The canonical bundle of a hermitian manifold

Gil Bor Luis Hernández-Lamoneda ^{∗}

Centro de Investigación en Matemáticas, A.C. (CIMAT) Guanajuato, México

November 4, 1998

Abstract

This note contains a simple formula (Proposition 1 in Section 3) for the curvature of the canonical line bundle on a hermitian manifold, using the Levi-Civita connection (instead of the more usual hermitian connection, compatible with the holomorphic structure). As an immediate application of this formula we derive the following: the six-sphere does not admit a complex structure, orthogonal with respect to any metric in a neighborhood of the round one. Moreover, we obtain such a neighborhood in terms of explicit bounds on the eigen-values of the curvature operator. This extends a theorem of LeBrun.

Keywords: Hermitian manifold, almost-complex structure, canonical bundle, curvature. AMS Subject classification number: 53C55.

1 Introduction

First, some standard definitions. An almost-complex structure on an evendimensional manifold M^{2n} is a smooth endomorphism J : T M → T M, such that J^{2} = −Id. The standard example is M = ℂ^{n} with J given by the usual scalar multiplication by i. A holomorphic map between two almost-complex manifolds (M_{1}, J_{1}) and (M_{2}, J_{2}) is a smooth map f : M_{1} → M_{2} satisfying df ∘ J_{1} = J_{2} ∘df. An almost-complex structure is said to be integrable, or is called simply a complex structure, if it is locally holomorphicaly diffeomorphic to the standard example; in other words, for each x ∈ M there exist neighbourhoods U ⊂ M, x ∈ U, and V ⊂ ℂ^{n}, and a holomorphic diffeomorphism f : U → V.

∗Both authors received support from grants 0329P-E of CONACyT and E130.728 of CONACyT-CSIC

Given an even-dimensional manifold, how is one to decide if it admits a complex structure? There are some, more or less obvious, necessary conditions (e.g. the existence of an almost-complex structure, which can be tested by characteristic classes), but in general there is no known answer to this question. A well-known example, so far undecided, is the 6-sphere (this is the only interesting dimension, because in all other dimensions n ≠ 2, 6, the n-sphere does not admit even an almost-complex structure). This space admits a non-integrable almost-complex structure, but it is unknown as yet if it admits a complex structure.

A related question is that of the existence of an orthogonal complex structure. Here the set-up is the following: given an even-dimensional riemannian manifold (M, g), one is looking for an integrable almost-complex structure J which is orthogonal with respect to g; that is, g(X, Y) = g(JX, JY), for all X, Y ∈ T_{m}M and m ∈ M. One calls such a pair (g, J) a hermitian structure. The problem here is then that of extending a given riemannian structure to a hermitian one.

One way of analysing the problem of the existence of orthogonal complex structures is to consider the space of all orthogonal almost-complex structures. These are sections of a bundle over M, whose fiber at a point of the manifold consists of all (linear) orthogonal complex structures on the tangent space at that point. The total space Z of this bundle is called the twistor space associated to (M, g) and it admits a tautological almost-complex structure. Then the idea is to translate differential geometric problems on M to complex-geometric problems on Z. For example, an orthogonal almostcomplex structure on M is given, by definition, by a section of Z → M; it will be integrable if the section is holomorphic, thus embedding M as a complex sub-manifold of Z. In other words, the problem of orthogonal complex structures on M is translated into that of certain complex submanifolds of Z. This approach leads to the proof of C. LeBrun of non-existence of an orthogonal complex structure on S^{6} relative to the round metric [2]. The twistor space Z in this case turns out to be Kähler, so that an orthogonal complex structure on S^{6} would give an embedding of S^{6} as a complex submanifold of a Kähler one, thus inheriting a Kähler structure, which is clearly impossible for S^{6} (since H^{2}(S^{6}) = 0). For more information on this approach to orthogonal complex structures we recommend the survey article of S. Salamon [3].

Here we suggest a different construction, considerably more elementary. This is based on the observation that the curvature of a connection on a complex line bundle is a closed two-form (representing the first Chern class of the line bundle, up to a constant), so one can try to use the given data (g, J) on M to construct a line bundle with connection whose curvature twoform is non-degenerate, i.e. a symplectic form. On certain manifolds this might be impossible (e.g. on a compact manifold with H^{2} = 0), so if one uses a connection coming from the Levi-Civita connection on (M, g) then one obtains in this way a curvature obstruction for the existence of an orthogonal complex structure.

