aa r X i v : . [ m a t h . C V ] S e p The regular quantizations of certain holomorphic bundles
Zhiming Feng
School of Mathematical and Information Sciences, Leshan Normal University, Leshan, Sichuan 614000, P.R.ChinaEmail: [email protected]
Abstract
In this paper, we study the regular quantizations of K¨ahler manifolds by using the firsttwo coefficients of Bergman function expansions. Firstly, we obtain sufficient and necessary conditionsfor certain Hermitian holomorphic vector bundles and their ball subbundles to be regular quantiza-tions. Secondly, we obtain that some projective bundles over the Fano manifolds M admit regularquantizations if and only if M are biholomorphically isomorphism to the complex projective spaces.Finally, we obtain the balanced metrics on certain Hermitian holomorphic vector bundles and theirball subbundles over the Riemann sphere. Key words:
Bergman functions · Balanced metrics · Regular quantizations · Complex projectivespaces · Hermitian holomorphic vector bundles
Mathematics Subject Classification (2010): · · Let (
L, h, π ) be the quantum line bundle over a K¨ahler manifold (
M, g ) of complex dimension n , namely, ( L, h ) is a positive Hermitian holomorphic line bundle such that c ( L, h ) = ω , where ω denotes the K¨ahler form associated to the K¨ahler metric g , c ( L, h ) denotes the curvature of the Chernconnection on the Hermitian holomorphic bundle (
L, h ), and π : L → M is the bundle projection.The curvature c ( L, h ) is given by c ( L, h ) = − √− π ∂ ¯ ∂ log h ( σ ( z ) , σ ( z ))for a trivializing holomorphic section σ : U → L \{ } . The quantum line bundle ( L, h ) is also called ageometric quantization of the K¨ahler manifold (
M, ω ).For a given positive integer m , let ( L m , h m ) be the m -th tensor power of L with c ( L m , h m ) = mω ,and H ( M, L m ) be the space consisting of global holomorphic sections of L m . Define H m ( M ) := (cid:26) s ∈ H ( M, L m ) : k s k := Z M h m ( s ( z ) , s ( z )) ω n n ! < + ∞ (cid:27) . (1.1)If H m ( M ) = { } , the Bergman function on M with respect to the metric mg is defined by ǫ mg ( z ) = dim H m ( M ) X j =1 h m ( s j ( z ) , s j ( z )) , z ∈ M, (1.2)where { s j : 1 ≤ j ≤ dim H m ( M ) } is an orthonormal basis for the Hilbert space H m ( M ).For compact manifolds by Catlin [9] and Zelditch [40], and for non-compact manifolds by Ma-Marinescu [35, 36] and Engliˇs [13], ǫ mg ( z ) admits an asymptotic expansion as m → + ∞ ǫ mg ( z ) ∼ + ∞ X j =0 a ( g ) j ( z ) m n − j , (1.3)egular quantizations 2where expansion coefficients a ( g ) j are polynomials of the curvature and its covariant derivatives for themetric g . For compact manifolds by Lu [33] and for non-compact manifolds by Ma-Marinescu [35](Theorem 6.1.1) and Engliˇs [14], the expansion coefficients a ( g ) j have the same expression, in particular a ( g )0 , a ( g )1 and a ( g )2 are given by a ( g )0 = 1 ,a ( g )1 = k g ,a ( g )2 = △ k g + | R g | − | Ric g | + k g . (1.4)Here k g , △ g , R g and Ric g denote the scalar curvature, the Laplace, the curvature tensor and the Riccicurvature associated to the metric g , respectively. For graph theoretic formulas of coefficients a j , seeXu [39]. For the more references of the Bergman function expansions, refer to Berezin [4], Dai-Liu-Ma[10], Berman-Berndtsson-Sj¨ostrand [5], Ma-Marinescu [37], Hsiao [20] and Hsiao-Marinescu [21].The K¨ahler metric g on M is balanced if the Bergman function ǫ g ( z ) ( z ∈ M ) is a positive constanton M . Balanced metric plays an important role in the quantization of a K¨ahler manifold, see Berezin[4], Rawnsley [38], Cahen-Gutt-Rawnsley [7, 8], Engliˇs [12], Arezzo-Loi [2], Ma-Marinescu [35] andLui´c [34]. Balanced metrics play a central role when the polarized algebraic manifolds admit K¨ahlermetrics of constant scalar curvature, see Donaldson [11]. For the study of the balanced metrics, seealso Engliˇs [12, 15], Loi-Mossa [28], Loi-Zedda [30, 31], Loi-Zedda-Zuddas [32] and Arezzo-Loi-Zuddas[3].If there is a positive integer m such that Bergman functions ǫ mg ( z ) are positive constants on M for all m ≥ m , ( L, h ) is called a regular quantization of the K¨ahler manifold (
M, ω ), for regularquantizations of compact complex manifolds, see Cahen-Gutt-Rawnsley [7, 8]. Cahen-Gutt-Rawnsley[7] have shown that a geometric quantization (
L, h ) of a homogeneous and simply connected compactK¨ahler manifold (
M, ω ) is regular. If (
M, ω ) admits a regular quantization, Cahen-Gutt-Rawnsley [7]also have generalized Berezin’s method [4] to the case of compact K¨ahler manifolds and to obtain adeformation quantization of the K¨ahler manifold (
M, ω ), Loi in [26] has proved that there exists anasymptotic expansion on (
M, ω ) for ǫ mg ( z ) as (1.3) and (1.4), and all coefficients a ( g ) j are constants.For the study of the regular quantization, also refer to Arezzo-Loi [2] and Loi [27].For the case of non-compact manifolds, Loi-Mossa in [28] have shown that a bounded homogeneousdomain admits a regular quantization. For the nonhomogeneous setting, we give the existence ofregular quantizations on some Hartogs domains in [6] and [18].In [17], the author studied constant scalar curvature K¨ahler metrics such that the second coefficients a ( g )2 of the Bergman function expansions are constants on certain Hartogs domains, namely the part(I) of Theorem 2.2 below with λ = 1.In this paper, we use the coefficients of Bergman function expansions to study the existence ofregular quantizations of K¨ahler manifolds. Firstly, we use Theorem 2.2 to study regular quantizationsof trivial Hermitian holomorphic vector bundles and their ball subbundles, and get Theorem 2.3. Sec-ondly, we study regular quantizations of certain Hermitian holomorphic vector bundles and their ballsubbundles according to Theorem 2.3, we get Theorem 1.1. Thirdly, we study regular quantizations ofcompactification of certain Hermitian holomorphic vector bundles over compact complex manifolds,we obtain Theorem 1.2. Finally, using Theorem 1.1 we get the balanced metrics on certain Hermitianholomorphic vector bundles and their ball subbundles over the Riemann sphere CP , namely Corollary1.3. The main results of this article are described below.Before describing Theorem 1.1, we first define quantum line bundles over certain holomorphic vectorbundles. Let ( L ∗ , h ∗ , π ) be the dual bundle of the quantum line bundle ( L , h , π ) over a connectedK¨ahler manifold ( M , ω ) of complex dimension d . For a given positive integer r , let ( L ∗⊕ r , h := h ∗⊕ r )be the direct sum of r copies of ( L ∗ , h ∗ ), B ( L ∗⊕ r ) := { v ∈ L ∗⊕ r : h ( v, v ) < } be the ball subbundleegular quantizations 3of the Hermitian holomorphic vector bundle ( L ∗⊕ r , h ), p : L ∗⊕ r → M and Π : L := p ∗ L → L ∗⊕ r bethe bundle projections, where L := p ∗ L be pull-back of L under the map p , that is L = p ∗ L −−−−→ L y π y L ∗⊕ r p −−−−→ M , L = p ∗ L −−−−→ L y π y B ( L ∗⊕ r ) p −−−−→ M . (1.5)Let M := L ∗⊕ r or B ( L ∗⊕ r ). Given a real smooth function F with F (0) = 0, let h F ( · , · ) := e − F ( h (Π · , Π · )) × ( p ∗ h )( · , · ) (1.6)be a Hermitian metric on the line bundle L over M , where the Hermitian metric p ∗ h on line bundle L is pull-back of h under the map p . If ω F := c ( L, h F ) >
0, then (
L, h F , Π) is the quantum linebundle over (
M, ω F ). Let g F be a K¨ahler metric associated to the K¨ahler form ω F = c ( L, h F ) on M . Theorem 1.1.
