe product Tk,k(0),f−NTk,k(0),g−Nhas the asymptotic expansion
T(0),f
k,k−N◦T(0),g
k,k−N = Xℓ
p=0
T(0),Cp(f,g)
k,k−N k−p+O(k−ℓ−1+k3n−N2), k→ ∞, (1.21) locally in the L2operator norm. Moreover,
C0(f,g)=fg, C1(f,g)= − 1
2πh∂f ,∂giω, (1.22)
and therefore the commutator of two Toeplitz operators satisfies h
Tk,k(0),f−N,Tk,k(0),g−N
i=
√−1
k Tk,k(0),{f−N,g}+O(k−2+k3n−N2), k→ ∞, (1.23) where{f,g}is the Poisson bracket on(M(0), 2π ω).
We will give formulas for the coefficientsCj(f,g),j = 0, 1, 3, in Corollary7.5. They have the same form as those in the expansion of the Toeplitz operators acting on spaces of holo- morphic sections, see [17, (1.29)], [27, (0.20)]. Formula (1.23) represents the semiclassical correspondence principle between classical and quantum observables. Theorem1.4allows us to introduce a star-product on the set where a line positive is positive, see Remark6.5.
As an application of Theorems1.1and1.2, we obtain:
Theorem 1.5. Assume (A) and (B) are fulfilled and let N >2n. Then T(q),f
k,k−N(x,x)=kn(2π )−n
detR˙L(x)
f(x)IdetW∗(x)+O(kn−1),k→ ∞, (1.24) locally uniformly on M(q), for every D⋐M, there exists CD>0independent of k such that
T(q),f
k,k−N(x,x)
≤CDkn, ∀x∈D, (1.25)
and if1M(q)denotes the characteristic function of M(q), we have the pointwise convergence:
klim→∞k−nTk,k(q),f−N(x,x)=(2π )−nf(x)detR˙L(x)1M(q)(x)IdetW∗(x), x∈M. (1.26) SinceLkx⊠(Lkx)∗ ∼=C, we can identifyTk,λ(q),f(x,x)to an element of End(30,qx (T∗M). Then M∋x7−→Tk,λ(q),f(x,x)∈End(30,qx (T∗M)) (1.27) is a smooth section of End(30,q(T∗M)). Let TrTk,λ(q),f(x,x)denote the trace ofT(q),fk,λ (x,x)with respect toh ·,· i. WhenMis compact, we define
TrTk,λ(q),f := Z
M
TrTk,λ(q),f(x,x)dvM(x). (1.28)
Forλ=0, we setTk(q),f :=Tk,0(q),f,Tk(q),f(x,y):=Tk,0(q),f(x,y), TrTk(q),f(x,x):= TrTk,0(q),f(x,x), TrTk(q),f :=TrTk,0(q),f.
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 TrT(q),f
k,k−N =kn(2π )−n Z
M(q)
f(x)detR˙L(x)dvM(x)+o(kn), k→ ∞. (1.29) From Theorem1.6we deduce the following (see Section8).
Theorem 1.7. Under assumptions (A) and (B) suppose that M is compact and M(q−1)= ∅, M(q+1)= ∅. Then
klim→∞
k−nTk(q),f(x,x)−(2π )−nf(x)
detR˙L(x)
1M(q)(x)IdetW∗(x)
=0in L1(0,q)(M). (1.30)
In particular,
TrTk(q),f =kn(2π )−n Z
M(q)
f(x)
detR˙L(x)
dvM(x)+o(kn) as k→ ∞. (1.31)
Let’s considerq = 0 and f ≡ 1 in (1.31). IfM(1) = ∅, we obtain dimH0(M,Lk) = kn(2π )−nR
M(0)
detR˙L(x)dvM(x)+o(kn)ask → ∞. Therefore, dimH0(M,Lk) ∼ kn as k → ∞, providedM(0) 6= ∅andM(1) = ∅. This is a form of Demailly’s criterion for a line bundle to be big, which answers the Grauert–Riemenschneider conjecture, see [11], [24, Theorem 2.2.27].
