K ¨AHLER C-SPACES AND QUADRATIC BISECTIONAL CURVATURE Albert Chau & Luen-Fai Tam Abstract

94(2013) 409-468

K ¨AHLER C-SPACES AND QUADRATIC BISECTIONAL CURVATURE

Albert Chau & Luen-Fai Tam Abstract

In this article we give necessary and sufficient conditions for an irreducible K¨ahler C-space with b2 = 1 to have nonnegative or positive quadratic bisectional curvature, assuming the space is not Hermitian symmetric. In the cases of the five exceptional Lie groupsE6, E7, E8, F4, G2, the computer package MAPLE is used to assist our calculations. The results are related to two conjectures of Li-Wu-Zheng.

1. Introduction

Let (Mⁿ, g) be a K¨ahler manifold of complex dimension n and let o ∈M. M is said to have nonnegative quadratic orthogonal bisectional curvature at o if for any unitary frame e_i at o and real numbers ξ_i we have

(1.1) X

i,j

R_i¯ij¯j(ξⁱ−ξ^j)² ≥0.

Here R_i¯ij¯j =R(e_i,¯e_i, e_j,e¯_j). Recall that M is said to have nonnegative bisectional curvature atoif for any X, Y ∈To^(1,0)(M),R(X,X, Y,¯ Y¯)≥ 0, andM is said to have nonnegativeorthogonalbisectional curvature at oif R(X,X, Y,¯ Y¯)≥0 for all unitary pairsX, Y ∈T_o^(1,0)(M). Following [16] we abbreviate by QB ≥ 0 for nonnegative quadratic orthogonal bisectional curvature, B≥0 for nonnegative bisectional curvature, and B^⊥≥0 for nonnegative orthogonal bisectional curvature. It is obvious that B ≥0 ⇒B^⊥ ≥0⇒ QB ≥0. Note that in dimension n= 2, the conditionsB^⊥≥0 and QB≥0 are the same.

It is well-known that compact manifolds with B≥0 have been com- pletely classified by the works [18, 20,14, 1, 17]. By these works, we know that any compact simply connected irreducible K¨ahler manifold with B ≥ 0 is either biholomorphic to CPⁿ or is isometrically biholo- morphic to an irreducible compact Hermitian symmetric space of rank at least 2. While the condition B^⊥ ≥0 seems weaker, by the works of Chen [10] (see also [22]) and Gu-Zhang [13] we know that a compact

Received 3/23/2012.

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simply connected irreducible K¨ahler manifold withB^⊥≥0 is also either biholomorphic toCPⁿor is isometrically biholomorphic to an irreducible compact Hermitian symmetric space of rank at least 2. In this sense, no new compact complex manifolds are introduced when we weaken the condition B ≥0 to the condition B^⊥≥0.

The condition QB ≥ 0 was first considered by Wu-Yau-Zheng [24]

where they proved that on a compact K¨ahler manifold with QB ≥ 0 any class in the boundary of the K¨ahler cone can be represented by a smooth closed (1,1) form which is everywhere nonnegative. There are other interesting properties satisfied by compact K¨ahler manifolds withQB ≥0. A fundamental property of such manifolds, implicit from earlier works [3] (see [8] for additional references) is that all harmonic real (1,1) forms are parallel. Recently it has been proved in [8] that the scalar curvature of such a manifold must be nonnegative, and if the manifold is irreducible, then the first Chern class is positive.

The ultimate goal is to classify K¨ahler manifolds with QB ≥0. For the compact case, a partial classification of the de Rham factors of the universal cover of such a manifold is given in [8]. Hence it remains to study the structure of compact simply connected irreducible K¨ahler manifolds with QB ≥ 0. By the parallelness of real harmonic (1,1) forms mentioned above, such K¨ahler manifolds also have b₂ = 1 (see [14]). In view of the above results for B^⊥ ≥0, one may wonder if any new compact complex manifolds are introduced when we weaken the condition B^⊥ ≥ 0 to the condition QB ≥ 0. To address this, Li, Wu, and Zheng [16] constructed the first example of a simply connected irreducible compact K¨ahler manifold having QB ≥ 0, which does not support a K¨ahler metric having B^⊥ ≥ 0. Their example is (B₃, α₂), a classical K¨ahlerC-space with second Betti numberb₂= 1. It was further conjectured that all K¨ahler C-spaces with second Betti number b₂ = 1 must have QB≥0, and the following conjectures were raised in [16]:

Conjecture 1.1. (1) Any K¨ahler C-space with b₂ = 1 satisfies QB ≥0 everywhere.

(2) A compact simply connected irreducible K¨ahler manifold (Mⁿ, g) with QB≥0 is biholomorphic to a K¨ahler C-space with b₂= 1.

(3) In (2), if the manifold is not CPⁿ, then g is a constant multiple of the standard metric.

A K¨ahler C-space is a compact simply connected K¨ahler manifold such that the group of holomorphic isometries acts transitively on the manifold; see [21, 15]. There is a complete classification of K¨ahler C- spaces withb2 = 1, and this is associated with the classification of sim- ple complex Lie algebras which are just A_n=sl_n+1, B_n =so_2n+1, C_n= sp_2n, D_n=so_2nand the exceptional casesE₆, E₇, E₈, F₄, G₂. Motivated by the work [16], we establish the following theorems related to conjec- tures (1) and (3). For the classical types we have the following:

Theorem 1.1.

(i) The K¨ahler C-space (B_n, α_p), n≥3, 1< p < n satisfies QB ≥0 if and only if 5p + 1 ≤ 4n. Moreover, QB > 0 if and only if 5p+ 1<4n.