Under the assumptions above, further suppose that (i)
There exists a non-negative integer m such that H m ( M ) := (cid:26) s ∈ H ( M , L m ) : k s k := Z M h m ( s, s ) ω d d ! < + ∞ (cid:27) = { } for all m > m . (ii) There exists a dense open contractible subset U ⊂ M such that M − U is an analytic subset of M , thus the restriction of L to U is the trivial holomorphic line bundle. (iii) ( L, h F , Π) is the quantum line bundle over ( M, ω F ) . (iv) The fibre metric of g F is complete.Then we have the following conclusions. (I) There exists a positive integer m such that the Bergman functions ǫ mg F for ( L m , h mF ) are con-stants on M = B ( L ∗⊕ r ) for all m ≥ m if and only if F ( ρ ) = − A log(1 − ρ ) , A > ,ǫ mg = m + r − (1 + r ) A,ǫ mg F = Q rj =1 ( m − jA ) ,m ≥ m > max { m , (1 + r ) A } (1.7) for the case of d = 1 , and F ( ρ ) = − log(1 − ρ ) ,ǫ mg = Q dj =1 ( m − j ) ,ǫ mg F = Q nj =1 ( m − j ) ,m ≥ m > max { m , n } (1.8) for the case of d > , where ǫ mg are Bergman functions for ( L m , M , h m ) , and n = d + r denotes thedimension of M . egular quantizations 4(II) There exists a positive integer m such that the Bergman functions ǫ mg F are constants on M = L ∗⊕ r for all m ≥ m if and only if d = 1 ,F ( ρ ) = cρ, c > ,ǫ mg = m + r,ǫ mg F = m r ,m ≥ m > m . (1.9) Theorem 1.2.
Let ( L , h , Π ) be the quantum line bundle over a connected compact K¨ahler manifold ( M , ω ) of complex dimension d with the first Chern class c ( M ) > when d > . Assume that thereexists a dense open contractible subset Ω ⊂ M such that M − Ω is an analytic subset of M , thus therestriction of L to Ω is the trivial holomorphic line bundle. Suppose that H m ( M ) := (cid:26) s ∈ H ( M , L m ) : k s k := Z M h m ( s, s ) ω d d ! < + ∞ (cid:27) = { } for all m ∈ N .Let E be the direct sum of r copies of L , i.e. E = L ⊕ r with an associated hermitian metric stilldenoted by h . Denote O M as the structure sheaf of M . The projective bundle M := P ( E ⊕ O M ) can be viewed as a compactification of E . The projection of the vector bundle E to M also denotedby Π : E → M .Let ω ( u ) = (Π ∗ ω )( u ) + √− π ∂∂F ( h ( u, u )) ( u ∈ E ) be a K¨ahler form on E with F (0) = 0 such that (i) ω can be extended across M − E (the extension of ω on M is still expressed as ω ); (ii) ( M, ω ) admits a geometric quantization ( L, h ) . (iii) Bergman functions ǫ mg for ( L m , h m ) are constants on M for all m ≥ , where g is a K¨ahlermetric associated with the K¨ahler form ω .Then M and M are biholomorphically isomorphism to the complex projective spaces CP d and CP d + r ,respectively, and F ( ρ ) = log(1 + cρ ) with c > . Denote O CP d (1) as the hyperplane line bundle of the d -dimensional complex projective space CP d , ω F S as the standard Fubini-Study form on CP d . Let ( O CP d (1) , h F S ) be the quantum line bundle overthe complex projective space ( CP d , ω F S ). For a given k ∈ N + , let L = O CP d ( k ) and h = h kF S bethe k -th tensor power of O CP d (1) and h F S , respectively. Write ( O CP n ( − k ) , h − kF S ) as the dual bundleof ( O CP n ( k ) , h kF S ). Let ( L , M , h ) = ( O CP ( k ) , CP , h kF S ) in Theorem 1.1, using ǫ mg = m + k , weobtain the following Corollary 1.3. Corollary 1.3.
For given positive integers k, r ∈ N + , let L := p ∗ O CP ( k ) −−−−→ L := O CP ( k ) Π y π y ⊕ ri =1 O CP ( − k ) p −−−−→ M := CP , where p : ⊕ ri =1 O CP ( − k ) → CP egular quantizations 5 and Π : p ∗ O CP ( k ) → ⊕ ri =1 O CP ( − k ) are the bundle projections. (I) For kr > , set h = ⊕ ri =1 h − kF S , λ = k ( r + 1) kr − ,M = { v ∈ ⊕ ri =1 O CP ( − k ) : h ( v, v ) < } and h F ( u, u ) = (1 − h (Π( u ) , Π( u ))) λ × ( p ∗ h kF S )( u, u ) , u ∈ L | M . Let g F be the K¨ahler metric associated to the K¨ahler form ω F := c ( L, h F ) on M , then ( L | M , h F ) isa quantum line bundle over ( M, g F ) , and metrics mg F are balanced metrics on M for all m ≥ r . (II) For c > , k = r = 1 , M = O CP ( − , let g F be the K¨ahler metric associated to the K¨ahlerform ω F := c ( L, h F ) on M , where h F ( u, u ) = e − c × h − F S (Π( u ) , Π( u )) × ( p ∗ h F S )( u, u ) , u ∈ L. Then ( L, h F ) is a quantum line bundle over ( M, g F ) , and metrics mg F are balanced metrics on M forall m ≥ . Remark 1.1.
Recently, Aghedu-Loi [1] has get the same conclusion as the part (II) of Corollary 1.3.
The organization of this paper is as follows. In Section 2, we study regular quantizations of trivialHermitian holomorphic vector bundles and their ball subbundles. In Section 3, Section 4 and Section5, we give proofs of Theorem 1.1, Theorem 1.2 and Corollary 1.3, respectively.
From Lemma 2.4 and Lemma 2.5 of [17], we have the following lemma.
Lemma 2.1.