We wish now to link the quantization scheme, we proposed above by using spectral spacesEq
k−N(M,Lk)to the more traditional quantization using holomorphic sections (or, more generally, harmonic forms). For this purpose we need the notion ofO(k−N)small spectral gap property introduced in [18, Definition 1.5]:
Definition 1.8. LetD⊂M. We say that2(q)k hasO(k−N)small spectral gap on Dif there exist constantsCD >0,N∈N,k0∈N, such that for allk≥k0andu∈0,q0 (D,Lk), we have
(I−P(q)k,0)u
k≤CDkN 2(q)k u
k.
Let D0,D1 ⊂ M be open sets and Ak,Ck : 0,q0 (D1) → 0,q(D0)be k-dependent continuous operators with smooth kernelsAk(x,y),Ck(x,y)∈C∞(D0×D1,30,q(T∗M))⊠ (30,q(T∗M))∗). We writeAk≡Ck mod O(k−∞)locally uniformly onD0×D1orAk(x,y)≡ Ck(x,y) modO(k−∞)locally uniformly onD0×D1if
∂xα∂yβ(Ak(x,y)−Ck(x,y))
=O(k−N) uniformly on every compact set inD0×D1, for allα,β ∈N2n
0 and everyN >1.
The following result describes the asymptotics of the kernels of Toeplitz operators corre- sponding toharmonic formsin the case of small spectral gap.
Theorem 1.9. Under the assumptions (A) and (B) let j ∈ {0, 1,. . .,n}and D0,D1 ⋐ M on which L is trivial. Suppose that one of the following conditions is fulfilled:
(i) D0⋐M(j)and j6=q,
(ii) D0⋐M(q),2(q)k has an O(k−N)small spectral gap on D0and D0T
D1= ∅. Then
Tk(q),f(x,y)≡0 modO(k−∞)locally uniformly on D0×D1,
(Tk(q),f ◦Tk(q),g)(x,y)≡0 modO(k−∞)locally uniformly on D0×D1. Assume that D0⋐M(q)and2(q)k has an O(k−N)small spectral gap on D0. Then,
T(q),fk (x,y)≡eik9(x,y)bf(x,y,k) modO(k−∞)locally uniformly on D0×D0,
(T(q),fk ◦Tk(q),g)(x,y)≡eik9(x,y)bf,g(x,y,k) modO(k−∞)locally uniformly on D0×D0, where bf(x,y,k),bf,g(x,y,k)∈Sn(1;D0×D0,30,q(T∗M))⊠(30,q(T∗M))∗
are as in(1.12) and(1.19), respectively, and9(x,y)∈C∞(D0×D0)is as in Theorem1.1.
There are several geometric situations when there exists a spectral gap. For example, if Lis a positive line bundle on a compact manifoldM, or more generally, ifLis uniformly positive on a complete manifold(M,2) with√
−1RK∗M and∂2bounded below, then the Kodaira Laplacian2(0)k has a “large" spectral gap onM, that is, there exists a constantC>0 such that for allkwe have inf{λ ∈ Spec(2(0)k );λ 6= 0} ≥ Ck, (see [24, Theorem 1.5.5], [24, Theorem 6.1.1, (6.1.8)]). Therefore, we can recover from Theorem 1.9 results about quantization of noncompact manifolds, such as [24, Theorem 7.5.1], [25, Theorem 5.3], [26, Theorem 2.30].
In this paper, as applications of Theorem1.9, we establish Berezin–Toeplitz quantization for semipositive and big line bundles. We assume now that(M,2)is compact and we set
Herm(L)=
singular Hermitian metrics onL , M(L)=
h∈Herm(L);his smooth outside a proper analytic set and the curvature current ofhis strictly positive .
Note that by Siu’s criterion [24, Theorem 2.2.27],Lis big under the hypotheses of Theorem 1.10below. By [24, Lemma 2.3.6],M(L)6= ∅. Set
M′ :=
p∈M;∃h∈M(L)withhsmooth nearp . (1.32) Theorem 1.10. Let(M,2)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) 6= ∅. Let f,g ∈ C∞(M)and let D0 ⋐ M(0)T
M′be an open set on which L is trivial. Then
Tk(0),f(x,y)≡eik9(x,y)bf(x,y,k) modO(k−∞)locally uniformly on D0×D0,
(T(0),fk ◦Tk(0),g)(x,y)≡eik9(x,y)bf,g(x,y,k) modO(k−∞)locally uniformly on D0×D0, where bf(x,y,k),bf,g(x,y,k) ∈ Sn(1;D0×D0)are as in(1.12)and(1.19), respectively, and 9(x,y)∈C∞(D0×D0)is as in Theorem1.1.
