Math. Z. (2006) 253: 821–853DOI 10.1007/s00209-006-0936-8 Mathematische Zeitschrift

O. Hijazi · S. Montiel · F. Urbano

Spinc geometry of Kähler manifoldsand the Hodge Laplacian on minimalLagrangian submanifolds

Received: 24 May 2005 / Accepted: 24 November 2005 / Published online: 28 March 2006© Springer-Verlag 2006

Abstract From the existence of parallel spinor fields on Calabi-Yau, hyper-Kähleror complex flat manifolds, we deduce the existence of harmonic differential formsof different degrees on their minimal Lagrangian submanifolds. In particular, whenthe submanifolds are compact, we obtain sharp estimates on their Betti numberswhich generalize those obtained by Smoczyk in [49]. When the ambient manifoldis Kähler-Einstein with positive scalar curvature, and especially if it is a complexcontact manifold or the complex projective space, we prove the existence of Kähle-rian Killing spinor fields for some particular spinc structures. Using these fields,we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangiansubmanifolds and give some estimates for their spectra. These results also gener-alize some theorems by Smoczyk in [50]. Finally, applications on the Morse indexof minimal Lagrangian submanifolds are obtained.

Keywords Spinc geometry · Dirac operator · Lagrangian submanifolds · HodgeLaplacian

Mathematics Subject Classification (1991) Differential Geometry

Research of S.M. and F.U. is partially supported by a spanish MEC-FEDER grant No. MTM2004-00109

O. HijaziInstitut Élie Cartan, Université Henri Poincaré, Nancy I, B.P. 239, 54506 Vandœuvre-Lès-NancyCedex, FranceE-mail: hijazi@iecn.u-nancy.fr

S. Montiel (B)Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, SpainE-mail: smontiel@goliat.ugr.es

F. UrbanoDepartamento de Geometría y Topología, Universidad de Granada, 18071 Granada, SpainE-mail: furbano@goliat.ugr.es

822 O. Hijazi et al.

1 Introduction

Recently, connections between the spectrum of the classical Dirac operator onsubmanifolds of a spin Riemannian manifold and its geometry were investigated.Even when the submanifold is spin, many problems appear. In fact, it is knownthat the restriction of the spin bundle of a spin manifold M to a spin submanifoldis a Hermitian bundle given by the tensorial product of the intrinsic spin bundleof the submanifold and certain bundle associated with the normal bundle of theimmersion ([2,3,6]). In general, it is not easy to have a control on such a Hermitianbundle. Some results have been obtained ([2,23,24]) when the normal bundle of thesubmanifold is trivial, for instance for hypersurfaces. In this case, the spin bundleof the ambient space M restricted to the hypersurface gives basically the intrinsicspin bundle of the hypersurface and the induced Dirac operator is the intrinsicclassical Dirac operator on the hypersurface.

Another interesting manageable case is the case where the ambient space M isa spin Kähler manifold and the spin submanifold is Lagrangian. In this case, thenormal bundle of the immersion is isomorphic, through the complex structure ofthe ambient space, to the tangent bundle of the submanifold. Hence, it is naturalto expect that the restriction of the spin bundle of the Kähler manifold should be awell-known intrinsic bundle associated with the Riemannian structure of the sub-manifold. In fact, Ginoux has proved in [16] that, in some restrictive cases, thisbundle is the complexified exterior bundle on the submanifold. On the other hand,Smoczyk [49] showed that non-trivial harmonic 2-forms on minimal Lagrangiansubmanifolds of hyper-Kähler manifolds are obtained as restrictions to the subman-ifold of some parallel exterior forms on the ambient space. These two works can beconsidered as the starting point of this paper which can be framed into a series ofresults where well-behaved geometric objects on submanifolds are obtained fromparallel objects on the ambient manifold. In this sense, such results should be seenas sophisticated versions of the old Takahashi theorem [51] (see also [2,47]).

Our goal is to find conditions on an ambient Kähler manifold and on a Lagrang-ian submanifold, in order to get information on the spectrum of the Hodge Lapla-cian acting on the bundle of differential forms of the submanifold. For this, themain tools are to make use of the spinc geometry of the Kähler ambient manifoldand the symplectic geometry of the submanifold. In fact, to consider spin Kählermanifolds is quite restrictive, because important examples, as complex projectivespaces of even complex dimension, are not spin. On the other hand, any Kählermanifold admits a spinc structure. This allows us to work with such spinc structureson Kähler manifolds, to look for suitable (parallel, Killing, Kählerian Killing,…)spinor fields for those structures and to investigate what do these fields induceon their Lagrangian submanifolds. In fact, they induce differential forms whichare eigenforms for the Hodge Laplacian, provided that a certain condition on thesymplectic geometry of the submanifold is satisfied. In this way we find an unex-pected connection between spinc geometry of the ambient Kähler manifold andsome invariants of the symplectic geometry of its Lagrangian submanifolds.

There are two different successful situations: when the ambient space is aCalabi-Yau manifold and when it is a Kähler-Einstein manifold of positive scalarcurvature. In the first case, we consider the trivial spin structure, which carries par-allel spinor fields, and the Lagrangian submanifold is assumed to be minimal. We

Spinc geometry and the Hodge Laplacian 823

see that the restrictions to the submanifold of parallel spinor fields of the Calabi-Yaumanifold are harmonic forms, which yield sharp topological restrictions on suchsubmanifolds (Theorem 22), improving those obtained in [49]. In the second case,we use the cone on the maximal Maslov covering of a Kähler-Einstein manifoldof positive scalar curvature (which are simply-connected Calabi-Yau manifolds)and to find Kählerian Killing spinors for different spinc structures on some of suchmanifolds: the complex projective space and the Fano complex contact manifolds(Theorem 17). In these cases, to assume the minimality of the Lagrangian sub-manifold is not enough to guarantee that the restriction of some spinc bundle isthe exterior complex bundle. The reason is that there exists a very nice relationbetween the spinc bundles on the Kähler-Einstein manifold which restricted to theLagrangian submanifold are the exterior bundle and certain integer number associ-ated with a minimal Lagrangian submanifold, studied by Oh [45] and Seidel [48],which has its origin in symplectic topology. This number will be called the level ofthe minimal Lagrangian submanifold. Since in the above references that numberis only studied for embedded Lagrangian submanifolds, in Sections 3 and 4 wegeneralize their construction to immersed totally real submanifolds (Propositions4 and 8), computing in Section 5 the level of certain minimal Lagrangian subman-ifolds of the complex projective space. In Theorem 24, we summarize the aboveresults obtaining eigenforms of the Hodge Laplacian on some minimal Lagrang-ian submanifolds of the complex projective space and of Fano complex contactmanifolds (twistor manifolds on quaternionic-Kähler manifolds).

Finally, as the Jacobi operator of a compact minimal Lagrangian submanifoldof a Kähler manifold is an intrinsic operator acting on 1-forms of the submanifolds[43], in the last section we obtain some applications to study the index of suchsubmanifolds (Corollaries 29 and 30).

2 Some bundles over almost-complex manifolds

Let M be an almost-complex manifold of dimension 2n and let J be its almost-com-plex structure. The space of complex valued p-forms, p = 0, . . . , 2n, decomposesas follows

�p(M)⊗ C =⊕

r+s=p

�r,s(M) =⊕

r+s=p

(�r,0(M)⊗�0,s(M)

), (2.1)

where the vector bundles�r,0(M) and�0,s(M) of holomorphic and antiholomor-

phic type forms have complex ranks

(nr

)and

(ns

), respectively. In particular,

the line bundle

KM = �n,0(M) (2.2)

is called the canonical bundle of the manifold M .We will deliberately not distinguish complex line bundles and S

1-principal bun-dles over the manifold M . Their equivalence classes of isomorphisms are param-etrized by the cohomology groups

c1 : H1(M, S1) ∼= H2(M,Z), (2.3)

824 O. Hijazi et al.

where the isomorphism is given by mapping each line bundle or S1-principal bundle

L onto its first Chern class c1(L).Suppose that the first Chern class c1(KM ) of the canonical bundle of the almost-

complex manifold M is a non-zero cohomology class. Let p ∈ N∗ be the greatest

number such that

α = 1

pc1(KM ) ∈ H2(M,Z).

When M is a Kähler manifold this natural number p is called the index of the man-ifold. Instead, in symplectic geometry, this number is usually called the minimalChern number of M (see [48]). We will use the first terminology in the generalcase. Take now an S

1-principal L → M bundle such that

c1(L) = α = 1

pc1(KM ).

Then, the isomorphism (2.3) shows that

Lp = L⊗ p· · · ⊗L = KM , (2.4)

that is, L is a p-th root of the canonical bundle KM and so it is a p-fold coveringspace of KM . Since, in the context of symplectic geometry, a q-fold Maslov cover-ing of M (see [48, Section 2.b.]) can be defined as a q-th root of the power (KM )

2,then the bundle L is a 2p-fold Maslov covering of M and, indeed it is a Maslovcovering with a maximal number of sheets.

Remark 1 It is well-known that the index of the complex projective space CPn ,endowed with its usual complex structure, is n+1. The only line bundle L with firstChern class 1

n+1 c1(KCPn ) is the universal bundle over CPn (see [17]). The cor-responding principal S

1-bundle L is nothing but the Hopf bundle S2n+1 → CPn .

Then, in this case, KCPn = Ln+1 = S2n+1/Zn+1 and, consequently, (KCPn )2 =

L2n+2 = S2n+1/Z2n+2.

Remark 2 A Kähler manifold M of dimension 2n = 4k + 2 is said to be a com-plex contact manifold when it carries a codimension 1 holomorphic subbundle ofT 1,0 M which is maximally non-integrable or, equivalently, when the Chern classc1(KM ) is divisible by k + 1 (see [33]). So, the index p of such a M must bea multiple of k + 1 = n+1

2 . For instance, the complex projective space CP2k+1

is a complex contact manifold and, in fact, it is the only complex contact mani-fold with complex dimension 2k + 1 and index 2k + 2. Any other such manifoldhas index k + 1 = n+1

2 (see [7, Proposition 2.12]). Of course, there are a lotof such Kähler manifolds (see [7] again). In particular, Bérard-Bergery [4] andSalamon [46] showed that the twistor space of any 4k-dimensional quaternionic-Kähler manifold is a (4k+2)-dimensional complex contact manifold which admitsa Kähler-Einstein metric of positive scalar curvature. Later LeBrun proved in [33,Theorem A] that any complex contact manifold admitting such a metric must bethe twistor space of a quaternionic-Kähler manifold.