A natural complex line bundle to consider, for a given complex structure, is the so-called canonical line bundle K := Λ^{n,0}(M) – the bundle of (n, 0)- forms, or the top exterior power of the holomorphic cotangent bundle. Now there are two natural ways to use the hermitian structure on M to equip K with a connection. First, the complex structure on M induces a holomorphic structure on K and the riemannian metric on M induces a hermitian metric on K; these two in turn determine uniquely a canonical hermitian connection (a metric-preserving connection whose (0, 1)-part coincides with the ¯∂-operator of the complex structure on M; see for example Griffiths and Harris [1], p. 73). The other choice of a connection on K comes from the Levi-Civita connection on T M, extended (by the Leibniz rule) to the bundle of exterior n-forms Λ^{n}(M), complexified, then projected orthogonally to the sub-bundle K ⊂ Λ_{ℂ}^{n}(M).

Unless the orthogonal complex structure happens to be Kähler (i.e. the Kähler 2-form ω = g(J·, ·) is closed), these two choices of a connection are different. We make here the second choice, the one coming from the Levi-Civita connection, as it seems to us more natural from a Riemannian geometric point of view, e.g. for relating the resulting curvature 2-form of the canonical bundle with the Riemann curvature tensor of (M, g).

The outcome then is a rather simple formula for the curvature of the canonical line bundle on a hermitian manifold (Proposition 1 of Section 3). From this formula it becomes obvious that a complex structure compatible with the round metric on the sphere will render the curvature 2-form of the corresponding canonical line bundle a symplectic form (in fact Kähler), and that this property will be maintained for nearby metrics (Corollaries 2 and 3 of Section 4). ^{1}

We shall now outline the details of the calculation indicated above. We need to recall first some standard terminology.

Let E → M be a complex hermitian vector bundle over a differentiable manifold, with a hermitian connection D : Γ(E) → Γ(T^{∗}(M) ⊗ E), i.e.

d⟨s_{1}, s_{2}⟩ = ⟨Ds_{1}, s_{2}⟩ + ⟨s_{1}, Ds_{2}⟩

for any two sections s_{1}, s_{2} ∈ Γ(E).

The curvature R of (E, D) is defined by first extending D to Γ(Λ^{k}(M) ⊗ E) → Γ(Λ^{k+1}(M) ⊗ E),

D(α ⊗ s) := dα ⊗ s + (−1)^{k}α ⊗ Ds,

then

R := D^{2} ∈ Γ(Λ^{2}(M) ⊗ End(E)).

If E_{0} ⊂ E is a sub-bundle then there is an induced hermitian connection on E_{0} as follows: let s_{0} be a section of E_{0}, and let (E_{0})^{⊥} be the orthogonal complement of E_{0} in E, then decompose orthogonally

Ds_{0} = D_{0}s_{0} + Φs_{0},

with

D_{0}s_{0} ∈ Γ(T^{∗}(M) ⊗ E_{0}), Φs_{0} ∈ Γ(T^{∗}(M) ⊗ (E_{0})^{⊥}).

One then verifies easily that D_{0} defines a hermitian connection on E_{0} and that Φ is “tensorial”, i.e. a section of T^{∗}(M) ⊗ Hom(E_{0}, (E_{0})^{⊥}), called the second fundamental form of E_{0} in E.

1Claude LeBrun has informed us recently that his proof also extends to metrics near the round one, but this requires embedding the usual twistor space inside a larger one. Also, after completing the work described here we found two articles ([4] and [5]) containing ideas close to ours.

Now there is a well-known formula for the curvature of (E_{0}, D_{0}) in terms of the curvature R of (E, D) and the second fundamental form Φ of E_{0} in E. It is given by

Ω = π_{0} ∘ R ∘ π_{0}^{∗} + Φ^{∗} ∧ Φ,  (1)

where π_{0} : E → E_{0} is orthogonal projection. The (easy) calculation can be found for example in [1], p.78.

In our case, starting with the Levi-Civita connection on Λ_{ℂ}^{n}(M) and projecting onto the canonical line-bundle K = Λ^{n,0}(M) ⊂ Λ_{ℂ}^{n}(M), we find out the following:

1. π_{0} ∘ R ∘ π_{0}^{∗} = iR̃(ω), where ω is the Kähler form and R̃ is the interpretation of the Riemann curvature tensor of M as an operator in End(Λ^{2}(M)) (see the corollary to Lemma 1 in Section 3).

