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[EQUATION] Locally we may also write [MATH] In the special case where [MATH] , we may use the frame [MATH] in a local holomorphic coordinate [MATH] . In the local frames [MATH] |
of [MATH] we may write sections as [MATH] and then [MATH] and then, [EQUATION] where [MATH] is the area form of [MATH] 3. Bochner Laplacians on line bundles |
In this section, we give explicit local formulae for Bochner Laplacians Bochner Laplacians [MATH] on [MATH] equipped with the data |
[EQUATION] where [MATH] is a Hermitian holomorphic line bundle, [MATH] is a metric on [MATH] [MATH] is a connection on [MATH] . In a local frame [MATH] of [MATH] , with [MATH] , the inner product [MATH] on [MATH] takes the form, |
[EQUATION] The inner product on [MATH] has the form, [EQUATION] With no loss of generality, we fix [MATH] on [MATH] and assume that [MATH] is a Kähler metric with [MATH] Then [MATH] . There is only one metric coefficient, [MATH] . It is a Hermitian metric on [MATH] |
and is compatible with [MATH] We also denote the Riemannian volume form by [MATH] {rema} Notational remark: We use [MATH] rather than [MATH] or [MATH] for the dual co-metric on [MATH] -forms, because it is a convenient notation for later variations. |
The Bochner Laplacian is the Laplacian on [MATH] determined by the quadratic form, [EQUATION] Throughout we assume that [MATH] is [MATH] -compatible. In a local frame [MATH] of [MATH] , with [MATH] [MATH] |
and with [MATH] and the quadratic form is given by [EQUATION] The adjoints are taken with respect to the volume form [MATH] We give local formulae for [MATH] under several assumptions on [MATH] and in correspondingly adapted frames (equivalently, choosing a gauge for [MATH] ): |
(i) [MATH] is [MATH] -compatible (see Section 3.1 ); in this case, we compute in a local unitary frame. Fixing [MATH] is equivalent to fixing the principal [MATH] bundle [MATH] , and varying the connection [MATH] -forms |
[MATH] on [MATH] (with fixed [MATH] ). (ii) [MATH] is compatible with a fixed complex structure [MATH] on [MATH] (see Section 3.2 ); in this case we compute in a local holomorphic frame. In the next section we fix [MATH] and vary [MATH] (with fixed [MATH] ). |
(iii) [MATH] is compatible with both [MATH] , hence is the Chern connection; see Section 3.3 The [MATH] -part of the connection has the form [MATH] and is parameterized by the Hermitian metric [MATH] on [MATH] ; in the next section we consider its variation with [MATH] (with fixed [MATH] ). |
(iv) When [MATH] is the canonical line bundle, we let [MATH] be the Hermitian metric induced by [MATH] and let [MATH] be the Levi-Civita connection. This is a special case of an [MATH] -compatible connection but is special because [MATH] is induced by [MATH] . Moreover, the Riemannian connection w.r.t. [MATH] is the Ch... |
[MATH] . In the next section, we vary this connection by varying [MATH] on [MATH] There exist many formulae for Bochner Laplacians in the literature (see for instance |
), but they often make assumptions on the compatibility of the connection with other data (the Hermitian metric or complex structure) and we need explicit dependence on the compatibility conditions so that we can perturb some of the data while holding others fixed. We therefore go through the calculations with explicit... |
We also recall the general identities, [MATH] Note that [MATH] Hence, [MATH] since [MATH] 3.1. Calculation in a unitary frame In this section we assume that [MATH] is compatible with [MATH] We recall from Section that on a Hermitian line bundle [MATH] , the set [MATH] of connections on [MATH] |
which are compatible with the Hermitian metric is the affine space [MATH] where [MATH] is a fixed background connection and [MATH] are the real [MATH] -forms on [MATH] . The Hermitian metric determines the principal [MATH] |
bundle [MATH] of unitary frames of [MATH] and as before [MATH] determines a connection [MATH] -form [MATH] on [MATH] . On the base [MATH] , the connection [MATH] -form is [MATH] -valued in a unitary frame and we write it as [MATH] with a real-valued [MATH] |
Proposition 3 Let [MATH] be a Riemannian manifold and let [MATH] be a Hermitian line bundle with [MATH] -compatible connection [MATH] . Let |
