Degeneration of Kaehler structures and half-form quantization of toric varieties
aa r X i v : . [ m a t h . DG ] D ec Degeneration of K¨ahler structuresand half-form quantization of toric varieties
William D. Kirwin ∗ , Jos´e M. Mour˜ao and Jo˜ao P. NunesSeptember 3, 2018 Center for Mathematical Analysis, Geometry and Dynamical SystemsandDepartment of MathematicsInstituto Superior T´ecnicoAv. Rovisco Pais1049-001 Lisbon, Portugal
Abstract
We study the half-form K¨ahler quantization of a smooth symplectic toric manifold (
X, ω ),such that [ ω/ π ] − c ( X ) / ∈ H ( X, Z ) and is nonnegative. We define the half-form correctedquantization of ( X, ω ) to be given by holomorphic sections of a certain hermitian line bundle L → X with Chern class [ ω/ π ] − c ( X ) /
2. These sections then correspond to integral pointsof a “corrected” polytope P L with integral vertices. For a suitably translated moment polytope P X for ( X, ω ), we have that P L ⊂ P X is obtained from P X by a one-half inward-pointing normalshift along the boundary.We use our results on the half-form corrected K¨ahler quantization to motivate a definitionof half-form corrected quantization in the singular real toric polarization. Using families ofcomplex structures studied in [BFMN11], which include the degeneration of K¨ahler polariza-tions to the vertical polarization, we show that, under this degeneration, the half-form corrected L -normalized monomial holomorphic sections converge to Dirac-delta-distributional sectionssupported on the fibers over the integral points of P L , which correspond to corrected Bohr–Sommerfeld fibers. This result and the limit of the corrected connection, with curvature sin-gularities along the boundary of P X , justifies the direct definition we give for the correctedquantization in the singular real toric polarization. We show that the space of quantum statesfor this definition coincides with the space obtained via degeneration of the K¨ahler quantization.We also show that the BKS pairing between K¨ahler polarizations is not unitary in general.On the other hand, the unitary connection induced by this pairing is flat. ∗ current address: Mathematics Institute, University of Cologne, Weyertal 86 – 90, 50931 Cologne, GERMANY ontents X . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 The K¨ahler structure of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 Line bundles and sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.4 The canonical bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 K Ever since ´Snyaticki proposed cohomological wave functions to construct the quantum Hilbert spacecorresponding to geometric quantization in real polarizations [´S75], the question of how to addressthe case of real polarizations with singular fibers has resisted full treatment. In [Ham07], Hamiltonproposed the extension of ´Snyaticki’s definition to the case with singular fibers by also consideringthe higher cohomology of the same sheaf of polarized smooth sections of the prequantization bundle.His results show, however, that the formalism will have to be modified in order to obtain theexpected quantization even in the case of the harmonic oscillator. Indeed, for singularities ofelliptic type (like in the case of toric varieties) Hamilton obtains states corresponding only to non-singular Bohr–Sommerfeld leaves. In the toric case, these correspond to interior integral pointsof the moment polytope. If one doesn’t take into account the half-form correction, however, oneexpects the quantization to include all states corresponding to the integral points of the polytope,including those on the boundary. Only in this way, for the compact case, does one get the samedimension of the space of quantum states as for the holomorphic polarizations.In [BFMN11], a solution of this problem was proposed within the context of toric varietieswithout the half-form correction. The real polarized sections are defined directly as distributionalsolutions of the equations of covariant constancy and can also be obtained by degenerating appro-priately normalized K¨ahler polarized sections. These normalized holomorphic sections converge,under the degeneration, to Dirac-delta-distributional sections supported on the Bohr–Sommerfeld2bers which correspond to integral points of the moment polytope, including the ones on the bound-ary. The corresponding Bohr–Sommerfeld orbits are increasingly singular (lower dimensional) asthe codimension of the face of the polytope on which they are increases.On the other hand, one would expect these quantum states not to be present in a quantizationin the real toric polarization correctly reproducing the “vacuum energy shift” of the harmonicoscillator. We show that this expected behavior of the quantum states is precisely achieved byour definition of the half-form corrected K¨ahler quantizations deforming continuously to the realpolarization.An immediate obstacle to defining the half-form quantization in a K¨ahler polarization is thefact that the canonical bundle K X of a toric variety may not admit a square root, for instance for CP n . (See the appendix for a discussion of the existence of √ K X in terms of the fan of X .) InSection 3, we consider K¨ahler quantization of a compact toric manifold X with symplectic structure ω such that [ ω ]2 π − c ( X )2 ∈ H ( X, Z ) and is nonnegative. (This integrality condition has also beenproposed in [C78].) In the case when c ( X ) is even, so that K X admits a square root, one is thenreduced to the usual setting for half-form quantization. Let L → X be an hermitian line bundlewith connection of curvature given by − iω + i ρ , where ρ is the Ricci form for the K¨ahler metric on X , so that [ ρ/ π ] ∈ c ( X ). When a √ K X exists, this corresponds to taking the usual prequantumconnection plus one-half the Chern–Levi–Civita connection on K X , which gives a connection on √ K X .The condition [ ω ]2 π − c ( X )2 ∈ H ( X, Z ) allows us to choose the moment polytope (see equations(3.6) and (3.7) in Section 3.2), P X = µ ( X ), P X = { x ∈ R n : ℓ j ( x ) = ν j · x + λ j ≥ , j = 1 , . . . , r } , with all λ j ’s half-integral λ j ∈
12 + Z , j = 1 . . . , r, where x are action coordinates and ν j is the primitive inward pointing normal vector to the j -thfacet. With this choice, there are no integral points in the boundary of P X and to all integralpoints inside P X there will correspond K¨ahler polarized states, that is holomorphic sections of L .Unlike Hamilton’s case however, the integral points start at lattice distance 1 / idθ jv in (4.2) corresponds to a connection with curvature supported on the inverse imageof the j -th facet by the moment map. From the point of view of the real polarization thesesingular connections are responsible for the vacuum energy shifts (which correspond to shifted Bohr–Sommerfeld conditions) as they prevent the existence of covariantly constant sections supported onthe boundary. Quantization in the real singular toric polarization is then defined directly in termsof this limit connection. This provides an approach for defining half-form corrected quantizationin real singular polarizations. By finding the type of singularities of the half-form corrected limitconnection, one finds corrected equations for the real polarized sections. In the toric case, thisdirect approach for the definition of the half-form corrected quantization in the singular toric realpolarization gives the same results as the degeneration of K¨ahler polarizations (Theorems 4.7 and4.15).In [BFMN11], the convergence to delta-distributions was achieved by taking L -normalizedsections. In the present paper, however, the half-form correction ensures the nice behavior of the3 -normalized sections in the limit of degenerating complex structure. This is in agreement withother examples such as finite-dimensional vector spaces [KW06] and abelian varieties [BMN10].One of the primary motivations for including the half-form correction is that it allows for acanonical pairing between quantizations associated to different complex structures. This pairingis known as the Blattner–Konstant–Sternberg (BKS) pairing. The BKS pairing between quantiza-tions associated to two K¨ahler complex structures is nondegenerate, and hence (since the K¨ahlerquantizations of a compact toric manifold are finite dimensional) induces an isomorphism betweenthem. One does not, though, expect in general that the BKS pairing provides a unitary identi-fication of quantizations associated to different complex structures. In several common cases, forexample for symplectic vector spaces equipped with translation invariant K¨ahler structures, and forcomplex Lie groups equipped with certain families of K¨ahkler structures (which include the canon-ical K¨ahler structure), the BKS pairing is unitary (see [Hal02], [KW06],[FMMN05],[FMMN06]). Ina few other cases, the BKS pairing is known to be not unitary, for example for T ∗ S [Raw79]. Inmost cases, it is not known whether the BKS pairing is unitary, and conditions for unitarity donot yet seem to be well understood. We show that the BKS pairing between half-form correctedquantizations of compact toric varieties is not unitary in general.We will also consider another method for comparing quantizations associated to different com-plex structures. Namely, one can construct a (finite-rank) Hilbert bundle over the space of toriccomplex structures on X with the fiber at a point being the quantum Hilbert space associatedto that complex structure. When the half-form correction is not included, the quantum Hilbertbundle is a subbundle of a trivial bundle, and hence carries a canonical connection obtained byorthogonal projection of the trivial connection. This connection is called the quantum connection. It was first introduced and studied by Axelrod, della Pietra and Witten in [APW91] and, froma slightly different point of view, by Hitchin in [Hit90]. See also [AGL07] for a treatment whichincludes the half-form correction.For linear complex structures on a symplectic vector space, the quantum connection turns outto be projectively flat, which means that up to a constant, one may identify all K¨ahler quantizationsat once. To extend the connection to the boundary of the space of complex structures, and thusstudy their degenerations and relate real quantizations to K¨ahler quantizations, one must introducethe half-form correction; parallel transport of the resulting corrected quantum connection, stillin the case of linear complex structures, was studied by the first author and Wu in [KW06],where it was found that parallel transport along geodesics with internal endpoints is just rescaledBergman projection, while transport along geodesics with one or two endpoints on the boundaryyields the well-known Segal–Bargmann and Fourier transforms, respectively. These results wererecently extended by Wu to the case of linear quantization of fermions [Wu10]. In the usual(bosonic) case, one may then quotient by the action of Z n , as done by Baier and the secondtwo authors in [BMN10], to study degenerations of complex structures on abelian varieties at thelevel of ϑ -functions. In a different direction, in [FMMN05] and [FMMN06], Florentino, Matias andthe second two authors studied the corrected quantum connection on a one-dimensional family ofcomplex structures on the complexification of a compact Lie group which degenerates to the verticalpolarization of the cotangent bundle of the underlying real Lie group; here, again, parallel transportwith respect to the quantum connection yields the (generalized) Segal–Bargmann–Hall transform.In related work, Lempert and Sz˝oke have recently studied the bundle of quantizations associated toa family of adapted-type complex structures on Grauert tubes of compact, real-analytic Riemannianmanifolds [LS10], although they use a Chern-type connection rather than the BKS constructionconsidered here.In Subsection 5.3, we show that the quantum connection on the quantum Hilbert bundle inducedby the BKS pairing is flat, so that the quantizations associated to different torus-invariant complexstructures can be canonically identified. 4 Preliminaries.