(ii) The K¨ahler C-space (C_n, α_p), n≥3, 1< p < n satisfies QB ≥0 if and only if 5p ≤ 4n+ 3. Moreover, QB > 0 if and only if 5p <4n+ 3.

(iii) The K¨ahler C-space (D_n, α_p), n ≥ 4, 1 < p < n−1 satisfies QB ≥0 if and only if5p+ 3≤4n. Moreover, QB >0 if and only if 5p+ 3<4n.

For the exceptional cases, we have the following:

Theorem 1.2.

(i) The K¨ahler C-space (G2, α2) satisfies QB >0.

(ii) The K¨ahler C-space (F₄, α_p), 1 ≤ p ≤ 4 satisfies QB ≥ 0 iff p= 1,2,4, in which cases QB >0.

(iii) The K¨ahler C-space (E₆, α_p), 2 ≤ p ≤ 5 satisfies QB ≥ 0 iff p= 2,3,5, in which cases QB >0.

(iv) The K¨ahler C-space (E7, αp), 1 ≤ p ≤ 6 satisfies QB ≥ 0 iff p= 1,2,5, in which cases QB >0.

(v) The K¨ahler C-space (E8, αp), 1 ≤ p ≤ 8 satisfies QB ≥ 0 iff p= 1,2,8, in which cases QB >0.

We only consider K¨ahlerC-spaces that are not Hermitian symmetric.

According to Itoh [15], Theorem 1.1 and 1.2 include all such K¨ahlerC- spaces withb₂ = 1. Here QB >0 means that (1.1) is a strict inequality unless allξ_iare the same. Note that ifQB >0, then a small perturbation of the K¨ahler metric will still satisfy QB > 0; see Lemma 2.6 (and Remark 2.1). Hence conjecture (1) for the classical types is true only under some restrictions mentioned in Theorem 1.1, while conjecture (3) is too strong. Conjecture (2), however, may still be true in general.

Theorems 1.1 and 1.2 give more information on the curvature prop- erties of K¨ahler C-spaces with b2 = 1. It is well-known that CPⁿ has B >0, and Hermitian symmetric spaces with rank at least 2 haveB≥0 but notB >0. All other K¨ahlerC-spaces which are non-Hermitian sym- metric spaces do not have B ≥0 or even B^⊥≥ 0. On the other hand, Itoh [15] proved that a K¨ahler C-space with b₂ = 1 is a Hermitian symmetric space if and only if its curvature operator has at most two distinct eigenvalues. Our results show that as far as the sign of curva- ture is concerned, K¨ahlerC-spaces withb₂ = 1 which are not Hermitian symmetric are further divided into two groups: some of them satisfy QB ≥0 and others do not have such a property.

We give here an idea of the proof and refer to§2 for details. Consider a Lie algebra as above, and a corresponding system ∆⊂Rⁿof root vec- tors in Rⁿ where the induced Killing form is induced by the standard Euclidean inner product. Then each associated K¨ahler C-space corre- sponds to a certain subset of m⁺ ⊂∆ representing a unitary frame in which curvature approximation reduces to taking sums and inner prod- ucts of the vectors inm⁺⊂∆ (we can calculate exact values in the case of bisectional curvatures). We combine this with symmetry, counting, and eigenvalue estimate arguments to obtain Theorem 1.1. For the five exceptional cases in Theorem 1.2, the computer package MAPLE was used to assist our calculations and details are provided in the appendix.

Theorem 1.1 was proved in an earlier version of this article [9], and while similar in spirit, our proof here is somewhat simpler, eliminating the need for many calculations from the appendix of [9].

The organization of the paper is as follows. In§2 we will state basic properties and formulae for K¨ahlerC-spaces that will be used through- out the paper. We will discuss the conditions QB ≥0 and QB >0 in general, then in relation to the K¨ahlerC-spaces. In§3 we prove Theorem 1.1 for the classical K¨ahlerC-spaces; details for some of the calculations in these sections can be found in the appendices of [9], which is an ear- lier version of this article. In§4 we present the details of our results on Theorem 1.2 for the exceptional K¨ahler C-spaces, with details of our use of MAPLE provided in the appendix.

Acknowledgments.The authors would like to thank F. Zheng for valuable comments and interest in this work.

The first author’s research was partially supported by NSERC grant no. #327637-11. The second author’s research was partially supported by Hong Kong RGC General Research Fund #CUHK 403011.

2. Basic facts

2.1. The K¨ahler C-spaces and curvature formulae.Consider a compact K¨ahler C-space (M, ω) with transitive holomorphic isometry group G, and suppose b₂(M) = 1. Then any real (1,1) form ρ on M is given by ρ = cω +√

−1∂∂f for some constant c and function f where ω is the K¨ahler form. Now if ρ is G invariant, then ∆gf is also G invariant and hence constant on M. Thus f is constant on M and ρ =cω. In particular,g is the unique G invariant K¨ahler metric onM and it is K¨ahler Einstein. For more discussions on K¨ahler C-space, see [2,15,21,16].

K¨ahler C-spaces with second Betti number b2 = 1 are obtained as follows (see [4,5,6,15,16,21]). LetGbe a simply connected, complex Lie group, and let g be its Lie algebra with Cartan subalgebra h and corresponding root system ∆⊂h^∗. Theng=h⊕L

α∈∆CE_α, whereE_α is a root vector of α. Letl= dimChand fix a fundamental root system

α₁, . . . , α_l ⊂∆. This gives an ordering of roots in ∆. Let ∆⁺and ∆⁻be the set of positive and negative roots, respectively. LetK be the Killing form for g. Then we may choose root vectors {E_α}, α∈∆ such that

K(E_α, E_−α) =−1, α∈∆⁺; [E_α, E_β] =n_α,βE_α+β

such that n_α,β = n_−α,−β ∈ R with n_α,β = 0 if α+β is not a root.