Spinc geometry and the Hodge Laplacian 825

3 Totally real submanifolds

Let L be an n-dimensional manifold immersed into the 2n-dimensional almost-complex manifold M . This immersed submanifold is called totally real when itstangent spaces have no complex lines, that is, when

Tx L ∩ Jx (Tx L) = {0}, ∀x ∈ L . (3.1)

The following result quotes one of the more relevant basic facts of this class ofsubmanifolds:

Lemma 3 Let ι : L → M be a totally real immersion of an n-dimensional man-ifold L into an almost-complex 2n-dimensional manifold M. Then the pull-backmaps

ι∗ : �r,0(M)|L → �r (L)⊗ C, ι∗ : �0,s(M)|L → �s(L)⊗ C

are vector bundle isomorphisms.

Proof It suffices to check it at each point x ∈ L and only for the map on theleft side. Since the complex vector spaces �r,0(M)x and �r (L)x have the same

dimension

(nr

), it is sufficient to prove that ι∗x has trivial kernel. Suppose that

ω ∈ �r,0(M)x and that

ω(ui1, . . . , uir ) = 0

for a given basis u1, . . . , un of Tx L . As ω is of (r, 0) type, we have

ω(ui1 − i Jui1, ui2 , . . . , uir ) = 2ω(ui1, . . . , uir ).

Then, by repeating the same argument at each entry of ω, we have

ω(ui1 − i Jui1, . . . , uir − i Juir ) = 2rω(ui1, . . . , uir ) = 0.

But condition (3.1) implies that the vectors u1 − i Ju1, . . . , un − i Jun form a basisfor T 1,0

x (M) and so ω = 0. �As a consequence of Lemma 3, if the submanifold L is orientable, then the

restricted line bundle KM |L = �n,0(M)|L = �n(L)⊗C is a trivial bundle and, inall cases, the same occurs for the tensorial 2-power (KM )

2|L since the line bundle�n(L)⊗�n(L) is trivial on each n-dimensional manifold L . Hence, the restrictionto the submanifold L of the maximal Maslov covering L over M verifies that L2p|Lis trivial. So, it is clear that the immersion ι can be factorized through L2p, that is,we have a lift

(KM )2 = L2p = L/Z2p

↗ ↓L

ι→ M.

826 O. Hijazi et al.

As ι is an immersion, then such any lift is another immersion which is trans-verse to the fundamental vector field (see [30]) produced by the S

1-action on L2p.Note that each lift is a section of (KM )

2|L and so, from Lemma 3, it provides anon-zero n-form, determined up to a sign, on the submanifold. Reciprocally, eachnon-trivial section of (�n(M)⊗ C)2 yields a lift .

Fix a lift : L → L2p of the totally real immersion ι: L → M . For each positiveinteger q such that q|2p, denote by Pq,2p : Lq → L2p the corresponding 2p

q -foldcovering map. We may consider the composition

π1(L)∗−→ π1(L2p) −→ π1(L2p)

(Pq,2p)∗(π1(Lq))∼= Z 2p

q. (3.2)

Moreover, denote by n(L , ) the smallest integer q such that this composition ofgroup homomorphisms is zero. Then we have n(L , )|2p and that can be liftedto Ln(L ,) with the property cannot be lifted to Lq for any q < n(L , ). Note that,when L is orientable, then any lift factorizes through Lp and then n(L , )|p.

The above facts can be summarized as:

Proposition 4 Let ι : L → M be a totally real immersion of an n-dimensionalmanifold into an almost-complex 2n-dimensional manifold M with index p ∈ N

∗.Then ι may be lifted to L2p, where L is any maximal Maslov covering of M. Foreach such lift : L → L2p, there exists a positive integer n(L , )|2p such that can be lifted to Ln(L ,) and this is the smallest index power for which this liftis possible. That is, there is an immersion : L → Ln(L ,) transverse to thefundamental field of the bundle such that

Ln(L ,)

↗ ↓L L2p

‖

↗ ↓L

ι→ M

is commutative. Moreover n(L , )|p if and only if L is orientable.

Remark 5 (Cf. [48, Lemmae 2.3 and 2.6]) Let q be an integer with 1 < q|n(L , ).Since Ln(L ,)|L is trivial, the restriction L

n(L ,)q |L trivializes over a non-trivial q-fold

covering of L . Hence we have

1 < q, q|n(L , ) ⇒ H1(L ,Zq) �= 0.

Moreover, as H1(L ,Zr ) is a subgroup of H1(L ,Zs), provided that r |s, we havethat

1 < n(L , )|q ⇒ H1(L ,Zq) �= 0.

As a consequence, as n(L , )|2p, we have that

q|2p, H1(L ,Zq) = 0 ⇒ n(L , )|2p

q.

Spinc geometry and the Hodge Laplacian 827

In particular, if H1(L ,Z2p) = 0, then we must have that n(L , ) = 1 for any lift of the immersion ι. In other words, each may be lifted to the maximal Maslovcovering L.

Remark 6 Since the principal S1-bundle Ln(L ,)|L is trivial, we have from the

isomorphism (2.3) that c1(Ln(L ,)|L) = 0, that is, n(L , )ι∗α = 0 in the groupH2(L ,Z). As a consequence, if ι∗α �= 0, the order of ι∗α ∈ H2(L ,Z) dividesn(L , ). Note that in the case ι∗α = 0 we have that the immersion ι : L → Mmay be lifted to the maximal Maslov covering L, but we have not n(L , ) = 1necessarily for every (if H1(L ,Z2p) �= 0) as we can see in Example 3 of Section5. This same example shows that n(L , ) depends on the lift fixed and not onlyon the totally real immersion ι.

4 Lagrangian submanifolds

Suppose now that the 2n-dimensional almost-complex manifold M is endowed witha Riemannian metric 〈 , 〉. Then the totally real submanifold L is another Riemann-ian manifold with the metric induced from M through the immersion ι : L → M .If x ∈ L , we have an n-form ωx defined, up to a sign, on Tx L by means of the con-ditionωx (e1, . . . , en) = 1 for each orthonormal basis {e1, . . . , en}. Thenω⊗ω is aglobal nowhere vanishing section of the line bundle�n(L)⊗�n(L) on L . When Lis orientable,ω is nothing but the volume form of its metric. Hence, using Lemma 3,we obtain a nowhere vanishing section of the restricted line bundle (�n,0(M))2|L .This means that, in this case, we have a privileged lift 0 : L → L2p = (KM )

2 forthe totally real immersion ι.

We will write nL for the integer n(L , 0) given in Proposition 4 where 0 isthe squared volume form of the metric induced on L and we will call it thelevel of the totally real submanifold L .

It is clear from (3.2) that this level is invariant for deformations of the immersion ιthrough totally real immersions, that is, it is an invariant of the (totally real) isotopyclass of ι.

Remark 7 One can see that, according to the above definition, the level nL coin-cides with the quotient 2p

NL, where NL is the integer defined by Seidel in his work

on grading Lagrangian submanifolds of symplectic manifolds (see [48]). We pointout that in [48], this integer is called the minimal Chern number of the pair (M, L)while it is called minimal Maslov number in [9]. In fact, the integer NL is thepositive generator of the subgroup µ(H1(L ,Z)) ⊂ Z, where µ ∈ H1(L ,Z) isthe Maslov class of L (see for example [10]), if the subgroup is non-zero, andNL = 2p if it vanishes. Note that these authors deal with embedded submani-folds and, consequently, they employ cohomology of pairs, while in this paper weconsider immersed submanifolds.

From now on, we assume that the almost-complex ambient manifold is Kähler.Then we can dispose of the Levi-Cività connection ∇ on any tensorial power bundleLq of the maximal Maslov covering of M . Thus it is natural to inquire under whichconditions the lift 0 to L2p of the immersion ι is horizontal with respect to this

828 O. Hijazi et al.

Levi-Cività connection. We will give an answer when the metric of M is Einstein.This answer will point out the importance of a fashionable family of totally realsubmanifolds.

Proposition 8 Let ι : L → M be a totally real immersion of an n-dimensionalmanifold L into a 2n-dimensional non scalar-flat Kähler-Einstein manifold M ofindex p ∈ N

∗ and 0 : L → L2p = (KM )2 the lift determined by the squared

volume form of the induced metric. Then 0 (or any other higher lift of 0) is hori-zontal with respect to the Levi-Cività connection of M if and only if ι is a minimalLagrangian immersion.

Proof Our assertion is an improvement of Proposition 2.2 in [45] (see also [8]) andthe proof follows from the corresponding one there. In fact, if 0 is horizontal, onehas a global nowhere vanishing section for the restricted line bundle (KM )

2|L onthe submanifold L . So the Levi-Cività connection on the restriction of the canon-ical bundle KM must be flat. But it is well known that the curvature form of thisconnection projects onto the Ricci form of the Kähler metric of M . Since this met-ric is Einstein and non scalar-flat, we have that ι∗�M = 0, where �M stands forthe Kähler two-form on M . Hence, the immersion is Lagrangian. But now, fromProposition 2.2 of [45] (take into account that Oh uses a different convention thatours for the mean curvature and that we work on the squared KM ), we have in thiscase

∇u0 = −2n√−1 〈J H, u〉 0 ∀u ∈ T L , (4.1)

where H is the mean curvature vector field of the immersion ι. Then 0 is parallelif and only if H is identically zero, that is, ι is minimal. �Remark 9 From Proposition 8 above, one deduces that if the totally real subman-ifold L is Lagrangian and minimal, then the restricted line bundle (KM )

2|L isgeometrically trivial, that is, it is flat and has trivial holonomy. The minimality ofthe submanifold L is not necessary in order to geometrically trivialize (KM )

2|L ,because one might find another parallel section different from 0. Using (4.1), onesees that a suitable modification of 0 through a change of gauge f : L → S

1 isparallel if and only if id log f coincides with the 1-form given by

u ∈ T L �−→ 2n〈J H, u〉.This means that the squared line bundle (KM )

2 is geometrically trivial on thesubmanifold L if and only if the so called mean curvature form

η(u) = n

π〈J H, u〉 ∀u ∈ T L ,

which is a closed form (see [11]), has integer periods, that is, the cohomologyclass [η] ∈ H1(L ,R) belongs to the image of the subgroup H1(L ,Z). With thesame effort, one can check in fact that Lq |L is geometrically trivial if and onlynL |q|2p and q

2p [η] ∈ H1(L ,Z). In particular, this happens when L is minimalor when the mean curvature form is exact (see [43]). This result is also true whenthe ambient manifold is Ricci-flat, provided that the submanifold is supposed to beLagrangian and not only totally real. In fact, in Ricci-flat ambient manifolds, thismean curvature class coincides with the Maslov class (see [34,10] and notice thatthere the mean curvature is not normalized) which is defined as an integer class.

