Eigenvalue Estimates of the ${\rm spin}^c$ Dirac Operator and Harmonic Forms on K\"ahler-Einstein Manifolds

We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact K\"ahler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of K\"ahlerian Killing ${\rm spin}^c$ spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a K\"ahlerian Killing ${\rm spin}^c$ spinor field vanishes. This extends to the ${\rm spin}^c$ case the result of A. Moroianu stating that, on a compact K\"ahler-Einstein manifold of complex dimension $4\ell+3$ carrying a complex contact structure, the Clifford multiplication between an effective harmonic form and a K\"ahlerian Killing spinor is zero.


Introduction
The geometry and topology of a compact Riemannian spin manifold (M n , g) are strongly related to the existence of special spinor fields and thus, to the spectral properties of a fundamental operator called the Dirac operator D [1,26]. It is a first order differential operator acting on spinors. An interesting tool when examining the Dirac operator is the Schrödinger-Lichnerowicz formula [26]. Using this formula, A. Lichnerowicz [26] proved, under the weak condition of the positivity of the scalar curvature, that the kernel of the Dirac operator is trivial. This fact, combined with the Atiyah-Singer index theorem [1], provides a topological obstruction for the existence of positive scalar curvature metrics. We denote by " · " the Clifford multiplication and by ∇ the spinorial Levi-Civita connection on the spinor bundle ΣM . A spinor field ϕ satisfying for all vector fields X, ∇ X ϕ = a X · ϕ, is called parallel if a = 0 and a (real) Killing spinor if a ∈ R * . The existence of parallel spinors characterizes Ricci flat metrics with special holonomy [17,37]. Killing spinors force the underlying metric to be Einstein of scalar curvature 4n(n − 1)a 2 . The classification of complete simply-connected Riemannian spin manifolds with real Killing spinors done by C. Bär [2] gives examples (other than the sphere) which are relevant to physicists in general relativity. Killing spinors are also related to the spectrum of the Dirac operator. In fact, on a compact Riemannian spin manifold (M n , g), Th. Friedrich [6] proved the following lower bound for the first eigenvalue λ of D: where S denotes the scalar curvature, assumed to be nonnegative. Equality holds if and only if the corresponding eigenspinor is parallel (if λ = 0) or a Killing spinor of Killing constant − λ n (if λ = 0). Finally, we point out that useful geometric information has been also obtained by restricting parallel and Killing spinors to hypersurfaces [3,9,10,11,12,13]. O. Hijazi proved the following: Theorem 1.1. [14,Proposition 6.2] The Clifford multiplication between a harmonic k-form β (k = 0, n) and a Killing spinor vanishes.
In fact, he proved that if β · ψ is non-zero, then β · ψ would be an eigenspinor for the eigenvalue (−1) k λ(1 − 2k n ), which has absolute value strictly less than Friedrich's bound λ = n 4(n−1) inf M S. Thus β · ψ = 0. In particular, the equality case in (1) cannot be attained on a Kähler spin manifold, since the Clifford multiplication between the Kähler form and a Killing spinor is never zero. It is natural then to refine (1) and define the analogue of Killing spinors on Kähler compact manifolds (M 2m , g, J) of complex dimension m and complex structure J. On (M 2m , g, J), K.-D . Kirchberg [18] showed that the first eigenvalue λ of the Dirac operator satisfies: Kirchberg's estimates rely essentially on the decomposition of ΣM under the action of the Kähler form Ω. In fact, we have ΣM = ⊕ m r=0 Σ r M , where Σ r M is the eigenbundle corresponding to the eigenvalue i(2r − m) of Ω. The limiting manifolds of (2) are also characterized by the existence of spinors satisfying a certain differential equation similar to the one fulfilled by Killing spinors. More precisely, in odd complex dimension m = 2ℓ + 1, it is proved in [20,19,15] that the metric is Einstein and the corresponding eigenspinor ϕ of λ is a Kählerian Killing spinor, i.e. ϕ = ϕ ℓ + ϕ ℓ+1 ∈ Γ(Σ ℓ M ⊕ Σ ℓ+1 M) and it satisfies: for any vector field X. We point out that the existence of spinors of the form (3), implies that m is odd and they lie in the middle, i.e. l ′ = m−1 2 . If the complex dimension is even, m = 2ℓ, the limiting manifolds are characterized by constant scalar curvature and the existence of socalled anti-holomorphic Kählerian twistor spinors ϕ ℓ−1 ∈ Γ(Σ ℓ−1 M), i.e. satisfying for any vector