Together with a suitable basis in h, they form a Weyl canonical basis forg. Now for any 1≤r≤l, let

∆⁺_r(k) ={X

i

n_iα_i ∈∆⁺|n_r=k}, ∆⁺_r = [

k>0

∆⁺_r(k).

Let P be the subgroup whose Lie algebra is h⊕L

α∈∆\∆⁺r CE_α. Then G/P is a complex homogeneous space having b₂ = 1. Now let

m⁺_k = M

α∈∆⁺r(k)

CEα, m⁻_k = M

α∈∆⁻r(k)

CEα, t=h⊕ M

α∈∆⁺(0)

(CEα⊕CE₋α).

Then m⁺=L

k>0m⁺_k can be identified with the tangent space ofG/P. As given in [4, 15, 16], the G-invariant K¨ahler form on G/P is given by:

Lemma 2.1.

(i) In a Weyl canonical basis, letω^α,ω^α¯ be the dual of Eα andE¯α :=

−E_−α, α∈∆⁺_r. The Ginvariant K¨ahler form on G/P is g= 2X

k>0

k X

α∈∆⁺r(k)

ω^α·ω^α¯ =X

k>0

(−kK)|_m⁺

k×m⁻_k.

(ii) [t,m^±_k]⊂m^±_k, [m^±_k,m^±_l ]⊂m^±_k+l, [m⁺_k,m⁻_k]⊂ t. If k > l > 0, then [m⁺_k,m⁻_l ]⊂m⁺_k−l, [m⁻_k,m⁺_l ]⊂m⁻_k−l.

The K¨ahler C space thus obtained is denoted as (g, α_r). Conversely, every K¨ahler C space with b₂= 1 can be obtained by the construction.

Thus the set {e_α := 1/√

kE_α}; α ∈ ∆⁺_k, k ≥ 1 forms a unitary basis for the tangent space of (g, αr) in the metricg. We call this basis as a Weyl frame. To compute the curvature tensor in this frame, we have the following from Li-Wu-Zheng [16, Proposition 2.1], using the method in [15]. For the sake of completeness, we give a proof.

Proposition 2.1. [Li-Wu-Zheng] Let Xⁱ ∈ m⁺_i , Y^j ∈ m⁺_j , Z^k ∈ m⁺_k, W^l ∈m⁺_l . Suppose i+k=j+l. Then

R(Xⁱ,Y¯^j,Z^k,W¯^l) =

(k−j)ξ_k−j− kl i+k

K([Xⁱ, Z^k],[ ¯Y^j,W¯^l])

+ (−(k−j)ξ_k−j+kξ_i−j+lξ_j−i+lδ_ijδ_kl)K([Xⁱ,Y¯^j],[Z^k,W¯^l]).

(2.1)

R(Xⁱ,Y¯^j, Z^k,W¯_l) = 0 if i+k6=j+l.

Here ξ_q= 1 if q >0 andξ_q= 0 if q≤0.

Proof. Note that g(U,V¯) =−kK(U,V¯), onm⁺_k ×m⁻_k etc.

Then [15, p. 43]

R(Xⁱ,Y¯^j, Z^k,W¯_l) =g(R(Xⁱ,Y¯^j)Z^k,W¯^l)

=g([Λ(Xⁱ),Λ( ¯Y^j)]Z^k,W¯^l)−g(Λ([Xⁱ,Y¯^j]_m)Z^k,W¯^l)

−g([[Xⁱ,Y¯^j]_t, Z^k],W¯^l), (2.2)

where Λ(U)V =n^′/(n+n^′)[U, V]_m⁺ if U ∈m⁺_n, V ∈m⁺_n′, and Λ( ¯U)V = [ ¯U , V]_m+,for all U, V ∈ m⁺; see [15, p. 45]. Here [U, V]_m+ is the com- ponent of [U, V] in m⁺. Now ifi+k=j+l,

[Λ(Xⁱ),Λ( ¯Y^j)]Z^k= Λ(Xⁱ)Λ( ¯Y^j)−Λ( ¯Y^j)Λ(Xⁱ) Z^k

=Λ(Xⁱ)([ ¯Y^j, Z^k]_m⁺)− k

i+kΛ( ¯Y^j)([Xⁱ, Z^k])

=(k−j)

l ξ_k−j[Xⁱ,[ ¯Y^j, Z^k]]− k

i+k[ ¯Y^j,[Xⁱ, Z^k]]_m⁺. (2.3)

Here each term is inm⁺_l . Hence g([Λ(Xⁱ),Λ( ¯Y^j)]Z^k,W¯^l)

=−(k−j)ξ_k−jK([Xⁱ,[ ¯Y^j, Z^k]],W¯^l) + kl

i+kK([ ¯Y^j,[Xⁱ, Z^k]],W¯^l)

=−(k−j)ξ_k−jK(Xⁱ,[[ ¯Y^j, Z^k],W¯^l]) + kl

i+kK([ ¯Y^j,[Xⁱ, Z^k]],W¯^l)

= (k−j)ξ_k−jK(Xⁱ,[[ ¯W^l,Y¯^j], Z^k] + [[Z^k,W¯^l]],Y¯^j) + kl

i+kK([ ¯W^l,Y¯^j],[Xⁱ, Z^k]])

=−(k−j)ξ_k−jK([Xⁱ,Y¯^j],[Z^k,W¯^l]) +

(k−j)ξ_k−j− kl i+k

K([Xⁱ, Z^k],[ ¯Y^j,W¯^l]).