[MATH] with [MATH] in a unitary frame [MATH] . Then, [EQUATION] where [MATH] is the scalar Laplace operator. Proof 3.1 In a unitary frame, [MATH] and this factor drops out. We leave it in until the last step for purposes of later comparison to other frames. Since |
[MATH] , and by ( 3.1 ), [EQUATION] Note that [EQUATION] is the Hermitian norm-squared, so [EQUATION] where in the last line we use that [MATH] in a unitary frame. Since [MATH] is real-valued, |
[EQUATION] Recall that [MATH] . Replacing [MATH] by [MATH] and integrating the [MATH] by parts gives [EQUATION] Thus, we get [EQUATION] |
3.2. Holomorphic line bundles: [MATH] -compatible connections. In this section we give a local formula for the Bochner Laplacian when [MATH] is a holomorphic line bundle and [MATH] is compatible with the complex structure. Thus, complex structures [MATH] on [MATH] and [MATH] on [MATH] are fixed. In a holomorphic frame,... |
The Bochner-Kodaira identity relates [MATH] to [MATH] , where [EQUATION] The analogue of Proposition is Proposition 4 If [MATH] is compatible with [MATH] with connection [MATH] -form [MATH] with |
[MATH] of type [MATH] in the holomorphic frame [MATH] , then [EQUATION] Proof 3.2 The proof is similar to that of Proposition , with two differences: (i) We use a holomorphic frame rather than a unitary frame and [MATH] is not equal to [MATH] ; (ii) [MATH] is of type [MATH] rather being |
[MATH] -valued. Note that [EQUATION] By ( 3.1 ), and integrating by parts the [MATH] term, and with [MATH] denoting the Hermitian metric, we get |
[EQUATION] Further, [EQUATION] We simplify the [MATH] term using that [MATH] so that [MATH] Integrating the [MATH] by parts gives |
[EQUATION] Combining with the term [MATH] , we get [EQUATION] 3.3. Chern connection In this section we assume [MATH] is both [MATH] -compatible and [MATH] -compatible, i.e., that it is the Chern connection with connection [MATH] -form [MATH] One can then compute [MATH] using the relation |
[EQUATION] between the Kodaira and Bochner Laplacians. Note that [MATH] is real and [MATH] , so [MATH] and [MATH] above. Also, [MATH] so the terms [MATH] cancel and from the preceding Proposition we get |
[EQUATION] We now prove this directly. Proposition 5 Let [MATH] be the Chern connection for [MATH] . Then, [EQUATION] Proof 3.3 Using ( 3.2 ) and [MATH] , we have |
[EQUATION] We rewrite [MATH] term using that [MATH] so that [MATH] Integrating the [MATH] by parts gives [EQUATION] Adding the curvature term adds [MATH] |
{rema} Proposition and ( 3.2 ) are consistent by the following calculation: If [MATH] is a Chern connection [MATH] -form, then [EQUATION] |
Indeed, in terms of the Hermitian inner product, [EQUATION] 3.4. Canonical bundle: [MATH] and [MATH] is the Riemannian connection |
Let [MATH] be a local holomorphic coordinate and let [MATH] be the associated section of [MATH] Differentials of type [MATH] are sections of [MATH] , the [MATH] -th power of the canonical bundle. The Riemannian metric on [MATH] induces a Hermitian metric [MATH] on [MATH] , namely |
[MATH] where [MATH] is the co-metric. [MATH] and at [MATH] [MATH] The metric [MATH] on [MATH] endows a Hermitian metric [MATH] on [MATH] and the associated Riemannian connection [MATH] |
is the Chern connection with connection [MATH] -form [MATH] in the frame [MATH] . For simplicity of notation we write [MATH] . It induces connections and Hermitian metrics on [MATH] with connection [MATH] -forms [MATH] . The associated Bochner Laplacian [MATH] on [MATH] corresponds to the quadratic form |
[EQUATION] Note that [MATH] and the Laplacian on scalar functions is given by [MATH] Proposition 6 Let [MATH] be the Chern connection for [MATH] . Then, |
[EQUATION] Proof 3.4 This follows from Proposition . We give a direct proof. By the Bochner-Kodaira formula ( 3.2 ), it suffices to prove |
[EQUATION] where [MATH] As above, we calculate the adjoint to be [EQUATION] It follows that [EQUATION] where we used [MATH] 4. Perturbation theory and genericity |
