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1 Basic satisfied by holomorphic spinors.

Let M 2m be a spin K¨ahler manifold of complex dimension m with complex structure J, K¨ahler g and spinor bundle S. Then the K¨ahler form Ω defines a canonical splitting

(1) S = S S S 0 ⊕ 1 ⊕···⊕ m 0,r m into holomorphic subbundles Sr = Λ So (r = 0, 1,...,m, ) of rank , where ∼ ⊗ r ! m,0 S0 = √Λ is the square root of the canonical bundle representing the spin structure of M 2m (see [6] or [3], Section 3.4.). Considering Ω as an endomorphism of S the action of Ω on S is just the multiplication by i(m 2r). Let p(X) := 1 (X iJX) r − 2 − andp ¯(X) := 1 (X + iJX) for any real vector field X and let ψ Γ(S ). Then we 2 ∈ r have p(X) ψ Γ(S ) andp ¯(X) ψ Γ(S − ). Furthermore, let (X ,...,X ) be · ∈ r+1 · ∈ r 1 1 2m any local frame of vector fields and (ξ1,...,ξ2m) the corresponding coframe. Using kl −1 k kl the notations gkl := g(Xn, Xl), (g ) := (gkl) , X := g Xl (Einstein’ s convention of summation) and g(X) := g(X, ) the holomorphic and antiholomorphic part of · the covariant ψ of any spinor ψ Γ(S) are locally defined by ∇ ∈

1,0ψ := ξk ψ = g(¯p(Xk)) ψ, ∇ ⊗∇p(Xk) ⊗∇p(Xk) (2) 0,1ψ := ξk ψ = g(p(Xk)) ψ, ∇ ⊗∇p¯(Xk) ⊗∇p¯(Xk) respectively. This yields the decomposition

(3) ψ = 1,0ψ + 0,1ψ ∇ ∇ ∇ with 1,0ψ Γ(Λ1,0 S) and 0,1ψ Γ(Λ0,1 S). ψ is called to be holomorphic ∇ ∈ ⊗ ∇ ∈ ⊗ (antiholomorphic) iff 0,1ψ = 0 ( 1,0ψ = 0). ∇ ∇ The Dirac operator D and its K¨ahler twist D˜ are locally given by

(4) D = Xk , D˜ = J(Xk) . ·∇Xk ·∇Xk 1 In the following the operators D and D− appear defined by D± := (D iD˜). + 2 ∓ Then it holds

2 D = p(Xk) = p(Xk) = Xk , + ·∇Xk ·∇p¯(Xk) ·∇p¯(Xk) (5) D− =p ¯(Xk) =p ¯(Xk) = Xk ·∇Xk ·∇p(Xk) ·∇p(Xk) and there are the operator equations

(6) D = D+ + D− ,

2 2 (7) D+ = D− = 0,

2 (8) D = D+D− + D−D+.

Moreover, for ψ Γ(S ), we have D±ψ Γ(S ± ). ∈ r ∈ r 1 Proposition 1: Let ψ Γ(S) be any holomorphic (antiholomorphic) spinor. Then ψ ∈ satisfies the equation

(9) D+ψ = 0 (D−ψ = 0) and, moreover, the equation

1 D−ψ + p¯(Ric(X)) ψ = 0 ∇p¯(X) 2 · (10) ( D ψ + 1 p(Ric(X)) ψ = 0) ∇p(X) + 2 · for each real vector field X, where Ric is the Ricci tensor.

Proof: By definition, ψ Γ(S) is holomorphic (antiholomorphic) iff there is the ∈ equation

(11) ψ = 0 ( ψ = 0) ∇p¯(X) ∇p(X) for each real vector field X. Thus, for example, we have 0 = Xk ψ = D ψ ·∇p¯(Xk) + and hence (9). For any ϕ Γ(S) and any X Γ(T M 2m), there is the well-known ∈ ∈ relation 1 (12) Xk C(X , X)ϕ = Ric(X) ϕ, · k 2 · where C is the curvature tensor of S. Hence, since M 2m is K¨ahler, it holds

1 1 1 p¯(Ric(X)) ϕ = Ric(¯p(X)) ϕ = Xk C(X , p¯(X))ϕ = 2 · 2 · 2 · k = Xk C(p(X ), p¯(X))ϕ =p ¯(Xk) C(p(X ), p¯(X))ϕ = · k · k =p ¯(Xk) C(X , p¯(X))ϕ. · k Thus, (12) implies in K¨ahler case the relations