Let g φ be a K¨ahler metric on a domain Ω ⊂ C d (that’s the connected open set of C d )associated with the K¨ahler form ω φ = √− π ∂∂φ , where φ is globally defined the K¨ahler potential on thedomain Ω . For = λ ∈ R , set M := n ( z, w ) ∈ Ω × C d : k w k < e − λφ ( z ) o or Ω × C d . Let g F be a K¨ahler metric on the domain M associated with the K¨ahler form ω F = √− π ∂∂ Φ F , here Φ F ( z, w ) := φ ( z ) + F ( λφ ( z ) + log k w k ) . Let k g , ∆ g , Ric g and R g be the scalar curvature, the Laplace, the Ricci curvature and the curvaturetensor with respect to the metric g = g φ or g F , respectively. Put a ( g φ )1 = 12 k g φ , a ( g φ )2 = 13 △ g φ k g φ + 124 | R g φ | − | Ric g φ | + 18 k g φ and a ( g F )1 = 12 k g F , a ( g F )2 = 13 △ g F k g F + 124 | R g F | − | Ric g F | + 18 k g F . Let t = λφ ( z ) + log k w k , x = F ′ ( t ) , ϕ ( x ) = F ′′ ( t ) , egular quantizations 6 σ = (cid:0) (1 + λx ) d x d − ϕ (cid:1) ′ (1 + λx ) d x d − and χ = d ( d − x − (cid:0) (1 + λx ) d x d − ϕ (cid:1) ′′ (1 + λx ) d x d − . We have the following conclusions. (i) k g F = 11 + λx k g φ + d ( d − x − (cid:0) (1 + λx ) d x d − ϕ (cid:1) ′′ (1 + λx ) d x d − , (2.1) | Ric g F | = | Ric g φ | − λσk g φ + dλ σ (1 + λx ) + ( d − (cid:18) σ − d x (cid:19) + ( σ ′ ) , (2.2) △ g F k g F = 1(1 + λx ) ( △ g φ k g φ ) − (cid:0) λ (1 + λx ) d − x d − ϕ (cid:1) ′ (1 + λx ) d x d − k g φ + (cid:0) (1 + λx ) d x d − ϕχ ′ (cid:1) ′ (1 + λx ) d x d − (2.3) and | R g F | = 1(1 + λx ) | R g φ | − λ ϕ (1 + λx ) k g φ + 2 d ( d + 1) λ ϕ (1 + λx ) + 4 dλ (cid:18)(cid:18) ϕ λx (cid:19) ′ (cid:19) + (cid:0) ϕ ′′ (cid:1) + ( d − ( dλ (cid:18) ϕx (1 + λx ) (cid:19) + 4 (cid:18)(cid:16) ϕx (cid:17) ′ (cid:19) + 2 d (cid:18) ϕ − xx (cid:19) ) . (2.4)(ii) a ( g F )1 = a ( g φ )1 λx + d ( d − x − (cid:0) (1 + λx ) d x d − ϕ (cid:1) ′′ λx ) d x d − (2.5) and a ( g F )2 = a ( g φ )2 (1 + λx ) + (cid:26) χ λx ) + λ ϕ (1 + λx ) (cid:27) a ( g φ )1 + 124 (cid:26) ϕχ ′ ) ′ + 8 (cid:18) dλ λx + d − x (cid:19) ϕχ ′ + 3 χ − σ ′ ) + ( ϕ ′′ ) +4 dλ (cid:18)(cid:18) ϕ λx (cid:19) ′ (cid:19) − dλ (1 + λx ) σ + 2 d ( d + 1) λ (1 + λx ) ϕ ) + d − ( dλ ϕ x (1 + λx ) + (cid:18)(cid:16) ϕx (cid:17) ′ (cid:19) + d ϕ − x ) x − ( σ − d ) x ) . (2.6)(iii) If ϕ ( x ) = x + λx , then a ( g F )1 = − n ( n + 1) λ λx (cid:18) d ( d + 1) λ a ( g φ )1 (cid:19) . (2.7)egular quantizations 7 a ( g F )2 = ( n − n ( n + 1)(3 n + 2) λ
24 + 11 + λx ( n − n + 2) λ (cid:18) d ( d + 1) λ a ( g φ )1 (cid:19) + 1(1 + λx ) (cid:18) a ( g φ )2 + ( d − d + 2) λ a ( g φ )1 + ( d − d ( d + 1)(3 d + 10) λ (cid:19) . (2.8) So, both a ( g F )1 and a ( g F )2 are constants if and only if a ( g φ )1 = − d ( d +1) λ ,a ( g φ )2 = ( d − d ( d +1)(3 d +2) λ . (2.9) Remark 2.1.
For other proofs of the formula (2.1) , see [22] and [19].Proof.
We only give a proof for the part (i), because the part (ii) and the part (iii) are derived directlyfrom the part (i).For the case of λ >
0, let e φ = λφ, e F = λF, ω e φ = λω φ , ω e F = λω F , e x = e F ′ , e ϕ = e F ′′ , e σ = (cid:0) (1 + e x ) d e x d − e ϕ (cid:1) ′ (1 + e x ) d e x d − , e χ = d ( d − e x − (cid:0) (1 + e x ) d e x d − e ϕ (cid:1) ′′ (1 + e x ) d e x d − . Then k g F = λk g e F , | Ric g F | = λ | Ric g e F | , △ g F k g F = λ △ g e F k g e F , | R g F | = λ | R g e F | , (2.10) k g φ = λk g e φ , | Ric g φ | = λ | Ric g e φ | , △ g φ k g φ = λ △ g e φ k g e φ , | R g φ | = λ | R g e φ | (2.11)and e x = λx, e ϕ ( e x ) = λϕ ( x ) , e σ ( e x ) = σ ( x ) , e χ ( e x ) = 1 λ χ ( x ) , e σ ′ ( e x ) = 1 λ σ ′ ( x ) , e χ ′ ( e x ) = 1 λ χ ′ ( x ) . (2.12)where g e φ and g e F are K¨ahler metrics associated with the K¨ahler form ω e φ and ω e F , respectively.By Lemma 2.4 and Lemma 2.5 of [17], we have k g e F = 11 + e x k g e φ + d ( d − e x − (cid:0) (1 + e x ) d e x d − e ϕ (cid:1) ′′ (1 + e x ) d e x d − , (2.13) | Ric g e F | = 1(1 + e x ) | Ric g e φ | − e σ (1 + e x ) k g e φ + ( e σ ′ ) + d (cid:18) e σ e x (cid:19) + ( d − (cid:18) e σ − d e x (cid:19) , (2.14) △ g e F k g e F = 1(1 + e x ) ( △ g e φ k g e φ ) − (cid:0) e ϕ (1 + e x ) d − e x d − (cid:1) ′ (1 + e x ) d e x d − k g e φ + (cid:0) e ϕ e χ ′ (1 + e x ) d e x d − (cid:1) ′ (1 + e x ) d e x d − (2.15)and | R g e F | = 1(1 + e x ) | R g e φ | − e ϕ (1 + e x ) k g e φ + 2 d ( d + 1) e ϕ (1 + e x ) + 4 d (cid:18)(cid:18) e ϕ e x (cid:19) ′ (cid:19) + (cid:0) e ϕ ′′ (cid:1) + ( d − ( d (cid:18) e ϕ e x (1 + e x ) (cid:19) + 4 (cid:18)(cid:18) e ϕ e x (cid:19) ′ (cid:19) + 2 d (cid:18) e ϕ − e x e x (cid:19) ) . (2.16)egular quantizations 8The above (2.13), (2.14), (2.15) and (2.16) combine with (2.10), (2.11) and (2.12), we get (2.1), (2.2),(2.3) and (2.4).For the case of λ <
0, since k g F , | Ric g F | , △ g F k g F and | R g F | are rational functions in λ under givenany ( z, x ) ∈ M , it follows that (2.1), (2.2), (2.3) and (2.4) still hold for λ < M, g F ) to be constants. Theorem 2.2.
Under assumptions of Lemma 2.1, let F ( −∞ ) = 0 and n = d + d . (I) For M = n ( z, w ) ∈ Ω × C d : k w k < e − λφ ( z ) o , if the fibre metric of g F is complete, then both a ( g F )1 and a ( g F )2 are constants on M if and only if (i) For d = 1 , F ( t ) = − A log(1 − e t ) , A > ,a ( g φ )1 = d λ − nA, λ > ,a ( g φ )2 = 0 . (2.17)(ii) For d > , F ( t ) = − λ log(1 − e t ) , λ > ,a ( g φ )1 = − d ( d + 1) λ,a ( g φ )2 = ( d − d ( d + 1)(3 d + 2) λ . (2.18)(II) For M = Ω × C d , if the fibre metric of g F is complete, then both a ( g F )1 and a ( g F )2 are constantson M if and only if d = 1 ,F ( t ) = ce t , c > ,a ( g φ )1 = d λ, λ > ,a ( g φ )2 = 0 . (2.19)(III) For M = Ω × C d , if the fibre metric of g F is incomplete, then both a ( g F )1 and a ( g F )2 are constantson M if and only if (i) For d = 1 , F ( t ) = − A log(1 + ce t ) , A < , λ ≥ A, c > ,a ( g φ )1 = d λ − nA,a ( g φ )2 = 0 . (2.20)egular quantizations 9(ii) For d > , F ( t ) = − λ log(1 + ce t ) , λ < , c > ,a ( g φ )1 = − d ( d + 1) λ,a ( g φ )2 = ( d − d ( d + 1)(3 d + 2) λ . (2.21) Proof.