In this section we prove generic properties of the eigenvalues and eigensections of Bochner Laplacians [MATH] on complex holomorphic Hermitian line bundles [MATH] . Our ultimate goal is to deduce generic properties of Kaluza-Klein Laplacians on the principal [MATH] frame bundles [MATH] associated to [MATH] . First we d... |
[MATH] as a regular value). This immediately implies that for the associated Kaluza-Klein Laplacians [MATH] on [MATH] , all joint eigenfunctions of the [MATH] |
action and [MATH] have simple joint spectrum and have [MATH] as a regular value. In Section 4.6 , we discuss the multiplicity of the spectrum of [MATH] , hence proving a part of Theorem 1.3 |
The main result of this section is: {theo} For generic ‘admissible data’ described below, and for every [MATH] , the spectrum of each Bochner Laplacians [MATH] on [MATH] is simple and all of its eigensections have zero as a regular value. Moreover, if we lift sections to equivariant eigenfunctions [MATH] then [MATH] an... |
The generic admissible data is of the following kinds: (i) We fix [MATH] and vary the connection [MATH] in [MATH] Fixing [MATH] is equivalent to fixing the principal [MATH] bundle [MATH] , and varying the connection [MATH] -forms. |
(ii) We fix [MATH] and vary both [MATH] and [MATH] , assuming that [MATH] is compatible with [MATH] on [MATH] but not necessarily with [MATH] |
(iii) We fix [MATH] and vary [MATH] assuming that [MATH] is compatible with both [MATH] , hence is the Chern connection of [MATH] |
(iv) We fix [MATH] and also fix [MATH] and vary [MATH] in the conformal class associated to [MATH] . We assume that [MATH] is the Hermitian metric induced by [MATH] and that [MATH] is the Levi-Civita connection. |
The proofs in each of the cases are given in separate sections. Note that the functions relevant to this article are smooth sections of a complex line bundle [MATH] and may locally be represented as complex valued functions [MATH] . We will prove that |
[MATH] has zero as a regular value, i.e., that [MATH] is surjective. It follows that [MATH] are independent and nowhere vanishing on their zero sets, and that each has zero as a regular value |
4.1. The Uhlenbeck framework To study generic properties of the spectrum, we follow and work with [MATH] spaces of metrics and connections. We use the following notation: |
We denote by [MATH] the Banach space of [MATH] metrics on [MATH] Since [MATH] is a surface and we usually fix the complex structure [MATH] , we only work with [MATH] metrics in the associated conformal class [MATH] and represent them in the usual Weyl gauge [MATH] |
relative to a fixed background metric [MATH] . Thus, we may identify [MATH] . We may also fix the area of the metrics with no loss of generality and then [MATH] |
may be identified with the space [MATH] of Kähler metrics on [MATH] in a fixed cohomology class. This is simply a different choice of gauge in which we write the Kähler forms as [MATH] and use the potentials [MATH] rather than the Weyl gauge [MATH] to parameterize metrics. |
We denote by [MATH] the Banach space of [MATH] Hermitian metrics on [MATH] . Once we fix a local frame [MATH] we may identify [MATH] with the function [MATH] such that [MATH] , and [MATH] is then equivalent to [MATH] except of course that the identification is frame dependent and the frame is only local (defined on the... |
We denote by [MATH] the space of connections with [MATH] connection forms. As before, we also denote by [MATH] , resp. [MATH] , the |
[MATH] -compatible (resp. [MATH] -compatible) [MATH] connections. We denote by [MATH] the [MATH] sections of [MATH] . We also denote by [MATH] the Sobolev space of sections with [MATH] derivatives in [MATH] |
We define [EQUATION] by [EQUATION] Here, the eigenvalue parameter [MATH] in the domain is allowed to be complex even though at zeros of [MATH] it is always real. This does not change the arguments in |
but is needed so that [MATH] spans the eigenspace when [MATH] is an eigensection. In the eigenfunctions were real-valued, so this issue did not arise. |
Recall that a linear map between Banach spaces is Fredholm if it has closed image and finite dimensional kernel and cokernel. The index of a Fredholm operator is the difference of the dimensions of its kernel and cokernel. A nonlinear map [MATH] of Banach manifolds is Fredholm if its derivative [MATH] is Fredholm for e... |
