Spectral Properties of Kähler manifolds

On the Spectrum of geometric operators on Kähler manifolds

Abstract.

On a compact Kähler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace-Beltrami operator. Because of the high degree of symmetry the Laplace-Beltrami operator on forms can not be quantum ergodic. We show that after taking these symmetries into account quantum ergodicity holds for the Laplace-Beltrami operator and for the Spin-Dirac operators if the unitary frame flow is ergodic. The assumptions for our theorem are known to be satisfied for instance for negatively curved Kähler manifolds of odd complex dimension.

Key words and phrases:
Dirac operator, eigenfunction, frame flow, quantum ergodicity, Kähler manifold
2000 Mathematics Subject Classification:
Primary: 81Q50 Secondary: 35P20, 37D30, 58J50, 81Q005
The first author was supported by NSERC, FQRNT and Dawson fellowship.

1. Introduction

Properties of the spectrum of the Laplace-Beltrami operator on a manifold are closely related to the properties of the underlying classical dynamical system. For example ergodicity of the geodesic flow on the unit tangent bundle of a compact Riemannian manifold implies quantum ergodicity. Namely, for any complete orthonormal sequence of eigenfunctions to the Laplace operator with eigenvalues one has (see [Shn74, Shn93, Zel87, CV85, HMR])

(1)

for any zero order pseudodifferential operator , where integration is with respect to the normalized Liouville measure on the unit-cotangent bundle , and is the principal symbol of . Quantum ergodicity is equivalent to the existence of a subsequence of counting density one such that

(2)

In particular, might be a smooth function on and the above implies that the sequence

(3)

converges to the normalized Riemannian measure in the weak topology of measures. For bundle-valued geometric operators like the Dirac operator acting on sections of a spinor bundle or the Laplace-Beltrami operator the corresponding Quantum ergodicity for eigensections is known in a precise way to relate to the ergodicity of the frame flow on the corresponding manifold [JS]; see also [BoG04, BoG04.2, BO06].

This paper deals with a situation in which the frame flow is not ergodic, namely the case of Kähler manifolds. In this case the conclusions in [JS] do not hold since there is a huge symmetry algebra acting on the space of differential forms. This algebra is the universal enveloping algebra of a certain Lie superalgebra that is generated by the Lefschetz operator, the complex differentials and their adjoints. On the level of harmonic forms this symmetry is responsible for the rich structure of the cohomology of Kähler manifolds and can be seen as the main ingredient for the Lefschetz theorems. Here we are interested in eigensections with non-zero eigenvalues, that is in the spectrum of the Laplace-Beltrami operator acting on the orthogonal complement of the space of harmonic forms. The action of the Lie superalgebra on the orthogonal complement of the space of harmonic forms is much more complicated than on the space of harmonic forms where it basically becomes the action of . In this paper we classify all finite dimensional unitary representations of this algebra and determine the asymptotic distribution of these representations in the eigenspaces. Since the typical irreducible representation of the algebra decomposes into four irreducible representation for this shows that eigenspaces to the Laplace-Beltrami operator have multiplicities. An important observation in our treatment is that the universal enveloping algebra of this Lie superalgebra is generated by two commuting subalgebras, one of which is isomorphic to the universal enveloping algebra of . This -action is generated by an operator and its adjoint which is going to be defined in section 3.1. This operator can be interpreted as the Lefschetz operator in the directions of the frame bundle which are orthogonal to the frame flow. However, is not an endomorphism of vector bundles, but it acts as a pseudodifferential operator of order zero.

Guided by this result we tackle the question of quantum ergodicity for the Laplace-Beltrami operator on -forms. Unlike in the case of ergodic frame flow it turns out that there might be different quantum limits of eigensections on the space of co-closed -forms because of the presence of the Lefschetz operator. Our main results establishes quantum ergodicity for the Dirac operator and the Laplace Beltrami operator if one takes the Lefschetz symmetry into account and under the assumption that the -frame flow is ergodic. For example our analysis shows that in case of an ergodic -frame flow for any complete sequence of co-closed primitive -forms there is a density one subsequence which converges to a state which is an extension of the Liouville measure and can be explicitly given. For the Spin-Dirac operators we show that quantum ergodicity does not hold for Kähler manifolds of complex dimension greater than one. Thus, negatively curved Spin-Kähler manifolds provide examples of manifolds with ergodic geodesic flow where quantum ergodicity does not hold for the Dirac operator. Our analysis shows that there are certain invariant subspaces for the Dirac operator in this case and we prove quantum ergodicity for the Dirac operator restricted to these subspaces provided that the -frame flow is ergodic.