field X: ∇ X ϕ ℓ−1 = − 1 2m (X + iJX) · Dϕ ℓ−1 . The limiting manifolds for Kirchberg's inequalities (2) have been geometrically described by A. Moroianu in [27] for m odd and in [29] for m even. In [35], this result is extended to limiting manifolds of the so-called refined Kirchberg inequalities, obtained by restricting the square of the Dirac operator to the eigenbundles Σ r M . When m is even, the limiting manifold cannot be Einstein. Thus, on compact Kähler-Einstein manifolds of even complex dimension, Kirchberg [21] improved (2) to the following lower bound: Equality is characterized by the existence of holomorphic or anti-holomorphic spinors. When m is odd, A. We extend this result to any odd complex dimension by using Spin c geometry instead of spin geometry. The setting is more general but there are more difficulties because the connection on the Spin c bundle, hence its curvature, the Dirac operator and its spectrum will not only depend on the geometry of the manifold, but also on the connection of the auxiliary line bundle associated with the Spin c structure.
Spin c geometry became an active field of research with the advent of Seiberg-Witten theory, which has many applications to 4-dimensional geometry and topology [36,38]. Several theorems arising from Donaldson's theory found an elementary proof [5]. C. LeBrun [24,25] obtained topological obstructions on 4-dimensional Einstein manifolds and together with M. Gursky [7], they gave lower and upper estimates of the Yamabe invariant for connected sums of certain complex surfaces.
From an intrinsic point of view, almost complex, Sasaki and some classes of CR manifolds carry a canonical Spin c structure. In particular, every Kähler manifold is Spin c but not necessarily spin. For example, the complex projective space CP m is spin if and only if m is odd. Moreover, from the extrinsic point of view, it seems that it is more natural to work with Spin c structures rather than spin structures [16,33,34]. For instance, on Kähler-Einstein manifolds of positive scalar curvature, O. Hijazi, S. Montiel and F. Urbano [16] constructed Spin c structures carrying Kählerian Killing Spin c spinors, i.e. spinors satisfying (3), where the covariant derivative is the Spin c one. The restriction of these spinors to minimal Lagrangian submanifolds provides topological and geometric obstructions on these submanifolds.
In [8], M. Herzlich and A. Moroianu extended Friedrich's estimate (1) to compact Riemannian Spin c manifolds. This new lower bound involves only the conformal geometry of the manifold and the curvature of the auxiliary line bundle associated with the Spin c structure. The limiting case is characterized by the existence of a Spin c Killing or parallel spinor, such that the Clifford multiplication of the curvature form of the auxiliary line bundle with this spinor is proportional to it. In [8], the classification of low dimensional limiting manifolds is given. Spin c manifolds carrying parallel or Killing spinors have been classified by A. Moroianu [30].
In this paper, we give an estimate for the eigenvalues of the Spin c Dirac operator, by restricting ourselves to compact Kähler-Einstein manifolds endowed with particular Spin c structures. More precisely, we consider (M 2m , g, J) a compact Kähler-Einstein manifold of Maslov index p ∈ N * . We endow M with a Spin c structure whose auxiliary line bundle is a tensorial power L q of the Maslov p-th root L of the canonical bundle, where q ∈ Z, p + q ∈ 2Z and |q| ≤ p. Then, the first eigenvalue λ of the Spin c Dirac operator satisfies cf. Theorem 3.7: Indeed, this is a consequence of more refined estimates for the eigenvalues of the square of the Spin c Dirac operator restricted to the eigenbundles Σ r M of the spinor bundle (see Theorem 3.5). The proof of this result is based on a refined Schrödinger-Lichnerowicz Spin c formula (see Lemma 3.4) written on each such eigenbundle Σ r M , which uses the decomposition of the covariant derivative acting on spinors into its holomorphic and antiholomorphic part. This formula has already been used in literature, for instance by K.-D. Kirchberg [21]. The limiting manifolds of (5) are characterized by the existence of Kählerian Killing Spin c spinors in a certain subbundle Σ r M . In particular, this gives a positive answer to the conjectured relationship between Spin c Kählerian Killing spinors and a lower bound for the eigenvalues of the Spin c Dirac operator, as stated in [16,Remark 16].