(2.4)

Now [Xⁱ,Y¯^j]m is inm⁺_i−j if i > j, andm⁻_j−i ifj > i, and is 0 ifi=j. So g(Λ([Xⁱ,Y¯^j]m)Z^k,W¯^l) =−(kξi−j+lξj−i)K

[Xⁱ,Y¯^j],[Z^k,W¯^l] . Also, [Xⁱ,Y¯^j]_t = 0 unless i = j. If i = j, then [[Xⁱ,Y¯^j]_t, Z^k] ∈ m⁺_k. Hence

g([[Xⁱ,Y¯^j]t, Z^k],W¯^l) =δ_ijδ_klg([[Xⁱ,Y¯^j], Z^k],W¯^l)

= −lδijδ_klK

[Xⁱ,Y¯^j],[Z^k,W¯^l] .

Also, R(Xⁱ,Y¯^j, Z^k,W¯_l) = 0 if i+k6=j+l. q.e.d.

Lemma 2.2. Same notations as in Proposition2.1. AssumeX, Z, W are canonical Weyl basis vectors and Z 6=W; then

R(X,X, Z,¯ W¯) = 0.

Proof. Since i = j, the lemma is true if k 6= l by Proposition 2.1.

Hence we assumek=l. We first assume thatk≤i. Then R(X,X, Z,¯ W¯) =− k²

i+kK([X, Z],[ ¯X,W¯]) +kK([X,X],¯ [Z,W¯]).

Now letX=E_α,Z =E_β,W =E_γ withβ 6=γ. Note that ¯E_α=−E_−α. Then [X, Z] =n_α,βE_α+β, [X, W] =n_α,γE_α+γ. Hence

K([X, Z],[ ¯X,W¯]) =−n_α,βn_α,γK(E_α+β, E_−α−β).

If α +β or α +γ is not a root, then n_α,β = 0 or n_α,γ = 0 and K([X, Z],[ ¯X,W¯]) = 0. Otherwise, both E_α+β and E_α+γ are canonical Weyl basis vectors and are inm⁺_i+k by Lemma 2.1. Since β6=γ, and K is proportional togonm⁺_i+k×m⁻_i+k, we also haveK([X, Z],[ ¯X,W¯]) = 0.

On the other hand, by the fact that K([x, y], z) = K(z,[y, z]), we have

(2.5) K([X,X],¯ [Z,W¯]) =K(X,[ ¯X,[Z,W¯]]).

Now [Z,W¯] =n_β,−γE_β−γ, [ ¯X,[Z,W¯]] =n_{−α,β−γ}n_β,−γE_{−α+β−γ}. Ifβ−γ or −α+β−γ is not a root, then as before we haveK(X,[ ¯X,[Z,W¯]]) = 0. Otherwise, by Lemma 2.1, [Z,W¯] ∈ t and [ ¯X,[Z,W¯]] ∈ m⁻_i . Since

−α+β−γ6=−α, so as before

K([X,X],¯ [Z,W¯]) =K([X,[ ¯X,[Z,W¯]) = 0.

Hence the lemma is true when k≤i.

Supposei < k. Then it is equivalent to proveR(X,Y , Z,¯ Z¯) = 0, but assumingi > k and X6=Y. In this case,

R(X,Y , Z,¯ Z) =¯ − k²

i+kK([X, Z],[ ¯Y ,Z]) +¯ kK([X,Y¯],[Z,Z¯]).

The previous argument implies the lemma is true in this case as well.

q.e.d.

To use the formula in Proposition 2.1, we need to compute the Lie bracket and Killing form in the given Weyl basis. Now the Killing form K is negative definite onhand thus induces a positive definite bilinear form, denoted also byK, on the dualh^∗. We can then identifyh^∗withR^l (or a subspace of some Rⁿ) so that K becomes the standard Euclidean inner product and the root system is represented by a subset ∆ ⊂R^l. It turns out that a corresponding Weyl basis{E_α}α∈∆⁺ exists in which the Lie bracket and Killing form are computed in terms of Euclidean

inner products and addition of the vectors α. We describe this in more detail below.

Let g be a semi simple Lie algebra, and let ∆ = {α, β, . . .} ⊂ Rⁿ be a corresponding root system with standard inner product (·,·) cor- responding to the induced Killing form K. To the positive roots there corresponds a Chevalley basis {X_α, X_−α, H_α}α∈∆⁺ where for each α, Xα, X₋α are root vectors for α,−α respectively, Hα ∈ h, and the fol- lowing relations are satisfied (see [19, p. 51]):

[X_α, X_−α] =H_α, [X_α, X_β] =

N_α,βX_α+β, if α+β is a root, N_α,β =−N_−α,−β ; 0, if α+β 6= 0 is not a root.

N_α,β =±(p+ 1), pis the largest integer so thatβ−pαis a root.

[H_α, X_β] =β(H_α)X_β. (2.6)

We also have:

(a) β(H_α) = 2^(β,α)_(α,α), [11, p. 337];

(b) K(H_α, H_α) = 2K(X_α, X_−α), [11, p. 207].

By these and [11, p. 207–208], we have the formulas K(H_α, H_−α) = 4

(α, α), K(X_α, X_−α) = 2

(α, α), K(H_α, H_β) = 4(α, β)

|α|²|β|², (2.7)

where in the last equation we have used the first equation of (2.6), (2.5), (a), and the second formula in (2.7).

Lemma 2.3. For positive roots α, let (2.8) E_α = √|α|

2X_α, E_−α =−√|α| 2X_−α. Then for positive roots α, β

K(E_α, E_−α) =−1, [Eα, E_β] =n_α,βE_α+β, [E_−α, E_−β] =n_−α,−βE_−α−β,

[E_α, E_−β] =n_α,−βE_α−β,if α−β 6= 0, (2.9)

where

n_−α,−β =n_α,β =

( √|α||β|

2|α+β|N_α,β, if α+β is a root, 0, if α+β is not a root,

and

n_α,−β =







|α||β|

√2|α−β|N_α,−β, if α−β is a positive root,

−^√|₂^α_|α^||β₋^|_β|N_α,−β, if α−β is a negative root, 0, if α−β is not a root.