2. Kähler manifolds

Let be a Kähler manifold of real dimension . Let be the metric, be the hermitian metric, and the symplectic form. As usual let be the complex structure. A -frame for the cotangent space at some point is called unitary if it is unitary with respect to the hermitian inner product induced by . Hence, a -frame is unitary iff is orthonormal with respect to . A unitary -frame at a point is an ordered orthonormal basis for viewed as a complex vector space.

Clearly, the group acts freely and transitively on the set of unitary -frames. The bundle of unitary -frames is therefore a -principal fiber bundle. Let be the unit cotangent bundle with bundle projection . Then projection onto the first vector makes a principal -bundle over .

Transporting covectors parallel with respect to the Levi-Civita connection extends the Hamiltonian flow on to a flow on which we call the -frame flow (in the literature it is also referred to as the restricted frame flow). This is indeed a flow on since is covariantly constant and therefore unitary frames are parallel transported into unitary ones. This flow is the appropriate replacement for the -frame flow for Kähler manifolds as it can be shown to be ergodic in some cases, whereas the -frame flow never is ergodic for Kähler manifolds

Suppose that is a negatively-curved Kähler manifold [Bor]. We summarize results that can be found in [Br82, BrG80, BrP74]. We refer the reader to [BuP03, JS, Br82] and references therein for discussion of frame flows on general negatively-curved manifolds. Note that the frame flow is not ergodic on negatively-curved Kähler manifolds, since the almost complex structure is preserved. This is the only known example in negative curvature when the geodesic flow is ergodic, but the frame flow is not. In fact, given an orthonormal -frame , the functions are first integrals of the frame flow.

However, the following proposition was proved in [BrG80]:

Proposition 2.1.

Let be a compact negatively-curved Kähler manifold of complex dimension . Then the -frame flow is ergodic on when , or when is odd.

3. The Hodge Laplacian and the Lefschetz decomposition

Let be the complex vector bundle , where is the complexification of the co-tangent bundle. Then the Lefschetz operator is defined by exterior multiplication with the Kähler form , i.e. . Its adjoint is then given by interior multiplication with . Is is well known that

(4)

where is the orthoprojection onto , and define a representation of which commutes with the Laplace operator . The decomposition into irreducible representations on the level of harmonic forms is called the Lefschetz decomposition. We will refer to this decomposition as the Lefschetz decomposition in general. Note that since the Lefschetz operator commutes with each eigenspace decomposes into a direct sum of irreducible subspaces for the action.

The operators satisfy the following relations (see e.g. [B])

(5)

Thus, the operators form a Lie superalgebra with central element (see also [FrGrRe99]). Let the inverse of the Laplace operator on the orthocomplement of the kernel of . We view this as an operator defined in by defining it to be zero on and write slightly abusing notation.

3.1. The transversal Lefschetz decomposition

The operator is a partial isometry with initial space and final space . Hence, is the orthoprojection onto and is the orthoprojection onto . From the above relations one gets

(6)
(7)
(8)

from which one finds that

(9)

We define the transversal Lefschetz operator by

(10)

Then clearly and one gets that

(11)
(12)
(13)

and hence, also the transversal Lefschetz operators defines an action of on . Unlike the Lefschetz operator the transversal Lefschetz operator commutes with the holomorphic and antiholomorphic codifferentials.

Denote by the Lie-superalgebra generated by and relations

(14)

The subspace of odd elements is spanned by , the subspace of even elements is spanned by and . In the following we will denote by the universal enveloping algebra of this Lie-superalgebra viewed as a unital -algebra, i.e. the unital -algebra generated by the symbols and the above relations.

The relations (3.1) are obtained from the relations (5) by sending to and to . Therefore, we obtain a -representation of the Lie-superalgebra on the orthogonal complement of the kernel of .

3.2. The representation theory of

The calculations in the previous section used the relations in only. Hence, they remain valid if we regard and as elements in the abstract -algebra . Hence, generate a subalgebra in which is canonically isomorphic to the universal enveloping algebra of and which we therefore denote by . Note that commutes with and . Since is generated by two commuting subalgebras the representation theory for is very simple. The -subalgebra generated by and has the following canonical representation on . For an orthonormal basis of define the action of by exterior multiplication by , and the action of by exterior multiplication by . It is easy to see that all non-trivial finite dimensional irreducible -representations of are unitarily equivalent to this representation.

Note that the equivalence classes of finite dimensional irreducible -representations of are labeled by the non-negative integers. Denote the Verma-module for the Spin- representation by and the distinguished highest weight vector in by . Remember that is spanned by vectors of the form with and we have and .

Now define an action of on by

(15)

Clearly, this defines a -representation of on .

Theorem 3.1.

The representations are irreducible and pairwise inequivalent. Any non-trivial finite dimensional irreducible -representation of is unitary equivalent to some .

Proof.

Since is generated by two commuting subalgebras and any irreducible -representation of is also an irreducible -representation of . If it is finite dimensional it is therefore a tensor product of two finite dimensional irreducible representations of and . ∎

Corollary 3.2.