3 1 p¯(Ric(X)) ϕ =p ¯(Xk) C(X , p¯(X))ϕ, 2 · · k (13) 1 p(Ric(X)) ϕ = p(Xk) C(X ,p(X))ϕ. 2 · · k Now, let P M 2m be any and let (X ,...,X ) be a frame on a neighbour- ∈ 1 2m hood of P such that

(14) ( X ) = 0 (k = 1, , 2m). ∇ k P · · · Using (5), (11), (13) and (14) we have at P

1 D−ψ + p¯(Ric(X)) ψ = D−ψ +p ¯(Xk) C(X , p¯(X))ψ = ∇p¯(X) 2 · ∇p¯(X) · k = (¯p(Xk) ψ) +p ¯(Xk)( ψ ψ ψ)= ∇p¯(X) ∇Xk ∇Xk ∇p¯(X) −∇p¯(X)∇Xk −∇[Xk,p¯(X)] k k k = p¯(X ) ψ = p¯(X ) ∇ ψ = p¯(X ) ∇ ψ = 0. − ∇[Xk,p¯(X)] − ∇ Xk p¯(X) − ∇p¯( Xk X)

This yields (10). ✷

Corollary 2: Let ψ Γ(S) be holomorphic (antiholomorphic). Then there is the ∈ equation

R + i (15) D + D − ψ = ψ ρ ψ, − 4 ( ) 2 · ( ) (+) − where R is the and ρ the Ricci form.

Proof: Using (10) and the first one of the well-known relations

Xk p¯(Ric(X )) = R iρ, · k − 2 − (16) Xk p(Ric(X )) = R + iρ, · k − 2 we have

k 1 k 0 = X D−ψ + X p¯(Ric(X )) ψ = ·∇p¯(Xk) 2 · k · R i = D D−ψ ψ ρ ψ. ✷ + − 4 − 2 ·

Let ∗ be the Bochner Laplacian on Γ(S) locally given by ∇ ∇ ∗ kl (17) = g ( ∇ ). ∇ ∇ − ∇Xk ∇Xl −∇ Xk Xl Then there is the well-known relation (see [12])

4 ∗ R (18) = D2 ∇ ∇ − 4 Using (8), (9), (15) and (18) we immediately obtain

Corollary 3: If ψ Γ(S) is holomorphic (antiholomorphic), then ψ satisfies the ∈ equivalent equations

R + i (19) D2ψ = ψ ( ) ρ ψ, 4 − 2 ·

+ i (20) ∗ ψ =( ) ρ ψ. ∇ ∇ − 2 · 2 Holomorphic spinors and limiting manifolds.

Theorem 4: Let M 2m be a closed spin K¨ahler manifold of odd complex dimension m m,0 2m with positive scalar curvature and spin structure S0 = √Λ . Then M is a limit- m 2m 0, +1 ing manifold iff M is Einstein and the bundle Λ 2 S admits a holomorphic ⊗ 0 section.

Proof: We recall that there are canonical unitary isomorphisms

∼ (21) α : Λ0,r S S (r = 0, 1,...,m) r ⊗ 0 −→ r − r defined by α (ω ψ ) := 2 2 ω ψ (see [7], Prop. 4). From Section 4 in [9] we r ⊗ 0 · 0 know that limiting manifolds of odd complex dimension m are Einstein. Moreover, m 2 m+1 there is a holomorphic eigenspinor ψ Γ(S +1 ) to the first eigenvalue λ1 = 4m R ∈ 2 −1 m 2 0, +1 of D . Thus, α m+1 ψ is a holomorphic section of Λ 2 S0. Conversely, 2 ◦ ⊗ m 2m 0, +1 let M be Einstein  and let ϕ 0 be a holomorphic section of Λ 2 S . Then 6≡ ⊗ 0 ψ := α m+1 ϕ Γ(S m+1 ) is holomorphic. Hence, using Corollary 3, the Einstein 2 ◦ ∈ 2 condition ρ = R Ω and Ωψ = i(m 2 m+1 )ψ = iψ we have 2m − · 2 − R i R R m + 1 D2ψ = ψ + ρψ = ψ + ψ = Rψ. 4 2 4 4m 4m Thus, M 2m is a limiting manifold. ✷