We begin with some facts about complex line bundles. Let E → X be a complex line bundle on amanifold X and let E = E \ { zero section } be its frame bundle. The isomorphism( | · | , arg ) : C ∗ ∼ = R + × U (1) c (cid:18) | c | , c | c | (cid:19) , defines a canonical isomorphism ([Wei04], page 6) E ∼ = | E | ⊗ E U (1) , where the complex line bundles | E | , E U (1) are associated to the principal C ∗ -bundle E , via thehomomorphisms C ∗ ∋ c
7→ | c | ∈ R + and C ∗ ∋ c arg ( c ) = c | c | ∈ U (1) respectively. Following[Wei04] we call the line bundle E U (1) the unitarization of E .This isomorphism is given explicitly by E p ∋ l (cid:26) | l | ⊗ l U (1) , l = 00 , l = 0 , where p ∈ X, | l | = [( l, |·| = [( lc − , | c | )] |·| ∈ | E | = E × ( C ∗ , |·| ) C and l U (1) = [( l, arg =[( lc − , c | c | )] arg ∈ E u (1) = E × ( C ∗ ,arg ) C , c ∈ C ∗ .For simplicity, we will identify E with | E | ⊗ E U (1) and write 0 = l = | l | ⊗ l U (1) = | l | l U (1) andthus, also, l U (1) = l | l | .Let { g αβ } be the transition functions for E associated to local trivializations for some opencover { U α } of X . Then, for the same open cover { U α } , the complex line bundle E U (1) has U (1)-valued transition functions { g αβ / | g αβ |} , and the complex line bundle | E | has R + -valued transitionfunctions {| g αβ |} .This decomposition of E = | E | ⊗ E U (1) induces an associated splitting of connections. Let ∇ be a connection on E with connection form Θ associated to a local trivializing section s , ∇ s = Θ s .Since, at the level of Lie algebras, the isomorphism C ∗ ∼ = R + × U (1) gives C ∼ = R ⊕ i R , we have ∇ = ∇ | E | + ∇ E U (1) where Θ | E | = Re Θ and Θ E U (1) = i Im Θ are the connection forms for | E | , E U (1) associated to the local trivializing sections | s | , s U (1) respectively: ∇ | E | | s | = Re Θ | s |∇ E U (1) s U (1) = i Im Θ s U (1) . Now let E have an hermitian structure h . Then | E | has a global trivializing section µ h definedas follows. Let s be any local trivializing section of E over an open set U ⊂ X . Over U , define µ h = | s | p h ( s, s ) . (2.1)Notice that, since h is an hermitian structure, µ h is independent of the choice of the local trivializingsection s and therefore extends to a global trivializing section of | E | . Let Γ( E ) denote the space of smooth sections of E . A connection ∇ on ( E, h ) → X is saidto be compatible with the hermitian structure if for any section s ∈ Γ( E ), one has dh ( s, s ) = h ( ∇ s, s ) + h ( s, ∇ s ). Let || s || = h ( s, s ). This property is equivalent to d || s || = Re Θ || s || which is,in turn, equivalent to ∇ | E | µ h = 0 . emark 2.1 The above isomorphism of | E | with the trivial bundle also defines, since E = | E | ⊗ E U (1) , an isomorphism of E with E U (1) given by l l √ h ( l,l ) | l | , l ∈ E . ♦ Remark 2.2 If X is a complex manifold and if E has an holomorphic structure, then given aglobal nonzero meromorphic section of E , s , one has µ h = | s | √ h ( s,s ) away from the divisor of s . Thisexpression extends uniquely to µ h on the whole of X . ♦ Any line bundle E U (1) has a canonical hermitian structure defined by e h ( zl U (1) , z ′ l U (1) ) = z ¯ z ′ .This hermitian structure is independent of the choice of representative l and is therefore well defined. Remark 2.3
An hermitian line bundle (
E, h ) → X can then be decomposed into smooth her-mitian line bundles ( E, h ) = ( | E | , b h ) ⊗ ( E U (1) , e h ). The hermitian structure on | E | is defined by b h ( z | l | , z ′ | l | ) = h ( l, l ) z ¯ z ′ , so that h ( zl, z ′ l ) = z ¯ z ′ b h ( | l | , | l | ) · e h ( l U (1) , l U (1) ) = z ¯ z ′ b h ( | l | , | l | ) , z, z ′ ∈ C , l ∈ E. ♦ Remark 2.4
Note that under the isomorphism | E | ≃ X × C defined by µ h , the hermitian form b h becomes the standard hermitian product on C . ♦ We recall the following standard results (see, for instance, Proposition 4.2.14 in [Huy05]):
Lemma 2.5 If ( E, h ) → X is a complex hermitian vector bundle, then there exists a compatibleconnection ∇ . Moreover, ∇ ′ is another compatible connection if and only if there exists a (global)real-valued -form β such that ∇ ′ = ∇ + iβ. If E → X is a holomorphic line bundle, then ¯ ∂ is a well-defined operator on Γ( E ). A connection ∇ on E is said to be compatible with the holomorphic structure if ∇ , = ¯ ∂ . Lemma 2.6
Let ( E, h ) → X be a hermitian holomorphic line bundle. There exists a uniqueconnection ∇ , called the Chern connection, which is compatible with both the hermitian structure h and the holomorphic structure of E . Moreover, if { U α , s α } is a holomorphic trivialization of E ,then ∇ s α = ( ∂ log h ( s α , s α )) s α . Then, in the holomorphic local trivialization { U α , s α } , we have F ∇ = d ( ∂ log h ( s α , s α )) = − ∂ ¯ ∂ log h ( s α , s α )that is, − log h ( s α , s α ) is a local potential for the curvature 2-form and, on the open set U α , [ i∂ ¯ ∂ ( − log h ( s α , s α ))] ∈ π · c ( E ) . The induced connections on | E | and E U (1) are then given by ∇ | E | | s | = 12 d log h ( s, s ) | s | , ∇ E U (1) s U (1) = 12 (cid:0) ∂ log h ( s, s ) − ¯ ∂ log h ( s, s ) (cid:1) s U (1) . If X is K¨ahler with integral symplectic form ω , and L is an hermitian holomorphic line bundlewith the curvature of the Chern connection given by − iω , then in a local holomorphic trivializationone has that κ = − log h ( s, s ) is a local K¨ahler potential.6 .2 Toric Manifolds Let (
X, ω ) be a compact smooth symplectic toric manifold with symplectic form ω , moment map µ : X → Lie ( T n ) ∗ ≃ R n and moment polytope P X = µ ( X ) with associated fan Σ. The K¨ahlerstructure of X , which connects the symplectomorphism class of X determined by P X to the biholo-morphism class of X determined by Σ, is fixed by choosing a so-called symplectic potential. Wewill find both descriptions, as well as the relation between them, essential for our work, and so wedescribe them briefly here (though we refer the interested reader to [Gui94], [Abr03], [CLS11] and[DP09] for details). X Let ˇ P X denote the interior of the moment polytope P X . On ˇ X = µ − ( ˇ P X ) ∼ = ˇ P X × T n consideraction-angle coordinates ( x, θ ), so that µ ( x, θ ) = x = t ( x , . . . , x n ). The symplectic form ω in thiscoordinate chart is simply ω | µ − ( ˇ P X ) = n X j =1 dx j ∧ dθ j . (2.2)The moment polytope P X is a Delzant polytope (see [D88] or page 698 of [DP09]) determinedby a set of inequalities { ℓ j ( x ) ≥ } j =1 ,...,r , where r is the number of facets of P X and for each j = 1 , . . . , r , ℓ j ( x ) = ν j · x + λ j where ν j is the (inward pointing) primitive integral vector normal to the j -th facet of P X , and λ j ∈ R . We now describe the coordinate chart associated to a vertex v ∈ P X . Since we assume X issmooth, the polytope is regular; that is, there are n facets adjacent to each vertex, with normalvectors forming a Z -basis of Z n . Reorder (if necessary) the inequalities so that the first n correspondto the facets adjacent to v . Then ℓ ( v ) = ℓ ( v ) = · · · = ℓ n ( v ) = 0 . Let A v ∈ GL n ( Z ) be thematrix whose rows are the vectors ν j , and let λ v = t ( λ , . . . , λ n ). Define new (vertex action-angle)coordinates x v on R n and θ v on T n by x v := A v x + λ v , and θ v := t A − v θ. (2.3)The image of the polytope P X under x v in (2.3) is also a Delzant polytope P vX , P vX = A v P X + λ v , (2.4)with the vertex v mapped to the origin and the codimension one faces meeting at the origincontained in the coordinate hyperplanes. Given a different vertex v ′ , with associated matrix A v ′ ∈ GL n ( Z ), and vector λ v ′ , the transition functions between the corresponding vertex action anglecoordinates read x v ′ = A v ′ A − v ( x v − λ v ) + λ v ′ θ v ′ = t A − v ′ t A v θ v . (2.5)For a face F ⊂ P X (that is, a linear boundary component of any codimension, including the polytopeitself, the facets and the edges), denote by ˇ F the interior of F , with the convention that ˇ { v } = { v } for vertices. The vertex chart neighborhood at v is defined to be the following T n -invariant openset U v := µ − [ faces F of P X adjacent to v ˇ F .