Spinc geometry and the Hodge Laplacian 829

Remark 10 As another consequence of Proposition 8,

The level nL of a minimal Lagrangian immersion ι : L → M into a nonscalar-flat Kähler-Einstein manifold of index p, defined at the beginningof Section 4, is nothing but the smallest number q|2p such that ι admits ahorizontal (also called Legendrian) lift to the Maslov covering Lq .

In fact, such a horizontal lift followed by the covering map Pq,2p : Lq → L2p =(KM )

2 is also a horizontal map and so differs from 0 only by a constant change ofgauge. Taking into account that, according to [10], the minimal Maslov number iscomputable in terms of symplectic energy for minimal Lagrangian submanifolds,our level nL coincides also with the integer defined by Oh in [45], although hisdefinition, made in terms of cohomology of pairs, was valid only for embeddings.In particular, the line bundles LqnL |L with q = 1, . . . , 2p

nLare not only trivial bun-

dles but trivial as flat bundles too. In other words, they are flat bundles with trivialholonomy.

5 Some examples in CPn

Now we are going to compute the levels of the most relevant examples of minimalLagrangian submanifolds of the complex projective space (see details in [41,42]).

1. Consider the totally geodesic embedding of the n-dimensional unit sphere

(x1, . . . , xn+1) ∈ Sn ⊂ R

n+1 �→ (x1, . . . , xn+1) ∈ S2n+1 ⊂ C

n+1 ≡ R2n+2.

If π : S2n+1 → CPn is the Hopf projection, then ι = π ◦ : S

n → CPn

is a totally geodesic Lagrangian immersion of Sn into CPn endowed with the

Fubini-Study metric of holomorphic sectional curvature 4 (see [30]) and is ahorizontal lift of ι. As a conclusion nSn = 1.

2. The immersion ι : Sn → CPn above factorizes through the totally geodesic

Lagrangian embedding

ι2 : {x1, . . . , xn+1} ∈ RPn �−→ [x1, . . . , xn+1] ∈ CPn .

It is easy to see that 2 : RPn → RP2n+1 defined by

2 : {x1, . . . , xn+1} ∈ RPn �−→ {x1, . . . , xn+1} ∈ RP2n+1

is a horizontal lift of ι2. As a consequence, we have that nRPn |2. But, on theother hand, we have that c1(L|Sn/Z2) = (ι2)

∗α and that

(ι2)∗ : H2(CPn,Z) ∼= Z −→ H2(RPn,Z) ∼= Z2

is surjective. Then the line bundle L|RPn is topologically non-trivial and hencethe level of ι2 cannot be 1. As a conclusion nRPn = 2.

830 O. Hijazi et al.

3. Consider the (n + 1)-dimensional flat torus

Tn+1 = S

1(

1√n + 1

)× n+1· · · ×S

1(

1√n + 1

)⊂ S

2n+1.

The action of the circle S1 on the sphere S

2n+1 fixes Tn+1 and induces a minimal

Lagrangian embedding ι : Tn+1/S1 → CPn , where the domain is isometric to

a certain flat torus Tn . We define a map : T

n → S2n+1/Zn+1 by

{z1, . . . , zn+1} ∈ Tn+1

S1

�→{

1n+1√

z1 · · · zn+1(z1, . . . , zn+1)

}∈ S

2n+1

Zn+1.

One can check that is a horizontal lift of the embedding ι and so it coincideswith the horizontal lift 0 determined by the volume form of T

n up to productby an (n + 1)-th root of the unity. On the other hand, it is not difficult to seethat the induced group homomorphism

∗ : π1(Tn) ∼= Z⊕ n· · · ⊕Z −→ π1

(S

2n+1/Zn+1) ∼= Zn+1

is surjective. Then, from definition (3.2), n(Tn, 0) = n(Tn, ) = n + 1.Hence nTn = n + 1. It is important to notice that, in this case, the cohomologyhomomorphism ι∗ : H2(CPn,Z) −→ H2(Tn,Z) is trivial, and so the restric-tion of any line bundle Lq = S

2n+1/Zq to the torus Tn is trivial too. That is,

the embedding ι may be lifted to the sphere S2n+1 but not horizontally (see

Remark 6).4. For n ≥ 3, consider the minimal Lagrangian immersion

(z, x) ∈ S1 × S

n−1 ι�−→[

1√n + 1

(√n

xn+1√

z,

n+1√

zn

)]∈ S

2n+1/S1 = CPn

which is the projection of a horizontal immersion : S1×S

n−1 → S2n+1/Zn+1

defined changing the squared brackets denoting homogeneous coordinates bythe curly brackets representing the equivalence classes in the quotient of thesphere S

2n+1. In a way similar to Example 3 above, one easily sees that theinduced homomorphism

∗ : π1(S1 × S

n−1) ∼= Z −→ π1(S

2n+1/Zn+1) ∼= Zn+1

is surjective. Consequently we have nS1×Sn−1 = n + 1.The immersion ι above yields a minimal Lagrangian embedding

ι : (S1 × Sn−1)/Z2 −→ CPn

where Z2 denotes the group spanned by the diffeomorphism (z, x) ∈ S1 ×

Sn−1 �→ (−z,−x) ∈ S

1 × Sn−1 (see [41, Lemma 6.2]). When n is even, the

horizontal lift of ιmay be induced on the quotient (S1×Sn−1)/Z2, but when n

is odd this is impossible. In this case, the same expression defines an alternativelift

: {(z, x)} ∈ S1 × S

n−1

Z2�→

{1√

n + 1

(√n

xn+1√

z,

n+1√

zn

)}∈ S

2n+1

Z2n+2.

Spinc geometry and the Hodge Laplacian 831

This is also a horizontal lift inducing a surjective homomorphism between thecorresponding fundamental groups. Hence

n(S1×Sn−1)/Z2=

{n + 1 if n is even2n + 2 if n is odd.

Of course, in this last case, the quotient (S1 × Sn−1)/Z2 is not orientable.

5. The special unitary group SU(m) can be immersed into the unit sphere of thevector space gl(m,C) of complex m-matrices as follows

: A ∈ SU(m) �−→ 1√m

A ∈ S2m2−1 ⊂ gl(m,C).

Naitoh saw in [42] that is Legendrian and so it induces a minimal Lagrang-ian immersion ι = π ◦ : SU(m) −→ CPm2−1 whose level is a fortiorinSU(m) = 1.This immersion yields a minimal Lagrangian embedding

ι : {A} ∈ SU(m)/Zm �−→[

1√m

A

]∈ CPm2−1.

But it is obvious that

: {A} ∈ SU(m)/Zm �−→{

1√m

A

}∈ S

2m2−1/Zm

is a horizontal lift of that embedding inducing an isomorphism

∗ : π1 (SU(m)/Zm) ∼= Zm −→ π1

(S

2m2−1/Zm

) ∼= Zm

between their fundamental groups. This shows that nSU(m)/Zm = m.6. The minimal Lagrangian immersions of the following symmetric spaces

SU(m)

SO(m)→ CP

12 (m

2+m−2),SU(2m)

Sp(m)→ CP2m2−1,

E6

F4→ CP26

studied in [42] provide minimal Lagrangian embeddings

SU(m)

SO(m) · Zm⊂ CP

12 (m

2+m−2),SU(2m)

Sp(m) · Z2m⊂ CP2m2−1,

E6

F4 · Z3⊂ CP26.

Reasoning as in Example 5, it can be shown that

nSU(m)/SO(m) = 1, nSU(2m)/Sp(m) = 1, nE6/F4 = 1

and also that

nSU(m)/SO(m)·Zm ={

m if m is oddm2 if m is even

nSU(2m)/Sp(m)·Z2m ={

2m if m is oddm if m is even. ,

nE6/F4·Z3 = 3.

832 O. Hijazi et al.

6 Structures on the Maslov coverings and their cones

Suppose now that the Kähler-Einstein 2n-dimensional manifold M of index p ∈ N∗

is compact and has positive scalar curvature. Up to a change of scale, we will con-sider that its value is exactly 4n(n+1), just the value taken on the complex projectivespace CPn with constant holomorphic sectional curvature 4. A result by Kobayashi(see [29]) implies that, in this case, M is simply-connected. Then, a suitable useof the Thom-Gysin and the homotopy sequences for the principal circle bundleL → M established in [3], that the maximal Maslov covering L of the manifold Mis also simply-connected. Moreover, we already pointed out that the Levi-Civitàconnection on M yields a corresponding S

1-connection on the principal S1-bundle

L → M , given by a 1-form iθ ∈ (�1(L)⊗ iR). With the help of the 1-form θ ,one defines a unique Riemannian metric on L by

〈 , 〉L = π∗〈 , 〉 + p2

(n + 1)2θ ⊗ θ (6.1)

such that the projection π : L → M is an oriented submersion and that the fun-damental vector field V defined by the free S

1-action on L has constant lengthp/(n + 1), that is, the fibers are totally geodesic circles of length 2pπ/(n + 1).Thus the tangent spaces of the Kähler manifold M can be seen as subspaces of thetangent spaces of L, as follows

TzL = Tπ(z)M ⊕ 〈Vz〉, ∀z ∈ L. (6.2)

This metric was originally considered by Hatekeyama [19]. Indeed, the bundle Lis nothing but the Boothby-Wang fibration on M , which is under our hypotheses, aHodge manifold. The main properties of the Riemannian manifold L and the sub-mersion π will be listed in the following theorem of [3] (see also [6, Proposition2.39 and Lemma 9.10]).