Hence {E_α} form a Weyl canonical basis.

Proof. It is easy to see that K(E_α, E_−α) = −1. If α+β is a root, then

[E_α, E_β] =|α||β|

2 [X_α, X_β]

=N_α,β|α||β| 2 X_α+β

= |α||β|

√2|α+β|N_α,βE_α+β =n_α,βE_α+β

[E_−α, E_−β] =|α||β|

2 [X_−α, X_−β]

=N_−α,−β|α||β| 2 X_−α−β

=N_α,β√|α||β|

2|α+β|E_−α−β =n_α,βE_−α−β where n_α,β = √^|α||β|

2|α+β|N_α,β. Here we have used the fact that N_α,β =

−N_−α,−β and X_−α−β = −_|α+β^√2_|E_−α−β. If α +β is not a root, then [E_,a, E_β] = 0.

If α−β 6= 0 and is a positive root, then [E_α, E_−β] = |α||β|

2 [X_α, X_−β] =N_α,−β|α||β| 2 X_α−β

= |α||β|

√2|α−β|N_α,−βE_α−β. If α−β 6= 0 and is a negative root, then

[E_α, E_−β] = |α||β|

2 [X_α, X_−β] =N_α,−β|α||β| 2 X_α−β

=−√|α||β|

2|α−β|N_α,−βE_α−β.

q.e.d.

Now let η ∈ ∆⁺, and consider the K¨ahler C-space (g, η) with cor- responding Weyl frame (unitary frame for (g, η)) e_α = √¹

kE_α for α ∈

∆⁺(k). For any positive roots α, β, define (2.10)

( Ne_α,β= _|^|_α+β^α||β|_|N_α,β;

Ne_α,−β = _|^|_α^α₋^||β_β^|_|N_α,−β, if α−β 6= 0.

We can now combine Lemma 2.2 and Proposition 2.1 with Lemma 2.3, (2.6), and (2.7) to obtain the following from [15, Proposition 2.4].

In the above setting, we denote the curvature tensor R(e_α,¯e_β, e_γ,e¯_δ) also by R(α,β, γ,¯ δ) or¯ R_α,β,γ,¯ ¯δ.

Lemma 2.4. Letα∈∆⁺(i),β ∈∆⁺(j), withi≤j, and letR_α¯αββ¯= R(e_α,e¯_α, e_β,e¯_β). Then

(2.11) R_α¯αββ¯= 1 j

(α, β) +1 2

i i+jNe_α,β²

.

Next let us consider R(α,β, γ,¯ δ) =¯ R(e_α,e¯_β, e_γ,¯e_δ) with α − β, γ−δ6= 0.

Lemma 2.5. Let e_α ∈ ∆⁺(i), e_β ∈ ∆⁺(j), e_γ ∈ ∆⁺(k), and e_δ ∈

∆⁺(l).

1) If α−β6=δ−γ, then R(α,β, γ,¯ δ) = 0.¯ 2) If α−β=δ−γ 6= 0, then

R(α,β, γ,¯ δ¯) =− 1 2√

ijkl

(k−j)ξ_k−j− kl i+k

Neα,γNe_β,δ

+ 1

2√ h ijkl

(−(k−j)ξ_k−j+kξ_i−j+lξ_j−i+lδ_ijδ_kl)Ne_α,−βNe_γ,−δ.i

=:R₁(α,β, γ,¯ δ) +¯ R₂(α,β, γ,¯ δ).¯ (2.12)

Proof. (1) follows from Lemma 2.2, and the fact thatK(E_α, E_β) = 0 unless α+β = 0.

(2) Note that ¯e_α =−e_−α, etc. First assume thatα+γ and α−β are both roots. By Lemma 2.3,

[eα, eγ] = 1

√ik[Eα, Eγ]

=n_α,γ 1

√ikE_α+γ

=N_α,γ|α||γ|

√2ikE_α+γ. Similarly,

[e_−β, e_−δ] =N_β,δ|√γ||δ|

2jlE_−β−δ,

We may assume that α−β is a positive root, thenγ−δ is a negative root.

[e_α, e_−β] =N_α,−β|√α||β| 2ijE_α−β, [e_γ, e_−δ] =−N_γ,−δ|γ||δ|

√2klE_γ−δ.

Since α+γ =β+δ and K(E_σ, E_−σ) =−1, we see that (2) is true by Proposition 2.1. The cases where α+γ or α−β is not a root can be proved similarly.

q.e.d.

2.2. The condition QB ≥ 0.We first discuss the conditionQB ≥ 0 on a K¨ahler manifold (M, ω) with K¨ahler formω. We will also consider the conditionQB >0 atp, which we define as:QB ≥0 atpwith strict inequality in (1.1) provided not all ξ_i^′s are the same. Now define the following bilinear forms on the space Ω^1,1_R (M) of real (1,1) forms on M:

F(η, σ) = X

i,j,k,l

R_i¯jk¯lρ^i¯lσ^k¯j = X

i,j,k,l

R_i¯lk¯jρ^i¯lσ^k¯j

G(η, σ) = 1

2(R_i¯jg_k¯l+R_k¯lg_i¯j)ρ^i¯lσ^k¯j

whereρ^i¯l, σ^k¯j are the local components ofρ, σwith indices raised. Clearly, G and F are well defined real symmetric bilinear forms on Ω^1,1_R (p) for any p. Now let θ_A be a unitary frame at any p with co-frame η_A and let a_A be real numbers. Take X = P

A

√−1a_Aη_A∧η_AΩ^1,1_R (p). Then a simple calculation gives

G(X, X)−F(X, X) =X

A

R_AA¯a²_A−X

A,B

R_AAB¯ B¯a_Aa_B

= 1 2

X

A,B

R_AAB¯ B¯(aA−aB)². (2.13)

The following was observed by Yau [26].