Any non trivial finite dimensional irreducible - representation of decomposes into a direct sum of equivalent modules for the action defined by .

If is a highest weight vector of then the kernel of in the representation is given by . Using the unitary basis for as before we see that the vectors

are in the kernel of . Moreover,

(16)
(17)
(18)

Therefore, in the decomposition of into irreducibles of the action defined by the representations occur with multiplicity at least and the representation occurs with multiplicity at least . The vector has weight and therefore, there must be another representation of highest weight greater or equal than occurring. Since

this shows that as a module for the action defined by we have .

Corollary 3.3.

Every non-trivial finite dimensional irreducible - representation of is as a module for the action defined by unitarily equivalent to the direct sum . By convention .

Corollary 3.4.

Let and be two finite dimensional modules. Then and are unitarily equivalent if and only if they are equivalent as modules for the action defined by .

3.3. The model representations

There is another very natural representation of the -algebra which is important for our purposes. This representation will be referred to as the model representation and can be described as follows. Let us view as a real vector space with complex structure . Let be the standard unitary basis in . Then in the complexification we define

(19)
(20)

We define to be the operator of exterior multiplication by on the space . Let be its adjoint, namely the operator of interior multiplication by . Let be the operator of exterior multiplication by and be the operator of exterior multiplication by . The operators and are defined as the adjoints of and . This defines a representation of on . This representation decomposes into a sum of irreducibles. Note that is given by exterior multiplication by . The restriction of to the two subalgebras generated by and define representations of . Since the maximal eigenvalue of is , only representations of highest weight with can occur in the decomposition of the model representation with respect to the -action by . Consequently, by Cor 3.3 in the decomposition of the model representation into irreducible representations only the representations with can occur.

4. Asymptotic decomposition of Eigenspaces

Since the action of commutes with the Laplace operator on forms each eigenspace

with is a -module and can be decomposed into a direct sum of irreducible -modules. In the previous section we classified all irreducible -representations of and found that they are isomorphic to for some non-negative integer . Therefore, we may define the function as

(21)

so that

(22)
Theorem 4.1.

Let be any compact Kähler manifold of complex dimension . Then in the decomposition of the eigenspaces of the Laplace-Beltrami operator into irreducible representations of the proportion of irreducible summands of type in is in average the same as the proportion of such irreducibles in the model representation of on :

(23)

where and is the spectral projection of the Laplace-Beltrami operator .

We recall that for the Laplacian on a bundle of rank over a manifold of real dimension . Note that apart from the fact that we are not dealing with a group but with a Lie superalgebra the action is neither on , nor on , but rather on the total space of the vector bundle . The action there leaves the fibers invariant and therefore it is rather different from a group action on the base manifold. The above theorem thus falls outside the scope of the equivariant Weyl laws of articles such as [BH1, BH2, GU, HR, TU]. In fact its conclusion is rather different from the conclusions in these articles as in our case only a fixed number of types of irreducible representations may occur.

Proof.

For a compact Kähler manifold acts by pseudodifferential operators on . Therefore, the symbol map defines an action of on each fiber of the bundle . The representation of on each fiber is easily seen to be equivalent to the model representation. Since the maximal eigenvalue of , acting on , is , only representations of highest weight with can occur in the decomposition of into irreducible subspaces with respect to the -action by . Again, by Cor 3.3 types with cannot occur in the decomposition with respect to the -action. Let be the orthogonal projection onto the type in . Then is actually a pseudodifferential operator of order . Namely, the quadratic Casimir operator of the -action by given by

(24)

is a pseudodifferential operator of order . On a subspace of type it acts like multiplication by . Therefore, if is a real polynomial that is equal to at and equal to at for any integer between and it follows that . Thus, is a pseudodifferential operator of order and its principal symbol at projects onto the subspace in the fiber which is spanned by the representations of type . Therefore, for every :

(25)

Applying Karamatas Tauberian theorem to the heat trace expansion

(26)

gives

(27)

After dividing by this reduces to the statement of the theorem. ∎

Remark 4.2.

A natural question is whether, for generic Kähler metrics, the eigenspaces of the Laplace-Beltrami operator are irreducible representations of the Lie superalgebra and of complex conjugation. Such irreducibility is suggested by the heuristic principle of ‘no accidental degeneracies’, i.e. in generic cases, degeneracies of eigenspaces should be entirely due to symmetries (see [Zel90] for some results and references). Cor. 5.3 would then suggest that for a generic Kähler manifold the spectrum of on the space of primitive co-closed -forms should be simple for fixed and .