Proposition 11 [3, Example 1, page 84] Let L be the maximal Maslov cover-ing of a Kähler-Einstein 2n-dimensional compact manifold with scalar curvature4n(n + 1) and index p ∈ N

∗. Then L is an orientable simply-connected compact(2n + 1)-manifold and there exist a unique orientation and a unique Riemannianmetric on L such that the projection L → M is an oriented Riemannian submer-sion and the fibers are totally geodesic circles of length 2pπ/(n + 1). This metricis Einstein with scalar curvature 2n(2n + 1) and supports a Sasakian structureon L whose characteristic field is the normalized fundamental vector field of theS

1-fibration.

Let U be the normalized fundamental field of the fibration, that is, the unit ori-ented field in the direction of the fundamental field V on L. The Sasakian structureis given by the fact that the vector field U is a Killing field for the metric (6.1) andthe sectional curvature of any section containing U is one. For other better knowndefinitions and more details on the notion of a Sasakian structure on a Riemannianmanifold one can consult [5] and [7]. Of course, the simplest example of a Sasaki-an manifold is provided by the standard odd-dimensional sphere S

2n+1, which isobtained, via the Proposition 11 above, from CPn endowed with the Fubini-Study

Spinc geometry and the Hodge Laplacian 833

metric of holomorphic sectional curvature 4. Probably, the most geometric defi-nition of a Sasakian structure (and also the most useful for our purposes) is thefollowing: Let C be the cone R

+ × L over the bundle L endowed with the metric

〈 , 〉C = dr2 + r2〈 , 〉L. (6.3)

Then L is isometrically embedded in C as the hypersurface r = 1 and we have

TrzC = TzL ⊕ 〈∂r 〉, ∀r ∈ R+, z ∈ L. (6.4)

Using (6.2) and (6.4), we have that

TrzC = Tπ(z)M ⊕ 〈{Uz, ∂r }〉.This orthogonal decomposition allows us to define an almost complex structureJrz on TrzC by

Jrz|Tπ(z)M = Jπ(z), Jrz (∂r ) = Uz .

The fact that L with the metric (6.1) is Sasakian, is equivalent to the fact thatthe cone C, with the metric (6.3) and the almost structure above, is an (2n + 2)-dimensional Kähler manifold (see [1, Lemma 4] or [7, Definition-Proposition 2]).Moreover, the fact that L is Einstein is equivalent to C being Ricci-flat. Since L,and so C, is simply-connected, we conclude that the holonomy group of the cone Cis contained in SU(n + 1), that is, C is a Calabi-Yau manifold. We may summarizeall these assertions in the following:

Proposition 12 [1,6,7] Let L be the maximal Maslov covering of a Kähler-Einstein 2n-dimensional compact manifold with scalar curvature 4n(n + 1) and Cthe corresponding cone. Then C is a simply-connected Calabi-Yau (2n +2)-dimen-sional manifold.

7 Spinc structures

We have seen that the cone C over the maximal Maslov covering L of M is aCalabi-Yau (2n + 2)-dimensional simply-connected manifold. Hence C is a spinmanifold and it has, up to an isomorphism, a unique spin structure (see any of [3,6,13,31] for generalities about spin structures). We will denote by �C the cor-responding complex spinor bundle. This is a Hermitian vector bundle with rank2n+1 endowed with a Levi-Cività spin connection ∇C and a Clifford multiplica-tion γ C : T C → End (�C). It is well-known that, if u1, . . . , u2n+2 is an arbitrarypositively oriented orthonormal basis in T C, then

ωC = in+1γ C(u1) · · · γ C(u2n+2)

does not depend on the chosen basis and it is called the spin volume form on C.Hence, we can put

ωCr z = in+1γ C(e1)γ

C(Je1) · · · γ C(en)γC(Jen)γ

C(∂r )γC(Urz), (7.1)

834 O. Hijazi et al.

where e1, Je1 . . . , en, Jen is any oriented orthonormal basis of Tπ(z)M and r ∈ R+,

z ∈ L. It is also known that ∇CωC = 0 and (ωC)2 = 1 and that it provides anorthogonal and parallel decomposition into ±1-eigensubbundles, that is, into theso-called subbundles of positive or negative chirality,

�C = �+C ⊕�−C.The Kähler form �C , acting by means of the Clifford multiplication on �C in thefollowing way

�C =n∑

i=1

γ C(ei )γC(Jei )+ γ C(∂r )γ

C(U ), (7.2)

induces another orthogonal decomposition into parallel subbundles

�C = �0C ⊕ · · · ⊕�rC ⊕ · · · ⊕�n+1C, (7.3)

where each�rC is the eigenbundle corresponding to the eigenvalue (2r − n − 1)i ,

0 ≤ r ≤ n + 1, and has rank

(n + 1

r

). One easily checks that the spin volume

form ωC and the Kähler form �C commute and so they can be simultaneouslydiagonalized. In fact, we have

�+C =⊕

r even�rC, �−C =

⊕

r odd�rC.

The S1-bundle L is the hypersurface r = 1 in the cone C. We consider the ori-

entation determined by the orthonormal bases of the form e1, Je1, . . . , en, Jen,Utangent to L where e1, Je1, . . . , en, Jen are tangent to M . This orientation deter-mines a unique spin structure on the hypersurface L from that of C (see [3,2,23,53]) and each one of the restrictions �+C|L and �−C|L may be identified with acopy of the intrinsic spinor bundle�L of L. From now on we will choose this one

�L = �+C|L. (7.4)

With this identification, the Clifford multiplication and the spin Levi-Cività con-nection of the spin structure on L are

γL = −γ Cγ C(∂r ), ∇L = ∇C + 1

2γ Cγ C(∂r ) = ∇C − 1

2γL. (7.5)

If we denote by ωL the complex spin volume form on L, we have

ωL = in+1γL(e1)γL(Je1) · · · γL(en)γ

L(Jen)γL(U ). (7.6)

Using (7.5), the commutativity relations defining the Clifford algebras and (7.1),we obtain the equality ωL = ωC |L. Hence

ωL = +1 on �L. (7.7)

Spinc geometry and the Hodge Laplacian 835

It is not difficult to see that the endomorphism of �L, given by

ω = inγL(e1)γL(Je1) · · · γL(en)γ

L(Jen)

is independent on the choice of the J -bases tangent to M and thatωL = iωγL(U ).Hence we have

ω = iγL(U ), ω2 = 1. (7.8)

We will think of the action of the endomorphism ω on the spinors of L as aconjugation and will write

ωψ = ψ, ∀ψ ∈ (�L).Also, the Kähler form on the holomorphic distribution determined by (6.2) on L,acts on the spinor fields of L by means of the Clifford multiplication as

� =n∑

i=1

γL(ei )γL(Jei ).

Indeed, taking into account (7.2) and the definition (7.5), we get the followingrelation between the Kähler forms � and �C

� = �C |L − γL(U ) = �C |L + iω. (7.9)

Since the endomorphisms ω and � commute on �L and the eigenvalues of ω areclearly ±1, we see that � has eigenvalues (2s − n)i for s = 0, . . . , n and obtainan orthogonal decomposition into eigenbundles

�L = �0L ⊕ · · · ⊕�sL ⊕ · · · ⊕�nL,

where each �sL has rank

(ns

). More precisely, the following relations between

the eigenspaces of � and that of �C hold:

�0C|L = �0L, �n+1C|L = �nL (if n + 1 is even)

�rC|L = �rL ⊕�r−1L, 0 < r < n + 1, r even. (7.10)

The first two relations can be checked easily. When r = 0 or r = n + 1, �0C and�n+1C are line bundles which can be characterized as follows (see [13, Section5.2])

γ C ◦ J |�0C = iγ C |�0C, γ C ◦ J |�n+1C = −iγ C |�n+1C .

Then, from definition (7.5), one obtains

γL(Ju)ψ = iγL(u)ψ, γL(Ju)ϕ = −iγL(u)ϕ

where u ∈ T L, u ⊥ U , ψ ∈ (�0C|L) and ϕ ∈ (�n+1C|L).It is important to point out that the above decomposition into eigenbundles of

� is not parallel with respect to the connection ∇L defined in (7.5). The reason isthat the almost-complex structure J on the holomorphic distribution on L is not

836 O. Hijazi et al.

parallel with respect to the Levi-Cività connection, although it is parallel when itis induced on the Kähler manifold M . This is why we will consider an inducedClifford multiplication γ and a modified connection ∇ on the spinor bundle �L,working only for tangent directions in the holomorphic distribution, which in somesense, reproduces the Levi-Cività connection of M . For any u a horizontal vectorfield according to the decomposition (6.2), i.e., u ∈ T L, u ⊥ U , set

γ (u) := γL(u), (7.11)

and

∇u := ∇Lu + 1

2γL(Ju)γL(U ) = ∇L

u − i

2γ (Ju)ω. (7.12)

Now one can check that, this connection ∇ leaves invariant each eigensubbundle�sL, with 0 ≤ s ≤ n.

From [25], it is known that any Kähler manifold, and so M , admits a spinc

structure because its second Stiefel-Whitney class w2(M) ∈ H2(M,Z2) is thereduction modulo 2 of the corresponding first Chern class c1(M) ∈ H2(M,Z).Moreover, the set of spinc structures on M is parametrized by the cohomology clas-ses β ∈ H2(M,Z) such that β ≡ c1(M) mod 2 (see also [13, Proposition, p. 53]).Hence, having in mind the isomorphism (2.3), this set is classified by the equiv-alence classes of principal S

1-bundles on M whose first Chern classes verify theabove relation. The line bundle β is usually called the auxiliary or determinant linebundle of the given spinc structure. Under this correspondence, the spinc structureswith auxiliary trivial line bundle, that is with β = 0, are just the spin structures.Moreover, given a spinc structure on M with auxiliary line bundle β ∈ H2(M,Z),the oriented Riemannian submersion π : L → M induces a spinc structure on Lwhose auxiliary line bundle is π∗(β) ∈ H2(L,Z) (one can see [6, Lemma 2.40]).On the other hand, the Thom-Gysin sequence associated with that fibration

0 −→ H0(M,Z) ∼= Z∪(c1(L)=α)−→ H2(M,Z)

π∗−→ H2(L,Z) −→ 0

implies that ker π∗ = Z(α). This means that the spinc structures on the Kählermanifold M whose auxiliary line bundles are the tensorial powers Lq , for q ∈ Z,are exactly those inducing on L, through the projection π , its unique spin structure.But, among these powers Lq , only those satisfying the condition

c1(Lq) ≡ c1(M) mod 2

provide us a spinc structure. Since c1(L) = α and c1(M) = −c1(KM ) = −pα,this is equivalent to the equation

(q + p)α ≡ 0 mod 2 in H2(M,Z),

where p is the index of M . Now, as α is an indivisible class in H2(M,Z), thislatter occurs if and only if

p + q ∈ 2Z.