Lemma 2.6. At any pointp we have (a) QB ≥0 if and only if G−F ≥0.

(b) QB >0 if and only if G−F >0 on Ω^1,1_R (p)\Rω(p).

HereΩ^1,1_R (p)\Rω(p)are the real(1,1)forms atpwhich are not multiples of the K¨ahler form.

Proof. We first prove (a). The fact that G−F ≥ 0 implies QB ≥ 0 follows immediately from (2.13) and the fact that θ_A and a_A are arbitrary. Conversely, supposeQB ≥0 and letXbe any real (1,1) form at p. Then we can always diagonalizeX. Namely, there exists a unitary frame e_A with co-frame η_A such that X = P

A

√−1a_Aη_A∧η_A. Now

(2.13), and the assumption QB ≥ 0, immediately implies G(X, X)− F(X, X)≥0.

Now we prove (b). The proof is basically the same in part (a) once we observe that X ∈ Rω(p) if and only if: for every unitary frame e_A at p with co-frame η_A we have X = cP

A

√−1η_A∧η_A for some real constant c. The fact thatG−F >0 on Ω^1,1_R (p)\Rω(p) impliesQB >0 now follows immediately from (2.13) and the fact that θA and aA are arbitrary. Conversely, suppose QB > 0 and let X ∈ Ω^1,1_R (p)\Rω(p).

Then there exists a unitary frame e_A with co-frame η_A such that X = P

A

√−1aAηA ∧ηA with a^′_As not all the same. Now (2.13) and the assumption QB >0 immediately impliesG(X, X)−F(X, X)>0.

This concludes the proof of the lemma. q.e.d.

Remark 2.1. Thus QB > 0 if and only if G−F is positive in the orthogonal complement of Rω. In particular, if (M, g) is a compact K¨ahler manifold with QB > 0, then a K¨ahler metric which is a small perturbation of g will also satisfy QB >0.

Remark 2.2. Viewed as an endomorphism on Ω^1,1_R (M),G−F is in fact the curvature term in the Weitzenb¨ock identity for real (1,1) forms:

∆_g−∆ is given byG−F up to a positive constant multiple where ∆_g is the Bochner Laplacian with respect togand ∆ is the Laplace-Beltrami operator. The standard Bochner technique and Lemma 2.6 then give:all real harmonic (1,1) forms onM are parallel providedQB≥0; moreover, dim(H_R^1,1(M)) = 1providedQB >0 whereH_R^1,1(M) is the space of real harmonic (1,1) forms on M. See §1 for a reference to these facts and their implicit appearance in earlier works.

By Lemma 2.6, to check whetherQB ≥0 (orQB >0), it is sufficient to consider G−F ≥0 in a unitary frame of our choice. In the case of K¨ahler C-spaces, the natural choice is a Weyl frame. By Lemmas 2.2 and 2.6, we have:

Corollary 2.1. On a K¨ahler C-space, let Ric = µg and let eA be a Weyl frame. Then QB ≥ 0 if and only if the largest eigenvalues of the quadratic forms P

A,BR_AAB¯ B¯x_Ax_B, with x_A’s real, and P

A,B,C,D;

A6=B,C6=D

R_ABC¯ D¯xABxCD, with xAB = xBA, are at most µ. QB >0 if QB ≥ 0 and the eigenvalue µ of P

A,BR_AAB¯ B¯x_Ax_B is simple and the largest eigenvalue of P

A,B,C,D;

A6=B,C6=D R_ABC¯ D¯x_ABx_CD is less thanµ.

The following simple fact will be used throughout the paper to esti- mate the largest eigenvalue of a quadratic form.

Lemma 2.7. [row sums] Letx₁, . . . , x_n, a₁, . . . , a_n, and λbe real or complex numbers. Suppose |x_k|= max{|x_i| 1≤i≤n}>0 and

λx_k= Xn

i=1

a_ix_i.

Then

|λ| ≤ Xn

j=1

|a_i|.

In particular, if λ is an eigenvalue of ann×n matrix (a_ij), then

|λ| ≤max

i



 Xn

j=1

|aij|



.

We also note the following modification of Lemma 2.7 which will only be needed in a few exceptional cases.

Lemma 2.8. [weighted row sums] Letλbe an eigenvalue of ann×n matrix A = (a_ij) such that |λ| > 0. Let µ > 0 be a positive number.

Define b^s_j inductively: b⁽⁰⁾_j = 1, and b^(s+1)_j = min(1,X

l

|a_jl|b^(s)_l µ ).

Then for all s≥0,

(2.14) |λ| ≤max{max

i



 Xn

j=1

|a_ij|b^(s)_j



, µ}. In particular, if for some s≥0,

(2.15) max

i



 Xn

j=1

|a_ij|b^(s)_j



< µ, then |λ|< µ.