Let us consider a singular Hermitian holomorphic line bundle(L,h) → M (see e.g., [24, Definition 2.3.1]). We assume thathis smooth outside a proper analytic set6and the curvature current ofhis strictly positive. The metrichinduces singular Hermitian metrics hk on Lk. We denote by I(hk) the Nadel multiplier ideal sheaf associated to hk and by H0(M,Lk⊗I(hk))⊂H0(M,Lk)the space of global sections of the sheafO(Lk)⊗I(hk)(see (2.12)), whereH0(M,Lk) :=
u∈C∞(M,Lk);∂ku=0 . We denote by(·,·)kthe natural inner products onC∞(M,Lk⊗I(hk))induced byhand the volume formdvM onM(see (2.11) and see also (2.10) for the precise meaning ofC∞(M,Lk⊗I(hk))). The (multiplier ideal) Bergman kernel ofH0(M,Lk)I(hk))is the orthogonal projection
P(0)k,I :L2(M,Lk)→H0(M,Lk⊗I(hk)). (1.33) Letf ∈C∞(M). The multiplier ideal Berezin–Toeplitz operator is
Tk,(0),fI :=P(0)k,I ◦f ◦P(0)k,I :L2(M,Lk)→H0(M,Lk⊗I(hk)) (1.34)
where we denote byf the multiplication operator onL2(M,Lk)byf. LetTk,(0),fI(x,y)be the distribution kernel ofTk,(0),fI . Note thatTk,(0),fI (x,y)∈C∞((M\6)×(M\6),(Lk)∗⊠Lk).
Theorem 1.11. Let(L,h)be a singular Hermitian holomorphic line bundle over a compact Hermitian manifold(M,2). We assume that h is smooth outside a proper analytic set6and the curvature current of h is strictly positive. Let f,g∈C∞(M). Let D0⊂M\6be an open set on which L is trivial. Then
Tk,(0),fI (x,y)≡eik9(x,y)bf(x,y,k) modO(k−∞)locally uniformly on D0×D0,
(T(0),fk,I ◦Tk,(0),gI )(x,y)≡eik9(x,y)bf,g(x,y,k) modO(k−∞)locally uniformly on D0×D0, where bf(x,y,k),bf,g(x,y,k) ∈ Sn(1;D0×D0)are as in(1.12)and(1.19), respectively, and 9(x,y)∈C∞(D0×D0)is as in Theorem1.1.
The paper is organized as follows. In Section2, we collect terminology, definitions and statements we use throughout. In Sections3and4prove the off-diagonal decay for the kernels P(q)
k,k−N(·,·) andT(q),f
k,k−N(·,·). In Section 5, we establish the full asymptotic of the Berezin–
Toeplitz kernelsT(q),f
k,k−N(·,·)and prove Theorem1.1. Section6is devoted to the expansion of the composition of two Toeplitz operators and contains the proof of Theorems1.3,1.4, and1.9–1.11. In Section7, we calculate the leading coefficients of the various expansions we established. Finally, in Section8, we prove Theorems1.2and1.7.
2. Preliminaries
Some standard notations.We denote byN = {0, 1, 2,. . .}the set of natural numbers and byRthe set of real numbers. We use the standard notationswα,∂xα for multi-indicesα = (α1,. . .,αm)∈Nm,w∈Cm,∂x=(∂x1,. . .,∂xm).
Letbe aC∞paracompact manifold equipped with a smooth density of integration. We letTandT∗denote the tangent bundle ofand the cotangent bundle of, respectively.
The complexified tangent bundle ofand the complexified cotangent bundle ofwill be denoted byCT := T⊗RCandCT∗ := T∗⊗RC, respectively. We writeh ·,· ito denote the pointwise duality betweenTandT∗. We extendh ·,· ibilinearly to(T⊗R C)×(T∗⊗RC).