The corresponding holomorphic characterization of limiting manifolds in case of even complex dimension m is more complicated since such manifolds are not Einstein for m 4. In case m = 2 the complete list of limiting manifolds is given by S2 S2 ≥ × and S2 T 2 (see [2]). In case m = 2l 4 it is known that the scalar curvature R × ≥ is a (positive) constant and that there is a holomorphic eigenspinor ψ Γ(S m ) to 2 2 m 2 ∈ the first eigenvalue λ1 = 4(m−1) R of D which additionally satisfies the eigenvalue equation

R (22) ρ ψ = i ψ · − 2m 2 −

5 (see [9], Section 4). A previous result is

Proposition 5: Let M 2m be a closed spin K¨ahler manifold of even complex di- mension m 4 with positive scalar curvature R. Then M 2m is a limiting manifold ≥ iff R is constant and S m admits a holomorphic section satisfying equation (22). 2

Proof: Let 0 ψ Γ(S m ) be holomorphic and also a solution of equation (22) and 6≡ ∈ 2 let R> 0 be constant. Then, by Corollary 3, it holds R i R R m D2ψ = ψ + ρψ = ψ + Rψ = Rψ. 4 2 4 4(m 1) 4(m 1) − − Hence, M 2m is a limiting manifold. The converse is true by the preceeding remarks. ✷

In the following we replace the condition (22) by a more geometrical one. We use the result of A. Moroianu that the Ricci tensor of a limiting manifold M 2m with m = 2l 4 has exactly the two constant eigenvalues R and 0 of multiplicity ≥ 2m−2 2m 2 and 2, respectively (see [16]). This property is called the condition (Ric). − Lemma 6: Let M 2m be a spin K¨ahler manifold satisfying the condition (Ric) and let r 1,...,m 1 . Then there is an orthogonal splitting ∈ { − } (23) S = S S r r,0 ⊕ r,1 m 1 into complex subbundles with rankC(Sr,ε) = − (ε = 0, 1) such that iρ r ε − − ! acts on S as multiplication by R (m 1 2(r ε)). r,ε 2m−2 · − − − Proof: Let P M 2m be any point. The Ricci form ρ defines an endomorphism ∈ ρ : S S being compatible with the splitting S = (S ) (S ) . P P → P P 0 P ⊕···⊕ m P If we look at the spin representation, then there is an identification such that

m m (24) Ω = e − e , ρ = ρ (P )e − e , P 2k 1 · 2k P k 2k 1 · 2k kX=1 kX=1 where ρ (P ), . . . , ρ (P ) are the eigenvalues of the Ricci tensor at P and (e , , e ) 1 m 1 · · · 2m is the canonical of R2m corresponding to a suitable orthonormal frame of T M 2m. Let u ε , , ε 1, 1 be the standard basis of the spin mod- P { ε1...εm | 1 · · · m ∈ { − }} ule. Then there are the relations

(25) e − e u = iε u (k = 1, ,m). 2k 1 · 2k · ε1...εm k · ε1...εm · · · This yields

m

(26) ρP uε1...εm = i εkρk(P ) uε1...εm . · ! · kX=1 In case of condition (Ric) we can assume that R ρ (P )= = ρ − (P )= , ρ (P ) = 0. Then (26) implies 1 · · · m 1 2m−2 m

6 − R m 1 ρP uε1...εm = i εk uε1...εm . · 2m 2 ! · − kX=1 C Now, (Sr)P corresponds to the over spanned by all uε1...εm for which exactly r of the ε are equal to 1. Hence, corresponding to the two possibilities k − ε = 1 and ε = 1 the endomorphisms ρ restricted to (S ) has exactly the two m m − P r P R R m 1 eigenvalues i 2m−2 (m 1 2r) and i 2m−2 (m 1 2(r 1)) of multiplicity − − − − − − r ! m 1 and − , respectively. ✷ r 1 − ! Lemma 6 shows that in case of condition (Ric) a spinor ψ Γ(S m ) satisfies the ∈ 2 equation (22) iff ψ is a section of the subbundle S m of S m . Now we will see how 2 ,0 2 this subbundle can be constructed with help of the Ricci tensor. The condition (Ric) provides an orthogonal J- decomposition