Our first goal is to show that ϕ is a polynomial if both a ( g F )1 and a ( g F )2 are constants.By (2.5) and (2.6), if both a ( g F )1 and a ( g F )2 are constants, then a ( g φ )1 and a ( g φ )2 are constants, thus | R g φ | − | Ric g φ | and | R g F | − | Ric g F | also are constants. From (2.5), ϕ ( x ) can be written as ϕ ( x ) = X j =0 A j x j + d − X j =1 B j x j + d X j =1 C j (1 + λx ) j . (2.22)Substituting (2.22) into (2.4) and (2.2), we have | R g F | − | Ric g F | = p ( d − , d − B d − x d − + p ( d, d ) λ C d (1 + λx ) d +4 + · · · , where p ( d, k ) = 2 d ( d + 1) + 4 d ( k + 1) + k ( k + 1) − d + ( k + 1) )( d − k ) . For d >
1, from p ( d − , d − > p ( d, d ) >
0, we have B d − = C d = 0, so | R g F | − | Ric g F | = p ( d − , d − B d − x d − + p ( d, d − λ C d − (1 + λx ) d − + · · · , which follows that B d − = 0 for d > C d − = 0 for d >
1. Thus from (2.22), ϕ ( x ) can bewritten as ϕ ( x ) = X j =0 A j x j + d − X j =1 B j x j + d − X j =1 C j (1 + λx ) j . (2.23)Let S := { j : B j = 0 , ≤ j ≤ d − } , S := { j : C j = 0 , ≤ j ≤ d − } . If S = ∅ , or S = ∅ , let k j := max S j , j = 0 , . Substituting (2.23) into (2.5), we obtain a ( g F )1 = − q ( d − , k ) B k x k +2 + · · · or a ( g F )1 = − q ( d, k ) λ C k λx ) k +2 + · · · for k > k >
0, respectively. Here q ( d, k ) = ( d − k )( d − k − . egular quantizations 10Then B k = 0 or C k = 0, this conflicts with the definitions of k j , j = 0 ,
1. Namely, B j = 0 for1 ≤ j ≤ d − C j = 0 for 1 ≤ j ≤ d − g F is defined at t = −∞ , it follows that ϕ (0) = 0 and ϕ ′ (0) = 1. Hence ϕ ( x ) = x + Ax .Substituting ϕ ( x ) = x + Ax into (2.5), we have2 a ( g F )1 = − A ( d + d + 1)( d + d ) + 2 a ( g φ )1 + d (2 Ad + 2 Ad − dλ − d λ + λ )1 + λx − d ( d − A − λ )(1 + λx ) , (2.24)it follows that d = 1 ,ϕ ( x ) = x + Ax ,a ( g φ )1 = d λ − nA,a ( g F )1 = − n ( n + 1) A (2.25)and d > ,ϕ ( x ) = x + λx ,a ( g φ )1 = − d ( d + 1) λ,a ( g F )1 = − n ( n + 1) λ, (2.26)where n = d + d .For the case of d = 1, substituting (2.25) into (2.2) and (2.4), we have | R g F | − | Ric g F | = − n ( n + 1)(2 n + 1) A + | R g φ | − | Ric g φ | + 12( d λ − nA ) (1 + λx ) , (2.27)thus | R g φ | − | Ric g φ | = − d λ − nA ) , | R g F | − | Ric g F | = − n ( n + 1)(2 n + 1) A . (2.28)So a ( g φ )2 = 0 ,a ( g F )2 = ( n − n ( n + 1)(3 n + 2) A . (2.29)For the case of d >
1, by (2.26), (2.2) and (2.4), we get | R g F | − | Ric g F | = − n ( n + 1)(2 n + 1) λ + | R g φ | − | Ric g φ | + 2 d ( d + 1)(2 d + 1) λ (1 + λx ) , (2.30)then | R g φ | − | Ric g φ | = − d ( d + 1)(2 d + 1) λ , | R g F | − | Ric g F | = − n ( n + 1)(2 n + 1) λ . (2.31)egular quantizations 11Therefore a ( g φ )2 = ( d − d ( d + 1)(3 d + 2) λ ,a ( g F )2 = ( n − n ( n + 1)(3 n + 2) λ . (2.32)Using (3.9) of [16], it follows that F ( t ) = − A log | ce t | , A = 0 ,ce t , A = 0 , c >
0; (2.33)For the case of M = n ( z, w ) ∈ Ω × C d : k w k < e − λφ ( z ) o . The completeness of the fibre metric of g F requires Z − p F ′′ ( t ) dt = + ∞ . Using (2.33), we have F ( t ) = − A log (cid:0) − e t (cid:1) . From g F is a K¨ahler metric on M , it follows that 1 + λF ′ ( t ) > F ′′ ( t ) > −∞ < t <
0. Soby λ = 0, we get A > λ > M = Ω × C d . If A = 0, then d = 1 and F ( t ) = ce t , it is easy to see that the fibremetric of g F is complete at this time. If A = 0, since 1 + ce t = 0 for −∞ ≤ t < + ∞ , it follows that c >
0. By (2.33), we have F ( t ) = − A log (cid:0) ce t (cid:1) , c > . As λ = 0, 1 + λF ′ ( t ) > F ′′ ( t ) > −∞ < t < + ∞ , it follows that A < A ≤ λ .The following we give sufficient and necessary conditions for some trivial Hermitian holomorphicvector bundles and their ball subbundles to be regular quantizations. Theorem 2.3.
Let g φ be a K¨ahler metric on a domain Ω ⊂ C d associated with the K¨ahler form ω φ = √− π ∂∂φ , where φ is a globally defined real analytic K¨ahler potential on Ω . For = λ ∈ R , define M := n ( z, w ) ∈ Ω × C d : k w k < e − λφ ( z ) o or Ω × C d . Set g F is a K¨ahler metric on the domain M associated with the K¨ahler form ω F = √− π ∂∂ Φ F , here Φ F ( z, w ) := φ ( z ) + F ( e λφ ( z ) k w k ) , F (0) = 0 . Suppose that there exists a set E ⊂ [0 , + ∞ ) such that H α (Ω) := ( f ∈ Hol (cid:0) Ω (cid:1) : Z Ω | f | e − αφ ω dφ d ! < + ∞ ) = { } for α ∈ E , where Hol(Ω) denotes the space of holomorphic functions on Ω , and E satisfies E = E + λ N := { α + λk : α ∈ E, k ∈ N } egular quantizations 12 for λ > .Let K α ( z, ¯ z ) be the reproducing kernels of H α (Ω) , the Bergman functions ǫ αg φ for (Ω , αg φ ) definedby ǫ αg φ = e − αφ ( z ) K α ( z, ¯ z ) . In the same way, define the Bergman functions ǫ αg F for ( M, αg F ) . (I) Given a number α ∈ E , assume that the fibre metric of g F is complete, then Bergman functions ǫ αg F are constants on the domain M = n ( z, w ) ∈ Ω × C d : k w k < e − λφ ( z ) o for all α ∈ E ∩ [ α , + ∞ ) if and only if (i) For the case of d = 1 , F ( ρ ) = − A log(1 − ρ ) , A > ,ǫ αg φ = α + d λ − nA, λ > ,ǫ αg F = Q nj =1 ( α − jA ) ,α ∈ ( nA, + ∞ ) ∩ [ α , + ∞ ) ∩ E, (2.34) where n = d + d . (ii) For the case of d > , F ( ρ ) = − λ log(1 − ρ ) , λ > ,ǫ αg φ = Q dj =1 ( α − jλ ) ,ǫ αg F = Q nj =1 ( α − jλ ) ,α ∈ ( nA, + ∞ ) ∩ [ α , + ∞ ) ∩ E. (2.35)(II) Let λ = 1 and α ∈ E , suppose that the fibre metric of g F is complete, then Bergman functions ǫ αg F are constants on the domain M = Ω × C d for all α ∈ E ∩ [ α , + ∞ ) if and only if d = 1 ,F ( ρ ) = cρ, c > ,ǫ αg φ = α + d ,ǫ αg F = α n ,α ∈ (0 , + ∞ ) ∩ [ α , + ∞ ) ∩ E. (2.36)(III) For d > , λ = − and E = N , let α ∈ N , then Bergman functions ǫ αg F are constants on thedomain M = Ω × C d for all α ∈ N ∩ [ α , + ∞ ) if and only if F ( ρ ) = log(1 + cρ ) , c > ,ǫ αg φ = Q dj =1 ( α + j ) ,ǫ αg F = Q nj =1 ( α + j ) ,α ∈ N ∩ [ α , + ∞ ) . (2.37)egular quantizations 13 Proof.