2. The second fundamental form Φ ∈ Λ^{1}(M) ⊗ Hom(Λ^{n,0}, (Λ^{n,0})^{⊥}) is of type (1, 0), hence Φ^{∗} ∧Φ is non-positive (see Lemmas 2 and 3 in Section 3; see next section, Definition 2, for the sign convention).

The first fact does not require even the integrability of the orthogonal almost-complex structure, i.e. it holds also for almost-hermitian manifolds. The second one does depend on the integrability (in fact, it can be shown to be equivalent to the integrability of the almost-complex structure).

We use these two basic results to deduce rather easily the non-degeneracy of the 2-form Ω in the proof of the above mentioned theorem of LeBrun, as well as its extension to metrics which are nearby the round one (see Section 4).

2 Some definitions and notation

First, to make sense of Formula (1) in the Introduction, we need to review some terminology.

Let V be a real 2n-dimensional vector space with a euclidean inner product (·, ·) and a linear orthogonal almost-complex structure J. We extend the inner product (·, ·) on V in the usual way to the real exterior algebra Λ^{∗}(V^{∗}), by declaring the k-forms {η_{i_{1}} ∧ … ∧ η_{i_{k}}|1 < i_{1} < … < i_{k} < 2n} an orthonormal basis of Λ^{k}(V^{∗}), where {η_{1}, …, η_{2n}} is the dual basis of an orthonormal basis of V.

We denote also by (·, ·) the complex-linear extension of the euclidean inner product (·, ·) to the complexified vector spaces Λ_{ℂ}^{k}(V^{∗}) = Λ^{k}(V^{∗}) ⊗ ℂ. The hermitian inner-product on these spaces is thus given by ⟨ϕ, ψ⟩ = (ϕ, ψ̄).

Next, let W be a complex vector space with an hermitian inner product ⟨·, ·⟩ and denote by End_{ℂ}(W) the complex-linear endomorphisms of W. Denote by End(V) the real endomorphisms of V.

All tensor products, unless denoted otherwise, are over the reals.

Definition 1 Let V and W be as above, and α, β ∈ V^{∗} ⊗ End_{ℂ}(W) two endomorphism-valued 1-forms.

1. The wedge product α ∧ β ∈ Λ^{2}(V^{∗}) ⊗ End_{ℂ}(W) is defined by

α ∧ β(X, Y) := α(X) ∘ β(Y) − α(Y) ∘ β(X).

Equivalently, if α = a ⊗ A, β = b ⊗ B, where a, b ∈ V^{∗} and A, B ∈ End_{ℂ}(W), then α ∧ β := (a ∧ b) ⊗ (A ∘ B).

2. The adjoint α^{∗} ∈ V^{∗} ⊗ End_{ℂ}(W) is defined by

α^{∗}(X) = [α(X)]^{∗}.

Equivalently, for α = a ⊗ A, α^{∗} = a ⊗ A^{∗}.

Note that when extending the notation to complex forms in V_{ℂ}^{∗}⊗_{ℂ}End_{ℂ}(W), one has that α^{∗}(Z) = [α(Z̄)]^{∗}, Z ∈ V_{ℂ}, so that if α = ϕ ⊗ A, where ϕ ∈ V_{ℂ}^{∗}, then α^{∗} = ϕ̄ ⊗ A^{∗}. (Proof: if Z = X + iY, then α^{∗}(Z) = α^{∗}(X) + iα^{∗}(Y) = [α(X)]^{∗} + i[α(Y)]^{∗} = [α(X) − iα(Y)]^{∗} = [α(Z̄)]^{∗}.) Hence if α is of type (1, 0) then α^{∗} is of type (0, 1) etc.

Next, we need to make some convention concerning positivity (watch for a confusing error in [1], pp. 29 & 79, around this definition).

Definition 2 1. A 2-form ω ∈ Λ^{2}(V^{∗}) is called positive, ω > 0, if B(X, Y) = ω(X, JY) is a symmetric positive bilinear form. Equivalently: ω is positive if it is a real 2-form of type (1, 1) (that’s the “symmetric” requirement) and ω(X′, X̄′)/i > 0 for all non-zero X′ ∈ V^{1,0}, where V_{ℂ} = V^{1,0} ⊕ V^{0,1} is the decomposition of the complexification of V into ±i eigen-spaces of J. Obviously, a positive (or negative) 2-form is non-degenerate.