5. Quantum ergodicity for the Laplace-Beltrami operator

We will now investigate the question of quantum ergodicity for the Laplace-Beltrami operator on a compact Kähler manifold and we keep the notations from the previous sections. As shown in [JS] this question is intimately related to the ergodic decomposition of the tracial state on the -algebra . The transversal Lefschetz decomposition plays an important role here.

5.1. Ergodic decomposition of the tracial state

On the space of -forms denote by the projection onto the space of transversally-primitive forms, i.e. onto the kernel of . Let be the projection onto the range of . The operators

(28)
(29)
(30)
(31)

are projections onto the ranges of the corresponding operators. We have

(32)

where is the finite dimensional projection onto the space of harmonic forms. Using the transversal Lefschetz decomposition we obtain a further decomposition

(33)

where each of the subspaces onto which projects is invariant under the Laplace operator.

Note that the principal symbols of these projections are invariant projections in and the above relation gives rise to a decomposition of the tracial state on defined by

(34)

into invariant states. Thus, the tracial state is not ergodic. However, if the -frame flow is ergodic this decomposition turns out to be ergodic.

Proposition 5.1.

Suppose that the -frame flow on is ergodic. Let be one of the projections

Then the state on defined by is ergodic. Here .

Proof.

The bundle can be naturally identified with the associated bundle , where is the representation of on

obtained from the canonical representation on . The pull back of can analogously be identified with the associated bundle

(35)

where is the restriction of to the subgroup . Since the first vector in is invariant under the action of we have the decomposition

into invariant subspaces. The projections onto these subspaces in each fiber is exactly given by the principal symbols of the projections . The representation of on may still fail to be irreducible. However, it is an easy exercise in representation theory (c.f. [FuHa91], Exercise 15.30, p. 226) to show that the kernel of in each fiber is an irreducible representation of . Thus, projects onto a sub-bundle of that is associated with an irreducible representation of , i.e.

(36)

This identification intertwines the -frame flow on and the flow . To show that the state is ergodic it is enough to show that any positive -invariant element in is proportional to (see [JS], Appendix). Under the above identification gets identified with a function which satisfies

(37)

If such a function is invariant under the -frame flow it follows from ergodicity of the -frame flow that it is constant almost everywhere. So almost everywhere , where is a matrix. By the above transformation rule commutes with . Since is irreducible it follows that is a multiple of the identity matrix. Thus, is proportional to the identity and consequently, is proportional to . ∎

Applying the abstract theory developed in [Zel96] the same argument as in [JS] can be applied to obtain

Theorem 5.2.

Let be one of the projections

and let be an orthonormal basis in with

(38)

If the -frame flow on is ergodic, then quantum ergodicity holds in the sence that

(39)

for any .

Since for co-closed forms primitivity and transversal primitivity are equivalent there is a natural gauge condition that manages without the above heavy notation.

Corollary 5.3.

Let be a complete sequence of primitive co-closed -forms such that

(40)

Then, if the -frame flow on is ergodic, quantum ergodicity holds in the sence that

(41)

for any , where is the orthogonal projection onto the space of primitive co-closed -forms.

6. Quantum ergodicity for Spin-Dirac operators

In this section we consider the quantum ergodicity for Dirac type operators rather than Laplace operators. The complex structure on Kähler manifolds gives rise to the so-called canonical and anti-canonical Spin- structures. The spinor bundle of the latter can be canonically identified with the bundle in such a way that the Dirac operator gets identified with the so-called Dolbeault Dirac operator. Other Spin- structures (e.g. the canonical one) can then be obtained by twisting with a holomorphic line bundle. Let us quickly describe the construction of the twisted Dolbeault operator.

Let be a holomorphic line bundle. Then the twisted Dolbeault complex is given by

This is an elliptic complex and the twisted Dolbeault Dirac operator is defined by

(42)

As mentioned above this operator is the Dirac operator of a Spin-structure on where the spinor bundle is identified with . Note that Spin structures on are in one-one correspondence with square roots of the canonical bundle , i.e. with holomorphic line bundles such that . In this case the Dirac operator is exactly the twisted Dolbeault Dirac operator.

The twisted Dolbeault Dirac operator is a first order elliptic formally self-adjoint differential operator. It is therefore self-adjoint on the domain of sections in the first Sobolev space. As is a first order differential operator its spectrum is unbounded from both sides.

The Dolbeault Laplace operator is given by and will be denoted by . The Hodge decomposition is

(43)

Note that the Dirac operator leaves invariant since it commutes with . Moreover, maps to and to . Therefore, the subspaces

(44)

are invariant subspaces for the Dirac operator. The orthogonal projections onto the closures of are clearly zero order pseudodifferential operators which commute with the Dirac operator.

Let be the norm closure of the -algebra of zero order pseudodifferential operators in . Then the symbol map extends to an isomorphism

(45)

By theorem 1.4 in [JS] is invariant under the automorphism group and the induced flow on