An example of compact K\"ahler manifold with nonnegative quadratic bisectional curvature

We construct a compact K\"ahler manifold of nonnegative quadratic bisectional curvature, which does not admit any K\"ahler metric of nonnegative orthogonal bisectional curvature. The manifold is a 7-dimensional K\"ahler C-space with second Betti number equal to 1, and its canonical metric is a K\"ahler-Einstein metric of positive scalar curvature


Introduction
In recent years, the condition nonnegative quadratic bisectional curvature (which we will denote by QB ≥ 0) has drawn more and more attentions; see for example [17], [9], [6], [18], and [13]. At a point p on a Kähler manifold (M n , g), this condition is defined by for any unitary tangent frame {e 1 , . . . , e n } at p and any real numbers x 1 , . . . , x n .
Note that when the bisectional curvature is nonnegative (denoted as B ≥ 0 from now on), namely, when R XXYȲ ≥ 0 for any two type (1, 0) tangent vectors X, Y at p, then QB ≥ 0 at p.
A condition slightly weaker than B ≥ 0 is the so-called nonnegative orthogonal bisectional curvature, denoted as B ⊥ ≥ 0, which requires that R XXYȲ ≥ 0 for any two type (1, 0) tangent vectors X, Y at p which satisfy X ⊥ Y . Clearly, B ⊥ ≥ 0 already implies QB ≥ 0, since the diagonal terms in the summation vanish, and when n = 2, B ⊥ ≥ 0 and QB ≥ 0 coincide. However, when n ≥ 3, QB ≥ 0 does not have to have all the orthogonal bisectional curvature terms to be nonnegative, thus it is weaker than B ⊥ ≥ 0, at least from the algebraic point of view. It is however a totally different question whether there will be any compact Kähler manifold (M n , g) (n ≥ 3) with QB ≥ 0 everywhere such that M n does not admit any Kähler metric with B ⊥ ≥ 0 everywhere.
The purpose of this note is exactly to demonstrate the existence of such a manifold.
This topic is of course closely related to the Frankel-Hartshorne conjectures, which attempts to understand the elliptic end of the high dimensional uniformization theory. The famous solution of Mok [14] to the generalized Frankel conjecture states that any compact simply-connected Kähler manifold with B ≥ 0 everywhere must be biholomorphic to a compact Hermitian symmetric space. Recently, using the Ricci flow technique and earlier work of X. X. Chen [7] and Brendle-Schoen [4,5], Gu-Zhang [10] proved the following result: Theorem (Gu-Zhang). Let (M n , g) be a simply-connected compact Kähler manifold with B ⊥ ≥ 0 everywhere. Then M is biholomorphic to a compact Hermitian symmetric space.
In other words, the condition B ⊥ ≥ 0, although algebraically weaker than B ≥ 0, do not generate any new examples, since for any compact Hermitian symmetric space, its canonical Kähler metric has B ≥ 0 everywhere.
Here we avoided the discussion of non-simply connected cases, since splitting theorems are already known, in the generalized Frankel case by the classic slitting theorem of Howard-Smyth-Wu [11], [16], and in the generalized Hartshorne case by Demailly-Peternell-Schneider [8].
The generalized Hartshorne conjecture seeks to understand all Fano manifolds with numerically effective tangent bundles. This class includes all the Kähler C-spaces, namely, all the compact simply-connected homogeneous Kähler manifolds. The conjecture, in its narrowest sense, states that any compact simply-connected Kählerian manifold M n with numerically effective tangent bundle must be biholomorphic to a Kähler C-space (see, for example, [19, p. 218]).
Note that for a Kähler C-space M n , its canonical Kähler-Einstein metric (which is unique up to a constant multiple) has B ≥ 0 everywhere if and only if M n is Hermitian symmetric. When M n is not Hermitian symmetric, any Kähler metric on M n cannot have B ≥ 0 everywhere by Mok's Theorem. In fact, it cannot have B ⊥ ≥ 0 everywhere by the recent theorem of Gu and Zhang. So one has to tolerate some mild negativity of bisectional curvature in the quest of generalized Hartshorne conjecture, at least via differential geometric approach. In light of this, the condition QB ≥ 0 comes into play, and it is natural to ask whether this condition will give a good differential geometric description of Kähler C-spaces.
By the splitting results of H. Wu et. al., we know that under the condition QB ≥ 0 everywhere, any harmonic (1, 1) form must be parallel, thus M n will admit de Rham decomposition if the second betti number b 2 > 1. For this reason we should restrict ourselves to Kähler C-spaces with b 2 = 1. This class includes all irreducible compact Hermitian symmetric spaces.
Let M n be an n-dimensional Kähler C-space with b 2 = 1. The homogeneous Kähler metric on M n is unique up to a constant multiple, and it is Kähler-Einstein. Let us denote this metric by g 0 . Let C be the set of all Kähler C-spaces with b 2 = 1 excluding all irreducible compact Hermitian symmetric spaces. We raise the following is a compact simply-connected and locally irreducible Kähler manifold with QB ≥ 0 everywhere, then M n is biholomorphic to a Kähler C-space with b 2 = 1.
3). In 2), g is actually isometric to (a constant multiple of ) g 0 , if the manifold M n is not P n .
In this note, we give an explicit calculation of one special example (M 7 , g 0 ) in C, and show that it indeed has QB ≥ 0, thus confirming that the condition is genuinely more tolerant than B ≥ 0 or B ⊥ ≥ 0. Our computation is brute-force, due to our lack of knowledge in algebra. We suspect that a more representation-theoretic computation would establish 1).
Let G be a simply-connected, simple complex Lie group, and g its Lie algebra. Fix a Cartan subalgebra h of g. Let l = dim C h, and let ∆ be the root system of g with respect to h. Fix a fundamental root system {α 1 , . . . , α l } of ∆. It determines an ordering of the root system ∆ = ∆ + ∪ ∆ − . We have where g α is the root space corresponding to α, satisfying [g α , g β ] ⊆ g α+β for any two roots α, β ∈ ∆. Now let us fix an integer r with 1 ≤ r ≤ l. Denote by Let P ⊆ G be the subgroup whose Lie algebra is then P is a parabolic subgroup, namely, the complex manifold M = G/P is compact. This gives a Kähler C-space with b 2 = 1. Conversely, any Kähler C-space with b 2 = 1 is given this way. We will denote this space by (g, α r ). Table 1 on page 55 of [12] gives the list of all Kähler C-spaces with b 2 = 1, using the Dynkin diagrams. The double circled ones are the Hermitian symmetric ones. From the table, we see that the simplest non-symmetric example would be M 7 = (B 3 , α 2 ), which will be our example in this note.
In the remainder of this section, we will follow Itoh's notations and calculations [12], and collect the necessary formulas that we need later for computing the curvature of Kähler C-spaces with b 2 = 1. Let M n = (g, α r ) be a Kähler C-space with b 2 = 1. Denote The space m + can be identified with the holomorphic tangent space of M n . Let K be the Killing form of g. Let E α be a Weyl canonical basis of g, namely, E α ∈ g α for each α ∈ ∆, and The canonical metric g 0 = , is given by for any X, Y ∈ m + k , where the complex conjugation on M is determined bȳ E α = E −α for each α. In the following, we will assume that where i, j, k, l are any positive integers, and compute the curvature component R XȲ ZW of the canonical metric g 0 . From [12], we have Like in [12], a straight forward computation yields the following formulas.
Proposition 2.1. On a Kähler C-space (M n , g 0 ) with b 2 = 1, then for any , the curvature components are given by
Note that by the invariance of K and the Jacobi identity, we see that the first (or the third) term on the right hand side of (2.3) can be expressed as a linear combination of the other two terms.
In the following, we will assume that g satisfies the condition This condition is satisfied by all four classical sequences A, B, C, D for all r, and for some of the exceptional cases. Note that the case m + = m + 1 corresponds exactly to all the irreducible Hermitian symmetric cases.
For the sake of simplicity, we will assume that M n = (g, α r ) is of contact type, namely, the condition (2.4) holds and ∆ + r (2) consists of only one element. Applying Proposition 2.1 to this contact case, we get 3. The curvature tensor of (B 3 , α 2 ) Let us now consider the specific case of M 7 = (B 3 , α 2 ), where B 3 = so 7 (C) is the Lie algebra of the Lie group SO 7 (C). We will regard B 3 as the space of all 7×7 complex matrices A = (a ij ) such that a ij = −a j ′ i ′ for all 1 ≤ i, j ≤ 7, where here and from now on we will write i ′ = 8 − i. A Cartan subalgebra is given by all the diagonal matrices h = diag(a 1 , a 2 , a 3 , 0, −a 3 , −a 2 , −a 1 ) | a i ∈ C, 1 ≤ i ≤ 3 .