(27) T M 2m = E F, ⊕ where the fibres of the subbundles E and F at any point P M 2m are given ∈ by E := ker(Ric R ) and F := ker(Ric ), respectively. This implies the P P − 2m−2 P P decomposition

(28) T 0,1∗M 2m = E0,1∗ F 0,1∗ ⊕ 0,1∗ 0,1∗ with rankC(E )= m 1 and rankC(F ) = 1. Thus, we have −

Λ0,r : = Λr(T 0,1∗M 2m) = Λr(E0,1∗ F 0,1∗)= ⊕ = (ΛrE0,1∗) ((Λr−1E0,1∗) F 0,1∗). ⊕ ⊗ Hence, using the isomorphisms (21) we obtain

∗ − ∗ ∗ (29) (ΛrE0,1 ) S (Λr 1E0,1 ) F 0,1 S = S . ⊗ 0 ⊕ ⊗ ⊗ 0 ∼ r By construction, it holds

r 0,1∗ m 1 rankC((Λ E ) S0)= − , ⊗ r ! (30)

r−1 0,1∗ 0,1∗ m 1 rankC((Λ E ) F S )= − . ⊗ ⊗ 0 r 1 − ! Lemma 7: Let r 1,...,m 1 . Then the isomorphism (29) induces isomor- ∈ { − } phisms

(ΛrE0,1∗) S = S , ⊗ 0 ∼ r,0 (31) (Λr−1E0,1∗) F 0,1∗ S = S . ⊗ ⊗ 0 ∼ r,1

7 Proof: By Lemma 6 and (30), it is sufficient to show that α ((ΛrE0,1∗) S )= S . r ⊗ 0 r,0 Hence, we have to prove that ρ is the multiplication by i R (m 1 2r) on 2m−2 − − α ((ΛrE0,1∗) S ). For any local frame (X ,...,X ),n := 2m, the Ricci form ρ r ⊗ 0 1 n acts on S by 1 (32) ρ = J(Xk) Ric(X ) 2 · k

(see [9], (54)). Now, let (X1,...,Xn) be orthonormal such that

(33) J(X2k)= X2k−1 (k = 1,...,m),

R (34) Ric(X )= X (k = 1,...,m 1), Ric(X ) = 0. 2k 2m 2 2k − n − Then we have

1 n (35) Ω= J(X ) X 2 k · k kX=1 and, moreover, by (32) and (34),

− R n 2 (36) ρ = J(X ) X . 2(n 2) · k · k − kX=1 Let us consider the 2-form η := Ω n−2 ρ. Then, by (33), (35) and (36), we obtain − R 1 1 η = (J(X − ) X − + J(X ) X )= ( X X − + X − X )= X − X . 2 n 1 · n 1 n · n 2 − n · n 1 n 1 · n n 1 · n

Using this we determine the action of ηP on the space

α ((ΛrE0,1∗) S ) = ω ψ ω (ΛrE0,1∗) , ψ (S ) r ⊗ 0 P { · 0| ∈ P 0 ∈ 0 P } ∗ for any P M 2m. We remark that ω (ΛrE0,1 ) implies X − yω = X yω = 0. ∈ ∈ p n 1 n Hence, using (25) and the properties of Clifford multiplication we calculate

r η (ω ψ )= X − X (ω ψ ) = ( 1) X − (ω (X ψ )) = · · 0 n 1 · n · · 0 − n 1 · · n · 0

r r = ( 1) ( 1) ω (X − X ψ )= iω ψ . − · − · n 1 · n · 0 · 0 Thus, η is the multiplication by i on α (ΛrE0,1∗ S ). This proves the assertion r ⊗ 0 since ρ = R (Ω η) and Ω is the multiplication by i(m 2r) on S . ✷ n−2 − − r

Let M 2m be a spin K¨ahler manifold of even complex dimension m satisfying the condition (Ric). Then the subbundle

m m 0,1∗ 0, (Λ 2 E ) S Λ 2 S ⊗ 0 ⊂ ⊗ 0 constructed with help of the Ricci tensor only is not holomorphic in general. (Clearly, it is holomorphic if Ric is parallel). But also in case of a limiting manifold we can