In other terms, we will be interested in the powers Lq such that Lq ⊗ KM has asquare root.

Spinc geometry and the Hodge Laplacian 837

We will denote by�q M the spinor bundle of the corresponding spinc structurewith auxiliary line bundle Lq , q ∈ −p + 2Z. All these spinor bundles are identifi-able, via the projection π , with the spinor bundle �L of the spin manifold L (see[6, Lemma 2.40]), that is,

π∗�q M = �L ∀q ∈ Z. (7.13)

As M is Kähler, the spinor bundle �q M associated with any of its consideredspinc structures also decomposes into eigensubbundles corresponding to the spi-norial action of its Kähler form �M

�q M = �q0 M ⊕ · · · ⊕�

qs M ⊕ · · · ⊕�

qn M.

Each �qs M is associated with the eigenvalue (2s − n)i , 0 ≤ s ≤ n and has rank(

ns

). It is easy to see that π∗�M = � and so

π∗�qs M = �sL 0 ≤ s ≤ n.

Now, as in Section 4, for each integer q ∈ Z, we will consider the Levi-Civitàconnection iθq ∈ (�1(Lq)⊗ iR) on the line bundle Lq , induced from the Kählermetric on M . It is not difficult to see that all these 1-forms are mapped, via thepull-backs of the q-fold coverings P1,q : L → Lq , on integer multiples of thesame 1-form on L. In fact, we have

P∗1,qθ

q = qθ1

where the connection θ1 = θ was already used in Section 6 to construct the metricon L. We will denote by ∇q the Hermitian connection on the spinor bundle�q M ,of any of the spinc structures that we are considering on M , constructed from theLevi-Cività connection of M and from the connection θq on the auxiliary line bun-dle Lq . Then (see for example [6, Theorem 2.41]), if we use the definitions (7.11)and (7.12), we conclude that all the ∇q are identified with ∇ via the projection π ,that is,

π∗∇quψ = ∇u∗π∗ψ, ∀u ∈ T M, ψ ∈ (�q M).

where u∗ ∈ T L means the horizontal lift of vectors with respect to the decompo-sition (6.2).

As a consequence from (7.13), we see that the spinor fields ψ ∈ (�q M)of the spinc structure on M with auxiliary bundle Lq can be identified with cer-tain spinor fields ϕ = π∗ψ ∈ (�L) on the Maslov covering L. These fieldsare usually called q-projectable spinor fields and they have of course a particularbehaviour with respect to the vertical direction of the submersion. This can be seen,for instance, in the second equation of [6, Theorem 2.41]. We have in fact

∇LUϕ + q

2iθ(U )ϕ = ∇L

Uϕ + q(n + 1)

2piϕ = 1

2�ϕ

for every ϕ ∈ (�L), q-projectable. This is a necessary condition for the spinorfields ϕ on the manifold L to be obtained, via the projection π : L → M , fromspinor fields ψ ∈ (�q M) on M . It is not difficult to see that, in fact, this is alsoa sufficient condition.

838 O. Hijazi et al.

Proposition 13 A spinor field ϕ ∈ (�L) on the total space of the oriented Rie-mannian submersion π : L → M is projectable onto a spinor field ψ ∈ (�q M)corresponding to the spinc structure of the Kähler manifold M with auxiliary linebundle Lq , q ∈ −p + 2Z, if and only if

∇LUϕ + q(n + 1)

2piϕ = 1

2�ϕ,

where p is the index of the Kähler manifold M, the endomorphism� is the Kählerform of the holomorphic distribution on L and U is the normalized fundamentalvector field of the principal S

1-fibration.

8 Parallel and Killing spinor fields

M. Wang classified in [55] the simply-connected spin Riemannian manifolds sup-porting non-trivial parallel spinor fields. Hitchin had already pointed out in [25]that they must be Ricci-flat and Wang proved that the only possible holonomies are0, SU(n), Sp(n), Spin7 and G2. Among them we find the Calabi-Yau manifolds,whose holonomy is contained in SU(n). Henceforth, that holonomy must be 0, andthe manifold is flat, or Sp(n/2) with n even, and the manifold is hyper-Kähler, orjust SU(n).

We already know that the cone C constructed over the maximal Maslov cov-ering L of the Kähler-Einstein manifold M is a Calabi-Yau (2n + 2)-dimen-sional manifold. Hence it carries a non-trivial parallel spinor field ψ ∈ (�C).Since the eigensubbundles (7.3) of �C for the Kähler form �C are parallel, eachr -component of ψ is also parallel and so, we will suppose that ψ ∈ (�rC), forsome 0 ≤ r ≤ n + 1. That is

∇Cψ = 0, �Cψ = (2r − n − 1)iψ.

Suppose now that r is even. Then the identification (7.4) allows to see therestriction ξ = ψ |L as a section of (�L), that is, as a spinor field on L which isalso non-trivial since ψ has constant length. Taking into account (7.5) and (7.10),we have that

ξ ∈ (�rL ⊕�r−1L), ∇Lξ = −1

2γLξ,

with the convention �−1L = �n+1L = L × {0}. In particular, ξ is a real Killingspinor field on the spin manifold L (see [3] for a definition and references). Usingthis Killing equation satisfied by ξ for horizontal vectors v ∈ T L, we obtain from(7.8), (7.11) and (7.12)

∇vξ = −1

2γ (v)ξ − i

2γ (Jv)ξ ∀v ∈ T L, v ⊥ U.

Since the connection ∇ stabilizes each �sL and the Clifford multiplication γ sat-isfies

γ (T L)(�sL) ⊂ �s−1L ⊕�s+1L,

Spinc geometry and the Hodge Laplacian 839

if 0 < r < n + 1 and one of the two components ξr−1 or ξr of ξ vanished, then wewould have ∇ξ = 0 and γ (Jv)ξr = iγ (v)ξr or γ (Jv)ξr−1 = −iγ (v)ξr−1for anyhorizontal v. But this latter would imply that�ξr = −niξr or�ξr−1 = niξr−1 andso r = 0 in the first case or r − 1 = n in the second one, which is a contradiction.Then, when ξ has two components, its two components are non-trivial. On theother hand, taking now the unit vertical direction U ∈ T L and using (7.9), one has

∇LU ξ = i

2ξ = 1

2�ξ − 2r − n − 1

2iξ.

This last equality, according to Proposition 13, means that ξ is a q-projectablespinor field on L with

q = p

n + 1(2r − n − 1) = 2r p

n + 1− p.

From this, we will be able to get spinor fields of a spinc structure of M providedthat q + p is even. In fact, only in this case, there will exist a spinc structure onM whose auxiliary line bundle is Lq . We now show that q + p = 2r p

n+1 is divisibleby 2. If the holonomy of C is 0, then C = C

n+1, L = S2n+1 and M = CPn .

Then p = n + 1 and we have a parallel spinor for each 0 ≤ r ≤ n + 1. But then2r pn+1 = 2r and we conclude. If the holonomy group of the Calabi-Yau cone C is

Sp(k + 1) with 2k + 1 = n, then C is a hyper-Kähler manifold. This implies thatL is 3-Sasakian and M is a complex contact manifold. In this case the index of Mis exactly p = k + 1 (see, for example, Remark 2 and [7, Proposition 2.12]) andthere are parallel spinor fields on C only for 0 ≤ r ≤ n + 1 even (see [55]). Butthen 2r p

n+1 = r is even and we also conclude. The last case is when the holonomy ofC is just SU(n + 1). In this case we only know that p < n + 1 but the only parallelspinor fields carried by C are for r = 0, n + 1 (see [55]) and then 2r p

n+1 = 0, 2p isalso even.

If the parallel spinor field ψ ∈ (�rC) has degree r odd, we would not workwith the identification (7.4), but with

�L = �−C|L.In this case, one can check that all equalities involved are still true except (7.8) and(7.9) where the sign of ω has to be changed. Hence, only some minor changes areobtained in the conclusion. We will summarize this information in the followingstatement.

Theorem 14 Let C be the cone over the maximal Maslov covering L of an 2n-dimensional Kähler-Einstein compact manifold M with scalar curvature4n(n + 1) and index p ∈ N

∗. Each non-trivial parallel spinor field ψ ∈ (�rC)with degree r , 0 ≤ r ≤ n + 1 induces two non-trivial spinor fields ξr ∈ (�q

r M)

and ξr−1 ∈ (�qr−1 M) of the spinc structure on M with auxiliary bundle Lq ,

where q = pn+1 (2r − n − 1) ∈ Z, satisfying the first order system

{∇qv ξr = − 1

2γ (v)ξr−1 + i2γ (Jv)ξr−1

∇qv ξr−1 = − 1

2γ (v)ξr − i2γ (Jv)ξr ,

840 O. Hijazi et al.

for all v ∈ T M. Such coupled spinor fields on the Kähler manifold M are usuallycalled Kählerian Killing spinor fields.

Remark 15 Kählerian Killing spinor fields on Kähler manifolds play a rôle analo-gous to what of Killing spinors on Riemannian spin manifolds (see [27,22]). Bothtwo classes appear when one studies the so-called limiting manifolds, that is, thoseattaining the equality for the standard (Friedrich ([12]), Hijazi ([21]), Kirchberg([26]), Herzlich-Moroianu ([20,38,39])) lower estimates for the eigenvalues of theDirac operator. In the case of Kähler manifolds (see [27,35–37], a usual strategy toclassify manifolds admitting these particular fields is to construct the cone over asuitable principal S

1-bundle in order to get manifolds carrying parallel spinor fieldsand to apply the corresponding results. In the proof of Theorem 14 we adopted theopposite approach. Also, details of the proof are given since we need an exactcontrol on the behaviour of the Kähler form on the Kählerian Killing spinor fieldsand on the spinc structure.