The following we only prove necessary conditions, sufficiency conditions are obvious.As ǫ αg F are constants on M for all α ∈ E ∩ [ α , + ∞ ), thus ǫ αg F ∼ α n + a ( g F )1 α n − + a ( g F )2 α n − + · · · , α → + ∞ , where a ( g F )1 and a ( g F )2 are constants, and a ( g F )1 = 12 k g F , a ( g F )2 = 13 △ g F k g F + 124 | R g F | − | Ric g F | + 18 k g F . By Theorem 2.2, we have F ( ρ ) = − A log (1 − ρ ) , d = 1 , − λ log (1 − ρ ) , d > M = (cid:8) ( z, w ) ∈ Ω × C d : k w k < e − λφ ( z ) (cid:9) , and F ( ρ ) = cρ, d = 1 , c > , log (1 + cρ ) , d > , c > M = Ω × C d .Put H α ( M ) := (cid:26) f ∈ Hol (cid:0) M (cid:1) : 1 π n Z M | f ( Z ) | e − α Φ F ( Z ) det( ∂ Φ F ∂Z t ∂Z )( Z ) dm ( Z ) < + ∞ (cid:27) , where n = d + d , Z = ( z, w ), and dm ( Z ) denotes the Lebesgue measure.Let f ∈ H α ( M ), then f ( z, w ) = X m f m ( z ) w m , (2.38)where m = ( m , m , . . . , m d ), m i ∈ N , 1 ≤ i ≤ d , w m = Q d i =1 w m i i .From Lemma 2.1 of [17], we get e − α Φ F ( Z ) det( ∂ Φ F ∂Z t ∂Z )( Z ) = e − ( α − λd ) φ ( z ) det( ∂ φ∂z t ∂ ¯ z )( z ) H ( α, ρ ) , here H ( α, ρ ) = e − αF ( ρ ) ( F ′ ( ρ )) d − ( F ′ ( ρ ) + ρF ′′ ( ρ ))(1 + λρF ′ ( ρ )) d . Using the measures e − α Φ F ( Z ) det( ∂ Φ F ∂Z t ∂Z )( Z ) dm ( Z ) are invariant under transformations( z, w , w , . . . , w d ) (cid:16) z, e √− θ w , e √− θ w , . . . , e √− θ d w d (cid:17) , ∀ θ i ∈ R , ≤ i ≤ d , we have 1 π n Z M | f ( Z ) | e − α Φ F ( Z ) det( ∂ Φ F ∂Z t ∂Z )( Z ) dm ( Z )= 1 π n X m Z M | f m ( z ) | w m w m e − α Φ F ( Z ) det( ∂ Φ F ∂Z t ∂Z )( Z ) dm ( Z )= X m π d Z Ω | f m ( z ) | e − ( α + λ | m | ) φ ( z ) × det( ∂ φ∂z t ∂ ¯ z )( z ) dm ( z ) × I m , (2.39)egular quantizations 14here | m | = P d i =1 m i and I m = 1 π d Z B d w m w m H ( α, k w k ) dm ( w ) or 1 π d Z C d w m w m H ( α, k w k ) dm ( w ) . (2.40)Let dσ be the Euclidean invariant measure on the sphere S d − , using integral formulas Z S d − w m w m dσ ( w ) = 2 π d Q d i =1 Γ(1 + m i )Γ( | m | + d ) , (2.41)we obtain I m = Q d i =1 Γ(1 + m i )Γ( | m | + 1) ψ ( α, | m | ) , (2.42)where ψ ( α, k ) = Γ( k + 1)Γ( k + d ) Z u k + d − H ( α, u ) du or Γ( k + 1)Γ( k + d ) Z + ∞ u k + d − H ( α, u ) du. (2.43)According to (2.39) and (2.42), we get the orthogonal direct sum decompositions H α ( M ) = M I m < + ∞ ,α + λ | m |∈ E H α + λ | m | (Ω) ⊗ { cw m : c ∈ C } for α ∈ E . Then the reproducing kernels of H α ( M ) can be expressed as K α ( Z, Z ) = X I m < + ∞ ,α + λ | m |∈ E K α + λ | m | ( z, ¯ z ) 1 ψ ( α, | m | ) Γ( | m | + 1) Q d i =1 Γ(1 + m i ) w m w m = X k ∈ N ,α + λk ∈ E K α + λk ( z, ¯ z, ) 1 ψ ( α, k ) k w k k , (2.44)where K α ( z, ¯ z ) are the reproducing kernels of H α (Ω).Applying K α ( z, ¯ z ) = e αφ ( z ) ǫ αg φ ( z ), K α ( z, w, z, w ) = e α Φ F ( z,w ) ǫ αg F ( z, w ) and (2.44), we have ǫ αg F ( z, w ) = e − αF ( ρ ) X k ∈ N ,α + λk ∈ E ǫ ( α + λk ) g φ ( z ) ψ ( α, k ) ρ k , (2.45)where ρ = e λφ ( z ) k w k . Talking w = 0 in (2.45), using F (0) = 0, we get ǫ αg F ( z,
0) = ǫ αg φ ( z ) ψ ( α, , (2.46)by ǫ αg F ( z, w ) are constants on M , it follows that ǫ αg φ ( z ) are constants on Ω.Substituting (2.46) into (2.45), we have e αF ( ρ ) = X k ∈ N ,α + λk ∈ E ǫ ( α + λk ) g φ ǫ αg φ ψ ( α, ψ ( α, k ) ρ k , α ∈ [ α , + ∞ ) ∩ E. (2.47)(I) For M = n ( z, w ) ∈ Ω × C d : k w k < e − λφ ( z ) o , egular quantizations 15using F ( ρ ) = − A log (1 − ρ ) , d = 1 , − λ log (1 − ρ ) , d > , then (2.47) becomes + ∞ X k =0 ǫ ( α + λk ) g φ ǫ αg φ ψ ( α, ψ ( α, k ) ρ k = (1 − ρ ) − αA , d = 1 , (1 − ρ ) − αλ , d > , (2.48)where ψ ( α, k ) = Γ( k +1)Γ( αA − n ) × ( α + λk + d λ − nA ) A n Γ( αA + k ) , α ∈ ( nA, + ∞ ) ∩ E, d = 1 , Γ( k +1)Γ( αλ − n ) λ d Γ( αλ + k − d ) , α ∈ ( nλ, + ∞ ) ∩ E, d > . (2.49)Using (2.48) and (1 − u ) − α = + ∞ X k =0 Γ( α + k )Γ( α )Γ( k + 1) u k , | u | < , we get ǫ ( α + λk ) g φ ǫ αg φ ψ ( α, ψ ( α, k ) = Γ( αA + k )Γ( αA )Γ( k +1) , d = 1 , Γ( αλ + k )Γ( αλ )Γ( k +1) , d > . Which combines with (2.49), we obtain ǫ ( α + λk ) g φ ǫ αg φ = α + λk + d λ − nAα + d λ − nA , d = 1 , Γ( αλ + k )Γ( αλ + k − d ) Γ( αλ − d )Γ( αλ ) , d > . (2.50)Combining (2.50) and lim k → + ∞ ǫ ( α + λk ) g φ ( λk ) d ǫ αg φ = 1 ǫ αg φ , it follows that ǫ αg φ = α + d λ − nA, d = 1 , λ d Γ( αλ )Γ( αλ − d ) , d > . (2.51)Using (2.45), (2.49) and (2.51), we have ǫ αg F = A n Γ( αA )Γ( αA − n ) , d = 1 , λ n Γ( αλ )Γ( αλ − n ) , d > α ∈ ( nA, + ∞ ) ∩ [ α , + ∞ ) ∩ E .(II) For M = Ω × C d , d = 1, c > λ = 1, substituting F ( ρ ) = cρ into (2.47), we obtain + ∞ X k =0 ǫ ( α + λk ) g φ ǫ αg φ ψ ( α, ψ ( α, k ) ρ k = e cαρ , (2.52)egular quantizations 16where ψ ( α, k ) = Γ( k + 1) × ( α + λk + λd ) c k α k + d +1 , α ∈ E ∩ (0 , + ∞ ) . (2.53)From (2.52) and (2.53), we infer that ǫ ( α + λk ) g φ ǫ αg φ = α + λk + λd α + λd , α ∈ E ∩ (0 , + ∞ ) ∩ [ α , + ∞ ) , (2.54)By ǫ αg φ ∼ α, α → + ∞ , we get ǫ αg φ = α + λd = α + d , which combines with (2.45) and (2.53), it follows that ǫ αg F = α d , α ∈ E ∩ (0 , + ∞ ) ∩ [ α , + ∞ ) . (III) For M = Ω × C d and λ = −