Proof. First we show that b^(s+1)_j ≤ b^(s)_j for all j. Note that by defi- nition 1 ≥ b^(s)_j ≥ 0. It is obviously true that b⁽¹⁾_j ≤ 1 = b⁽⁰⁾_j . Suppose b^(s+1)_j ≤b^(s)_j for allj; then

b^(s+2)_j = min(1,X

l

|a_jl|b^(s+1)_l /µ)≤min(1,X

l

|a_jl|b^(s)_l /µ) =b^(s+1)_j . To prove the lemma: If |λ| ≤ µ, then the lemma is true. Suppose

|λ|> µ. Letxibe the components of an eigenvector ofAwith eigenvalue λ. Suppose, without loss of generality, that max_i|x_i|= 1. We claim that for all s≥0,

|x_i| ≤b^(s)_i for all i≥1. For s= 1, then, for anyj,

λxj =X

l

a_jlx_l

and

|x_j| ≤ 1

|λ| X

l

|a_jlx_l| ≤ 1

|λ| X

l

|a_jl|.

So |x_j| ≤b⁽¹⁾_j because|λ|> µ and |x_j| ≤1. Now suppose|x_j| ≤b^(s)_j for all j. Then as before,

|λ||x_j| ≤X

l

|a_jl||x_l| ≤X

l

|a_jl|b^(s)_l and

|x_j| ≤X

l

|a_jl|b^(s)_l µ . Hence |x_j| ≤b^(s+1)_j . Hence the claim is true.

Now we may assume without loss of generality that |x₁|= 1. (2.14) is true fors= 0 by the previous lemma. Fors≥1,

|λ|=|λx₁| ≤X

l

|a_1l|x_l| ≤X

l

|a_1lb^(s)_l . Hence (2.14) is also true in this case.

If (2.15) is true, then it is still true if µis replaced byµ−ǫforǫ >0 small enough. Then by (2.14),|λ| ≤µ−ǫ < µ.

q.e.d.

3. K¨ahler C-spaces of classical type

According to [15], the K¨ahler C-spaces withb₂ = 1 of classical type which are not Hermitian symmetric spaces are (Bn, αp), with n ≥ 3, 1 < p < n, (C_n, α_p), withn≥3, 1< p < n, and (D_n, α_p), withn≥4, 1 < p < n−1. For each Lie algebra B_n, C_n, D_n below, we assume an identification has been made between h^∗, the dual Cartan subalgebra, and V = Rⁿ so that the induced Killing form corresponds to the Eu- clidean inner product (·,·). We will then present the corresponding root system ∆ as a set of vectors in V =Rⁿ. We refer to [7] for details.

3.1. The spaces (B_n, α_p).We first consider the space (B_n, α_p), with n ≥3, 1< p < n. Let V =Rⁿ and let εi be the standard basis on V. The root system for B_nis

(3.1) ∆ ={±εi±εj|1≤i, j≤n, i6=j} ∪ {±εi|1≤i≤n} Simple positive roots are

(3.2) α₁=ε₁−ε₂, α₂=ε₂−ε₃, . . . , α_n−1=ε_n−1−ε_n, α_n=ε_n. Positive roots are

(3.3) ∆⁺={ε_i+ε_j}i<j ∪ {ε_i−ε_j}i<j ∪ {ε_i}.

In terms of the α_i’s, the positive roots are εi =αi+· · ·+αn

εi+εj =αi+· · ·+αj−1+ 2αj+· · ·+ 2αn, i < j ε_i−ε_j =α_i+· · ·+α_j−1, i < j.

(3.4)

Let 1< p < n. Recall that

∆⁺_p(k) =



α∈∆⁺|α=kα_p+X

i6=p

m_iα_i, m_i ≥0, m_i ∈Z



. By (3.3) and (3.4), we have

∆⁺_p(1) ={ε_a|1≤a≤p}[

{ε_a+ε_i|1≤a≤p, p+ 1≤i≤n} [{ε_a−ε_i|1≤a≤p, p+ 1≤i≤n},

(3.5)

∆⁺_p(2) ={ε_a+ε_b|1≤a < b≤p}. (3.6)

(3.7) ∆⁺_p(k) =∅

fork≥3. The dimension of (B_m, α_p) is ¹₂p(4n−3p+ 1). We denote the elements of the ∆⁺_p(k)’s by: X_ai = ε_a−ε_i, Y_ai = ε_a+ε_i, 1 ≤ a ≤ p, p+ 1≤ i≤ n; Ua = εa, W_ab =εa+ε_b, 1 ≤ a, b≤ p. In the following a, b, . . . will range from 1 to p and i, j, . . . will range from p+ 1 to n.

Thus

∆⁺_p(1) ={X_ai}1≤a≤p;p+1≤i≤n

[{Y_ai}1≤a≤p;p+1≤i≤n

[{U_a}1≤a≤p,

∆⁺_p(2) ={W_ab}1≤a<b≤p.

Now recall that N_α,β = ±(p+ 1) where p is the largest integer, so that β−pαis a root, and also the definition of Ne_α,β in (2.10).

Lemma 3.1. Let α, β be positive roots in (B_n, α_p); then Ne_α,β =

√2 sgn(N_α,β). If α−β 6= 0, then Ne_α,−β =√

2 sgn(N_α,−β).

Proof. Note that if σ is a root, then either |σ|² = 1 or |σ|² = 2. We begin by proving the first part of the lemma. Letα, β be positive roots.

We may assume α+β is a root; otherwise the first part of the lemma is obviously true.

Suppose|α|²=|β|²= 1 and suppose|α+β|² = 1; then (α, β) =−¹₂, and this is impossible because one can see that (α, β) is an integer.

Hence |α+β|² = 2 and (α, β) = 0. So α−β is also a root [11, p. 324].

α−2β is not a root because|α−2β|² = 5. HenceN_α,β=±2. Therefore, by the definition ofNe_α,β in (2.10), Ne_α,β =√

2 sgn(N_α,β).

Suppose|α|² = 1 and|β|² = 2. As before, one can prove that (α, β) =

−1 andN_α,β =±1. Hence Ne_α,β =√

2 sgn(N_α,β).