In section 4, we give a geometric application of (5). We extend Theorem 1.2 to the Spin c context with respect to the particular Spin c structures introduced above (see Acknowledgment. The first named author gratefully acknowledges the financial support of the Berlin Mathematical School (BMS) and would like to thank the University of Potsdam, especially Christian Bär and his group, for their generous support and friendly welcome during summer 2013 and summer 2014. The first named author thanks also the Faculty of Mathematics of the University of Regensburg for its support and hospitality during his two visits in July 2013 and July 2014. The authors are very much indebted to Oussama Hijazi and Andrei Moroianu for many useful discussions.

Preliminaries and Notation
In this section, we set the notation and briefly review some basic facts about Spin c and Kähler geometries. For more details we refer to the books [4], [23] and [31].
Let (M n , g) be an n-dimensional closed Riemannian Spin c manifold and denote by ΣM its complex spinor bundle, which has complex rank equal to 2 [ n 2 ] . The bundle ΣM is endowed with a Clifford multiplication denoted by "·" and a scalar product denoted by ·, · . Given a Spin c structure on (M n , g), one can check that the determinant line bundle det(ΣM ) has a root L of index 2 [ n 2 ]−1 . This line bundle L over M is called the auxiliary line bundle associated with the Spin c structure. The connection ∇ A on ΣM is the twisted connection of the one on the spinor bundle (induced by the Levi-Civita connection) and a fixed connection A on L. The Spin c Dirac operator D A acting on the space of sections of ΣM is defined by the composition of the connection ∇ A with the Clifford multiplication. For simplicity, we will denote ∇ A by ∇ and D A by D. In local coordinates: where {e j } j=1,...,n is a local orthonormal basis of T M . D is a first-order elliptic operator and is formally self-adjoint with respect to the L 2 -scalar product. A useful tool when examining the Spin c Dirac operator is the Schrödinger-Lichnerowicz formula: where ∇ * is the adjoint of ∇ with respect to the L 2 -scalar product and F A is the curvature (imaginary-valued) 2-form on M associated to the connection A defined on the auxiliary line bundle L, which acts on spinors by the extension of the Clifford multiplication to differential forms.
We recall that the complex volume element ω C = ı [ n+1 2 ] e 1 ∧ . . . ∧ e n acts as the identity on the spinor bundle if n is odd. If n is even, ω 2 C = 1. Thus, under the action of the complex volume element, the spinor bundle decomposes into the eigenspaces Σ ± M corresponding to the ±1 eigenspaces, the positive (resp. negative) spinors.
Every spin manifold has a trivial Spin c structure, by choosing the trivial line bundle with the trivial connection whose curvature F A vanishes. Every Kähler manifold (M 2m , g, J) has a canonical Spin c structure induced by the complex structure J. The complexified tangent bundle decomposes into T C M = T 1,0 M ⊕ T 0,1 M, the ieigenbundle (resp. (−i)-eigenbundle) of the complex linear extension of J. For any vector field X, we denote by X ± := 1 2 (X ∓ iJX) its component in T 1,0 M , resp. T 0,1 M . The spinor bundle of the canonical Spin c structure is defined by and its auxiliary line bundle is The line bundle L has a canonical holomorphic connection, whose curvature form is given by −iρ, where ρ is the Ricci form defined, for all vector fields X and Y , by ρ(X, Y ) = Ric(JX, Y ) and Ric denotes the Ricci tensor. Similarly, one defines the so called anti-canonical Spin c structure, whose spinor bundle is given by Λ * ,0 M = ⊕ m r=0 Λ r (T * 1,0 M ) and the auxiliary line bundle by K M . The spinor bundle of any other Spin c structure on M can be written as: where L 2 = K M ⊗ L and L is the auxiliary line bundle associated with this Spin c structure. The Kähler form Ω, defined as Ω(X, Y ) = g(JX, Y ), acts on ΣM via Clifford multiplication and this action is locally given by: for all ψ ∈ Γ(ΣM ), where {e 1 , . . . , e 2m } is a local orthonormal basis of TM. Under this action, the spinor bundle decomposes as follows: if and only if r is even (resp. r is odd). Moreover, for any X ∈ Γ(T M ) and where L is the auxiliary line bundle associated with the Spin c structure. For example, when the manifold is spin, we have (Σ 0 M ) 2 = K M [17,18]. For the canonical Spin c structure, since L = (K M ) −1 , it follows that Σ 0 M is trivial. This yields the existence of parallel spinors (the constant functions) lying in Σ 0 M , cf. [30].