1, using F ( ρ ) = log(1 + cρ ) , c > , we infer from (2.47) that α X k =0 ǫ ( α − k ) g φ ǫ αg φ ψ ( α, ψ ( α, k ) ρ k = (1 + cρ ) α , α ∈ E ∩ [ α , + ∞ ) , (2.55)where ψ ( α, k ) = Γ( k + 1)Γ( α − k + d + 1) c k Γ( α + n + 1) , α ∈ E, ≤ k ≤ α. (2.56)Using (2.55) and (2.56), we have ǫ ( α − k ) g φ ǫ αg φ = Q dj =1 ( α − k + j ) Q dj =1 ( α + j ) , α ∈ E ∩ [ α , + ∞ ) , ≤ k ≤ α. Namely ǫ kg φ ǫ αg φ = Q dj =1 ( k + j ) Q dj =1 ( α + j ) , α ∈ E ∩ [ α , + ∞ ) , ≤ k ≤ α. (2.57)According to (2.57) and lim k → + ∞ ǫ αg φ α d = 1 , we have ǫ kg φ = d Y j =1 ( k + j ) . So from (2.45) and (2.56), we obtain ǫ αg F = n Y j =1 ( α + j ) , α ∈ E ∩ [ α , + ∞ ) . egular quantizations 17 In order to prove Theorem 1.1, we need the following lemma.
Lemma 3.1.
Let ( L, h ) be a Hermitian holomorphic line bundle over a K¨ahler manifold ( M, ω ) ofcomplex dimension n . Let f be a holomorphic section on M − Y such that Z M − Y h ( f ( z ) , f ( z )) ω n n ! < + ∞ , where Y is an analytic set of M . Then f extends to a (unique) global holomorphic section, namelythere exists s ∈ H ( M, L ) such that s ( z ) = f ( z ) for all z ∈ M − Y . Lemma 3.1 can be proved by using Skoda’s lemma (see e.g. Lemma 2.3.22 of [35]) and its application([35], (ii) in Lemma 6.2.1), for details refer to Lemma 4.1 of [29].
Proof of Theorem
The part (II) can be proved in the same way as the part (I), so only theproof of the part (I) is given.Suppose U is biholomorphically equivalent to a domain Ω ⊂ C d , and θ : L ⊃ π − ( U ) → U × C is an analytic homeomorphism such that for every z ∈ U the map θ : π − ( z ) → { z } × C → C is a linear isomorphism.Let τ ( z ) := θ − ( z,
1) ( z ∈ U ) be a trivializing holomorphic section, then π − ( U ) = { v = uτ ( z ) ∈ L : z ∈ U, u ∈ C } , and there exists a globally defined real function φ on U such that h ( v , v ) = e − φ ( z ) | u | , v = uτ ( z ) ∈ π − ( U ) . For the vect bundle L ∗⊕ r over M , since θ p : L ∗⊕ r ⊃ p − ( U ) −→ U × C r , ( z, wτ ∗ ( z )) ( z, w )is an analytic homeomorphism, it follows that θ p ◦ Π : L ⊃ ( p ◦ Π) − ( U ) −→ p − ( U ) × C −→ U × C r × C , ( z, wτ ∗ ( z ) , vτ ( z )) ( z, wτ ∗ ( z ) , v ) ( z, w, v )is also an analytic homeomorphism, where τ ∗ ( z ) are the duals of τ ( z ). So h ( x , x ) = e φ ( z ) k w k , x = ( z, wτ ∗ ( z )) ∈ p − ( U ) ,M ∩ p − ( U ) = B ( L ∗⊕ r ) ∩ p − ( U ) = n ( z, wτ ∗ ( z )) : z ∈ U, w ∈ C r , e φ ( z ) k w k < o ,h F ( y , y ) = e − F ( e φ ( z ) k w k ) e − φ ( z ) | v | , y = ( z, wτ ∗ ( z ) , vτ ( z )) ∈ ( p ◦ Π) − ( U )egular quantizations 18and ω F | M ∩ p − ( U ) = √− π ∂∂ Φ F , Φ F ( z, wτ ∗ ( z )) = φ ( z ) + F ( e φ ( z ) k w k ) . For any s ∈ H ( M, L m ), then s | M ∩ p − ( U ) ( x ) = f ( z, w ) τ m ( z ) , x = ( z, wτ ∗ ( z )) ∈ p − ( U ) , where f is a holomorphic function on a domain e U = n ( z, w ) ∈ U × C r : e φ ( z ) k w k < o . Let H m M ) = ( s ∈ H ( M, L m ) : Z M h mF ( s, s ) ω d + rF ( d + r )! < + ∞ ) and H m ( e U ) = ( f ∈ Hol (cid:0) e U (cid:1) : Z e U | f ( Z ) | e − m Φ F ( Z ) ̟ d + rF ( Z )( d + r )! < + ∞ ) , where ̟ F = √− π ∂∂ e Φ F , e Φ F ( Z ) = φ ( z ) + F ( e φ ( z ) k w k ) for Z = ( z, w ) ∈ e U , and Hol (cid:0) e U (cid:1) stands for thespace of holomorphic functions on e U .Since U is a dense open contractible subset of M , it follows that the correspondence e U ∋ ( z, w ) ( z, wτ ∗ ( z )) ∈ M sets up a bijection between the Hartogs domain e U and a dense open subset of M . By p − ( M − U ) isan analytic set and Lemma 3.1, there exists an isometric isomorphism between H m ( M ) and H m ( e U ),that is s ∈ H m ( M ) f ∈ H m ( e U )if s | M ∩ p − ( U ) ( x ) = f ( z, w ) τ m ( z ), for x = ( z, wτ ∗ ( z )) ∈ p − ( U ). Thus ǫ mg F ( z, wτ ∗ ( z )) = ǫ m e g F ( z, w ) , ( z, w ) ∈ e U , (3.1)here e g F is a K¨ahler metric on e U associated to the K¨ahler form ̟ F = √− π ∂∂ e Φ F . Similarly, we have ǫ mg ( z ) = ǫ mg φ ( z ) , z ∈ U, (3.2)where g φ is a K¨ahler metric on U associated to the K¨ahler form ω φ = √− π ∂∂φ .As ǫ mg and ǫ mg F are continuous functions, by U and M ∩ p − ( U ) are dense open subsets of M and M , respectively, it follows that both ǫ mg φ and ǫ m e g F are constants if and only if both ǫ mg and ǫ mg F are constants. So using Theorem 2.3, (3.1) and (3.2), we obtain (1.7) and (1.8). In this section, using Theorem 2.2, Theorem 2.3, the Hirzebruch-Riemann-Roch formula and Kobayashi-Ochiai’s characterization of the projective spaces, we give a proof of Theorem 1.2.egular quantizations 19
Proof of Theorem
Let n = d + r . Denote g as a K¨ahler metric associated with the K¨ahlerform ω .Because ( M, ω ) admits the regular quantization, from [26], we obtain ǫ mg ∼ + ∞ X j =0 a ( g ) j m n − j , m → + ∞ , and all a ( g ) j are constants on M . Thus all a ( g ) j are constants on E , using Theorem 2.2, it follows that a ( g ) j (1 ≤ j ≤
2) are constants on M , F ( ρ ) = − A log(1 + cρ ) , λ ≥ A, d = 1 , − λ log(1 + cρ ) , d > a ( g )1 = d λ − nA, d = 1 , − d ( d + 1) λ, d > , where c > λ = −
1. According to Lemma 4.1 of [19] and the explicit expression of function F , ω can be extended across M − E .For d = 1, the Ricci form Ric g > a ( g )1 is a constant on M and a ( g )1 = d λ − nA = − d − nA ≥ − d + n = d = 1 , it follows from [23] that M is simply connected. Using the uniformization theorem, we infer that M is biholomorphic to a complex projective space CP .Due to Ric g = 2 a ( g )1 c ( L , h ) , it follows that 2 ≤ a ( g )1 = R M Ric g R M c ( L , h ) = R M c ( M ) R M c ( L ) = 2 k , k ≥ , k ∈ N , where c ( L ) and c ( M ) denote Chern classes of L and M , respectively. So a ( g )1 = 1 and A = − d >