8 not say up to now whether this bundle must be holomorphic or not. By Proposition 5, Lemma 6 and Lemma 7 we immediately obtain

Theorem 8: Let M 2m be a closed spin K¨ahler manifold of even complex dimension m 4 and positive scalar curvature. Then M 2m is a limiting manifold iff M 2m m ≥ 0, satisfies the condition (Ric) and Λ 2 S admits a holomorphic section which si- 0 m ⊗ 0,1∗ multaneously is a section of the subbundle (Λ 2 E ) S . ⊗ 0

We remark that the conditon (Ric) holds obviously if M 2m = N 2m−2 T 2, where − × N 2m 2 is a limiting manifold of odd complex dimension m 1 (see [9], Section 5). In − these cases the Ricci tensor is always parallel. Hence, if the theorem of Lichnerowicz is valid, then the Ricci tensor must be parallel for each limiting manifold of even complex dimension m 4. But Theorem 8 suggests the that there are ≥ examples with non-parallel Ricci tensor.

3 Holomorphic spinors on Einstein manifolds.

Let j : S S be the j-structure of the spinor bundle S which always exists in even → real . We recall that j is parallel and anti-linear, preserves the of spinors, commutes with Clifford multiplication by real vectors and has the properties

(37) jSr = Sm−r (r = 0, 1,...,m),

m m 2 ( +1) (38) j = ( 1) 2 . − Then we have obviously

(39) j ker( 1,0) = ker( 0,1) , j ker( 0,1) = ker( 1,0). ∇ ∇ ∇ ∇ 2 2 Furthermore, let λ be any eigenvalue of D and let Eλ(D ) denote the corresponding eigenspace. Then the relation

2 2 (40) jEλ(D )= Eλ¯(D ) is also obvious. E1,0(D2) := E (D2) ker( 0,1) E0,1(D2) := E (D2) ker( 1,0) λ λ ∩ ∇ λ λ ∩ ∇ is the corresponding holomorphic (antiholomorphic) eigenspace. Moreover, we say that λ is holomorphic (antiholomorphic) iff E1,0(D2) = 0 (E0,1(D2) = 0). From λ 6 λ 6 (39) and (40) we see that

1,0 2 0,1 2 0,1 2 1,0 2 (41) jEλ (D )= Eλ¯ (D ) , jEλ (D )= Eλ¯ (D ). Thus, λ is holomorphic iff λ¯ is antiholomorphic.

Proposition 9: Let M 2m be any spin K¨ahler-Einstein manifold. Then the sets of holomorphic eigenvalues of D2 coincides with the set of antiholomorphic eigenvalues and is contained in rR r = 0, 1,...,m . Moreover, it holds { 2m | }

9 1,0 2 E rR (D ) Γ(Sr), 2m ⊆ (42) (r = 0, 1,...,m) 0,1 2 E rR (D ) Γ(Sm−r). 2m ⊆

Proof: Let λ be any holomorphic eigenvalue of D2. Moreover, let 0 ψ E1,0(D2) 6≡ ∈ λ and ψ = ψ + ψ + + ψ the decomposition according to (1). Using the equation 0 1 · · · m (19) and the Einstein condition

R (43) ρ = Ω 2m we have R i R R 0= λψ D2ψ = λψ ψ ρψ = λ ψ iΩψ = 4 2 4 4m − − −  −  −

R R m m sR = λ ψ + (m 2s)ψ = λ ψ − 4 4m − s − 2m s   sX=0 sX=0   and hence (λ sR )ψ = 0 for s = 0, 1,...,m. Since ψ 0 there is an r with ψ 0. − 2m s 6≡ r 6≡ This implies λ = rR and ψ = 0 for s = r. ✷ 2m s 6

Proposition 10: Let M 2m be a spin K¨ahler-Einstein manifold, let r 0, 1,...,m ∈ { } and let ψ Γ(S ) be holomorphic (antiholomorphic). Then ψ satisfies the eigenvalue ∈ r equation

rR (m r)R (44) D2ψ = ψ (D2ψ = − ψ). 2m 2m Proof: For example, let ψ Γ(S ) be holomorphic. Then the equations (19), (43) ∈ r and Ωψ = i(m 2r)ψ imply (44). ✷. −

Theorem 11: If M 2m is a spin K¨ahler-Einstein manifold of scalar curvature R, then there are the decompositions

m 0,1 1,0 2 ker( )= E rR (D ), ∇ r=0 2m (45) L m 1,0 0,1 2 ker( )= E rR (D ). ∇ r=0 2m L Proof: Let ψ ker( 0,1) and ψ = ψ + ψ + + ψ the decomposition according ∈ ∇ 0 1 · · · m to (1). Since the splitting (1) is parallel, ψ is holomorphic iff each of its components 1,0 2 ψr is holomorphic. Thus, Proposition 10 implies ψr E rR (D ). ✷ ∈ 2m

From Proposition 10 and the structure (45) of the of holomorphic or anti- holomorphic spinors we immediately obtain

10 Theorem 12: Let M 2m be a spin K¨ahler-Einstein manifold and R its scalar curva- ture. Then the following holds:

(i) If rR is not an eigenvalue of D2 for an r 0, 1,...,m , then there is no 2m ∈ { } holomorphic section in the bundle Sr and no antiholomorphic section in Sm−r. (ii) If rR is not an eigenvalue of D2 for each r 0, 1,...,m , then there are no 2m ∈ { } holomorphic and no antiholomorphic spinors on M 2m.

4 Holomorphic spinors on closed manifolds.

Let Z,W be any complex vector fields on a spin K¨ahler manifold M 2m. Then we use the notation

:= ∇ ∇Z,W ∇Z∇W −∇ ZW for the corresponding second order derivative of spinors. We consider the K¨ahler- Bochner Laplacians on Γ(S) locally defined by

1,0∗ 1,0 := gkl , ∇ ∇ − ∇p¯(Xk),p(Xl) (46) 0,1∗ 0,1 := gkl . ∇ ∇ − ∇p(Xk),p¯(Xl) By definition, we have the inclusions

(47) ker( 1,0) ker( 1,0∗ 1,0) , ker( 0,1) ker( 0,1∗ 0,1). ∇ ⊆ ∇ ∇ ∇ ⊆ ∇ ∇ Proposition 13: There are the operator equations

2 1,0∗ 1,0 = D2 R + i ρ, ∇ ∇ − 4 2 (48) 2 0,1∗ 0,1 = D2 R i ρ. ∇ ∇ − 4 − 2 Proof: For example, we prove the first one of these equations. Let P M 2m be any ∈ point and (X ,...,X ) a frame in a neighbourhood of P with property ( X ) = 0 1 2m ∇ k P for k = 1,..., 2m. Using the formulas (5), (8), (13) and (16) we have at the point P

2 1,0∗ 1,0 = 2gkl = (XkXl + XlXk) = ∇ ∇ − ∇p¯(Xk)∇p(Xl) ∇p¯(Xk)∇p(Xl) l k = D D− + X X ( C(p(X ), p¯(X ))) = + ∇p(Xl)∇p¯(Xk) − l k l k = D D− + D−D X X C(p(X ), p¯(X )) = + + − l k 1 = D2 + Xlp(Xk)C(X ,p(X )) = D2 + Xlp(Ric(X )) = k l 2 l R i = D2 + ρ. − 4 2 ✷

In 1979 M.L. Michelsohn proved Weitzenb¨ock formulas which are very similar to (48). M.L. Michelsohn also showed that 0,1∗ 0,1 and 1,0∗ 1,0 are non-negative, ∇ ∇ ∇ ∇ elliptic and formally self-adjoint differential operators (see [14], Prop. 7.2, 7.6). We

11 prefer the equations (48) since the Ricci form ρ enters here explicitely being more convenient for applications considered here. For example, we remark that Corollary 3 also follows from (47) and (48) immediately.

Theorem14: Let M 2m be a closed spin K¨ahler manifold. Then a spinor ψ is holo- morphic (antiholomorphic) iff ψ satisfies equation (19) or (20).

Proof: In closed case the inclusions (47) are equalities (see [14], Prop. 7.2). ✷

Theorem 15: Let M 2m be a closed spin K¨ahler-Einstein manifold of scalar curvature R. Then we have the following:

(i) For any r 0, 1,...,m , ψ Γ(S ) is holomorphic (antiholomorphic) iff ψ ∈ { } ∈ r satisfies the eigenvalue equation (44).

(ii) In case R > 0 the bundles S with r m/2 (r m/2) do not admit any r ≤ ≥ holomorphic (antiholomorphic) section. In case R = 0 each holomorphic or antiholomorphic spinor is parallel.

0,1 1,0 2 1,0 0,1 2 (iii) It holds ker( )= E rR (D ) and ker( )= E rR (D ) . ∇ r>m/2 2m ∇ r>m/2 2m L L Proof: In Einstein case the equations (19) and (44) are equaivalent for ψ Γ(S ). ∈ r Thus, assertion (i) follows from Theorem 14 immediately. Moreover, using the equa- tion (20) and the Einstein condition (43) we see that ψ Γ(S ) is holomorphic ∈ r (antiholomorphic) iff ψ satisfies the equation

− ∗ m 2r ψ =(+) − Rψ ∇ ∇ 4m which implies

− ∗ m 2r ψ, ψ =(+) − R ψ 2. h∇ ∇ i 4m | | Integrating this equation we find

− m 2r ψ 2 =(+) − R ψ 2. ||∇ || 4m || || Let R > 0 and ψ 0. Then we obtain a contradiction for m > 2r (m < 2r). The 6≡ case m = 2r provides ψ = 0. But it is known that the existence of a parallel ∇ spinor implies Ric = 0 being a contradiction to our assumtion R > 0. Finally, in case R = 0 we always obtain ψ = 0. This proves (ii). The assertion (iii) follows ∇ from (ii) and Theorem 11. ✷

An essential generalization of Theorem 15, (ii) is

Theorem 16: Let M 2m be a closed non-Ricci-flat spin K¨ahler manifold and let ρ (P ) ρ (P ) ρ (P ) denote the eigenvalues of Ric at P M 2m. Then 1 ≥ 2 ≥ ··· ≥ m ∈ the bundle S (r 0, 1,...,m ) does not admit any holomorphic (antiholomorphic) r ∈ { } section if at each point P the condition

12 1 1 (49) ρ (P )+ + ρ (P ) R(P ) (ρ − (P )+ + ρ (P ) R(P )) 1 · · · r ≤ 4 m r+1 · · · m ≥ 4 is satisfied.

Proof: Since ρ (P )+ + ρ (P ) = 1 R(P ), we see from (26) that the set of all 1 · · · m 2 eigenvalues of the endomorphism iρ restricted to (S ) is given by − P r P 1 (50) R(P ) 2(ρi1 (P )+ + ρir (P )) 1 i1 < < ir m . 2 − · · · | ≤ · · · ≤  Thus, for any ψ Γ(S ), the condition (49) yields iρψ, ψ 0 ( iρψ, ψ 0). ∈ r h− i ≥ h− i ≤ On the other hand, if ψ Γ(S ) is holomorphic (antiholomorphic), then (20) implies ∈ r the equation

− 1 ψ 2 =(+) iρ ψ, ψ ||∇ || 2 2m h− · i ZM which provides a contradiction for iρ ψ, ψ 0 ( iρ ψ, ψ 0). ✷ h− · i ≥ h− · i ≤

We remark that the assertion of Theorem 16 is also valid if the condition (49) is not satisfied on a subset of M 2m of measure zero. Moreover, in case r = 0 the condition (49) simply reduces to R 0 (R 0). Hence, if M 2m is closed and non-Ricci-flat, ≥ ≤ the bundle S0 (Sm) does not admit any holomorphic (antiholomorphic) section in case R 0 (R 0). Theorem 16 immediately yields ≥ ≤ Corollary 17: Let M 2m be a closed non-Ricci-flat spin K¨ahler manifold of complex dimension m 3 with scalar curvature R 0 such that at each point P M 2m ≥ ≥ ∈ the Ricci tensor has the eigenvalues 1 R(P ) and 0 of multiplicity 2m 2 and 2m−2 − 0, respectively. Then the bundles S with r m−1 (r m+1 ) do not admit any r ≤ 2 ≥ 2 holomorphic (antiholomorphic) section.

By Theorem 8 and Corollary 17, we obtain

Proposition 18: If M 2m is a limiting manifold of even complex dimension m 4, − ≥ then the bundles S with r m 2 (r m+2 ) do not admit any holomorphic (anti- r ≤ 2 ≥ 2 holomorphic) section.

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