7e consider on U v coordinates { a jv , b jv } j =1 ,...,n related to the vertex action-angle coordinates { x jv , θ jv } j =1 ,...,n by a jv + ib jv = p x jv e iθ jv , j = 1 , . . . , n , on ˇ X (see, for example, Sections 3 and 4 of[DP09]). Since x v takes values in the polytope P vX ⊂ R n (2.4), (it is surjective to the polytopeminus the faces not containing the origin) and θ v ∈ R n / Z n , the image of U v under a v + ib v is abounded neighborhood of the origin in C n . The (non-holomorphic) transition functions betweencoordinate functions a v + ib v and a v ′ + ib v ′ for vertices v and v ′ can be obtained from (2.5) (seeSection 4 of [DP09]), a j ′ v ′ + ib j ′ v ′ = vuut n X j =1 ( A v ′ A − v ) j ′ j (cid:16) | a jv + ib jv | − λ jv (cid:17) + λ j ′ v ′ n Y j =1 a jv + ib jv | a jv + ib jv | ! ( t A − v ′ t A v ) j ′ j . We will also need the much simpler transition functions for holomorphic vertex coordinates (whichwill be introduced below in the section on K¨ahler structures).We note that the faces of P X correspond to points in X with nontrivial stabilizer as follows:suppose F is a face adjacent to v given by { x j s v = 0 } s =1 ,...,j F (so F is a codimension- j F face). Thenthe points in µ − ( F ) are fixed by the subtorus parameterized by the coordinates { ( θ j v , . . . , θ j F v ) } .Let V be the set of vertices of P X . We call { ( µ − ( ˇ P X ) , ( x, θ )) , ( U v , ( a v , b v )) : v ∈ V } the vertexatlas of X .The symplectic form in the vertex coordinate chart U v can be computed by pullback of (2.2)under the coordinate change (2.3) to be ω | U v = n X j =1 da jv ∧ db jv , and on U v ∩ ˇ X = ˇ X , ω | ˇ X = n X j =1 dx jv ∧ dθ jv . X In order to describe the toric K¨ahler structures on X , let us consider torus-invariant complexstructures on the symplectic toric manifold ( X, ω ) with moment polytope P X . Let g P X ∈ C ∞ ( ˇ P X )be g P X ( x ) = 12 r X j =1 ℓ j ( x ) log ℓ j ( x ) . (2.6) Definition 2.7
Let C ∞ P X ( P X ) be the set of smooth functions on P X such that ϕ ∈ C ∞ P X ( P X ) if Hess x ( g P X + ϕ ) is positive definite on ˇ P X and there exists a strictly positive function α ∈ C ∞ ( P X ) so that det(Hess x ( g P X + ϕ )) = α ( x ) r Y j =1 ℓ j ( x ) − (2.7) on ˇ P X . A torus-invariant complex structure on (
X, ω ) is determined by a symplectic potential g = g P X + ϕ, ϕ ∈ C ∞ P X ( P X ), see Section 4 of [Gui94] and Theorem 2.8 of [Abr03]. In the symplectic framedetermined by the action-angle coordinates ( x, θ ) on ˇ X , the toric complex structure I and themetric γ = ω ( · , I · ) tensors associated to the symplectic potential g are then I = (cid:18) − G − G (cid:19) and γ = (cid:18) G G − (cid:19) , (2.8)where G = Hess x g is the Hessian of g .Let us now relate these complex structures to the algebro-geometric description of toric mani-folds. By a standard construction, see, for example, Section 5 in [D88] or Definition 6.4.2 in [CdS],associated to the moment polytope P X there is an associated complete fan Σ. This fan definesa compact smooth toric variety Y diffeomorphic to X (see below) and with canonical complexstructure defined by Σ.The complex torus ( C ∗ ) n acts on Y with a dense open orbit (biholomorphic to ( C ∗ ) n ) which wehenceforth refer to as the open orbit . Let M denote the (integer lattice of) characters of ( C ∗ ) n , sothat after a choice of basis, M ≃ Z n . The characters of ( C ∗ ) n extend to meromorphic functionson Y with torus-invariant divisors.The toric variety Y has an atlas of holomorphic coordinates { ( V v , ˜ w v ) } v ∈ V , ˜ w v = ( ˜ w v , . . . , ˜ w nv ),where for each pair of vertices v, v ′ , over V v ∩ V v ′ the glueing conditions are given by˜ w v ′ = ˜ w A v A − v ′ v , (2.9)with A v A − v ′ interpreted as a row of multiindices; i.e. ˜ w jv ′ = Q nl =1 ( ˜ w lv ) ( A v A − v ′ ) lj . (See, for example,Section 5 of [DP09].)Denote the open orbit in Y by V . The symplectic potential g fixes a biholomorphism ( ˇ X, I ) ∼ =ˇ P X × T n ∼ = V ∼ = ( C ∗ ) n given byˇ P X × T n −→ V ≃ ( C ∗ ) n ( x, θ ) ˜ w = e y + iθ = ( e y + iθ , . . . , e y n + iθ n ) , (2.10)where y j = ∂g/∂x j . Note that this map is not a symplectomorphism with respect to the standardsymplectic structure on ( C ∗ ) n .The map x y = ∂g/∂x is a bijective Legendre transform. The inverse map is given by x = ∂h/∂y , where h is a K¨ahler potential given in terms of g by h := x · y − g .This biholomorphism extends uniquely to a biholomorphism ψ g : X → Y as follows. The complex structure associated to g defines holomorphic coordinates in the vertexcoordinate charts via the coordinate change (2.3) to yield U v −→ V v ( x v , θ v ) ˜ w v = e y v + iθ v , (2.11)where y jv := ∂g/∂x jv = P nk =1 (cid:0) A − v (cid:1) kj ∂g/∂x k . Using the observation that y v + iθ v = t A − v ( y + iθ ),one may verify that (2.9) is indeed satisfied. Similarly, on V we have˜ w = ˜ w A v v , (2.12) We will henceforth identify M ∼ = Z n and M ⊗ R ∼ = R n . I -dependent holomorphic vertex atlas { ( U v , w v ) } v ∈ V on X to be the pullback by ψ g of the holomorphic atlas { ( V v , ˜ w v ) } v ∈ V on Y . (We will also denote the pullback of the chart( V , ˜ w ) on Y to X by ( U , w ).) The I -dependent transition functions for the holomorphic coordinatecharts ( U v , w v ) , v ∈ V, ( U , w ) on X are therefore the pullbacks by ψ g of the corresponding transitionfunctions on Y in (2.9) and (2.12).Henceforth, we will assume that X is equipped with a K¨ahler structure determined by ω andby a symplectic potential g = g P X + ϕ, ϕ ∈ C ∞ P X ( P X ) . Since compact smooth toric varieties are simply connected, the Picard group of equivalence classesof holomorphic line bundles is isomorphic to H ( X, Z ), with isomorphism established by the firstChern class. In other words, fixing the first Chern class of a line bundle on the the complex toricmanifold X fixes the bundle up to isomorphism. (See the Corollary on p.64, on Section 3.4 of [F].)The linear equivalence classes of the torus-invariant divisors of X generate the Picard groupof X , and there is a one-to-one correspondence between irreducible torus-invariant divisors and1-cones in Σ (see, Part I, Chapter 4 of [CLS11]) . Denote the set of 1-cones in Σ by Σ (1) . The j -th1-cone in Σ (1) is generated by the primitive integral vector ν j normal to the j -th facet of P X . Then,the associated irreducible divisor D j = µ − ( { x ∈ P X : ℓ j ( x ) = ν j · x + λ j = 0 } ) is the inverse imageunder the moment map µ of that facet of P X . The Picard group is then generated by the linearequivalence classes of irreducible divisors D , . . . , D r . Consider a divisor D L = λ L D + · · · + λ Lr D r ,for λ L , . . . , λ Lr ∈ Z , defining a holomorphic line bundle L = O ( D L ) and a (unique up to constant)meromorphic section of L with divisor D L , σ D L .From [CLS11], the divisor of the (rational) function defined on the open orbit by w m , m ∈ Z n ,can be computed to be div( w m ) = r X j =1 h ν j , m i D j . (2.13)Then, we have H ( X, L ) = span C { w m σ D L : m ∈ Z n , div( w m σ D L ) ≥ } == span C (cid:8) w m σ D L : m ∈ Z n , h m, ν i i + λ Li ≥ , i = 1 , . . . , r (cid:9) . (2.14)Therefore, there is a natural bijection between a basis of H ( X, L ) whose elements are weightvectors for the action of the torus and the integral points of the Delzant polytope with integralvertices P L := { x ∈ R n : h x, ν j i + λ Lj ≥ , j = 1 , . . . , r } ⊂ R n . (2.15)For simplicity, let us assume that L is ample so that there is a canonical bijection betweenthe vertices of P L and the vertices of P X , defined by the equality of the set of normals of thefacets meeting at those vertices. (In fact, if L is ample there is a bijection between the facesof P X and of P L . See Section 3.2.1 of [CK].) Let us denote by the same symbol v a vertex of P L and the corresponding vertex of P X . The holomorphic section corresponding to the vertex v of P L will provide a local trivializing section on the open set U v , so that one obtains a globalsystem of local holomorphic trivializations for L . For such vertex v , we can order the inequalities { ℓ Lj ( m ) := h m, ν j i + λ Lj ≥ , j = 1 . . . r } so that ν , . . . , ν n are the normals to the facets of P L Note that we have different sign conventions for λ F than those used in [BFMN11]. We have chosen rather to followthe convention in [CLS11], as they seem to make certain equations more natural (for example, shifting λ F λ F + 1has the effect of shifting the facet F one unit along the outward pointing normal to F ). v ; this is the same ordering that we used in the definition of the vertex coordinates on X . Using this ordering, we set λ Lv = t ( λ Lv, , . . . , λ Lv,n ) := ( λ L , . . . , λ Ln ) . The holomorphic section corresponding to a vertex v of P L is given by v := w − λ Lv v σ D L . (2.16)Using (2.12) and (2.13), one obtains that the divisor of the meromorphic function w λ Lv v on U v isdiv U v ( w λ Lv v ) = (cid:0) λ Lv, D + · · · + λ Lv,n D n (cid:1) ∩ U v , and therefore div U v ( v ) = 0, so that v is a trivializing holomorphic section of L on U v = X \{∪ rj = n +1 D j } . We remark that these sections are determined up to a constant by their divisors andthey are therefore defined for every line bundle in the isomorphism class of L .For notational convenience, let = σ D L . Using (2.16), we may compute the transition functionsfor L relative to the holomorphic vertex atlas obtaining, g Lv ′ v := v / v ′ = w λ Lv ′ v ′ /w λ Lv v and g L v := v / = w − λ Lv v . Combined with (2.9), the transition functions for O ( λ L D + · · · + λ Lr D r ) become g Lv ( w ) = w A − v λ Lv and (2.17) g Lv ′ v ( w v ′ ) = w λ Lv ′ − A v ′ A − v λ Lv v ′ . Remark 2.8
Note that the transition functions of L depend on the variation of complex structureon X through the symplectic potential g , since w = e ∂g∂x + iθ and w v = e ∂g∂xv + iθ v (see (2.10) and(2.11)). We will consider one-parameter families of symplectic potentials, g s = g P X + ϕ + sψ, ϕ, ψ ∈ C ∞ P X ( P X ) , s ∈ R + . The transition functions and therefore L depend smoothly on s . ♦ These relations define a holomorphic line bundle on X for any integral values of λ Li , even if thisline bundle is not ample. In this case, sections of the sheaf of holomorphic sections over U v aredefined as in (2.14) with X replaced by U v and the divisors D replaced by D ∩ U v .Using the transition functions (2.17) for L , we can give it a concrete realization as the followingequivariant line bundle L = (cid:16)G v ∈ V U v × C (cid:17) / ∼ , (2.18)where ( w, z ) ∼ ( w ′ , z ′ ) if w = w ′ ∈ U v ∩ U v ′ and z = g Lvv ′ ( w ) z ′ . We will assume that L = O ( D ), D = P rj =1 λ Lj D j , is the line bundle defined by (2.18). In each open set in the holomorphic vertexatlas, the trivializing sections v (or on the open orbit) defined above are given by v ( w ) := [( w, , w ∈ U v (or w ∈ U , for the open orbit) . For σ ∈ Γ( L ) denote by σ v , σ its components on the local frames given, respectively, by v , .For an integral point m ∈ P L ∩ Z n , we denote by σ m the holomorphic section with σ m = w m . Usingthe transition functions, we obtain expressions for σ m in the holomorphic vertex charts: σ mv ( w v ) = w ℓ v ( m ) v v , (2.19)where ℓ v ( x ) := A v x + λ Lv . We have 11 emma 2.9
Let L be the equivariant holomorphic line bundle defined by (2.17) and (2.18). Theunitarization L U (1) associated with L defined over a compact toric variety X has complex structureindependent transition functions on the vertex atlas. Proof.