1, it follows from Theorem 2.3 that + ∞ X j =0 a ( g ) j (cid:12)(cid:12)(cid:12) Ω m d − j = d Y j =1 ( m + j ) , m ∈ N . Thanks to Ω is a dense subset of M , then + ∞ X j =0 a ( g ) j m d − j = d Y j =1 ( m + j ) , m ∈ N . Now recall the Hirzebruch-Riemann-Roch formula. Let V be a holomorphic vector bundle on acompact complex manifold X . Let H j ( X, V ) denote the j-th cohomology of X with coefficients in thesheaf O ( V ) of germs of holomorphic sections of V . Let χ ( X, V ) := + ∞ X j =0 ( − j dim H j ( X, V )egular quantizations 20be the Euler-Poincar´e characteristic of a holomorphic vector bundle V over a compact complex man-ifold X . The Hirzebruch-Riemann-Roch formula is given by χ ( X, V ) = Z X Td( X )ch( V ) , where the total Todd class td( X ) is defined byTd( X ) := Y j ξ j − e − ξ j , if X j c j ( X ) x j = Y j (1 + ξ j x ) , and the total Chern character ch( E ) is given bych( V ) = X j e ζ j , if X j c j ( V ) x j = Y j (1 + ζ j x ) . The above c j ( V ) are Chern classes of V , and c j ( X ) are Chern classes of the holomorphic tangentbundle of X .For X = M and V = L m , from the Hirzebruch-Riemann-Roch formula, it follows that χ ( M , L m )is a polynomial in m . Since L is a positive holomorphic line bundle over the Fano manifold M , itfollows that c ( K M ⊗ L − m ) = c ( K M ) + c ( L − m ) = − c ( M ) − mc ( L ) < m ≥
0, where K M is the canonical line bundle of M . By the Kodaira vanishing theorem (e.g.,page 68 of [24]), we get dim H j ( M , L m ) = dim H d − j ( M , K M ⊗ L − m ) = 0for 1 ≤ j ≤ d and m ≥
0. So dim H ( M , L m ) is a polynomial of degree d in m for m ≥ ǫ mg ∼ + ∞ X j =0 a ( g ) j m d − j = d Y j =1 ( m + j ) , as m → + ∞ , we have dim H ( M , L m ) = Z M ǫ mg ω d d ! ∼ ∞ X j =0 a ( g ) j m d − j Z M ω d d ! as m → + ∞ . Hence dim H ( M , L m ) = d Y j =1 ( m + j ) Z M ω d d !for m ≥ H ( M , L m ) = 1 for m = 0, it follows that Z M ω d = Z M ( c ( L )) d = 1 and dim H ( M , L m ) = 1 d ! d Y j =1 ( m + j ) . According to Theorem 1.1 of [25], M is biholomorphic to the complex projective space CP d .Using Corollary 4.2 in [2], there is a positive integer j and an automorphism Υ ∈ Aut( M ) =PGL( d + 1 , C ) such that ω = j Υ ∗ ω F S . As a ( g )1 = d ( d + 1), consequently j = 1 and Ric g = ( d + 1) ω .Let φ be the K¨ahler potential of the K¨ahler form ω on the domain Ω ⊂ M such that the Hermitianmetric h on E | Ω can be written as h ( u, u ) = e − φ ( z ) k w k , u = wτ ( z ) ∈ E, z ∈ Ω . egular quantizations 21Thus ω ( z ) = √− π ∂∂φ ( z ) , z ∈ Ωand ω ( u ) = √− π ∂∂ (cid:16) φ ( z ) + log(1 + ce − φ ( z ) k w k ) (cid:17) , u = wτ ( z ) ∈ E, z ∈ Ω . So Ric g ( u ) = − √− π ∂∂ log (cid:16) c r e − rφ det( ∂∂φ )(1 + ce − φ ( z ) k w k ) − d − r − (cid:17) = √− π ∂∂ ( rφ − log det( ∂∂φ ) + ( n + 1) log(1 + ce − φ ( z ) k w k ))for u = wτ ( z ) ∈ E and z ∈ Ω.By Ric g | Ω = − √− π ∂∂ log det( ∂∂φ ) and Ric g = ( d + 1) ω , we get Ric g = ( n + 1) ω on E . Using ω can be extended across M − E , then Ric g = ( n + 1) ω on M . It follows from [25] that M isbiholomorphically isomorphism to the complex projective space CP n . Proof of Corollary
It is well-known that CP is defined by CP := ( C − { } ) / C ∗ = { [ z , z ] : | z | + | z | 6 = 0 } , where C ∗ acts by multiplication on C .The tautological line bundle O CP ( −
1) over CP is defined by O CP ( −
1) := { ([ z , z ] , w ) ∈ CP × C : w = λ ( z , z ) , λ ∈ C } , its dual line bundle, denoted by O CP (1). Set O CP ( − k ) := O CP ( − ⊗ k , O CP ( k ) := O CP (1) ⊗ k for a positive integer k .Let CP = U ∪ U be the standard open covering, where U := (cid:8) [ z , z ] ∈ CP : z = 0 (cid:9) = (cid:8) [1 , w ] ∈ CP : z ∈ C (cid:9) and U := (cid:8) [ z , z ] ∈ CP : z = 0 (cid:9) = (cid:8) [ w , ∈ CP : w ∈ C (cid:9) , then U i biholomorphically equivalent to the domain C .For the holomorphic line bundle O CP ( k ) over CP , there exists a trivialization θ α : O CP ( k ) ⊃ π − ( U α ) −→ U α × C ,ζ α τ α ( w α ) ( w α , ζ α )for α = 0 or 1, where τ α ( w α ) = θ − α ( w α , , w = 1 w , ζ = ζ w k . egular quantizations 22A smooth Hermitian metric h kF S of O CP ( k ) is determined by h kF S ( v , v ) := | ζ α | (1 + | w α | ) k , v = ζ α τ α ( w α ) ∈ π − ( U α ) ⊂ O CP ( k ) , α = 0 , , its the Chern curvature determined by ω := − √− π ∂∂ log 1(1 + | w α | ) k = k √− π ∂∂ log(1 + | w α | ) , α = 0 , . Let M := CP , L := O CP ( k ), h := h kF S and ω := ω . For m ≥
1, by L m = O CP ( mk ) and thespaces H ( M , L m ) of holomorphic sections of L m are equal to the spaces of homogeneous polynomialsof degrees mk on C , we have that there exist isometric isomorphisms between H m ( M ) = (cid:26) s ∈ H ( M , L m ) : k s k := Z M h m ( s, s ) ω < + ∞ (cid:27) and H m ( C ) := (cid:26) f ∈ Hol (cid:0) C (cid:1) : k f k := k √− π Z C | f ( z ) | | z | ) mk +2 dz ∧ d ¯ z < + ∞ (cid:27) . Now that the reproduce kernels of H m ( C ) are K m ( z, ¯ z ) = (cid:18) m + 1 k (cid:19) (1 + | z | ) mk , it follows that the Bergman functions for ( L m , M , h m ) are ǫ mg = m + k on U or U , so ǫ mg = m + k on M .The above shows that ( L , M , h ) = ( O CP ( k ) , CP , h kF S ) satisfies the conditions of Theorem 1.1,according to Theorem 1.1, Corollary 1.3 holds. Acknowledgments
The author would like to thank the referee for many helpful suggestions. Theauthor was supported by the Scientific Research Fund of Leshan Normal University (No. ZZ201818).