Associated to the complex structure J, one defines the following operators: which satisfy the relations When restricting the Dirac operator to Σ r M, it acts as Ker r denotes the kernel of the Clifford multiplication restricted to Σ r M, we have, as in the spin case, the following Weitzenböck formula relating the differential operators acting on sections of Σ r M: where T r is the so-called Kählerian twistor operator and is defined by This decomposition further implies the following inequality for ϕ ∈ Γ(Σ r M): Equality in (11) is attained if and only if T r ϕ = 0, in which case ϕ is called a Kählerian twistor spinor (see e.g. [35]). The Lichnerowicz-Schrödinger formula (6) yields the following: Lemma 2.1. Let (M 2m , g, J) be a compact Kähler manifold endowed with any Spin c structure. If ϕ is an eigenspinor of D 2 with eigenvalue λ, D 2 ϕ = λϕ, and satisfies for some real number j > 1, and (S + 2F A ) · ϕ = c ϕ, where c is a positive function, then Moreover, equality in (13) holds if and only if the function c is constant and equality in (12) holds at all points of the manifold.
Let {e 1 , . . . , e 2m } be a local orthonormal basis of M 2m . We implicitly use the Einstein summation convention over repeated indices. We have the following formulas for contractions that hold as endomorphisms of Σ r M: The Spin c Ricci identity, for any spinor ϕ and any vector field X, is given by: where R A denotes the Spin c spinorial curvature. For any vector field X parallel at the point where the computation is done, a straightforward computation yields the following commutator rules: On a Kähler manifold (M, g, J) endowed with any Spin c structure, a spinor of the form ϕ r + ϕ r+1 ∈ Γ(Σ r M ⊕ Σ r+1 M ), for some 0 ≤ r ≤ m, is called a Kählerian Killing Spin c spinor if there exists a non-zero real constant α, such that the following equations are satisfied, for all vector fields X,  (7). We briefly describe these Spin c structures here. If the first Chern class c 1 (K M ) of the canonical bundle of the Kähler M is a non-zero cohomology class, the greatest number p ∈ N * such that 1 p is called the Maslov index of the Kähler manifold. One can thus consider a p-th root of the canonical bundle K M , i.e. a complex line bundle L, such that L p = K M . In [16], O. Hijazi, S. Montiel and F. Urbano proved the following: Theorem 2.2 (Theorem 14, [16]). Let M be a 2m-dimensional Kähler-Einstein compact manifold with scalar curvature 4m(m + 1) and index p ∈ N * . For each 0 ≤ r ≤ m + 1, there exists on M a Spin c structure with auxiliary line bundle given by L q , where q = p m+1 (2r − m − 1) ∈ Z, and carrying a Kählerian Killing spinor ψ r−1 + ψ r ∈ Γ(Σ r−1 M ⊕ Σ r M ), i.e. it satisfies the first order system ∇ X ψ r = −X + · ψ r−1 , for all X ∈ Γ(T M ).
For example, if M is the complex projective space CP m of complex dimension m, then p = m + 1 and L is just the tautological line bundle. We fix 0 ≤ r ≤ m + 1 and we endow CP m with the Spin c structure whose auxiliary line bundle is given by L q where q = p m+1 (2r − m − 1) = 2r − m − 1 ∈ Z. For this Spin c structure, the space of Kählerian Killing spinors in Γ(Σ r−1 M ⊕ Σ r M ) has dimension m+1 r . A Kähler manifold carrying a complex contact structure necessarily has odd complex dimension m = 2ℓ + 1 and its Maslov index p equals ℓ + 1. We fix 0 ≤ r ≤ m + 1 and we endow M with the Spin c structure whose auxiliary line bundle is given by For this Spin c structure, the space of Kählerian Killing spinors in Γ(Σ r−1 M ⊕ Σ r M ) has dimension 1. In these examples, for r = 0 (resp. r = m + 1), we get the canonical (resp. anticanonical) Spin c structure for which Kählerian Killing spinors are just parallel spinors.