Suppose |α|² =|β|² = 2. As before, one can prove that (α, β) =−1, N_α,β =±1, and henceNe_α,β=√

2 sgn(N_α,β).

The case forNe_α,−β can be proved similarly. q.e.d.

By Lemmas 2.4, 2.5, and 3.1 and the fact that R(α,β, γ,¯ δ) =¯ R(α,δ, γ,¯ β), we have:¯

Corollary 3.1. Let α ∈ ∆⁺(i), β ∈ ∆⁺(j), γ ∈ ∆⁺(k), and δ ∈

∆⁺(l).

1)

R_α¯αββ¯=





(α, β) + ¹₂(sgn(N_α,β)²), i=j = 1,

1

2(α, β), i= 1, j = 2;

1

2(α, β), i=j = 2,

2) If α−β6=δ−γ, then R(α,β, γ,¯ δ) = 0.¯

3) If α−β=δ−γ 6= 0, then for(i, j, k, l) = (1,1,1,1), R(α,β, γ,¯ δ) =¯





1

2sgn(N_α,γ)sgn(N_β,δ), if α−β is not a root, sgn(N_α,−β)sgn(N_γ,−δ), if α+γ is not a root,

−¹₂sgn(N_α,γ)sgn(N_δ,β), if β−γ 6= 0 is not a root.

For other cases, R(α,β, γ,¯ δ) =¯





1

2sgn(N_α,−β)sgn(N_γ,−δ), if (i, j, k, l) = (1,1,2,2),

1

2sgn(N_α,−β)sgn(N_γ,−δ), if (i, j, k, l) = (2,2,2,2),

1

2sgn(N_α,−β)sgn(N_γ,−δ) if (i, j, k, l) = (1,2,2,1).

To compute the Ricci curvature, we know that Ric =µgand thus µ= Ric(W12,W¯12)

=X

a,i

R(W12,W¯12, Xai,X¯ai) +R(W12,W¯12, Yai,Y¯ai)

+X

a

R(W₁₂,W¯₁₂, U_a,U¯_a) +X

a<b

R(W₁₂,W¯₁₂, W_ab,W¯_ab)

=1

2[2(n−p) + 2(n−p)] + 1 + 1

2(p+ (p−2))

= 2n−p.

Lemma 3.2. Let λbe the largest eigenvalue of the quadratic form X

A,B

R_AAB¯ B¯x_Ax_B in the Weyl frame, where x_A are real.

(a) λ≤2n−p if and only if 5p+ 1≤4n.

(b) If 5p+ 1<4n, thenλ= (2n−p) iff the corresponding eigenvector satisfies x_A=x_B for all A, B.

(c) If5p+1 = 4n, then there is an eigenvector with eigenvalue(2n−p) such that x_A6=x_B for some A6=B.

Proof. We begin with the proof of (a). Letv= (x_A) be an eigenvector corresponding to the largest eigenvalueλfor the quadratic form. Assume the components satisfy max_A|x_A| = 1. Let us denote the components x_A more specifically byx_ai, y_ai, a≤p < i;u_a, a≤p;t_ab, a < b≤p, and let us denote R(X_ai,X¯_ai, X_bj,X¯_bj) by R(X_ai, X_bj) etc. Then P(v) = P

A,BR_AAB¯ B¯x_Ax_B is equal to:

P(v) = X

a,b≤p<i,j

R(X_ai, X_bj)x_aix_bj+ X

a,b≤p<i,j

R(Y_ai, Y_bj)y_aiy_bj

+ X

a,b≤p<i,j

R(X_ai, Y_bj)x_aiy_bj+ X

a,b≤p<i,j

R(Y_ai, X_bj)y_aix_bj

+ 2 X

a,c≤p<i

R(X_ai, U_c)x_aiu_c+ 2 X

a,c≤p<i

R(Y_ai, U_c)y_aiu_c

+ 2 X

a<b,c≤p<i

R(w_ab, X_ci)t_abx_ci+ 2 X

a<b,c≤p<i

R(w_ab, Y_ci)t_aby_ci

+ X

a,b≤p

R(U_a, U_b)u_au_b+ 2 X

a<b,c≤p

R(w_ab, U_c)t_abu_c

+ X

a<b≤p,c<d≤p

R(W_ab, W_cd)t_abt_cd. (3.8)

From Corollary 3.1, it is easy to see that R(X_ai, X_bj) =R(Y_ai, Y_bj), R(X_ai, Y_bj) = R(Y_ai, X_bj), R(X_ai, U_b) = R(Y_ai, U_b), R(X_ai, W_bc) = R(Yai, W_bc). We see that if we interchangexaiandyaifor alla, iand ob- tain a vector w, thenP(v) =P(w) and |v|=|w|. We may then assume that eitherx_ai=y_aifor alla, i, or by consideringv−w, thatx_ai=−y_ai and u_a=t_ab= 0 for all a, b.

Suppose|u_a|= 1 for somea. We may assume thatu_a= 1. By Corol- lary 3.1, R(U_a,U¯_a, x,x)¯ ≥0 because (U_a, x)≥0 for all x∈∆⁺_p(k), k= 1,2.

λu_a=X

b≤p

R(U_a, U_b)u_b+ X

b≤p<i

R(X_bi, U_a)x_bi+ X

b≤p<i

R(Y_bi, U_a)y_bi

+ X

c<d≤p

R(w_cd, U_a)t_cd. (3.9)

Notice that the coefficients are all nonnegative and the sum is just Ric(Ua,U¯a) = 2n−p. Hence λ ≤ 2n−p. Moreover, if λ = 2n−p, then we must in fact have

(3.10) x_a,i=y_a,i=u_b =t_cd= 1