References [1] Aghedu F.C., Loi A: The Simanca metric admits a regular quantization. arXiv: 1809.04431, 2018.[2] Arezzo C., Loi A.: Moment maps, scalar curvature and quantization of K¨ahler manifolds. Comm. Math.Phys. , 543-559 (2004)[3] Arezzo C, Loi A, Zuddas F. : On homothetic balanced metrics. Annals of Global Analysis and Geometry,41(4): 473-491 (2012).[4] Berezin, F.A.: Quantization, Math. USSR Izvestiya , 1109-1163 (1974)[5] Berman, R., Berndtsson, B., Sj¨ostrand, J.: A direct approach to Bergman kernel asymptotics for positiveline bundles. Ark. Mat., (2), 197-217 (2008)[6] Bi E.C., Feng Z.M., Tu Z.H.: Balanced metrics on the Fock-Bargmann-Hartogs domains. Ann. GlobalAnal. Geom., , 349-359 (2016)[7] Cahen, M., Gutt, S., Rawnsley, J.: Quantization of K¨ahler manifolds. I: Geometric interpretation ofBerezin’s quantization. J. Geom. Phys. , 45-62 (1990)[8] Cahen, M., Gutt, S., Rawnsley, J.: Quantization of K¨ahler manifolds II, Trans. Am. Math. Soc. , 73-98(1993) egular quantizations 23 [9] Catlin, D.: The Bergman kernel and a theorem of Tian. Analysis and geometry in several complex variables(Katata, 1997), Trends Math., Birkh¨auser Boston, Boston, MA, pp. 1-23 (1999)[10] Dai X., Liu K., Ma X.: On the asymptotic expansion of Bergman kernel. J. Differential Geom., , 1-41(2006)[11] Donaldson, S.: Scalar curvature and projective embeddings, I. J. Differential Geom. , 479-522 (2001)[12] Engliˇs, M.: Berezin Quantization and Reproducing Kernels on Complex Domains, Trans. Amer. Math.Soc. , 411-479 (1996)[13] Engliˇs, M.: A Forelli-Rudin construction and asymptotics of weighted Bergman kernels. J. Funct. Anal. , 257-281 (2000)[14] Engliˇs, M.: The asymptotics of a Laplace integral on a K¨ahler manifold. J. Reine Angew. Math. , 1-39(2000)[15] Engliˇs, M.: Weighted Bergman kernels and balanced metrics. RIMS Kokyuroku , 40-54(2006)[16] Feng Z.M.: On the first two coefficients of the Bergman function expansion for radial metrics. Journal ofGeometry and Physics, , 256-271 (2017)[17] Feng, Z.M.: The first two coefficients of the Bergman function expansions for Cartan-Hartogs domains.International Journal of Mathematics, (6), 1850043 (45 page) (2018)[18] Feng, Z.M., Tu, Z.H.: Balanced metrics on some Hartogs type domains over bounded symmetric domains.Annals of Global Analysis and Geometry, , 305-333 (2015)[19] Fu J., Yau S.T., and Zhou W.: On complete constant scalar curvature K¨ahler metrics with Poincar´e-Mok-Yau asymptotic property. Communications in Analysis and Geometry, (3): 521-557 (2016).[20] Hsiao C.Y.: On the coefficients of the asymptotic expansion of the kernel of Berezin-Toeplitz quantization.Ann. Global Anal. Geom. (2), 207-245 (2012)[21] Hsiao C.Y., Marinescu G.: Asymptotics of spectral function of lower energy forms and Bergman kernel ofsemi-positive and big line bundles. Comm. Anal. Geom. (1), 1-108 (2014)[22] Hwang A., Singer M. : A momentum construction for circle-invariant K¨ahler metrics. Transactions of theAmerican Mathematical Society, (6), 2285-2325 (2002)[23] Kobayashi S. On compact K¨ahler manifolds with positive definite Ricci tensor. Annals of Mathematics, (3), 570-574 (1961)[24] Kobayashi S. Differential geometry of complex vector bundles. Princeton University Press, 1987[25] Kobayashi S, Ochiai T.: Characterizations of complex projective spaces and hyperquadrics. Journal ofMathematics of Kyoto University, (1), 31-47 (1973)[26] Loi A.: Regular quantizations of K¨ahler manifolds and constant scalar curvature metrics. Journal of Ge-ometry and Physics, (3), 354-364 (2005).[27] Loi A.: Regular quantizations and covering maps. Geometriae Dedicata, 123(1): 73-78 (2006).[28] Loi, A., Mossa, R.: Berezin quantization of homogeneous bounded domains. Geometriae Dedicata, (1),119-128 (2012)[29] Loi, A., Mossa, R., Zuddas, F.: The log-term of the Bergman kernel of the disc bundle over a homogeneousHodge manifold. Annals of Global Analysis and Geometry, , 35-51 (2017)[30] Loi, A., Zedda, M.: Balanced metrics on Hartogs domains. Abhandlungen aus dem Mathematischen Sem-inar der Universit¨at Hamburg (1), 69-77 (2011)[31] Loi, A., Zedda, M.: Balanced metrics on Cartan and Cartan-Hartogs domains. Math. Z. , 1077-1087(2012)[32] Loi A., Zedda M., Zuddas F.: Some remarks on the K¨ahler geometry of the Taub-NUT metrics. Annals ofGlobal Analysis and Geometry (4), 515-533 (2012)[33] Lu, Z.: On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch. Amer. J. Math. (2),235-273 (2000)[34] Lui´c, S.: Balanced Metrics and Noncommutative K¨ahler Geometry. Symmetry, Integrability and Geometry:Methods and Applications , 069, 15 pages (2010)[35] Ma, X. and Marinescu, G.: Holomorphic Morse inequalities and Bergman kernels. Progress in Mathematics,Vol. 254, Birkhˇauser Boston Inc., Boston, MA (2007) egular quantizations 24 [36] Ma, X. and Marinescu, G.: Generalized Bergman kernels on symplectic manifolds. Adv. Math. (4),1756-1815 (2008)[37] Ma, X. and Marinescu, G.: Berezin-Toeplitz quantization on K¨ahler manifolds. J. reine angew. Math. ,1-56 (2012)[38] Rawnsley, J.: Coherent states and K¨ahler manifolds. Q. J. Math. Oxford (2), 403-415 (1977)[39] Xu, H.: A closed formula for the asymptotic expansion of the Bergman kernel. Commun. Math. Phys. ,555-585 (2012)[40] Zelditch, S.: Szeg¨o kernels and a theorem of Tian. Internat. Math. Res. Notices6