Eigenvalue estimates for the Spin c Dirac operator on Kähler-Einstein Manifolds
In this section, we give a lower bound for the eigenvalues of the Spin c Dirac operator on a Kähler-Einstein manifold endowed with particular Spin c structures. More precisely, let (M 2m , g, J) be a compact Kähler-Einstein manifold of Maslov index p ∈ N * and of positive scalar curvature S, endowed with the Spin c structure given by L q , where L is the Maslov p-th root of the canonical bundle and q + p ∈ 2Z (among all powers L q , only those satisfying p + q ∈ 2Z provide us a Spin c structure, cf. [16, Section 7]). The curvature form F A of the induced connection A on L q acts on the spinor bundle as q p iρ. Since (M 2m , g, J) is Kähler-Einstein, it follows that ρ = S 2m Ω, where Ω is the Kähler form. Hence, for each 0 ≤ r ≤ m, we have: Let us denote by c r := 1 − q p · 2r−m m and a 1 : {0, . . . , m} → R, a 1 (r) := r + 1 2r + 1 c r , a 2 : {0, . . . , m} → R, a 2 (r) := m − r + 1 2m − 2r + 1 c r .
Remark 3.2. The inequality (24) can be expressed more explicitly, by determining the maximum according to several possible cases. However, since in the sequel we will refine this eigenvalue estimate, we are only interested in the characterization of the limiting cases, which will be used later in the proof of the equality case of the estimate (5).
In order to refine the estimate (24), we start by the following two lemmas. Lemma 3.3. Let (M 2m , g, J) be a compact Kähler-Einstein manifold of Maslov index p ∈ N * and of positive scalar curvature S, endowed with a Spin c structure given by L q , where q + p ∈ 2Z. For any spinor field ϕ and any vector field X, the Spin c Ricci identity is given by: and it can be refined as follows:

Proof:
Since the curvature form F A of the Spin c structure acts on the spinor bundle as q p iρ = q p S 2m iΩ, (25) follows directly from the Ricci identity (16). The refined identities (26) and (27) follow then directly from the following identities of endomorphisms of the spinor bundle: X − iΩ = X − and X + iΩ = −X + . Lemma 3.4. Under the same assumptions as in Lemma 3.3, the refined Schrödinger-Lichnerowicz formula for Spin c Kähler manifolds for the action on each eigenbundle Σ r M is given by where ∇ 1,0 (resp. ∇ 0,1 ) is the holomorphic (resp. antiholomorphic) part of ∇. They are locally defined, for all vector fields X, by Proof: Using a local basis {e 1 , . . . , e 2m } which is parallel at the point where the computation is made, we compute: A similar computation yields the second formula.
Theorem 3.5. Let (M 2m , g, J) be a compact Kähler-Einstein manifold of Maslov index p ∈ N * and positive scalar curvature S, carrying the Spin c structure given by L q with q + p ∈ 2Z, where L p = K M . We assume that p ≥ |q|. Then, for each r ∈ {0, . . . , m}, any eigenvalue λ r of D 2 | Γ(Σr M) satisfies the inequality: Proof: First we notice that our assumption |q| ≤ p implies that the lower bound in (30) is non-negative and that 0 ≤ 1 + q p m 2 ≤ m. The formulas (28) and (29) applied to ϕ r yield, after taking the scalar product with ϕ r and integrating over M , the following inequalities: and equality is attained if and only if the corresponding eigenspinor ϕ r satisfies ∇ 1,0 ϕ r = 0, resp. ∇ 0,1 ϕ r = 0. Hence, for any 0 ≤ r ≤ m we obtain the following lower bound: Remark 3.6. Let us denote q p · m+1 2 + m−1 2 by b 1 . Comparing the estimate given by Theorem 3.5 with the estimate from Proposition 3.1, we obtain for r ≤ b 1 : Hence, for r ≤ b 1 : we have: e(r) > a 1 (r) and e(r) = a 1 (r) iff r = b 1 ∈ N. Similarly, for r ≥ b 1 + 1, we compute: Hence, for r ≥ b 1 + 1 we have: e(r) > a 2 (r) and e(r) = a 2 (r) iff r = b 1 + 1 ∈ N.