Remark 16 Consider now the Dirac operator Dq : (�q M) → (�q M) actingon smooth spinor fields of the spinc structure on M with auxiliary bundle Lq . Thisoperator is locally defined by

Dq =2n∑

i=1

γ (ei )∇qei

where e1, . . . , e2n is any orthonormal basis tangent to M . A direct consequencefrom Theorem 14 is that we can compute the images by Dq of the fields ξr ∈(�

qr M) and ξr−1 ∈ (�q

r−1 M) obtained from the parallel spinor ψ ∈ (�rC).We have

Dqξr = 2(n + 1 − r)ξr−1, Dqξr−1 = 2rξr .

Hence ξr and ξr−1 are eigenspinors for the same eigenvalue of the second orderoperator (Dq)2. In fact

(Dq)2ξr,r−1 = 4r(n + 1 − r)ξr,r−1

and so the eigenvalue λ1(Dq) of Dq with least absolute value satisfies λ1(Dq)2 ≤4r(n + 1 − r). But we can observe that

4r(n + 1 − r) = n + 1

4ninf S − (n + 1 − 2r)2 = n + 1

4ninf S − q2,

where S denotes the scalar curvature of M . If we take into account the generaliza-tion given in [20] to spinc manifolds of Friedrich’s estimate and the fact that themanifold M is Kähler, we could expect that a similar generalization would existfor Kirchberg’s inequalities in the context of Kähler manifolds. It is natural to thinkthat, in fact,

λ1(Dq)2 = 4r(n + 1 − r)

and the Kähler manifolds involved must be limiting manifolds for the Dirac oper-ator of the spinc structures with suitable auxiliary bundle Lq . In fact, q = p orq = −p when the holonomy of C is SU(n + 1); q = 2s − k − 1, 0 ≤ s ≤ k + 1,when the holonomy of C is Sp(k + 1) with n = 2k + 1; and q = 2r − n − 1,0 ≤ r ≤ n + 1 when the M is the complex projective space.

Spinc geometry and the Hodge Laplacian 841

If we bear in mind, like in Remark 16, the possible holonomies for the coneC, the possible degrees of its parallel spinor fields, the corresponding multiplici-ties and the possible spinc structures on the Kähler manifold M , we can rewriteTheorem 14 to show that it improves Theorem 9.11 in [6] (see also [39,38,28]).

Theorem 17 Let M be an 2n-dimensional Kähler-Einstein compact manifold withscalar curvature 4n(n + 1).

(1) The vector space of Kählerian Killing spinors carried by the canonical spinc

structure on M is one (and all of them have degree n).(2) The vector space of Kählerian Killing spinors carried by the anti-canonical

spinc structure on M is one (and all of them have degree 0).(3) If M has a complex contact structure and so n = 2k+1, then, for each 0 ≤ s ≤

k+1, the space of Kählerian Killing spinorsψ ∈ (�2s−k−12s−1 M ⊕�2s−k−1

2s M)has dimension one.

(4) If M is the complex projective space CPn, then, for each 0 ≤ r ≤ n + 1,the space of Kählerian Killing spinors ψ ∈ (�2r−n−1

r−1 M ⊕�2r−n−1r M) has

dimension

(n + 1

r

).

9 Lagrangian submanifolds and induced Dirac operator

Let us come back to the situation that we studied in Section 4, i.e., consider aLagrangian immersion ι : L → M from an n-dimensional manifold L into theKähler-Einstein manifold M . We will denote by �q L the restriction to the sub-manifold L of the spinor bundle �q M on M corresponding to the spinc structurewith auxiliary line bundle Lq , with q ∈ Z. That is,

�q L = �q M|L . (9.1)

It is clear that �q L is a Hermitian vector bundle over L whose rank is 2n and thatit admits an orthogonal decomposition

�q L = �q0 L ⊕ · · · ⊕�

qs L ⊕ · · · ⊕�

qn L , (9.2)

where each summand is the restriction �qs L = �

qs M|L , with 0 ≤ s ≤ n, hence

a Hermitian bundle with rank

(ns

). Since the immersion ι allows to view the

tangent bundle T L as a subbundle of T M , we can consider the restrictions ∇q :T L ⊗ �q L → �q L and γ : T L ⊂ T M → End(�q L) to the submanifold L ofthe connection and of the Clifford multiplication corresponding to the spinc struc-ture on M . The problem of these two restrictions is that they are compatible withrespect to the Levi-Cività connection of M , but not with respect to the Levi-Cività∇L of the induced metric on L . To avoid this difficulty, we consider a modifiedconnection, also denoted by ∇L on the bundle �q L . We put

∇q,Lv ψ = ∇q

v ψ − 1

2

n∑

i=1

γ (ei )γ (σ (v, ei ))ψ, v ∈ T L , (9.3)

842 O. Hijazi et al.

where σ is the second fundamental form of the immersion ι and e1, . . . , en is anybasis tangent to L . Now one can verify that

∇q,Lv γ (X)ψ = γ (∇L

X Y )ψ + γ (Y )∇q,Lv ψ, v ∈ T L , X ∈ (T L).

It is immediate to check that this new connection ∇q,L also parallelizes the Hermi-tian product because the endomorphism

∑ni=1 γ (ei )γ (σ (v, ei )) is skew-Hermitian

for every v ∈ T L . Even more interesting is the fact that ι being Lagrangian andminimal implies that ∇q,L fixes each subbundle�q

s L in (9.2). To see this, first onemust use the well-known property that the 3-form

〈σ(X, Y ), J Z〉, X, Y, Z ∈ (T L)

is symmetric (see [41], for example). In this way, one obtainsn∑

j=1

γ (e j )γ (σ (v, e j )) = 1

2i

n∑

j,k=1

〈σ(e j , ek), Jv〉γ (e j − i Je j )γ (ek + i Jek)

because one hasn∑

j,k=1

〈σ(e j , ek), Jv〉γ (e j )γ (ek) = n〈H, Jv〉 = 0.

Now recall that

γ (T 1,0 M)(�qr M) ⊂ �

qr−1 M, γ (T 0,1 M)(�q

r M) ⊂ �qr+1 M.

As a conclusion the Hermitian vector bundle �q L endowed with the metricconnection ∇q,L and the Clifford multiplication γ is a Dirac bundle (see any of[13,31]) over the Riemannian manifold L . Moreover the orthogonal decomposition(9.2) is preserved by ∇q,L and

γ (T L)(�qs L) ⊂ �

qs−1L ⊕�

qs+1L .

Let D : (�q L) → (�q L) be the Dirac operator of this Dirac bundle, calledinduced Dirac operator, defined locally, for any orthonormal basis e1, . . . , en tan-gent to L , by

D =n∑

i=1

γ (ei )∇q,Lei .

Of course, when the submanifold L is endowed with a given spin structure, thecorresponding intrinsic Dirac operator could be different from D. From the Gaußequation (9.3) one can deduce the following expression for D,

Dξ =n∑

i=1

γ (ei )∇qei ξ + n

2γ (H)ξ ∀ξ ∈ (�q L). (9.4)

A first obvious consequence from formula (9.4) is that, if the immersion ι isminimal, then parallel spinor fields on the Kähler-Einstein manifold M yield har-monic spinor fields on the submanifold L . Note that this restriction is non-trivial,since the spinor field is parallel. Hence, looking at Wang’s and Moroianu-Semmelmann’s classifications [55,40] of complete spin manifolds carrying non-trivial parallel spinors, we have the following result.

Spinc geometry and the Hodge Laplacian 843

Proposition 18 Let L be an n-dimensional minimal Lagrangian submanifoldimmersed into a complete Calabi-Yau manifold M of dimension 2n and let �Lbe the Dirac bundle induced over L from the spinor bundle of M.

(1) There exists a non-trivial harmonic spinor in (�0L) and another one in(�n L).

(2) If M has a hyper-Kähler structure, and so n = 2k, for each 0 ≤ s ≤ k, thereexists a non-trivial harmonic spinor in (�2s L).

(3) If M = Cn/, where is a discontinuously acting group of complex trans-

lations, then, for every 0 ≤ r ≤ n, the space of harmonic spinors in (�r L)

has dimension at least

(nr

).

The harmonicity of the spinor fields above is always referred to the induced Diracoperator D of the submanifold.

A second consequence from the same formula (9.4) is obtained when the spinorfield ξ = ξr−1,r is no longer parallel, but a Kählerian Killing spinor as those con-structed in Theorem 14, when the Kähler-Einstein manifold M is compact and haspositive scalar curvature. In this case, we have

n∑

j=1

γ (e j )∇qe j ξr = n

2ξr−1 + i

2�ξr−1 = (n + 1 − r)ξr−1

n∑

j=1

γ (e j )∇qe j ξr−1 = n

2ξr − i

2�ξr = rξr

and so, if the Lagrangian submanifold L is minimal, we obtain

Dξr = (n + 1 − r)ξr−1, Dξr−1 = rξr . (9.5)

Hence, the spinor fields ξr and ξr−1 are eigenspinors for the second order operatorD2, namely

D2ξr−1,r = r(n + 1 − r)ξr−1,r .

Theorem 19 Let L be an n-dimensional minimal Lagrangian submanifold im-mersed into a Kähler-Einstein compact manifold M of dimension 2n and scalarcurvature 4n(n + 1). Let�q L be the Dirac bundle induced over L from the spinorbundle of M corresponding to the spinc structure with auxiliary line bundle Lq ,q ∈ Z.

(1) There exists a non-trivial harmonic spinor in (�−p0 L) and another one in

(�pn L), where p is the index of M.

(2) If M has a complex contact structure, and so n = 2k + 1, for each 0 ≤ s ≤k + 1, there exists a non-trivial spinor in (�2s−k−1

2s L) and another one in(�2s−k−1

2s−1 L) which are eigenspinors of D2 associated with the eigenvalue4s(k + 1 − s).

844 O. Hijazi et al.

(3) If M = CPn, then, for every 0 ≤ r ≤ n + 1, the eigenspace of D2 in(�2r−n−1

r−1 L) or in(�2r−n−1r L), associated with the eigenvalue r(n+1−r),

have dimension at least

(n + 1

r

).

The harmonicity of the spinor fields mentioned in (1) above is referred to theinduced Dirac operator D of the submanifold L.

Proof It only remains to point out that the restrictions of ξr and ξr−1 in Theorem 14to the submanifold L cannot be trivial. In fact, from (9.5), if one of them vanishes,then both of them have to vanish. But this is impossible because ξ = ξr−1 + ξr hasconstant length along the manifold M . �

10 The Hodge Laplacian on minimal Lagrangian submanifolds

In this section, we show that the Dirac bundles�q L constructed over a Lagrangianimmersed submanifold from the spinor bundles �q M of different spinc structureson a Kähler-Einstein manifold M can be identified with standard bundles. In fact,it is well-known (see [13, Section 3.4] for an explicit expression) that, for eachpossible r , we have an isometry of Hermitian bundles

�qr M = �0,r (M)⊗�

q0 M (10.1)

and �q0 M is a square root of the tensorial product Lq ⊗ KM , where Lq is the

auxiliary line bundle of the spinc structure and KM is the canonical bundle of M .This fact was originally discovered by Hitchin [25] for spin structures. That is, thespinor fields of the spinc structure can be thought of as anti-holomorphic r -formstaking values on the line bundle �q

0 M . Under this isometric identification, onecan easily see that the connection ∇q constructed on �q M from the Levi-Civitàconnection of the Kähler metric on M and from the connection iθq on the linebundle Lq , satisfies

∇q |�qr M = ∇M ⊗ ∇q |�q

0 M , (10.2)

where ∇M is the usual Levi-Cività connection on the exterior bundle. This is aconsequence of the compatibility of ∇q and the Levi-Cività connection on M . Onecan also see (for example in [13, page 23] or in [31]) that the Clifford multiplication

γ : T M → End(�0,r (M)⊗�q0 M) can be viewed as follows

γ (v)(β ⊗ ψ) = √2((v� ∧ β)0,r+1 − v�β)⊗ ψ, (10.3)

for all v ∈ T M , β ∈ (�0,r (M)) and ψ ∈ (�q0 M).

Now we consider the corresponding restrictions to the Lagrangian submani-fold L . Definitions (9.1) and (9.2), the isometry (10.1) above and Lemma 3 implyanother isometric identification of Hermitian bundles over L , namely

�qr L = �r (L)C ⊗�

q0 L .

Spinc geometry and the Hodge Laplacian 845

The expressions above for the connection ∇q and the Clifford product, togetherwith definition (9.3) of the modified connection ∇q,L on �q L , give

∇q,L |�qr L = ∇L ⊗ ∇q,L |�q

0 L ,

where ∇L is the Levi-Cività connection on the exterior bundle over L , and

γ (v)(β ⊗ ψ) = (v� ∧ β − v�β)⊗ ψ,

for all v ∈ T L , β ∈ (�r (L)C) and ψ ∈ (�q0 L).

Hence, as a consequence of these last three identifications, if we could findsome situations in which the line bundle �q

0 L is topologically trivial and ad-mits a non-trivial parallel section, we would be able to identify the Dirac bundle(�q L ,∇q,L , γ ) in a very pleasant manner.

Proposition 20 Suppose that the restriction �q0 L to a Lagrangian submanifold

L of the 0-component �q0 M of the spinor bundle of a spinc structure of a Kähler

manifold M is topologically trivial and admits a non-zero parallel section. Thenthe Dirac bundle (�q L,∇q,L , γ ) is isometrically isomorphic to the complex valuedexterior bundle

(�∗(L)⊗ C,∇L , ·� ∧ −·�)with its standard (see [31]) Dirac bundle structure. Henceforth the Dirac operatorD is nothing but the Euler operator d +δ and its square D2 is the Hodge Laplacian� of L.

We will consider two situations where Proposition 20 above can be applied.The first one is when M is an n-dimensional Calabi-Yau manifold and q = 0, thatis, we consider a spin structure on M . More precisely, we choose the trivial spinstructure on M , in other words, the spin structure determined by taking as a squareroot of KM , the bundle KM itself. In this case, we have that �0 M = KM , whichis topologically trivial and admits a non-trivial parallel section � (with respectto ∇q ). Take the restriction ϒ = �|L ∈ (�0L). From (9.3), we have for eachv ∈ T L that

∇0,Lv ϒ = −1

2

n∑

j,k=1

〈σ(e j , ek), Jv〉γ (e j )γ (Jek)ϒ.

But � is a 0-spinor and then γ (Jv)� = iγ (v)� for every v ∈ T M . Then

∇0,Lv ϒ = − i

2

n∑

i, j=1

〈σ(ei , e j ), Jv〉γ (ei )γ (e j )ϒ = −ni

2〈J H, v〉ϒ.

Soϒ is a non-trivial parallel section of�0L if and only if L is a minimal subman-ifold.

846 O. Hijazi et al.

Remark 21 As a consequence, we get to trivialize the restricted bundle �0L , butthis is not a necessary condition in order to have this trivialization. In fact, fromthe equality above, we can conclude, as in Remark 9, that, when the ambientmanifold M is a Calabi-Yau manifold, �0L is geometrically trivial if and only if14 [η] ∈ H1(L ,Z), where η is the mean curvature form of the Lagrangian subman-ifold L , that in this case of Ricci-flat ambient space, coincides with the Maslovclass. This explains why, in case n = 1 invoked in [16], one obtains the desiredidentification for the Lagragian immersion z ∈ S

1 �→ z2 ∈ C and not for theembedding z ∈ S

1 �→ z ∈ C.

Using Proposition 20, the assertions in Proposition 18 on the spectrum of theDirac operator become now conditions on the Hodge Laplacian acting on differ-ential forms of the Lagrangian submanifold.

Theorem 22 Let L be an n-dimensional minimal Lagrangian submanifold im-mersed into a complete Calabi-Yau manifold M of complex dimension n.

(1) There exists a non-trivial harmonic complex n-form on L. So, if L is compact,then L is orientable. Hence L is a special Lagrangian submanifold (see [18])of M.

(2) If M has a hyper-Kähler structure, and so n = 2k, for each 0 ≤ s ≤ k, thereexists a non-trivial harmonic complex 2s-form on L. As a conclusion, if L iscompact then the even Betti numbers of L satisfy b2s(L) ≥ 1.

(3) If M = Cn/, where is a discontinuously acting group of complex trans-

lations, then, for every 0 ≤ r ≤ n, the space of harmonic complex r–forms

on L has dimension at least

(nr

). In particular, all the Betti numbers of a

compact minimal Lagrangian submanifold L immersed in a flat complex torusT

2n satisfy br (L) ≥ br (Tn).

Remark 23 We point out that part (2) in Theorem 22 is an improvement of theresult by Smoczyk obtained in [49]. He proved Theorem 22 for s = 1 withoutusing spin geometry. He showed that the equality b2(L) = 1 implies that L is acomplex submanifold for one of the complex structures of M . For the moment, weare not able to treat the case b2s(L) = 1 with s > 1.

There is a second situation where Proposition 20 could be applied. It is the case,extensively studied in Section 7, of a Kähler-Einstein compact n-dimensional mani-fold with positive scalar curvature, normalized to be 4n(n +1). For integers r with0 ≤ r ≤ n + 1 and such that r p

n+1 ∈ Z, where p ∈ N∗ is the index of M , we

have already considered the spinc structures on M whose auxiliary bundles are Lq ,q = 2r p

n+1 − p, where L is a fix p-th root of the canonical bundle KM of M . In thissituation, we have that the line bundle of 0-spinors of this spinc structure satisfies

(�q0 M)2 = Lq ⊗ KM = L 2r p

n+1 =(L r p

n+1

)2.

Recall that (see comments after (2.4)) the space of square roots of this last bundleis an affine space over H1(M,Z2). But this cohomology group is trivial since Mis simply-connected under our hypotheses. This implies that the line bundle �q

0 M

Spinc geometry and the Hodge Laplacian 847

is just L r pn+1 . Hence, the desired condition in Proposition 20 is that we can find

a non-trivial parallel section on the line bundle L r pn+1 |L over the submanifold L .

But this is equivalent to find a horizontal (or Legendrian) lift of the Lagrangianimmersion ι : L → M to the Maslov covering L r p

n+1 (see Section 3). In other terms,this means that the level nL (see Section 4) of the submanifold divides the integer

r pn+1 . On the other hand, just before Theorem 17, we considered the admissiblevalues, under our hypotheses, for the index p and the integers r . We will use thisinformation, together with Theorem 19, to translate the assertions about spinorsand Dirac operator in the induced Dirac bundle �q L into results about exteriorforms and Euler operator on L , when L is minimal.

Theorem 24 Let L be an n-dimensional minimal Lagrangian submanifold im-mersed into a Kähler-Einstein compact manifold M of dimension 2n and scalarcurvature 4n(n + 1). Let p ∈ N

∗ be the index of M and 1 ≤ nL ≤ 2p the level ofthis submanifold.

(1) If nL |p, then there exists a non-trivial complex harmonic n-form on L. In par-ticular, when L is compact, if nL |p, then L is orientable, as in Proposition 8.

(2) If M has a complex contact structure, and so n = 2k+1, for each 0 ≤ s ≤ k+1such that nL |s, there exists a non-trivial exact 2s-form and another non-trivialco-exact (2s − 1)-form on L which are eigenforms of the Hodge Laplacian�associated with the eigenvalue 4s(k + 1 − s).

(3) If M = CPn, then, for every 0 ≤ r ≤ n + 1 such that nL |r , the eigenspaceof � on the subspace of exact forms of (�r (L) ⊗ C) or on the subspace ofco-exact forms in(�r−1(L)⊗C) associated with the eigenvalue r(n +1−r)

have dimension at least

(n + 1

r

).

Using the computation of levels for the minimal Lagrangian submanifolds ofthe complex projective space made in Section 5, we obtain some information aboutthe spectra of the Hodge Laplacian on some symmetric spaces.

Corollary 25 1. For each r = 0, . . . ,m2, the numbers r(m2 −r) are in the spec-trum of the Hodge Laplacian for r-forms and (r − 1)-forms on the Lie group

SU(m) with multiplicity at least

(m2

r

).

2. For each r = 0, . . . , 12 m(m+1), the numbers r( 1

2 m(m+1)−r) are in the spec-trum of the Hodge Laplacian for r-forms and (r − 1)-forms on the symmetric

space SU(m)SO(m) with multiplicity at least

(12 m(m + 1)

r

).