[MATH] Our first goal, roughly speaking, is to prove that [MATH] is a Fredholm map of index [MATH] i.e., to prove surjectivity of the differentials [MATH] |
from tangent spaces of [EQUATION] to [MATH] . It is sufficient to pick the relevant types of frames and calculate the Bochner Laplacians in the frame as in Section |
Regarding the surjectivity, we need to prove density of the image and that the image is closed. Some care needs to be taken because sections of complex line bundles are ‘vector-valued’, i.e., have two real components. As explained in |
, there are pitfalls to avoid when generalizing the arguments of to the vector-valued case. But sections of line bundles are locally complex-valued functions and are essentially scalar functions, albeit with scalars in [MATH] |
4.2. Uhlenbeck’s argument We briefly review Uhlenbeck’s proof that for generic metrics on compact [MATH] Riemannian manifolds, all eigenvalues are simple and all eigenfunctions have [MATH] as a regular value. |
Her framework is quite general and therefore uses the notation [MATH] for the relevant space of metrics or other geometric data, and [MATH] |
for the Laplacian associated to [MATH] . The relevant functions are denoted by [MATH] and the space of such functions on a manifold |
[MATH] is denoted by [MATH] even though they could be sections of a bundle over [MATH] Then define [EQUATION] and put [MATH] [MATH] |
[MATH] Then, [EQUATION] We often write [EQUATION] Further, let [MATH] denote the derivative of [MATH] along [MATH] . Then, [EQUATION] |
Also define [MATH] to be the image of [MATH] [EQUATION] We use the following ‘abstract genericity’ result of 20 , Theorem 1] 20 , Lemmas 2.7-2.8] |
{theo} Assume that [MATH] is [MATH] and has zero as a regular value. Then the eigenspaces of [MATH] are one-dimensional. If additionally, [MATH] has zero as a regular value, then additionally |
[EQUATION] is residual in [MATH] The key proposition is the following procedure for verifying the first hypothesis of Theorem 4.2 (see 20 , Proposition 2.10] ). |
Proposition 7 Let [MATH] and assume that for [MATH] and [MATH] , the property [MATH] for all [MATH] implies [MATH] . Then [MATH] is [MATH] and has zero as a regular value. |
For the sake of completeness, we briefly review the main steps in proving Theorem 4.2 The main input are two transversality theorems. The first is: Let [MATH] |
be a [MATH] map where [MATH] are Banach manifolds. If [MATH] is a regular value of [MATH] and [MATH] is a Fredholm map of index [MATH] , then the set [MATH] is residual in [MATH] |
The second statement follows from 20 , Lemma 2.7] : Let [MATH] be a [MATH] Fredholm map of index [MATH] . Then if [MATH] is a [MATH] map for [MATH] sufficiently large and if [MATH] |
is transverse to [MATH] then [MATH] is residual in [MATH] Let [EQUATION] {lemm} The eigenfunctions of [MATH] have zero as a regular value if [MATH] is a regular value of [MATH] and if [MATH] is a regular value of |
[MATH] Eigenfunctions and eigenvalues move continuously under perturbations of the operator. So it is easy to show that the set of metrics with for which the [MATH] th eigenvalue is simple is open. The difficulty is to prove that this set is dense. |
To prove the first statement in Theorem we need to verify the hypotheses of Theorem 4.2 and therefore need to prove Proposition , i.e., to determine the range of [MATH] |
Proposition 8 For each of the admissible types of perturbation, [MATH] is surjective from [MATH] 4.3. Base metric variations In this section we fix [MATH] |
and vary only [MATH] . Equivalently, we consider Kaluza-Klein metrics on a fixed [MATH] bundle [MATH] with a fixed connection [MATH] and vary the base metric [MATH] |
Proposition 9 Suppose that [MATH] is a Hermitian holomorphic line bundle with [MATH] -compatible connection [MATH] . Let [MATH] with [MATH] in a unitary frame [MATH] Then for generic Riemannian metrics [MATH] in the conformal class of [MATH] , all of the eigenvalues of [MATH] are simple and all of the eigensections hav... |
Proof 4.1 By Proposition [EQUATION] where [MATH] is the scalar Laplace operator, where [MATH] Taking the variation [MATH] with respect to [MATH] (and designating the variation with a dot), |