LetEbe aC∞vector bundle over. We writeE∗to denote the dual bundle ofE. The fiber ofEatx∈will be denoted byEx. We denote by End(E)the vector bundle overwith fiber End(E)x=End(Ex)overx∈.
LetFbe a vector bundle over anotherC∞paracompact manifold′. We introduce the vector bundleF⊠E∗ =π1∗(F)⊗π2∗(E∗)over′×, whereπ1andπ2are the projections of
′×on the first and second factor (see [24, p. 337]). The fiber ofF⊠E∗over(x,y)∈′× consists of the linear maps fromEytoFx.
LetY ⊂ be an open set and take anyL2inner product onC∞
0 (Y,E). By using thisL2 inner product, in this paper, we will consider a distribution section ofEoverYis a continuous linear form onC∞
0 (Y,E). From now on, the spaces distribution sections ofEoverYwill be denoted byD′(Y,E). LetE′(Y,E)be the subspace ofD′(Y,E)whose elements have compact
support inY. Form∈R, we letHm(Y,E)denote the Sobolev space of ordermof sections of EoverY. Put
Hlocm (Y,E)=
u∈D′(Y,E);ϕu∈Hm(Y,E),ϕ∈C∞
0 (Y) ,
Hcompm (Y,E)=Hlocm(Y,E)∩E′(Y,E).
LetMbe a complex manifold of dimensionn. We always assume thatMis paracompact.
LetT1,0MandT0,1Mdenote the holomorphic tangent bundle ofMand the antiholomorphic tangent bundle ofM, respectively. Let31,0(T∗M)be the holomorphic cotangent bundle of M and let30,1(T∗M)be the antiholomorphic cotangent bundle of M. Forp,q ∈ N, let 3p,q(T∗M)=3p(31,0(T∗M))⊗3q(30,1(T∗M))be the bundle of(p,q)forms ofM.
For an open setD⊂Mwe letp,q(D)denote the space of smooth sections of3p,q(T∗M) overDand let0,q0 (D)be the subspace of0,q(D)whose elements have compact support inD. Similarly, ifEis a vector bundle overD, then we letp,q(D,E)denote the space of smooth sections of3p,q(T∗M)⊗EoverD. Letp,q0 (D,E)be the subspace ofp,q(D,E)whose elements have compact support inD.
For a multi-indexJ = (j1,. . .,jq) ∈ {1,. . .,n}qwe set|J| = q. We say thatJ is strictly increasing if 16j1<j2<· · ·<jq6n. Let{e1,. . .,en}be a local frame for30,1(T∗M)on an open setD⊂M. For a multi-indexJ=(j1,. . .,jq)∈ {1,. . .,n}q, we puteJ=ej1∧· · ·∧ejq. LetEbe a vector bundle overDand letf ∈0,q(D,E).f has the local representation
f|D = X
|J|=q
′fJ(z)eJ,
whereP′
means that the summation is performed only over strictly increasing multi-indices andfJ ∈C∞(D,E).
Metric data.Let(M,2)be a complex manifold of dimensionn, where2is a smooth positive (1, 1)form, which induces a Hermitian metrich ·,· ion the holomorphic tangent bundle T1,0M. In local holomorphic coordinatesz=(z1,. . .,zn), if2=√
−1Pn
j,k=12j,kdzj∧dzk, thenh∂∂zj|∂∂zki = 2j,k,j,k =1,. . .,n. We extend the Hermitian metrich ·,· itoTM⊗RC in a natural way. The Hermitian metrich ·,· ionTM⊗RCinduces a Hermitian metric on 3p,q(T∗M)also denoted byh ·,· i.
Let(L,h)be a Hermitian holomorphic line bundle overM, where the Hermitian metric on Lis denoted byh. Until further notice, we assume thathis smooth. Given a local holomorphic framesofLon an open subsetD⊂Mwe define the associated local weight ofhby
|s(x)|2h=e−2φ (x), φ ∈C∞(D,R). (2.1) Let RL = (∇L)2 be the Chern curvature of L, where∇L is the Hermitian holomorphic connection. ThenRL|D =2∂∂φ.
LetLk,k > 0, be thekth tensor power of the line bundleL. The Hermitian fiber metric on L induces a Hermitian fiber metric on Lk that we shall denote by hk. If s is a local trivializing holomorphic section ofL thenskis a local trivializing holomorphic section of Lk. For p,q ∈ N, the Hermitian metric h ·,· ion 3p,q(T∗M)andhk induce a Hermitian metric on3p,q(T∗M)⊗Lk, denoted byh ·,· ihk. Fors∈p,q(M,Lk), we denote the pointwise norm|s(x)|2hk := hs(x),s(x)ihk. We takedvM = dvM(x)as the induced volume form onM.
TheL2–Hermitian inner products on the spacesp,q0 (M,Lk)andp,q0 (M)are given by (s1,s2)k=
Z
Mhs1(x),s2(x)ihkdvM(x), s1,s2∈p,q0 (M,Lk), (f1,f2)=
Z
Mhf1(x),f2(x)idvM(x), f1,f2∈p,q0 (M). (2.2) ksk2k=(s,s)k, s∈p,q0 (M,Lk), kfk2:=(f,f), f ∈p,q0 (M).
LetAk : L2(0,q)(M,Lk) → L2(0,q)(M,Lk)be ak-dependent continuous operator with smooth kernelAk(x,y). Lets,bs be local trivializing holomorphic sections ofLonD0⋐M,D1⋐M, respectively,|s|2h =e−2φ,|bs|2h = e−2bφ, whereD0,D1are open sets. The localized operator of AkonD0×D1is given by
Ak,s,bs :0,q0 (D1)→0,q(D0), u7−→s−ke−kφ(Akbskekbφu), (2.3) and letAk,s,bs(x,y)∈C∞(D0×D1,30,q(T∗M))⊠(30,q(T∗M))∗)be the distribution kernel ofAk,s,bs. Fors=bs,D0=D1, we set
Ak,s:=Ak,s,s, Ak,s(x,y):=Ak,s,s(x,y). (2.4) A self-adjoint extension of the Kodaira Laplacian.We denote by
∂k:0,r(M,Lk)→0,r+1(M,Lk), ∂k∗:0,r+1(M,Lk)→0,r(M,Lk) (2.5) the Cauchy–Riemann operator acting on sections ofLkand its formal adjoint with respect to (· | ·)k, respectively. Let
2(q)k :=∂k∂k∗+∂k∗∂k:0,q(M,Lk)→0,q(M,Lk) (2.6) be the Kodaira Laplacian acting on (0,q)–forms with values in Lk. We extend ∂k to L2(0,r)(M,Lk)by
∂k: Dom∂k⊂L2(0,r)(M,Lk)→L2(0,r+1)(M,Lk), (2.7) where Dom∂k := {u ∈ L2(0,r)(M,Lk);∂ku ∈ L2(0,r+1)(M,Lk)}, where∂kuis defined in the sense of distributions. We also write
∂k∗: Dom∂k∗⊂L2(0,r+1)(M,Lk)→L2(0,r)(M,Lk) (2.8) to denote the Hilbert space adjoint of∂kin theL2space with respect to(·,·)k. Let2(q)k denote the Gaffney extension of the Kodaira Laplacian given by
Dom2(q)k =n
s∈L2(0,q)(M,Lk);s∈Dom∂k∩Dom∂k∗, ∂ks∈Dom∂k∗, ∂k∗s∈Dom∂k o
, (2.9) and2(q)k s=∂k∂k∗s+∂k∗∂ksfors∈Dom2(q)k . By a result of Gaffney [24, Proposition 3.1.2], 2(q)k is a positive self-adjoint operator. Note that ifMis complete, the Kodaira Laplacian2(q)k is essentially self-adjoint [24, Corollary 3.3.4] and the Gaffney extension coincides with the Friedrichs extension of2(q)k .
Consider a singular Hermitian metrichon a holomorphic line bundleLoverM. Ifh0is a smooth Hermitian metric onLthenh=h0e−2ϕfor some functionϕ∈L1loc(M,R). TheNadel