The root system with respect to this h is
Note that g = so 7 (C) has three simple roots {α 1 , α 2 , α 3 }, which are given by Let ∆ + and ∆ − be the subsets of ∆ consisting of positive and negative roots, respectively. We have (2) and M 7 is of contact type. For convenience, let us denote by e ij the square matrix whose (i, j)-entry is 1 and other entries are zero. Let Then, h is spanned by {F ii | 1 ≤ i ≤ 3} over C. Furthermore, the root vectors can be explicitly given by We compute the trace form Since so 7 (C) is a simple Lie algebra, any two invariant symmetric bilinear form on so 7 (C) are proportional. Thus, by rescaling some constant, we can assume that the Killing form on so 7 (C) satisfies Moreover, observe that We will form a unitary tangent frame {E 1 , . . . , E 6 ; E 7 } (in the order) by , and let their complex conjugation {Ē 1 , . . . ,Ē 6 ;Ē 7 } be In the following, for the benefit of calculations in the next section, we will let a, b, c, d be the indices from {1, 2} and i, j, k, l the indices from {3, 4, 5}. We will also let X = F ai , Y = F bj , Z = F ck , W = F dl be vectors in m + 1 and So by Proposition 2.2 and formula (3.1), we get from a straight forward computation that The result below follows immediately.

Nonnegativity of the quadratic bisectional curvature
In this section, we shall show that the 7-dimensional Kähler C-space (M 7 , g 0 ) given by (B 3 , α 2 ) indeed has nonnegative quadratic bisectional curvature. In other words, for any unitary tangent frame {e a : 1 ≤ a ≤ 7} and any real numbers x 1 , . . . , x 7 , the quantity QB := 7 a,b=1 R(e a ,ē a , e b ,ē b )(x a − x b ) 2 is always nonnegative.
Write e a = 7 i=1 T ai E i for some T ∈ U (7), where {E 1 , . . . , E 7 } is the Weyl basis mentioned in the §3. Since the metric g 0 is Einstein with Ricci curvature equal to 4, we have where P = t T Λ xT is a Hermitian symmetric matrix. Here we wrote Λ x = diag{x 1 , . . . , x 7 } and |P | 2 = i,j |P ij | 2 = a x 2 a . Write We have |P | 2 = |P ′ | 2 + 2|ξ| 2 + t 2 . Since under the Weyl frame {E i }, we have R 7777 = 1 and R ij77 = 1 2 δ ij for any 1 ≤ i, j ≤ 6, we know that Plugging these into the formula for QB, we get So QB ≥ 0 for any Hermitian matrix P is equivalent to for any 6 × 6 Hermitian matrix P ′ .