3. For each r = 0, . . . , 2m2, the numbers r(2m2 − r) are in the spectrum of the

Hodge Laplacian for r-forms and (r −1)-forms on the symmetric space SU(2m)Sp(m)

with multiplicity at least

(2m2

r

).

4. For each r = 0, . . . , 27, the numbers r(27 − r) are in the spectrum of theHodge Laplacian for r-forms and (r − 1)-forms on the symmetric space E6

F4

with multiplicity at least

(27r

).

848 O. Hijazi et al.

Remark 26 Part (3) in Theorem 24 above was partially proved in [50] when thesubmanifold was Legendrian in the sphere S

2n+1, that is, when the level of thesubmanifold verifies nL = 1. In fact, Smoczyk made a naïf mistake by calcu-lating some differentials and he got the wrong eigenvalue n + 1 − r instead of

r(n + 1 − r). Moreover, he obtained only

(nr

)as an estimate for the multiplicity

of this eigenvalue. Our estimate

(n + 1

r

)for this multiplicity is sharp.

Remark 27 Nicolas Ginoux started in his Ph. D. Thesis [15] the study of the spingeometry of Lagrangian spin submanifolds L in spin Kähler manifolds M and gotupper estimates for the eigenvalues of the Dirac operator of the restriction to thesubmanifold of the spinor bundle on the ambient manifold. Later, in [16], Ginouxfound some conditions in order to be able to identify this restriction with the exteriorbundle of L and the Dirac operator with the Hodge operator. Our Proposition 20also gives some answers to that problem of identification. When M is a Calabi-Yaumanifold endowed with the trivial spin structure, we have observed before that

�0 M = K12M and so �0L = K

12M |L . Hence, the identification works if and only if

14 [η] is an integer cohomology class (see Remarks 9 and 21), where η is the meancurvature form. Then, when L is minimal, we have always the identification of thebundles and the identification of the operators. When M is Kähler-Einstein withpositive scalar curvature and so index p ∈ N

∗, we have that M is spin if and onlyif p is even and, in this case, �0 M = L p

2 . Hence we have the identification ofbundles when and only when nL | p

2 and 14 [η] ∈ H1(L ,Z). The identification of

operators occurs if L is also minimal. In fact, we already knew (Proposition 8) thatnL |p if and only if L is orientable.

11 Morse index of minimal Lagrangian submanifolds

Let L be a compact n-dimensional minimal Lagrangian submanifold immersedinto a Kähler-Einstein manifold M of dimension 2n and scalar curvature c. Then Lis a critical point for the volume functional and it is natural to consider the secondvariation operator associated with such an immersion. This operator, known as theJacobi operator of L , is a second order strongly elliptic operator acting on sectionsof the normal bundle T ⊥L , and it is defined as

J = �⊥ + B + R,where �⊥ is the second order operator

�⊥ =n∑

i=1

{∇⊥ei

∇⊥ei

− ∇⊥∇ei ei},

the operators B and R are the endomorphisms

B(ξ) =n∑

i, j=1

〈σ(ei , e j ), ξ〉σ(ei , e j ), R(ξ) =n∑

i=1

(R(ξ, ei )ei

)⊥,

Spinc geometry and the Hodge Laplacian 849

where {e1, . . . , en} is an orthonormal basis tangent to L , σ is the second funda-mental form of the submanifold L , R is the curvature tensor of M and ⊥ denotesnormal component. Let Q(ξ) = − ∫

L〈Jξ, ξ〉dV be the quadratic form associatedwith the Jacobi operator J . We will denote by ind(L) and nul(L) the index andthe nullity of the quadratic form Q, which are respectively the number of negativeeigenvalues of J and the multiplicity of zero as a eingenvalue of J . The minimalsubmanifold L is called stable if ind(L) = 0.

In [44], Oh studied the Jacobi operator for minimal Lagrangian submanifolds,proving that it is an intrinsic operator on the submanifold and can be written interms of the Hodge Laplacian acting on 1-forms of L . In fact, if we consider theidentification

T ⊥L = �1(L), ξ �→ −(Jξ)�,then the Jacobi operator J is given by

J : (�1(L)) −→ (�1(L)), J = �+ c,

where � is the Hodge Laplacian. In particular,

Any compact minimal Lagrangian submanifold of a Kähler-Einstein man-ifold with negative scalar curvature (c < 0) is stable and its nullity iszero.

Also, when the scalar curvature of the ambient manifold is zero, we have

Any compact minimal Lagrangian submanifold of a Ricci flat Kähler man-ifold (c = 0) is stable and its nullity coincides with its first Betti number,

and when the scalar curvature is positive,

The index is the number of eigenvalues of � less than c, and the nullity isthe multiplicity of c as an eigenvalue of �.

As a consequence, if we denote by b1(L) the first Betti number of L , it followsthat, when c = 0, nul(L) ≥ b1(L) and that, when c > 0, ind(L) ≥ b1(L).

When the ambient space is the complex projective space CPn , Lawson andSimons (see [32]) proved that any compact minimal Lagrangian submanifold ofCPn is unstable and Urbano proved in [54] that the index of a compact orientableLagrangian surface of CP2 is at least 2 and only the Clifford torus T

2 (see Example3 in Section 5) has index 2. As a consequence of a result proved by Oh ([44]) wecan get a lower bound for the nullity of this kind of submanifolds.

Corollary 28 Let L be a compact minimal Lagrangian submanifold immersed inCPn. Then

nul(L) ≥ n(n + 3)

2,

and the equality holds if and only if L is either the totally geodesic Lagrangianimmersion of the unit sphere S

n or the totally geodesic Lagrangian embedding ofthe real projective space RPn (Examples 1 and 2 in Section 5).

850 O. Hijazi et al.

Proof It is clear that if X is a Killing vector field in CPn , then the normal compo-nent of its restriction to the submanifold L is an eigenvector of the Jacobi operatorJ with eigenvalue 0. These normal vectors fields are called Jacobi vector fields onL . If � denotes the vector space of Killing vector fields on CPn , we will denoteby �⊥ the vector space of the normal components of their restrictions to L . Then,Oh proved in [44, Proposition 4.1] that dim�⊥ ≥ n(n + 3)/2 and the equalityimplies that the submanifold is totally geodesic. Hence nul(L) ≥ n(n + 3)/2 andthe equality implies that L is totally geodesic. The proof is complete if we take intoaccount that 2(n + 1) is an eigenvalue of � (acting on 1-forms) with multiplicityn(n + 3)/2 when L is the totally geodesic Lagrangian sphere S

n or the totallygeodesic Lagrangian real projective space RPn (see [14]). �

For other Kähler-Einstein manifolds with positive scalar curvature only veryfew results are known. We mention one due to Takeuchi [52], which says

If M is a Hermitian symmetric space of compact type and L is a compacttotally geodesic Lagrangian submanifold embedded in M, then L is stableif and only if L is simply-connected.

As a consequence of Theorem 22, it follows:

Corollary 29 Let L be a compact minimal Lagrangian submanifold immersedinto a complex torus C

n/, where is a lattice of complex translations. Thennul(L) ≥ n.

Also, from Theorem 24, we have the following consequence.

Corollary 30 Let L be a compact minimal Lagrangian submanifold immersedin a Kähler-Einstein compact manifold M of dimension 2n and scalar curvature4n(n + 1). If M has a complex contact structure and the level of L is one, i.e.nL = 1, then L is unstable.

Proof In this case, the Jacobi operator is J = � + 2(n + 1). But from Theorem24, it follows that there exists a nontrivial 1-form on L which is an eigenform of� associated with the eigenvalue 2(n − 1). Hence we obtain a negative eigenvalueof J . So the corollary follows. �Corollary 31 Let L be a compact minimal Lagrangian submanifold immersed inthe complex projective space CPn. Then,

(1) If the level of L is one, i.e. nL = 1, then

ind(L) ≥ (n + 1)(n + 2)

2,

and equality holds if and only if L is the totally geodesic Lagrangian immersionof the n-dimensional unit sphere S

n (Example 1 in Section 5).(2) If the level of L is two, i.e. nL = 2, then

ind(L) ≥ n(n + 1)

2,

and equality holds if and only if L is the totally geodesic Lagrangian embeddingof the n-dimensional real projective space RPn (Example 2 in Section 5).

Spinc geometry and the Hodge Laplacian 851

Proof If nL = 1, then, taking r = 1, 2 in Theorem 24, we obtain two orthogonaleigenspaces of� in (�1(L)) of dimensions n + 1 and n(n + 1)/2 correspondingto the eigenvalues n and 2(n − 1) respectively. These spaces are also eigenspacesof the Jacobi operator J with eigenvalues −(n + 2) and −4 respectively. Hencethe inequality in (1) follows.

If nL = 2, then, taking r = 2 in Theorem 24, we obtain an eigenspace of � in(�1(L)) of dimension n(n +1)/2 corresponding to the eigenvalue 2(n −1). Thisspace is also an eigenspace of the Jacobi operator J with eigenvalue −4. Hencethe inequality in (2) follows.

To study equality, we need to use [54, Theorem 1]. Such a result states that thefirst eigenvalue ρ1 of � acting on 1-forms of any compact minimal Lagrangiansubmanifold of CPn satisfies ρ1 ≤ 2(n − 1) and that, if the equality holds, then Lis totally geodesic. As the index of the totally geodesic real projective space RPn

is n(n + 1)/2, this completes the proof of part (2).To study equality in part (1) we need a modified version of [54, Theorem 1].

As in this case nL = 1, L is orientable and hence from the Hodge decomposi-tion Theorem, we have that (�1(L)) = H(L)⊕ d C∞(L)⊕ δ (�2(L)), whereH(L) stands for harmonic 1-forms. To prove [54, Theorem 1], the author usedcertain test 1-forms that actually belong to H(L)⊕δ (�2(L)). So really this The-orem says that the first eigenvalue ρ1 of� acting on H(L)⊕ δ (�2(L)) satisfiesρ1 ≤ 2(n − 1), with equality implying that L is totally geodesic. Now, if equalityin (1) holds we have that the two first eigenvalues of� acting on 1-forms are n and2(n − 1) with multiplicities n + 1 and n(n + 1)/2 respectively. But the eigenspacecorresponding to n is included in dC∞(L). So we can use the modified version ofthe above result to get that L is totally geodesic. �

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