Theorem 3.5 implies the following global lower bound for the eigenvalues of the Spin c Dirac operator acting on the whole spinor bundle: Theorem 3.7. Let M 2m be a compact Kähler-Einstein manifold of Maslov index p and positive scalar curvature S, carrying the Spin c structure given by L q with q + p ∈ 2Z, where L p = K M . We assume that p ≥ |q| and the metric is normalized such that its scalar curvature equals 4m(m + 1). Then, any eigenvalue λ of D 2 is bounded from below as follows: Equality is attained if and only if b 1 ∈ N and there exists a Kählerian Killing Spin c spinor in Γ(Σ b1 M ⊕ Σ b1+1 M ).
Proof: Since the lower bound established in Theorem 3.5 decreases on (0, 1 + q p m 2 ) and increases on ( 1 + q p m 2 , m), we obtain the following global estimate: However, this estimate is not sharp. Otherwise, this would imply that 1 + q p m 2 ∈ N and the limiting eigenspinor would be, according to the characterization of the equality case in Theorem 3.5, both holomorphic and antiholomorphic, hence parallel and, in particular, harmonic. This fact together with the Lichnerowicz-Schrödinger formula (6) and the fact that the scalar curvature is positive leads to a contradiction.

Parallel forms on limiting Kähler-Einstein manifolds
In this section we give an application for the eigenvalue estimate of the Spin c Dirac operator established in Theorem 3.7. Namely, we extend to all odd complex dimensions Theorem 1.3. As above, M denotes a 2m-dimensional Kähler-Einstein compact manifold of Maslov index p ∈ N * and normalized scalar curvature 4m(m + 1), which carries the Spin c structure given by L q with q + p ∈ 2Z, where L p = K M . We call M a limiting manifold if equality in (31) is achieved on M , which is by Theorem 3.7 equivalent to the existence of a Kählerian Killing Spin c spinor in Σ r M ⊕ Σ r+1 M for r = q p · m+1 2 + m−1 2 ∈ N. Let ψ = ψ r−1 + ψ r ∈ Γ(Σ r−1 M ⊕ Σ r M ) be such a spinor, i.e. Ω · ψ r−1 = i(2r − 2 − m)ψ r−1 , Ω · ψ r = i(2r − m)ψ r and the following equations are satisfied: By (21), we have: Recall that a form ω on a Kähler manifold is called effective if Λω = 0, where Λ is the adjoint of the operator L : Λ * M −→ Λ * +2 M , L(ω) := ω ∧ Ω. More precisely, Λ is given by the formula: Λ = −2 2m j=1 e + j e − j . Moreover, one can check that Lemma 4.1. Let ψ = ψ r−1 + ψ r ∈ Γ(Σ r−1 M ⊕ Σ r M ) be a Kählerian Killing Spin c spinor and ω a harmonic effective form of type (k, k ′ ). Then, we have Proof: The following general formula holds for any form ω of degree deg(ω) and any spinor ϕ: (e j ω) · ∇ ej ϕ.
Again, for m is odd, the factor before ω 1 is non zero and we have ω 1 is parallel and effective, so zero by Theorem 4.3. If m even and 0 < k < m 2 , the factor before ω 1 is also non zero and so ω 1 is parallel and effective and hence zero by Theorem 4.3. By induction, we get ω j = 0, for all j ≥ 1. For (k, k)-forms with k > m 2 , the result then follows by applying the Hodge- * -operator.
Kähler-Einstein manifolds carrying a complex contact structure are examples of odd dimensional Kähler manifolds with Kählerian Killing Spin c spinors in Σ r−1 M ⊕ Σ r M for the Spin c structure (described in the introduction) whose auxiliary line bundle is given by L q and q = r − ℓ − 1, where m = 2ℓ + 1. Thus, as a special case of Theorem 4.4, we obtain the announced generalization of Theorem 1.3: