The Twisted Spin $^ C $ Dirac Operator on K\" Ahler Submanifolds of The

Home , Spin structure

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(d + 1)2 if d is odd, λ 6  (1)  d(d + 2) if d is even.  Here µ denotes the dimension of the space of Kählerian Killing spinors on M 2m. Recall that the normal bundle is endowed with the induced spin structure coming from both manifolds M and M. The idea consists in f computing the so-called Rayleigh-quotient applied to the Kählerian Killing f spinor restricted to the submanifold M. The upper bound is then deduced by using the min-max principle. This technique was also used by C. Bär in [1] for submanifolds in Rn+1, Sn+1 and Hn+1.

The complex projective space CP m is a spin manifold if and only m is odd. In this case, the sharpness of the upper bound (1) was studied in [6] for the canonical embedding CP d → CP m, where d is also odd. In fact, it is shown that for d = 1, the upper estimate is optimal for m = 3, 5, 7 while it is not for m ≥ 9.

Kähler manifolds are not necessary spin but every Kähler manifold has a canonical Spinc structure (see Section 2) and any other Spinc structure can be expressed in terms of the canonical one. Moreover, O. Hijazi, S. Montiel and F. Urbano [8] constructed on Kähler-Einstein manifolds with positive scalar curvature, Spinc structures carrying Kählerian Killing spinors. Thus one can consider the result of N. Ginoux for Spinc manifolds.

Section 2 is devoted to recall some basic facts on Spinc structures on Kähler manifolds. In Section 3, we extend the estimate (1) to the eigenvalues of the twisted Dirac operator for a Kähler submanifold of the complex projective space (see Theorem 3.1). Finally, we discuss the sharpness for the embedding CP d → CP m with different values of m and d.

2 K¨ahler Submanifolds of K¨ahler manifolds

Let (M 2m,g,J) be a Kähler manifold of complex dimension m. Recall that the complexified tangent bundle splits into the orthogonal sum T CM = C T1,0M ⊕ T0,1M where T1,0M (resp. T0,1M) denotes the eigenbundle of T M corresponding to the eigenvalue i (resp. −i) of the extension of J. Using this 0,r r ∗ r,0 decomposition, we define Λ M := Λ (T0,1M) (resp. Λ M) as being the

2 bundle of complex r-forms of type (0,r) (resp. of type (r, 0)). Recall also that every K¨ahler manifold has a canonical Spinc structure whose complex 0,∗ m 0,r spinorial bundle is given by ΣM =Λ M = ⊕r=0Λ M, where the auxiliary c −1 bundle of this Spin structure is given by KM . Here KM is the canonical m,0 bundle of M deﬁned by KM =Λ M [4, 11]. On the other hand, the spinor bundle of any other Spinc structure can be written as [4, 8]:

ΣM =Λ0,∗M ⊗ L,

2 c where L = KM ⊗ L and L is the auxiliary bundle associated with this Spin structure. Moreover, the action of the K¨ahler form Ω of M splits the spinor bundle into [4, 10, 9]: m ΣM = ⊕r=0ΣrM, where ΣrM denotes the eigensubbundle corresponding with the eigenvalue m i(2r − m) of Ω with complex rank . For any vector ﬁeld X ∈ Γ(T M) k and ψ ∈ Γ(ΣrM), we have the following property p±(X) · ψ ∈ Γ(Σr±1M), 1 where p±(X)= 2 (X ∓ iJX).

Let (M 2d,g,J) be an immersed Kähler submanifold in a Kähler manifold (M 2m,g,J) carrying the induced complex structure J (i.e. J(T M) = T M) and denote respectively by Ω , Ω and Ω the Kähler form of M, M and f M N of the normal bundle NM −→f M of the immersion. Since the manifolds f M and M 2n are Kähler, they carry Spinc structures with corresponding c auxiliary line bundles LM and LM . This induces a Spin structure on the f f bundle NM such that the restricted complex spinor bundle ΣM|M of M can be identified with ΣM ⊗ ΣN, where ΣM and ΣN are the spinor bundles of f f M and NM respectively ([1], [7]). Moreover, the auxiliary line bundle LN c −1 of this Spin structure on NM is given by LN := (LM ) ⊗ (LM )|M . Given f ΣM⊗ΣN connection 1-forms on LM and LM , they induce a connection ∇ := ∇ on ΣM ⊗ ΣN. Thus one can statef a Gauss-type formula for the spinorial Levi-Civita connections ∇ and ∇ on ΣM and ΣM ⊗ ΣN respectively [13]. That is, for all X ∈ T M and ϕ ∈ Γ(ΣM ), we have e f|M

fd 1 2 ∇ ϕ = ∇ ϕ + e · II(X, e ) · ϕ, (2) X X 2 j j Xj=1 e where (ej)1≤j≤2d is any local orthonormal basis of T M and II is the second fundamental form of the immersion. As a consequence of the Gauss formula, 2d the square of the auxiliary Dirac-type operator D := j=1 ej · ∇ej is related P b e 3 ΣN 2d to the square of the twisted Dirac operator DM := j=1 ej ·∇ej by [5, Lemme 4.1]: P

2d 2 ΣN 2 2 2 N D ϕ =(DM ) ϕ − d |H| ϕ − d ej ·∇ej H · ϕ, Xj=1 b 1 where H := 2d tr(II) is the mean curvature vector field of the immersion. In our case, the mean curvature vanishes which means that the operators D2 and (DΣN )2 coincide. M b In the sequel, take the manifold M as the complex projective space CP m endowed with its Fubini-Study metric of constant holomorphic sec- f tional curvature 4. In [8], the authors proved that for every q ∈ Z, such that q + m +1 ∈ 2Z, there exists a Spinc structure on CP m whose auxiliary line q C m bundle is given by Lm. Here Lm denotes the tautological bundle of P . In particular for q = −m − 1 (resp. q = m + 1), the Spinc structure is the canonical one (resp. anti-canonical) [12] and for q = 0 it corresponds to the unique spin structure if m is odd. Let us denote by ΣqCP m the spinor bundle of the corresponding Spinc structure with Lq as auxiliary line bundle. For any integer r in {0, ··· , m +1} such that q =2r − (m + 1), the bundle q m Σ CP carries a Kählerian Killing spinor field ψ = ψr−1 + ψr satisfying, for all X ∈ Γ(T CP m) [8]

∇X ψr = −p+(X) · ψr−1, ∇Xeψr−1 = −p−(X) · ψr, (3) e m +1 The space of Kählerian Killing spinors is of rank . We point out r that for r = 0 (resp. r = m+1) the Kählerian Killing spinor is a parallel spinor which is carried by the canonical structure (resp. anti-canonical). Moreover, m+1 for r = 2 , i.e. m is odd, the Kählerian Killing spinor is the usual one lying 0 m 0 m in Σ m−1 CP ⊕ Σ m+1 CP defined in [9, 10]. 2 2

3 Main result

In this section, we will establish the estimates for the eigenvalues of the twisted Dirac operator of complex submanifolds of the complex projective space. We will test the sharpness of Inequality (4) for the canonical embedding CP d → CP m. For more details, we refer to [6].

4 Theorem 3.1 Let (M 2d,g,J) be a closed K¨ahler submanifold of the complex projective space CP m. For r ∈ {0, ··· , m +1} and q = 2r − (m + 1), there m +1 are at least -eigenvalues λ of (DΣN )2 satisfying r M

−(q2 − (d + 1)2)+2|q|(m − d) − 1 if m − d is odd λ 6  (4)  −(q2 − (d + 1)2)+2|q|(m − d) if m − d is even.  Proof. The proof relies on computing the Rayleigh-quotient

ΣN 2 M Reh(DM ) ψ, ψivg R 2 M |ψ| vg R m applied to any non-zero K¨ahlerian Killing spinor ψ = ψr−1 + ψr on CP . A straightforward computation of the auxiliary Dirac operator leads to

Dψr−1 =(q + d + 1)ψr + iΩN · ψr. b Dψr = −(q − d − 1)ψr−1 − iΩN · ψr−1. Summing up the aboveb two equations, we deduce after using the fact that the auxiliary Dirac operator commutes with the normal K¨ahler form [6], that

D2ψ = −(q2 − (d + 1)2)ψ − 2iqΩN · ψ + ΩN · ΩN · ψ.

Taking theb Hermitian inner product with ψ and using the fact that the seond term can be bounded from above by 2|q|(m − d), we get our estimates after using |ΩN · ψ|≥|ψ| if m − d is odd and 0 otherwise.

In the following, we will treat the sharpness through the embedding CP d → CP m. Recall ﬁrst that the complex projective space CP d can be seen as the symmetric space SUd+1/S(Ud × U1) where S(Ud × U1) := B 0 { | B ∈ U }. The tangent bundle of CP d can be described 0 det(B)−1 d as a homogeneous bundle which is associated with the S(Ud × U1)-principal d bundle SUd+1 → CP via the isotropy representation

α : S(Ud × U1) −→ Ud B 0 7−→ det(B)B. 0 det(B)−1

5 For the canonical embedding CP d → CP m, the normal bundle T ⊥CP d is ∗ Cm−d C d isomorphic to Ld ⊗ where Ld is the tautological bundle of P . The bundle Ld is isomorphic to the homogeneous bundle which is associated with the S(Ud × U1)-principal bundle SUd+1 via the representation

ρ : S(Ud × U1) −→ U1 B 0 7−→ (det(B))−1. 0 det(B)−1

Thus the normal bundle is associated with the S(Ud × U1)-principal bundle d SUd+1 → CP via the representation

ρ : S(Ud × U1) −→ Um−d B 0 7−→ det(B)I . 0 det(B)−1 m−d

Now, we endow CP d with a Spinc structure whose auxiliary line bundle is ′ q ′ Z given by Ld for q ∈ . In this case, its spinor bundle is given by

′ ′ q +d+1 q C d 0,∗C d 2 Σ P =Λ P ⊗Ld .

The existence of Spinc structures on both CP d and CP m induces also a Spinc structure on the normal bundle of the embedding with auxiliary line bundle ′ ′ q q q−q is given by Lm|CP d ⊗Ld which is isomorphic to Ld . Therefore the Lie-group homomorphism

ρ : S(Ud × U1) −→ Um−d × U1 B 0 ′ 7−→ (det(B)I , det(B)−(q−q )) 0 det(B)−1 m−d

c can be lifted through the non-trivial two-fold covering map Spin2(m−d) −→ SO2(m−d) × U1 to the homomorphism

c ρ : S(Ud × U1) −→ Spin2(m−d) ′ B 0 q−q +m−d e 7−→ (det(B))− 2 j(det(B)I ), 0 det(B)−1 m−d

c where for any positive integer k, we recall that j : Uk −→ Spin2k is given on elements of diagonal form of Uk as

iλ iλ i k λ λ1 λk j(diag(e 1 , ··· , e k )) = e 2 (Pj=1 j )R ( ) ··· R ( ). e1,Je1 2 ek,Jek 2 e e 6 k Here J is the canonical complex structure on C and Rv,w(λ) = cos(λ)+ sin(λ)v · w ∈ Spin is deﬁned for any orthonormal system {v,w} ∈ R2k. We 2k e point out that the integer q − q′ + m − d is even. Following the similar proof as in [6, Corollary 4.4], the complex spinor bundle of T ⊥CP d splits into the orthogonal sum

m−d − ′ − m − d q q +m d −s Σ(T ⊥CP d) ∼= L 2 , s d Ms=0

m − d where for each s ∈ {0,...,m − d}, the factor stands the multi- s ′ q−q +m−d 2 −s plicity which the line bundle Ld appears in the splitting. This gives the following decomposition

− ′ − m−d q q +m d −s C d ⊥C d m − d C d 2 Σ P ⊗ Σ P ≃ ⊕ Σ P ⊗Ld s=0 s ′ − ′ − m−d d+1+q q q +m d −s m − d 0,∗C d 2 2 ≃ ⊕ Λ P ⊗Ld ⊗Ld s=0 s

m−d m+1+q s m − d 0,∗C d 2 − ≃ ⊕ Λ P ⊗Ld . s=0 s

We point out here that the above decomposition does not depend on the Spinc structure chosen on CP d, since no power in q′ appears. In [3], the authors proved that (see also [6, 2]):

Proposition 3.2 Let CP d be the complex projective space of constant holomorphic sectionnal curvature 4 endowed with a Spinc structure whose spinor bundle is given by Λ0,∗CP d ⊗Lv, for some v ∈ Z, i.e., whose auxiliary line 2v−(d+1) bundle is given by Ld . Then, the spectrum of the square of the Dirac operator is given by the eigenvalue 0 if v ≤ 0 or v ≥ d +1 and by

λ2 = 4(l + v)(l − k + d), where l ∈ N, l + v ≥ k +1 and 0 6 k 6 d − 1. Moreover, the multiplicity of λ2 is given by

2(l + d)!(l + v − k − 1+ d)!(2l + v − k + d) l!k!d!(l + v − k − 1)!(d − k − 1)!(l + v)(l + d − k)

(|v|+d)! 6 (v−1)! and the multiplicity of 0 by d!|v|! if v 0 and by d!(v−d−1)! if v ≥ d +1.

7 In order to ﬁnd the spectrum of the square of the twisted Dirac operator corresponding with the embedding CP d → CP m, one should replace v m+1+q in Proposition 3.2 by 2 − s and in this case, the eigenvalue of the m+1+q 6 square of the twisted Dirac operator is given by 0 if 2 − s 0 or if m+1+q m+1+q 6 6 2 − s ≥ d +1 and by 4(l + 2 − s)(l − k + d) for 0 s m − d, 6 6 m+q+1 0 k d − 1 and l + 2 − s ≥ k + 1.

Let us consider particular values for d, m and q in order to check the optimality. For d = 1, m = 2 and q = 1, by Theorem 3.1, there are at least 3 eigenvalues of the square of the twisted Dirac operator satisfying the estimate λ 6 4. The multiplicity of zero is 1 and the multiplicity of the eigenvalue 4 is 4 which means that the estimate is optimal. For d = 1, m = 3 and q = 2, by Theorem 3.1, there are at least 4 eigenvalues of the square of the twisted Dirac operator satisfying the estimate λ 6 8. The multiplicity of zero is 3. The multiplicity of the eigenvalue 4 is 4 and the multiplicity of the eigenvalue 8 is 6 which means that the estimate is not optimal. For d = 2, m = 3 and q = 2, there are at least 4 eigenvalues of the square of the twisted Dirac operator satisfying the estimate λ 6 8. The multiplicity of zero is 1 and the multiplicity of the eigenvalue 8 is 6 which means that the estimate is optimal.

Acknowledgment

The authors would like to thank Nicolas Ginoux and Oussama Hijazi for fruitful discussions during the preparation of this work. They also thank the Max Planck Institute for Mathematics and the University of Nancy for their support.

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