[EQUATION] But each term is conformal to that of [MATH] with conformal factor [MATH] Hence [EQUATION] If [MATH] then [EQUATION] To prove that the image of [MATH] is dense we argue by contradiction and suppose that there exists [MATH] such that |
[EQUATION] for all [MATH] But this implies that [MATH] for all [MATH] ,. then [MATH] . Write [MATH] so that the integral becomes, |
[MATH] for all [MATH] . This is only possible if [MATH] . But [MATH] and [MATH] can only vanish on a set of measure zero, so [MATH] almost everywhere. |
The image is closed because [MATH] is a Fredholm operator. 4.4. Varying the Hermitian connection In this section we fix [MATH] and vary [MATH] . In the application to Kaluza-Klein metrics, [MATH] is fixed and the base and vertical metrics are fixed and only the splitting into horizontal and vertical is varied. |
We recall from Section that some of the variations are ‘trivial’, i.e., are within a gauge equivalence class. Bochner Laplacians with gauge-equivalent connections are unitary equivalent by a gauge transformation, i.e., they have the same spectrum and their eigensections are related by a gauge transformations. Viewed in... |
Proposition 10 Suppose that [MATH] is a Hermitian holomorphic line bundle and let [MATH] be given by [MATH] with [MATH] in a unitary frame [MATH] Suppose that [MATH] is non-flat, or if it is flat, that [MATH] Then for generic gauge equivalence classes [MATH] , all of the eigenvalues of [MATH] are simple and all of the ... |
Proof 4.2 Again by Proposition [EQUATION] where [MATH] is the scalar Laplace operator. Taking the variation with respect to [MATH] gives, |
[EQUATION] If the image is not dense, there exists [MATH] so that [EQUATION] for all [MATH] We integrate [MATH] by parts to get, |
[EQUATION] We may assume that the frame [MATH] is unitary so that [MATH] If [MATH] and [MATH] for all [MATH] , then [MATH] . Indeed, we may consider [MATH] |
of the types [MATH] [MATH] separately to get orthogonality of the components [MATH] with [MATH] . This reduces matters to the fact that if [MATH] are complex-valued and [MATH] for all [MATH] , then [MATH] We conclude that |
[EQUATION] On any open set [MATH] where [MATH] we may divide by [MATH] and write the solution as, [EQUATION] This implies that [EQUATION] |
on a dense open set and since [MATH] , it is everywhere closed and hence the curvature of [MATH] is zero. This is impossible unless [MATH] is a topologically trivial line bundle, and the contradiction implies that [MATH] except when [MATH] |
4.5. Proof of Theorem Eigenfunctions move continuously under perturbations of the operator. So it is easy to show that the set of metrics with for which the [MATH] th eigenvalue is simple is open. The difficulty is to prove that this set is dense. |
To prove the first statement in Theorem we need to verify the hypotheses of Theorem 4.2 and therefore need to prove Proposition , i.e., to determine the range [MATH] of [MATH] |
To complete the proof of Theorem it suffices to prove: Proposition 11 For each [MATH] [MATH] is surjective to [MATH] Proof 4.3 Let [MATH] be the kernel of the Green’s function |
[MATH] for [MATH] for a given background metric [MATH] . As above, one may use the Hermitian metric [MATH] on [MATH] or the associated Kähler metric [MATH] as the parameter space of metrics. |
We need to show that for each [MATH] [EQUATION] has [MATH] as a regular value, i.e., that [EQUATION] is surjective to [MATH] , where [MATH] is the differential along [MATH] with [MATH] held fixed. Since [MATH] is fixed we may use a local coordinate [MATH] and frame [MATH] as above and identify local sections of [MATH] ... |
The constraint equation for [MATH] is [EQUATION] and we can solve for [MATH] as [EQUATION] By Proposition , the range of [MATH] , i.e., the set of functions [MATH] , spans [MATH] . Therefore, the image [MATH] spans |
[MATH] . It follows that the possible values of [MATH] are all functions of the form, [EQUATION] where [MATH] . Thus, [MATH] is surjective to [MATH] unless for all [MATH] either the real or imaginary parts of |
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