Recall from Section 2.1 that the unitarization of L is the line bundle L U (1) with localtrivializing sections U (1) v on the vertex charts and U (1)0 on the open orbit. The correspondingtransition functions are˜ g Lv ′ v := U (1) v / U (1) v ′ = e i ( λ Lv ′ − A v ′ A − v λ Lv ) · θ v ′ and ˜ g L v := U (1) v / U (1)0 = e − i ( A − v λ Lv ) · θ . (2.20)We see that, unlike those for L itself, these transition functions are complex structure independent.Recall also from Remark 2.1 that a hermitian structure on L defines an isomorphism between L and L U (1) . The complex structure I on ( X, ω ) (see (2.8)) defines the canonical holomorphic line bundle K I := V n ( T ∗ ) , , whose sections are ( n, U , the I -holomorphic( n, dZ = dz ∧ · · · ∧ dz n = dW/w , where z = t ( z , . . . , z n ) = ∂g/∂x + iθ (2.21)and dW := dw ∧ · · · ∧ dw n , so that dZ and dW are trivializing sections of K I | U . (Here, =(1 , . . . ,
1) so that w = w · · · w n .) Then Lemma 2.10 [CLS11, Sec. 8.2] The ( n, -form given on the open orbit by dZ extends to ameromorphic section of K I with divisor div ( dZ ) = − D − · · · − D r . On the holomorphic vertexchart U v this section is proportional to dW v /w v = dZ v . Since the w jv ’s are holomorphic coordinates on the chart U v , it follows that a system of localholomorphic trivializations for K I is given by { ( U v , dW v ) } . Relative to this system of trivializationsof K I , the transition functions are computed to be g K I v ′ v = w − + A v ′ A − v v ′ , (2.22)so that, as expected from the form of div ( dZ ), K I is isomorphic to a line bundle of the form of(2.18).From Section 2.1, we have that the equivariant hermitian holomorphic line bundle K I admitsa decomposition K U (1) I ⊗ | K I | . The unitarization K U (1) is trivialized by { ( U v , dW v | dW v | ) } v ∈ V (2.23)with corresponding transition functions˜ g K I v ′ v = e i ( − + A v A − v ′ ) · θ v ′ . (2.24)We see that, in accordance with Lemma 2.9, K U (1) I has I -independent transition functions on thevertex atlas. On the other hand notice that the line bundles K U (1) I depend on I because their fiberschange with I (see (2.23). In order to facilitate the study of the dependence of polarized sectionson the complex structure I , it will be convenient to consider a line bundle with the same transitionfunctions as K U (1) I but defined as in (2.18), so that this line bundle is I -independent (but has an I -dependent isomorphism to K U (1) I ). 12 efinition 2.11 Denote by e K U (1) the ( I -independent) equivariant line bundle defined as in (2.18)with U (1) -valued transition functions given by (2.24). In the remainder of this section, we continue to consider a fixed toric complex structure I ,obtained from a symplectic potential g , and will drop the subscript I for simplicity. K has acanonical hermitian structure given by comparison with the Liouville volume form, that is, for an( n, η , k η k K := η ∧ ¯ η (2 i ) n ( − n ( n +1) / ω n /n ! . Let ∇ K denote the Chern connection corresponding to this hermitian structure. In the abovetrivialization, we can compute the connection 1-form of the Chern connection (using Lemma 2.6)to be ∂ log k dZ k K . Lemma 2.12 k dZ k K = det G, where G = Hess x g . Hence, the Chern connection -form in theopen orbit U is Θ = ∂ log det G. Proof.
Since z j = ∂g/∂x j + iθ j , we see that dz = Gdx + idθ and similarly that d ¯ z = Gdx − idθ. We can express these in the matrix equation (cid:18) dzd ¯ z (cid:19) = (cid:18) G i G − i (cid:19) (cid:18) dxdθ (cid:19) whence k dZ k K = dZ ∧ d ¯ Z ( − i ) n ( dx ∧ · · · ∧ dx n ∧ dθ ∧ · · · ∧ dθ n ) = 1( − i ) n det (cid:18) G i G − i (cid:19) = det G. Similarly we have k dZ v k K = det G v , where G v = Hess x v g .The curvature of the Chern connection is easily computed, giving F ∇ K = ¯ ∂∂ log det G. (2.25)Let Ric γ denote the Ricci curvature tensor of the metric γ = ω ( · , I · ), and let ρ = Ric ( I · , · ) be thecorresponding Ricci form. Then by [Mor07, Prop. 11.4] we have F ∇ K = iρ. (2.26)Note that this implies c ( K ) = − c ( X ) = − [ ρ/ π ]. Lemma 2.13
The section dW v = w v dZ v is a nowhere vanishing holomorphic section of K on theholomorphic vertex chart U v and, in the induced trivializations, the Chern connection -forms on K U (1) and | K | are Θ K U (1) v = i n X k =1 dθ kv + i (cid:18) ∂∂x v log det G v (cid:19) · G − v dθ v , and (2.27)Θ | K | v = 12 (cid:18) ∂∂x v log det G v (cid:19) · dx v + G v dx v . emark 2.14 The following expressions for the connection 1-forms for the induced connections ∇ K U (1) and ∇ | K | over the open orbit will also be useful below:Θ K U (1) = i Im Θ = i (cid:18) ∂∂x log det G (cid:19) · G − d θ, and (2.28)Θ | K | = Re Θ = 12 (cid:18) ∂∂x log det G (cid:19) · dx. ♦ Proof. If f is real valued, thenIm ∂f = 12 (cid:18) ∂f∂x · G − dθ − ∂f∂θ · G dx (cid:19) (2.29)and Re ∂f = 12 (cid:18) ∂f∂x · dx + ∂f∂θ · dθ (cid:19) , (2.30)with similar formulas in the vertex charts (with x j and θ j replaced by x jv and θ jv ). With f = log det G (observing that f = f ( x )) we obtain, from Θ = ∂ log det G , the open orbit 1-forms (2.28).Next, by [CLS11], the “canonical” section dZ = dW/w (which has the same representation dW v /w v in the holomorphic vertex charts) has simple poles along each torus-invariant divisor. Toobtain a trivializing section on the holomorphic vertex chart U v , we multiply by a factor with simplezeroes along the divisors adjacent to v and thus arrive at the desired combination w v dZ v .To obtain the connection 1-forms in the chart U v with respect to w v dZ v , we first recall the norm (cid:13)(cid:13) w v dZ v (cid:13)(cid:13) = (cid:12)(cid:12) w v (cid:12)(cid:12) det G v . (2.31)From this, noting that det G v is a constant multiple of det G , we see that the Chern connection1-form on U v isΘ v = ∂ log (cid:16)(cid:12)(cid:12) w v (cid:12)(cid:12) det G v (cid:17) = ∂ log det G v + X j ∂ log w jv = ∂ log det G v + X j dz jv . Since dz v = G v dx v + idθ v , using (2.29) and that G depends only on x , we obtain the connection1-form Θ K U (1) v = i Im ∂ log det G v + X j dz jv = i (cid:18) ∂∂x log det G v (cid:19) · G − v dθ v + i n X j =1 dθ jv as desired.Similarly, using (2.30) we obtain the connection 1-formΘ | K | v = Re ∂ log det G v + X j dz jv = 12 (cid:18) ∂∂x v log det G v (cid:19) · dx v + G v dx v as desired.Note that although K U (1) I has complex structure independent transition functions, its Chernconnection depends on I . 14 Half-form corrected K¨ahler quantization
Suppose the square root √ K I of the corresponding canonical bundle exists so that, in particular, c ( X ) / ω/ π ] ∈ H ( X, Z ) and let ℓ → X be a (smooth)hermitian line bundle with compatible connection with curvature given by − iω , that is, ℓ is aprequantum line bundle. More specifically, let ℓ be an equivariant line bundle defined as in (2.18),with U (1)-valued transition functions on the vertex atlas (2.20)˜ g ℓvv ′ = e i ( A v ′ A − v λ v − λ v ′ ) · θ v ′ , and ˜ g ℓv = e i λ v · θ v , (3.1)with λ v ∈ Z suitably chosen to have c ( ℓ ) = [ ω π ]. Equip ℓ with the U (1)-connection ∇ ℓ given bythe connection forms Θ ℓ = ∇ ℓ U (1)0 U (1)0 = ix · dθ on ˇ X, Θ ℓv = ∇ ℓ U (1) v U (1) v = ix v · dθ v on U v , v ∈ V. (3.2)One may easily check that { Θ ℓ , Θ ℓv : v ∈ V } does indeed define a U (1)-connection on ℓ ; see thecomment following (3.10).Since the square of a (local) section η ∈ Γ( √ K I ) can be identified with a (local) section of K I ,the line bundle √ K I inherits a hermitian structure from that of K I given by [Woo91] k η k √ K I = s η ∧ ¯ η (2 i ) n ( − n ( n +1) / ω n /n ! . (3.3)This defines a Chern connection ∇ √ K I on √ K I as in Lemma 2.6. The curvature of ∇ √ K I isthen F ∇√ KI = i ρ I , where ρ I is the Ricci form on X .The quantum Hilbert space for the half-form corrected K¨ahler quantization of X is defined tobe H QI := n s ∈ Γ( ℓ ⊗ p K I ) : (cid:16) ∇ ℓ P I ⊗ ⊗ ∇ √ K I P I (cid:17) s = 0 o , where P I is the holomorphic polarization of X determined by I .Recall from (2.1) that the bundle |√ K I | has a trivializing covariantly constant section µ I = | dZ | k dZ k K I . (3.4)This defines an isomorphism H QI ∼ = B QI ⊗ µ I where B QI := (cid:26) s ∈ Γ (cid:18) ℓ ⊗ g √ K U (1) (cid:19) : (cid:18) ∇ ℓ P I ⊗ ⊗ ∇ g √ K U (1) P I (cid:19) s = 0 (cid:27) , (3.5)Note that, from Lemma 2.9, the unitarization ℓ ⊗ g √ K U (1) is a smooth complex line bundle in-dependent of I . In this way, using the I -dependent isomorphisms above, we can describe the15uantum Hilbert spaces H QI through the Hilbert spaces B QI which are subspaces of a fixed linearspace Γ (cid:18) ℓ ⊗ g √ K U (1) (cid:19) .We will now use this representation to motivate the definition of the half-form corrected quantumHilbert space in the more general situation when the canonical bundle of X may not admit a squareroot. Let (
X, ω, I ) be a compact smooth toric K¨ahler manifold with toric complex structure I and suchthat (cid:2) ω π (cid:3) − c ( X ) is an ample integral cohomology class. Let the moment polytope be P X = { x ∈ R n : ℓ j ( x ) = ν j · x + λ j ≥ , j = 1 , . . . , r } , (3.6)where we use the freedom of translating the moment polytope to choose the { λ j } j =1 ,...,r to behalf-integral and defined as follows. Consider an equivariant complex line bundle L ∼ = O ( λ L D + · · · + λ Lr D r ) as in (2.18) and with U (1)-valued transition functions (2.20) defined by { λ Lj } j =1 ,...,r ,such that c ( L ) = (cid:2) ω π (cid:3) − c ( X ). As in (2.15), the { λ Lj } j =1 , ··· ,r define a polytope with integralvertices, P L . The half-integral { λ j } j =1 ,...,r in (3.6) are then defined by λ j := λ Lj + 12 ∈
12 + Z , j = 1 , . . . , r, (3.7)in accordance with the fact that div ( dZ ) = − D · · · − D r (see Lemma 2.10). Note that P L isobtained from the moment polytope P X by shifts of along each of the integral primitive inwardpointing normals. (See Remarks 3.7 and 3.8 below for examples.) We will call P L ⊂ P X the corrected polytope .We equip L with a U (1) connection ∇ I with curvature F ∇ I = − iω + i ρ I . Since H ( X ) = 0this connection is unique up to isomorphism.Following the reasoning in the last section, and noticing that p | K I | and µ I (see (3.4)) existalways even if √ K I does not, and that the hermitian structure on p | K I | gives || µ I ||√ | K I | = 1, weset Definition 3.1
The quantum Hilbert space for the half-form corrected K¨ahler quantization of ( X, ω, L, I ) is defined by H QI = B QI ⊗ µ I , where B QI = { s ∈ Γ( L ) : ∇ I P I .s = 0 } . The inner product is defined by h σ ⊗ µ I , σ ′ ⊗ µ I i = h σ, σ ′ i = 1(2 π ) n Z X h L ( σ, σ ′ ) ω n n ! . (3.8)Now fix a choice of symplectic potential g for the complex structure I on X . We define theconnection ∇ I on L by (using Lemma 2.13 and (3.2))Θ v := ∇ I U (1) v U (1) v = − i x v · dθ v + i n X k =1 dθ kv + i (cid:18) ∂∂x v log det G v (cid:19) · G − v dθ v (3.9)= − i x v · dθ v + i ∂ log det G v + n X k =1 dz kv ! .
16n the open orbit ˇ X , the connection is then given byΘ := − i x · dθ + i (cid:18) ∂∂x log det G (cid:19) · G − dθ (3.10)= − i x · dθ + i ∂ log det G. One may check that Θ v − Θ v ′ = d log ˜ g Lv ′ v and Θ v − Θ = d log ˜ g L v so that { Θ , Θ v : v ∈ V } doesindeed define a U (1)-connection on L . Remark 3.2
Note that, even though dθ jv is singular as x jv →
0, (3.9) defines a non-singular 1-formon U v , as can be verified by studying the behavior of G v or using the coordinates { a jv , b jv } j =1 ,...,n . ♦ The complex structure I and the connection ∇ I combine to give a holomorphic structure on L which we can describe by giving the resulting I -holomorphic sections of L .Let h I ( x ) = x · ∂g/∂x − g and h Iv ( x v ) = x v · ∂g/∂x v − g . Also, note that det G v = (det A v ) − det G . Lemma 3.3 An I -holomorphic section of L , s ∈ B QI , is locally given by s | U v = s v U (1) v where thefunction s v ∈ C ∞ ( U v ) is of the form F v ( w v ) e − h Iv ( x v ) e − i / · θ v k dZ v k / K I . (3.11) On the orbit one then obtains s | U = s U (1)0 , where the function s ∈ C ∞ ( U ) is of the form F ( w ) e − h I ( x ) k dZ k / K I , (3.12) where F is holomorphic and F v ( w v ) = w λ v v F ( w A v v ) | det A v | . Remark 3.4
Since the λ v are generally only half-integer, the “functions” F v in the above theoremare not single valued. However, F v e − i · θ v is single valued. In fact, the collection { F , F v : v ∈ V } defines a ramified section of L . There are other possible geometric interpretations of such an object,for instance through the notion of Kawamata covering, but we will not pursue them here. ♦ Proof.
We compute first in the open orbit. From (2.21), we see that − i x · dθ (cid:18) ∂∂ ¯ z j (cid:19) = 12 x j . Recall that x can be expressed in terms of y via the Legendre transform x = ∂h I /∂y , where h I = x · y − g . Since ∂/∂z j = ( ∂/∂y j − i∂/∂θ j ), we have x j = 2 ∂h I /∂ ¯ z j so that − i x · dθ (cid:18) ∂∂ ¯ z j (cid:19) = ∂h I ∂ ¯ z j . (3.13)Next, from Lemma 2.13 we know that i (cid:18) ∂∂x log det G (cid:19) G − dθ = i ∂ log det G = 14 ( ∂ − ¯ ∂ ) log det G
17o that i (cid:18) ∂∂x log det G (cid:19) G − dθ (cid:18) ∂∂ ¯ z j (cid:19) = − ∂∂ ¯ z j log det G. (3.14)Combining (3.13) and (3.14), we see that a section s = f U (1)0 ∈ Γ U ( L ) is holomorphic if andonly if f satisfies the differential equation ∂f∂ ¯ z j + f ∂∂ ¯ z j (cid:18) h I −
14 log det G (cid:19) = 0 , for each j = 1 , . . . , n . We solve this easily to see that s is holomorphic if and only if f is of the form f = F ( w ) e − h I (det G ) / , where F is an holomorphic function in U . From Lemma 2.12, we recognize (det G ) / = k dZ k / K to obtain (3.12) as desired.We have computed that F ( w ) e − h I (det G ) / U (1)0 is holomorphic. Using the transition functions˜ g v , we conclude that F ( w ) e − h I (det G ) / ˜ g v ( w v ) U (1) v , when expressed in terms of w v , should beholomorphic. First, note that x · y = x v · y v − λ v · y v which implies h I = h Iv − λ v · y v . From (2.17) we therefore see that the holomorphic combination in U v should be F ( w ) e − h Iv + λ v · y v (det G ) / e iλ Lv · θ v U (1) v = F ( w ) e − h Iv e λ v · y v + iλ v · θ v e − i / · θ v (det G ) / U (1) v = F ( w ) w λ v v e − h Iv e − i / · θ v | det A v | k dZ v k / K I U (1) v . By (2.12) we see that if we set F v ( w v ) := F ( w A v v ) w λ v v | det A v | , we obtain (3.11) and the finalstatement of the lemma as desired. Remark 3.5
We can rewrite a local holomorphic section on U v ∩ U in a way similar to that in[BFMN11] as follows: F v ( w v ) e − h Iv ( x v ) e − / · θ v k dZ v k / K I = F ( w A v v ) w λ v v e − h Iv ( x v ) e − i / · θ v | det A v | k dZ v k / K I = F e − h Iv + λ v · y v e iλ Lv · θ v | det A v | k dZ v k / K I . The combination h Iv − λ v · y v corresponds to h m (for m = v ) in [BFMN11]. ♦ Theorem 3.6
The Hilbert space B QI of holomorphic sections of L has an orthogonal basis { σ m } m ∈ P L ∩ Z n where σ m is locally given by σ m = w m e − h I k dZ k / K I U (1)0 , and σ mv = w A v m + λ Lv v e − h Iv + / · y v | det A v | k dZ v k / K I U (1) v , over the open orbit and holomorphic vertex charts, respectively. The corresponding orthogonal basisfor the quantum Hilbert space H QI is given by { ˆ σ m := σ m ⊗ µ I } m ∈ P L ∩ Z n . roof. From Lemma 3.3 we can certainly find a basis for the space of holomorphic sections of L consisting of elements given locally over the open orbit by w m e − h I k dZ k / K I U (1)0 , where m ∈ Z n , with the corresponding expressions over the holomorphic vertex charts. Such asection will have poles unless m belongs to the corrected polytope P L . The fact that σ m and σ m ′ are orthogonal for m = m ′ follows immediately from integration along T n .Therefore, the space of half-form corrected holomorphic wave functions for the K¨ahler quanti-zation of X has a natural basis whose elements are labeled by the integral points of the correctedpolytope P L . These coincide also with the (interior) integral points of the moment polytope P X and they correspond to shifted nonsingular Bohr–Sommerfeld fibers of ( X, ω ). Remark 3.7
Pictured below is the moment polytope P X for X = CP , in the case [ ω π ] = c ( CP ) = 3 c ( O (1)). On the left, we show the more standard choice of moment polytope,with integral vertices. On the right, we show the moment polytope chosen in accordance with(3.6) and (3.7), that is such that λ , λ are half-integral. In this example, L ∼ = O (2), the correctedpolytope is P L = [0 ,
2] and the moment polytope is P X = [ − , ]. One has dim H Q = 3 . b b b b b b b − ♦ Remark 3.8
According to Theorem 3.6, we can count holomorphic sections of L by countingintegral points inside the moment polytope P X , which are exactly the integral points which occurin the corrected polytope P L . Pictured below is one of such polytopes when X = CP CP , thatis, CP blown up at a point. b b b b b bb b b b b bb b b b b bb b b b b b ♦ Remark 3.9
One can consider the case [ ω/ π ] = c ( X ) / , where L = O X is just the structuresheaf of X (which is, of course, not ample). In this case, there is only one integral point inside thepolytope P X corresponding to the constant function 1 ∈ H ( O X ). As we will see in Section 4.3,when we study degenerations of the complex structure, also in this case we have convergence toa Dirac delta distribution supported on the (shifted) Bohr–Sommerfeld fiber above that integralpoint. ♦ Half-form corrected quantization in the singular real toric po-larization
In order to study quantization in the real toric polarization of (
X, ω ), following [BFMN11], we con-sider distributional sections of L . Let us briefly recall how one can define covariant differentiation inthis case. Let L ( L ) denote the Hilbert space of L sections of L . Consider the rigged Hilbert space(see Sections 4.2 and 4.3 of [GV]) (Γ( L ) , L ( L ) , Γ( ¯ L ) ′ ), where Γ( ¯ L ) ′ is the space of distributionalsections of L given by the topological dual of Γ( ¯ L ). One has the continuous inclusionsΓ( L ) ⊂ L ( L ) ⊂ Γ( ¯ L ) ′ , where we embed σ ∈ Γ( L ) i ( σ ) ∈ Γ( ¯ L ) ′ via the Liouville volume form; i.e. i ( σ )(¯ τ ) := 1(2 π ) n Z X h L ( σ, τ ) ω n n ! . (In particular, we may view I -holomorphic sections of L as distributional sections.) We have then,for σ ∈ Γ( L ), i ( σ )(¯ τ ) = h σ, τ i L , ∀ τ ∈ Γ( L ) . Let ∇ L be a connection on L and let ∇ ∗ be the adjoint of the (unbounded) operator ∇ L on theHilbert space L ( X, L ), so that i ( ∇ L σ )(¯ τ ) = h∇ L σ, τ i L = h σ, ∇ ∗ τ i L , ∀ σ, τ ∈ Γ( L ) . We can now define covariant differentiation of distributional sections, which we will still denoteby ∇ L , by ∇ L ( σ )(¯ τ ) = σ ( ∇ ∗ τ ) , σ ∈ Γ( ¯ L ) ′ , τ ∈ Γ( L )so that, as distributions ∇ L ( iσ ) = i ( ∇ L σ ) , ∀ σ ∈ Γ( L ) . In the next sections, we will interpret holomorphic sections as distributional sections in thisway, and we will identify σ ∈ Γ( L ) with i ( σ ) ∈ Γ( ¯ L ) ′ . Recall that the real singular toric polarization is defined by P R ( p ) = span C ((cid:18) ∂∂θ i (cid:19) p , i = 1 , . . . , n ) , ∀ p ∈ X. In this section, we will define the half-form corrected quantization of X in this polarization directlyin terms of covariantly constant sections. Recall the families of toric complex structures consideredin [BFMN11]. For any smooth function ψ which is strictly convex on a neighborhood of P X , forany ϕ ∈ C ∞ P X ( P X ) and for any s ∈ R ≥ , the sum ϕ + sψ is in C ∞ P X ( P X ) and hence defines a K¨ahlerstructure on X with symplectic potential g s := g P X + ϕ + sψ. Denote the corresponding s -dependent complex structure by I s .If P is a polarization of ( X, ω ), denote by C ∞ ( P ) its space of smooth sections, C ∞ ( P ) = { ξ ∈ C ∞ ( T X ⊗ C ) : ξ ( p ) ∈ P p } , where C ∞ ( T X ⊗ C ) is the space of smooth sections of the complexified tangent bundle of X .20 heorem 4.1 ([BFMN11], Theorem 1.2 p. 415, Theorem 3.4, p. 429)Pointwise on the dense open orbit ˇ X , as vector fields, ∂∂ ¯ z js = 12 (cid:18) ∂∂y js + i ∂∂θ j (cid:19) → i ∂∂θ j , as s → ∞ . Therefore, at each point p ∈ ˇ X , the holomorphic polarizations P s , s ≥ of X , associated tothe complex structures I s , converge, as s → ∞ , to the real toric polarization, in the LagrangianGrassmannian of T p X ⊗ C and C ∞ ( lim s →∞ P s ) = C ∞ ( P R ) . (4.1) Remark 4.2
The equality in (4.1) is an equality of spaces of smooth sections of two differentpolarizations lim s →∞ P s and P R which coincide over ˇ X but not over X \ ˇ X . (See Theorem 3.4 of[BFMN11]). ♦ From the expressions for the half-form corrected connection in (3.9), we see that in the localtrivializations U (1) v − i ∇ I s ∂/∂ ¯ z jsv → ∇ R ∂/∂θ jv := ∂∂θ jv − ix jv + i , as s → ∞ , (4.2)in the sense that − i (cid:16) ∇ I s ∂/∂ ¯ z jsv σ (cid:17) (¯ τ ) → (cid:16) ∇ R ∂/∂θ jv σ (cid:17) ( τ ) , as s → ∞ , ∀ σ ∈ Γ( ¯ L ) ′ , ∀ τ ∈ Γ( L ). Similarly, from (3.10), on the open orbit in the trivialization U (1)0 we have − i ∇ I s ∂/∂ ¯ z js → ∇ R ∂/∂θ j = ∂∂θ j − ix j , as s → ∞ . (4.3)We will take the expressions on the right hand side of (4.2) and (4.3) to define a partial con-nection ∇ R on Γ( ¯ L ) ′ , along P R , which will be used to define the quantization in this polarization.Let B Q R = ker ∇ R = n \ j =1 ker ∇ R ∂/∂θ j ⊂ Γ( ¯ L ) ′ . (4.4) Remark 4.3
We note that additive term i in the right hand side of (4.2), corresponds to alimiting Chern connection on K I s which is flat on U and singular along ∪ ri =1 D i . The additionof this singular connection to the prequantum connection is at the core of our approach to thehalf-form quantization in the singular real toric polarization. We will describe in more detail thesingular behavior of the limiting connection at the end of this section. ♦ Remark 4.4
As explained in the previous section, we should think of L as the tensor product ofthe uncorrected bundle ℓ with the smooth bundle with U (1)-valued transition functions g √ K U (1) , though in general these may not exist individually. On the other hand, the geometric quantizationassociated to a polarization P is supposed to be the space of P -covariantly constant sections of thetensor product of the uncorrected bundle with the square root of the canonical bundle associated P . For the real polarization P R , the sections of the associated canonical bundle are n -forms of theform a ( x ) dx ∧ · · · ∧ dx n . Of course, the U (1)-part of this is hidden in the sections of L , and whatis missing is the modulus of the square root of the canonical bundle associated to P R . To put it another way, according to the standard procedures of geometric quantization, weshould actually define the quantization H Q R to be sections of L ⊗ p | K P R | . Some care must be21aken to interpret exactly what is meant by | K P R | (and hence what is meant by its square root).In the next section, we will see that sections of p | K P R | can be thought of as maps on the space of n -tuples of vector fields. Then, p | K P R | admits a canonical section dX := dx ∧ · · · ∧ dx n on U , and0 otherwise. Hence, we should define H Q R = B Q R ⊗ p | dX | . Of course, at this point, such a changeis merely cosmetic. On the other hand, we will see in the next section that such expressions arisenaturally when studying the degenerations of the complex structure on X to the real polarizationat the level of holomorphic sections. ♦ Definition 4.5
The vector space of quantum states for the half-form corrected quantization of ( X, ω, L ) in the toric polarization P R is defined by H Q R = B Q R ⊗ p | dX | , where B Q R was defined in (4.4). Remark 4.6
A natural Hilbert space structure in H Q R will be introduced in Section 4.3, via de-generation of K¨ahler quantizations of ( X, ω, L, I ). ♦ The open orbit U carries a free T n -action which lifts to L U via geometric quantization. Thisaction is generated by ∇ ∂∂θ + ix. Then, the trivializing section U (1)0 is T n -invariant and if τ ∈ Γ U ( L ) is given by τ = τ U (1)0 fora smooth function τ ∈ C ∞ ( U ), we can decompose it into Fourier modes with respect to the T n -action. Specifically, τ ( x, θ ) = X m ∈ Z n e − im · θ ˆ τ ,m ( x ) , where ˆ τ ,m ( x ) = π ) n R T n e im · θ τ ( x, θ ) dθ is the m -th Fourier mode of τ . For m ∈ P L ∩ Z n , let δ m ∈ Γ( ¯ L ) ′ be the distributional section defined by δ m (¯ τ ) = ¯ˆ τ m ( m ) = 1(2 π ) n Z T n e im · θ ¯ τ ( m, θ ) dθ, (4.5)for all τ ∈ Γ( L ). We have Theorem 4.7
The vector space H Q R is the finite-dimensional vector space generated by { δ m ⊗ p | dX |} m ∈ P X ∩ Z n . Therefore, quantization in the real polarization is also given by the integral points in the interiorof the moment polytope. Recall, from Section 3.2 and Remarks 3.7 and 3.8, that P L ⊂ P X isobtained from P X by a one-half shift along the inward pointing normals to the facets of P X . InSection 4.3, these distributional sections will be described as coming from holomorphic sectionsthrough degeneration of the complex structure. Proof of Theorem 4.7.
By acting with ∇ R ∂/∂θ j , j = 1 , . . . , n , in (4.3) on the distributions δ m in(4.5) we conclude that they belong to ker ∇ R . Moreover, any element of the kernel can be restrictedto the open orbit, by restricting it to sections of ¯ L with compact support contained in the openorbit. From, Proposition 3.1 in [BFMN11], we see that such restrictions can only have supportalong µ − ( ˇ P X ∩ Z n ), as these are the only fibers along which ∇ R has trivial holonomy, and oneeasily verifies that δ m is the unique (up to a constant) solution supported on µ − ( m ).22t remains to be shown that there are no more elements in the kernel of ∇ R . All we need toshow is that there are no solutions with support along µ − ( ∂P X ) . Let us consider a solution withsupport along µ − ( x jv = 0), for some fixed j = 1 , . . . , n . Let ˇ x v = ( x v , . . . , x j − v , x j +1 v , . . . , x nv ) andˇ θ v = ( θ v , . . . , θ j − v , θ j +1 v , . . . , θ nv ). In a neighborhood of the preimage by µ of the interior of the facet x jv = 0 of P X , we can take coordinates ( u, v, ˇ x v , ˇ θ v ) (see, for example [DP09], [BFMN11]), so that x jv = 0 ⇔ ( u, v ) = (0 ,
0) and ∇ R ∂∂θjv = − i (cid:18) − v ∂∂u + u ∂∂v (cid:19) + i , in that neighborhood. A solution with support along the facet will be of the form, σ = ∞ X k,l =0 α kl (ˇ x v , ˇ θ v ) δ ( k ) ( u ) δ ( l ) ( v ) U (1) v , where only a finite number of terms in the sum can be nonzero and where δ ( k ) denotes the order- k derivative of the Dirac δ distribution. (See Theorem 2.3.5 of [H¨or90].) Using xδ ( k ) ( x ) = − kδ ( k − ( x )and polynomial test sections of the form u k v l χ U (1) v , where χ is a cutoff function which is constantand equal to 1 in the neighborhood, t he condition ∇ R ∂∂θjv σ = 0 then implies that such a distributionalsection is zero. Therefore, no nonzero solutions of this form exist.As we will see in the next section, there is complete agreement between the direct approach tohalf-form corrected quantization in the real polarization and the approach based on degenerationof holomorphic sections. We note that the partial connection ∇ R “remembers” the degenerationprocedure due to the contribution of the Ricci-curvature term.It is possible to gain some more geometric intuition about the fact that the boundary of P X does not contribute to the kernel of ∇ R , unlike what happens without the half-form correction[BFMN11]. As s → ∞ , the piece of the connection ∇ I s coming from the Levi–Civita connectionon the canonical bundle of X develops curvature singularities outside of the open orbit U . Thisis behind the fact, shown above, that for the real polarization the half-form correction forbidssolutions of the covariant constancy equations supported on µ − ( ∂P X ). For completeness, let usdescribe these curvature singularities in more detail.Recall Lemma 2.13, which gives explicit expressions for the Chern connection 1-forms on K U (1) (induced from the Chern connection on K ) in the vertex atlas. The proof of the following Propo-sition is immediate. Proposition 4.8 lim s →∞ Θ K Is v = n X j =1 dθ jv , in the sense that for any closed curve C , with C ⊂ U v \ µ − ( ∂P X ) = U ∼ = ( C ∗ ) n , the holonomy ofthe singular connection, along C depends only on the homotopy class of C in U and, lim s →∞ I C Θ K Is v = i I C n X j =1 dθ jv . Therefore, in the limit s → ∞ , we obtain a singular connection on K U (1) , flat on µ − ( ˇ P X ),with curvature supported on µ − ( ∂P X ) = ∪ ri =1 D i and with nonvanishing monodromies around the23oric invariant divisors. In the vertex chart U v the curvature, in the limit s → ∞ , is given by thefollowing current 2 πi n X j =1 δ ( a jv ) δ ( b jv ) da jv ∧ db jv , where ( a v , b v , . . . , a nv , b nv ), with a jv = p x jv cos θ jv , b jv = p x jv sin θ jv , i = 1 , . . . , n , are coordinates on U v . In this section, we will obtain the degeneration, as s → ∞ , of the (appropriately L -normalized)elements ˆ σ ms of the orthogonal basis of H QI s , defined in Theorem 3.6, to the same distributionalsections δ m ⊗ p | dX | ∈ H Q R obtained in the previous section, see Definition 4.5, (4.5) and Theorem4.7. In particular, this will allow us to define a natural inner product in H Q R . We will first studythe degeneration of the basis elements σ ms of B QI s and then the degeneration of the sections µ I s of p | K I s | .As the complex structure I s varies with s , we can regard the spaces B QI s of I s -holomorphicsections, described in Definition 3.1, as finite-dimensional subspaces of the fixed infinite-dimensionalspace of distributional sections of L (see Section 4.1). That is, for all s , B QI s ⊂ Γ( L ) ⊂ Γ( ¯ L ) ′ . We will study (weak) convergence of I s -holomorphic sections in Γ( ¯ L ) ′ , as s → ∞ . Denote the I s -holomorphic section of L associated to m ∈ P L ∩ Z n by σ ms , as in Theorem 3.6. Suppose τ ∈ Γ( L ) isgiven locally by { τ U (1)0 , τ v U (1) v : v ∈ V } and let m ∈ P L ∩ Z n . Then since ˇ X and the holomorphicvertex charts U v are dense in X , we see from Theorem 3.6 that i ( σ ms )(¯ τ ) = 1(2 π ) n Z ˇ X w m e − h Is k dZ k / K ¯ τ ω n n ! (4.6)= 1(2 π ) n Z U v w ℓ v ( m ) · y sv − h Isv + / · y sv v | det A v | k dZ v k / K ¯ τ v ω n n ! , where y s v = ∂g s /∂x v .The next lemma, which we recall from [BFMN11], will allow us to use Laplace’s approximationto compute the asymptotics that we are interested in. Lemma 4.9 [BFMN11, Lemma 5.1] For m ∈ P X and any smooth function ψ which is strictlyconvex on a neighborhood of P X , let f m := ( x − m ) · ∂ψ/∂x − ψ. Then f m has a minimum value of − ψ ( m ) on P which is obtained at the unique point x = m. Moreover, (Hess f m ) ( m ) = (Hess ψ ) ( m ) . Remark 4.10
It is important to observe that the function f m in Lemma 4.9 has a unique minimumon the entire polytope P X , not just the interior, which implies that the leading order asymptoticsthat we will be interested in all arise from the behavior of the integrand at x = m. ♦ Recall, 24 emma 4.11 (Laplace’s Approximation) Suppose a function f ∈ C ( R ) on the closed region R ⊂ R n has a unique nondegenerate minimum the unique point x ∈ ˇ R in the interior of R ; so inparticular, Hess x f is positive definite. Then if g s is a continuous function on R such that g s ∼ s r g + O ( s r − ) , s → ∞ , we have Z R e − sf g s dx ∼ (cid:18) πs (cid:19) n/ e − sf ( x ) s r g ( x ) p det (Hess f ) ( x ) , s → ∞ . Lemma 4.12 As s → ∞ , the leading order asymptotic value of the L -norm of the family ofsections σ ms , s ∈ R > is k σ ms k L ∼ π n/ e g s ( m ) . Proof.
To compute the asymptotics we can restrict the integral to the open orbit, where we have k σ ms k L = 1(2 π ) n Z ˇ P X × T n | w m | e − h Is k dZ k K ω n n != 1(2 π ) n Z ˇ P X × T n e m · y s − x · y s − g s ) (det G s ) / ω n n != Z ˇ P X e − s (( x − m ) · ∂ψ/∂x − ψ ) e − x − m ) · y − g ) (det G s ) / dx, (4.7)where in the last line we used the fact that y s = ∂g s /∂x = ∂g /∂x + s∂ψ/∂x . Note that G s = G + s Hess ψ implies det G s ∼ s n det Hess ψ + O ( s n − ) . (4.8)We would like to apply Laplace’s approximation to the integral (4.7). By Lemma 4.9, the firstexponential has the correct behavior. The only remaining subtlety is to show that the remain-der of the integrand is continuous on P X , which is not immediate since g s is singular along ∂P X . Using the explicit expression (2.6) and the regularity conditions (2.7), we conclude that e − x − m ) · y − g ) (det G s ) / behaves like Π ri =1 ℓ i ( x ) ( ℓ i ( m ) − ) times a smooth function on P X . Therefore, it is continuous and goes to zero at the boundary of P X precisely when m belongs to the corrected polytope P L ⊂ P X . From another point of view,the integrand in (4.7) is the pointwise norm of the T n -invariant holomorphic section σ ms , which isnecessarily continuous on P X . Then, Laplace’s approximation (Lemma 4.11) yields k σ ms k L ∼ (cid:18) πs (cid:19) n/ e sψ ( m ) e g ( m ) s n/ p det Hess ψ ( m ) p n det Hess ψ ( m ) = π n/ e g s ( m ) as desired.Recall that G s = G + s Hess ψ , where G = Hess( g P + ϕ ). Theorem 4.13
For each τ ∈ Γ( L ) , the leading order asymptotic value of i ( σ ms / k σ ms k L ) on ¯ τ as s → ∞ is determined by i σ ms (det G s ) / k σ ms k L ! (¯ τ ) ∼ n/ π n/ ˆ¯ τ ,m ( m ) . hat is, in terms of the distributional sections δ m , m ∈ P X ∩ Z n , described in the Section 4.2,formula (4.5), lim s →∞ σ ms (det G s ) / k σ ms k L = 2 n/ π n/ δ m . Proof.
We will first compute the asymptotics of i ( σ ms )(¯ τ ), and then simply divide by the resultsof the previous lemma to obtain the desired expressions. To this end, compute first in the openorbit. From (4.6) we have i ( σ ms )(¯ τ ) = 1(2 π ) n Z ˇ X w m e − h Is k dZ k / K ¯ τ ω n n != Z ˇ P X e m · y s − ( x · y s − g − sψ ) (det G s ) / π ) n Z µ − ( x ) e im · θ ¯ τ dθ ! dx = Z ˇ P X e − s (( x − m ) · ∂ψ/∂x − ψ ) e − (( x − m ) · y − g ) (det G s ) / ˆ¯ τ ,m ( x ) dx. Let f m := ( x − m ) · ∂ψ/∂x − ψ. Then by an argument similar to that in the proof of Lemma 4.12and by Lemma 4.9, we can use Laplace’s approximation with equation (4.8) to obtain as s → ∞ that i ( σ ms )(¯ τ ) ∼ (cid:18) πs (cid:19) n/ e sψ ( m )+ g ( m ) s n/ (det Hess ψ ( m )) / ˆ¯ τ ,m ( m ) p det Hess ψ ( m )= (2 π ) n/ s − n/ e g s ( m ) (det Hess m ψ ) − / ˆ¯ τ ,m ( m ) . Using Lemma 4.12, we have(det G s ) / k σ ms k L ∼ s n/ (det Hess ψ ( m )) / π − n/ e − g s ( m ) , s → ∞ from which we see that as s → ∞ i σ ms (det G s ) / k σ ms k L ! (¯ τ ) ∼ (2 π ) n/ s − n/ e g s ( m ) (det Hess ψ ( m )) − / ˆ¯ τ ,m ( m ) × s n/ (det Hess ψ ( m )) / π − n/ e − g s ( m ) = 2 n/ π n/ ˆ¯ τ ,m ( m )as desired.Observe that the asymptotics of the normalized sections σ ms / k σ ms k L described by Theorem4.13 contain the additional term (det G s ) / . This extra factor is better understood in the contextof the degeneration of H QI s which we now study.To have the spaces Γ( p | K s | ), for all s ≥
0, as subspaces of a given fixed vector space, weconsider α ∈ Γ( p | K s | )( U ), for an open set U ⊂ X , as a map α : X ( U ) n → C ( U ), where X ( U )is the space of smooth complex vector fields on U and C ( U ) is the space of continuous complexvalued functions on U . Then, we definelim s →∞ α s = β ⇔ lim s →∞ α s ( X , . . . , X n ) = β ( X , . . . , X n ) , for all X , . . . X n ∈ X ( U ), where on the right hand side we consider pointwise convergence in C ( U ). An n -form β ∈ Ω n ( U ) ⊗ C then also defines a map p | β | : X ( U ) n → C ( U ), given by p | β | ( X , . . . , X n ) = | β ( X , . . . , X n ) | , for X , . . . , X n ∈ X ( U ). Consider now the global n -form dX = dx ∧ · · · ∧ dx n , vanishing on ∪ ri =1 D i , and the corresponding map p | dX | .26 emma 4.14 In the sense defined above, lim s →∞ µ I s (det G s ) / = p | dX | . Proof.
On the open orbit, using Lemma 2.12, we have µ I s (det G s ) / = p | dZ s |√ det G s . But dz = Gdx + idθ , which implies dZ s ∼ s n det (Hess x ψ ) dX , and det G s ∼ s n det (Hess x ψ ), solim s →∞ p | dZ s |√ det G s = lim s →∞ s n/ p det (Hess x ψ ) p | dX | s n/ p det (Hess x ψ )as desired. Note that on ∪ ri =1 D i both sides vanish.We are now ready to explain the factor of (det G ) / which appears in Theorem 4.13. As wesee, the same term appears in the denominator in Lemma 4.14. Recall the orthogonal basis of H QI s given by { ˆ σ ms = σ ms ⊗ µ I s } m ∈ P L ∩ Z n , and also that || ˆ σ ms || L = || σ m || L , since the hermitian structure on p | K I s | gives || µ I s || = 1. Combining Lemma 4.14 with Theorem4.13, we obtain the following theorem. Theorem 4.15 lim s →∞ ˆ σ ms k ˆ σ ms k L = lim s →∞ σ ms k σ ms k L ⊗ µ I s = 2 n/ π n/ δ m ⊗ p | dX | , in the sense that lim s →∞ i ( σ ms ) ⊗ µ I s k σ ms k L ( τ ; X , . . . , X n ) = 2 n/ π n/ δ m ( τ ) | dX ( X , . . . , X n ) | , for all test sections τ ∈ Γ( ¯ L ) and smooth complex vector fields X , . . . , X n ∈ X ( X ) . Remark 4.16
This Theorem justifies the definition of a natural inner product in H Q R defined bydeclaring { n/ π n/ δ m ⊗ p | dX |} m ∈ P X ∩ Z n to be an orthonormal basis. ♦ We note that the results of Theorem 4.15 are also valid when X is not compact provided thatthe growth of ψ at infinity is appropriately controlled, so that the function on ˇ P X e − c R t ( x − m ) · G s ( m + t ( x − m )) · ( x − m ) dt (det G s ) is bounded for some sufficiently small c >
0. This ensures the existence of the L -norms in questionand also that one can still apply the Laplace approximation to obtain the convergence to Diracdelta distributions. 27 The BKS pairing
Let K I and K J be the canonical bundles on X associated to two toric K¨ahler complex structures I and J . One may define a nondegenerate sesquilinear pairing Γ ( K I ) × Γ ( K J ) → C ∞ ( M ) bycomparison with the Liouville form. Specifically, for sections α ∈ Γ( K I ) and β ∈ Γ( K J ), define thepairing of α and β to be the function h α, β i := α ∧ ¯ β (2 i ) n ( − n ( n +1)2 ω n /n ! . Suppose for the moment that K I and K J admit square roots. Then the above pairing inducesa sesquilinear pairing Γ (cid:0) √ K I (cid:1) × Γ (cid:0) √ K J (cid:1) → C ∞ ( M ) via h µ, ν i := s µ ∧ ¯ ν (2 i ) n ( − n ( n +1)2 ω n /n ! , for sections µ ∈ Γ (cid:0) √ K I (cid:1) and ν ∈ Γ (cid:0) √ K J (cid:1) . Note that when I = J , the half-form pairing abovereduces to the canonical hermitian structure (3.3) on √ K I . Moreover, this pairing of half-formsinduces a pairing on the half-form corrected prequantizations of X which is known as the Blattner–Kostant–Sternberg (BKS) pairing: h s ⊗ µ, t ⊗ ν i BKS := Z X h ℓ ( s, t ) h µ, ν i ω n n ! . Let ℓ, L be the line bundles over X as in Section 3. The fact that ℓ ⊗ √ K ∼ = L ⊗ p | K | will motivatethe definition of the BKS pairing even in the case when √ K does not exist.Let us examine the half-form pairing on X in a little more detail. A short computation showsthat on the open orbit h dZ I , dZ J i = det (cid:18) G I + G J (cid:19) > , (5.1)since both G I and G J are positive definite. Notice that although in general one expects h dZ I , dZ J i to be complex, in the toric case it turns out to be real (and positive) since the unitarization of K I is equal to the unitarization of K J , whence the phases do not contribute to the pairing. To put itin another way, the pairing of K I and K J is entirely captured by the modulus bundles | K I | and | K J | . Consequently we define a pairing between sections of p | K I | and p | K J | by h µ I , µ J i := h dZ I , dZ J i k dZ I k K I k dZ J k K J , (5.2)where µ I , µ J are defined in (3.4). Then, in the general case when √ K may not exist, we define aBKS pairing by h ˆ σ I , ˆ σ J i := 1(2 π ) n Z X h L ( σ I , σ J ) h µ I , µ J i ω n n ! , (5.3)where ˆ σ I = σ I ⊗ µ I ∈ H QI , ˆ σ J = σ J ⊗ µ J ∈ H QJ . This pairing coincides with the inner product (3.8) in H QI when I = J . From (5.2), we also seethat it coincides with the standard BKS pairing in the case when √ K exists.Let h Im = ( x − m ) ∂g I ∂x − g I , where g I is the symplectic potential defining the complex structure I . From Theorem 3.6 and (5.1) it is straightforward to obtain28 roposition 5.1 In the orthogonal basis { ˆ σ mI } m ∈ P L ∩ Z n for H QI and { ˆ σ mJ } m ∈ P L ∩ Z n for H QJ , wehave h ˆ σ mI , ˆ σ m ′ J i = δ mm ′ Z P X e − h Im − h Jm s det (cid:18) G I + G J (cid:19) dx, (5.4) for m, m ′ ∈ P L ∩ Z n . Let I s denote the “simple” family of complex structures on X associated to the symplectic potentials g s = g P + ϕ + sψ for s ∈ [0 , ∞ ). (Recall that we could have more general deformations of I definedby ψ ( s ).) We can consider the BKS pairing for two values s, s ′ in the same simple family. As wewill see, even for these simple families, the BKS pairing is not unitary.From Proposition 5.1, we see that I - and J -holomorphic sections associated with differentintegral points are orthogonal. In the following we will therefore consider only one “Fourier” sectorat a time, that is, a one-dimensional subspace of the quantization space. Let m ∈ P L ∩ Z n , andconsider the corresponding monomial section ˆ σ ms ∈ H QI s .Since the BKS pairing h ˆ σ ms , ˆ σ ms ′ i BKS is real and positive (the integrand is positive), the unitarityof the BKS pairing map for the complex structures s, s ′ is equivalent to h ˆ σ ms , ˆ σ ms ′ i BKS = || ˆ σ ms || L · || ˆ σ ms ′ || L . For our choice of monomial sections, writing out the integral in Proposition 5.1 shows that for somefunction α ∈ C ∞ ( R ≥ ), h ˆ σ ms , ˆ σ ms ′ i BKS = α ( s + s ′ ) > . Unitarity then implies α ( s + s ′ ) = p α (2 s ) α (2 s ′ ). Putting α = e f , we get f (cid:18) s + s ′ (cid:19) = f ( s ) + f ( s ′ )2 , (5.5)and differentiating in s we obtain 12 f ′ (cid:18) s + s ′ (cid:19) = 12 f ′ ( s ) . Therefore,
Lemma 5.2
The BKS pairing between the I s - and I s ′ -holomorphic quantizations of X is unitaryif and only if for each m ∈ P L ∩ Z n and for each s ≥ , k ˆ σ ms k L = k ˆ σ m k L e sb , (5.6) for some constant b . Comparing with Lemma 4.12, we see that if (5.6) holds, then we have b = − ψ ( m ) and k ˆ σ m k L = π n/ e g ( m ) . Moreover, replacing ψ by ψ + const does not change the complex structure. Hence,we can assume that ψ ( m ) = 0 so that the BKS pairing is unitary if and only if k ˆ σ ms k L = k ˆ σ m k L ,that is, that the L -norm of ˆ σ ms is independent of s .By the argument above, the following theorem implies that the BKS pairing is not unitary alongthe simple family I s , s ≥
0. 29 heorem 5.3
For sufficiently large s , dds k ˆ σ ms k L = 0 . In particular, k ˆ σ ms k L is not constant, whence the BKS pairing is not unitary. Proof.
We have, using the identity dds (cid:12)(cid:12) s =0 det( A + sX ) = det( A ) tr ( A − X ) ,dds k ˆ σ ms k L = Z P X (cid:18) (cid:18) ψ − ( x − m ) · ∂ψ∂x (cid:19) + 12 tr (cid:0) G − s Hess ψ (cid:1)(cid:19) e − h Ism p det G s dx. Without loss of generality, assume that ψ is scaled so that ψ ( m ) = 0. Then by Lemma 4.9, (cid:16) ψ − ( x − m ) · ∂ψ∂x (cid:17) is strictly negative, so that for large s the integrand is also strictly negative,since tr (cid:0) G − s Hess ψ (cid:1) converges pointwise to zero as s → ∞ , which implies that dds k ˆ σ ms k L is notequal to zero. Let I be the set of compatible toric complex structures on X . This naturally corresponds tothe space of allowed Guillemin-Abreu symplectic potentials modulo additive constants, since twosymplectic potentials define the same holomorphic coordinates iff they differ by an additive constant.Note that if g I is a symplectic potential defining the complex structure I , then g I + sψ where s ∈ R , ψ ∈ C ∞ ( P X ) will also be an allowed symplectic potential provided that | s | is sufficientlysmall. Therefore, we can regard I as an open subset of the affine space g P X + ( C ∞ ( P X )) / R . Fix apoint p ∈ ˇ P X . In the following we will assume that for each complex structure I ∈ I a symplecticpotential g I was chosen such that g I ( p ) − g P X ( p ) = 0.One can define a vector bundle over I , H Q → I , with fiber H QI . The bundle H Q is the naturalsetting for studying the dependence of K¨ahler quantization on the choice of complex structure.As mentioned in the introduction, Axelrod, Della Pietra and Witten in [APW91] (see also therelated work [Hit90] of Hitchin), for the case where I is the space of linear complex structures on R n which are compatible with the standard symplectic form, but without including the half-formcorrection, introduced a natural unitary connection, which is called the quantum connection , onthe analogue of the quantum bundle H Q . This quantum connection is defined to be the projectionof the trivial connection in I × Γ( ℓ ) to H Q .In our case, the BKS pairing induces a connection, ∇ Q , on H Q → I as follows. Recall thatdifferent Fourier modes are orthogonal with respect to the BKS pairing, so that we can considereach Fourier mode labeled by m ∈ P L ∩ Z n separately. Consider the monomial sections ˆ σ mI ∈ H qI , I ∈ I , m ∈ P L ∩ Z n . Define ∇ Qψ by h∇ Qψ ˆ σ mI , ˆ σ mI i = dds | s =0 h ˆ σ mI + sψ , ˆ σ mI i , where ψ ∈ C ∞ ( P X ) with ψ ( p ) = 0 and where I + sψ denotes the complex structure defined by thesymplectic potential g I + sψ for | s | sufficiently small. Note that, by construction, ∇ Q is unitary.A final implication of the fact that all of the complex structures arising from points in I havethe same “phases” (that is, of Lemma 2.9) and of the consequent reality of the BKS pairing is thefollowing theorem. Theorem 5.4
The global frame for H Q → I given by { ˆ σ mI / || ˆ σ mI ||} m ∈ P L ∩ Z n is horizontal withrespect to ∇ Q . Therefore, the connection ∇ Q is flat. roof. We have, from the reality of the BKS pairing, for each Fourier mode m ∈ P L ∩ Z n and forany ψ ∈ C ∞ ( P X ) with ψ ( p ) = 0,0 = dds | s =0 h ˆ σ mI + sψ || ˆ σ mI + sψ || , ˆ σ mI + sψ || ˆ σ mI + sψ || i = 2 h∇ Qψ ˆ σ mI || ˆ σ mI || , ˆ σ mI || ˆ σ mI || i , whence ∇ Qψ ˆ σ mI || ˆ σ mI || = 0. K For toric varieties, the following proposition explains how the existence of √ K depends on thecombinatorics of the fan Σ associated to X . Note also that, in general, if a square root of thecanonical bundle exists then there may be many choices of square root, and they are parameterizedby H ( X, Z ). For toric varieties, H ( X, Z ) = { } , and so if it exists, √ K is unique.Let { ν j } j =1 ,...,r be primitive generators of the 1-dimensional cones in the fan Σ. Let L bea holomorphic line bundle on X . The divisor of a torus-invariant meromorphic section of L onany holomorphic vertex chart U v determines the section up to multiplicative constant. Indeed, atorus-invariant principal divisor is of the form r X i =1 h α, ν j i D j , for α ∈ R n . If, say, ν , . . . , ν n are the generators associated to U v , which form a basis of Z n , thenthe restriction of the principal divisor to U v determines it completely since α is fixed by its innerproducts with ν , . . . , ν n . Such a line bundle has a system of meromorphic locally trivializing sec-tions { Lv } on the holomorphic vertex charts, as in Section 2.2.3, defined uniquely (up to constants)by the property that div( Lv ) | Uv = 0.If c ( X ) is even then √ K exists and it has a system of holomorphic trivializations on theholomorphic vertex charts given by { ( U v , v ) } , where v := √ Kv . Then, { ( U v , v ⊗ v ) } gives asystem of trivializing sections for K and we have, up to an irrelevant constant, v ⊗ v = dW v .Therefore, the sections dW v have even divisors and we can write v = √ dW v where { ( U v , √ dW v ) } is a system of holomorphic trivializing sections on the holomorphic vertex charts for √ K . Wetherefore have Proposition 6.1 If c ( X ) is even, then the sections dW v have even divisors and { ( U v , √ dW v ) } isa system of holomorphic trivializations of √ K. As a consequence, we obtain the following useful criterion for the existence of √ K. For a vertex v , let us call vertex basis associated to v to a basis of Z n given by the primitive generators of the1-dimensional cones defining v . Proposition 6.2 K admits a square root if and only if for each 1-dimensional cone F in Σ thesum of the coordinates of the primitive generator ν F expressed in any one of the vertex basis is odd. Proof.
We have seen that the existence of √ K is equivalent to having a system of trivializingsections {√ dW v } where the dW v all have even divisors. Choose a vertex v and let ν , . . . ν n be avertex basis for v . Then, using Lemma 2.10 and the fact that dZ v is a constant multiple of dZ ,div( dW v ) = div( w v ) + div( dZ v ) = r X i = n +1 n X j =1 ν ji D i − r X i = n +1 D i , ν ji is the jth coordinate of the vector ν i in the vertex basis. Therefore, this divisor is eveniff P nj =1 ν ji is odd for all i . Clearly, this happens for any v and the proposition follows. Acknowledgements:
We wish to thank Thomas Baier for extensive discussions on the sub-ject of the paper. We also wish to thank the referee for several helpful comments which helpedimproving the clarity of the text. The authors are partially supported by the Center for Mathe-matical Analysis, Geometry and Dynamical Systems, IST, through the Funda¸c˜ao para a Ciˆencia ea Tecnologia (FCT / Portugal).
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