2. Now let Ω ∈ Λ^{2}(V^{∗}) ⊗ End_{ℂ}(W) be a 2-form on V with values in antihermitian endomorphisms on W, so iΩ is an hermitian-valued 2-form (we have in mind the curvature of a hermitian connection). Then Ω is called positive, Ω > 0, if ⟨iΩw, w⟩ is a positive 2-form for all non-zero w ∈ W. Equivalently, Ω > 0 if it is an End(W)-valued (1, 1)-form such that Ω(X′, X̄′) is a positive hermitian operator for all non-zero X′ ∈ V^{1,0}. We define similarly Ω ≥ 0, Ω < 0, etc.

A word of caution: According to the last definition, the Kähler form ω = (J·, ·) on V is a real positive 2-form, whereas iω is an imaginary negative form.

Definition 3 Let A ∈ End(V) be an antisymmetric endomorphism on V, i.e. (Av, w) = −(v, Aw) for all v, w ∈ V. Define

1. Â ∈ Λ^{2}(V^{∗}) by Â(v, w) = (v, Aw).

2. A^{∗} ∈ End(V^{∗}) by (A^{∗}η)(v) = η(Av), as well as its extension to Λ^{∗}(V^{∗}) as a derivation:

A^{∗}(α ∧ β) = (A^{∗}α) ∧ β + α ∧ (A^{∗}β).

We use throughout the article the shorthand notation Λ^{k}(M) for the bundle of alternating k-forms Λ^{k}(T^{∗}M).

Definition 4 1. Let R ∈ Λ^{2}(V^{∗}) ⊗ End(V) (we have in mind the curvature tensor of the Levi-Civita connection on a riemannian manifold). Define R̃ ∈ End(Λ^{2}(V^{∗})) as follows: if R = ∑_{j} α_{j} ⊗ A_{j}, where α_{j} ∈ Λ^{2}(V^{∗}) and A_{j} ∈ End(V), then

R̃(β) = ∑_{j} (Â_{j}, β)α_{j}, β ∈ Λ^{2}(V^{∗}).

2. Applying this definition to the curvature tensor of a riemannian manifold R ∈ Γ(Λ^{2}(M) ⊗ End(T M)), we obtain the so-called curvature operator R̃ ∈ Γ(End(Λ^{2}(T^{∗}M))).

Another word of caution concerning sign conventions: we have made the choice of signs in the above definitions so as to make R̃ coincide with the curvature operator as defined in riemannian geometry. Thus, for example, the round sphere has a positive curvature operator (in fact, it is the identity operator). This is also tied up with our definition R = D^{2}, where there seems to be a conflict in the literature. In complex geometry it is usual to define the curvature of a connection by D^{2}, as we did in the Introduction. Thus, the curvature of the canonical bundle of ℂP^{1} is i times the area form. In riemannian geometry on the other hand, probably for historical reasons, the curvature tensor of the Levi-Civita connection is defined by the formula ∇_{[X,Y]} − [∇_{X}, ∇_{Y}], which amounts to defining R = −D^{2}. Our sign choice in Definition 4 is made so as to reconcile this conflict.

3 Three lemmas on hermitian structures

The first lemma is quite simple, and has probably appeared elsewhere. The second is essentially in Griffiths and Harris ([1], p.79, after overcoming the positivity confusion). The third is a curious fact about the Levi-Civita connection on a hermitian manifold, probably known, though we could not find it in the literature.

Let V be, as in the last section, a euclidean 2n-dimensional real vector space with an orthogonal almost-complex structure J, and let ω = (J·, ·) denote the associated Kähler 2-form.

Lemma 1 Let A ∈ End(V) be an antisymmetric endomorphism of V and let π_{0} : Λ_{ℂ}^{n}(V^{∗}) → Λ^{n,0}(V^{∗}) denote orthogonal projection. Then

π_{0} ∘ A^{∗} ∘ π_{0}^{∗} = −i(Â, ω),

where A^{∗} ∈ End(Λ_{ℂ}^{n}(V^{∗})) and Â ∈ Λ^{2}(V^{∗}) are given above in Definition 3.

Proof. Choose a unitary basis θ_{1}, …, θ_{n} for (V^{∗})^{1,0}, so that

ω = i(θ_{1} ∧ θ̄_{1} + … + θ_{n} ∧ θ̄_{n}).

Now ψ = θ_{1} ∧ … ∧ θ_{n} is a unitary element of Λ^{n,0}(V^{∗}), hence

π_{0} ∘ A^{∗} ∘ π_{0}^{∗} = (A^{∗}ψ, ψ̄) =

= ((A^{∗}θ_{1}) ∧ θ_{2} ∧ … ∧ θ_{n} , θ̄_{1} ∧ θ̄_{2} ∧ … ∧ θ̄_{n}) +
+ (θ_{1} ∧ (A^{∗}θ_{2}) ∧ … ∧ θ_{n} , θ̄_{1} ∧ θ̄_{2} ∧ … ∧ θ̄_{n}) + …
= (A^{∗}θ_{1}, θ̄_{1}) + ⋯ + (A^{∗}θ_{n}, θ̄_{n}).

Now, given any α, β ∈ V^{∗}, one can check easily from our definition of Â that

(A^{∗}α, β) = (Â, α ∧ β),

hence

π_{0} ∘ A^{∗} ∘ π_{0}^{∗} = (Â, θ_{1} ∧ θ̄_{1} + ⋯ + θ_{n} ∧ θ̄_{n}) = −i(Â, ω),

as claimed. QED

Corollary 1 Let (M, g, J) be an almost-hermitian manifold and let π_{0} : Λ^{2}(M) ⊗ Λ_{ℂ}^{n}(M) → Λ^{2}(M) ⊗ Λ^{n,0}(M) denote orthogonal projection in the second factor. Then

π_{0} ∘ R ∘ π_{0}^{∗} = iR̃(ω),

where ω = g(J·, ·) is the Kähler form, R ∈ Γ(Λ^{2}(M) ⊗ End(Λ_{ℂ}^{n}(M)) is the curvature of the connection induced on Λ_{ℂ}^{n}(M) by the Levi-Civita connection on T M, and R̃ is the curvature operator associated to the riemannian metric (as in Definition 4 above).

Proof. The main point to notice is that if the curvature tensor of a connection on T M is given (locally) by ∑ α_{j} ⊗ A_{j}, where α_{j} ∈ Γ(Λ^{2}(M)) and A_{j} ∈ Γ(End(T M)), then the curvature tensor of the induced connection on Λ_{ℂ}^{n}(M) is given by − ∑ α_{j} ⊗ A_{j}^{∗}, with A_{j}^{∗} given by Definition 3. The result now follows immediately from the previous lemma and the definition of R̃. QED

Lemma 2 If Φ ∈ Λ^{1,0}(V^{∗}) ⊗_{ℂ} End_{ℂ}(W), where W is a hermitian vector space, then Φ^{∗} ∧ Φ ≤ 0.

Proof. As noted above (Section 2, after Definition 1), Φ^{∗} is of type (0, 1), hence Φ^{∗} ∧ Φ is of type (1, 1). Next, for any X′ ∈ V^{1,0},

(Φ^{∗} ∧ Φ)(X′, X̄′) = Φ^{∗}(X′)Φ(X̄′) − Φ^{∗}(X̄′)Φ(X′) =
= −Φ^{∗}(X̄′)Φ(X′) = −[Φ(X′)]^{∗}Φ(X′),

and the claim follows since A^{∗}A is a hermitian non-negative operator for any A ∈ End_{ℂ}(W). QED Lemma 3 Let M^{2n} be a riemannian manifold with an orthogonal complex structure (i.e. a hermitian manifold). Denote by ∇ the Levi-Civita connection on T M, as well as its extension to Λ_{ℂ}^{∗}(M) (using the Leibniz rule). Then the second fundamental form of the canonical bundle K = Λ^{n,0}(M) ⊂ Λ_{ℂ}^{n}(M), with respect to the Levi-Civita connection, is of type (1, 0) (as in the previous Lemma).

Proof. In fact, the statement is true for all the sub-bundles Λ^{k,0}(M) ⊂ Λ_{ℂ}^{k}(M), k = 1, 2, …, n, and follows from the case k = 1. To see this, let θ_{1}, …, θ_{n} be a local framing of Λ^{1,0}(M), and

∇θ_{i} = Σ_{j} (α_{ij} ⊗ θ_{j} + β_{ij} ⊗ θ̄_{j}) , α_{ij}, β_{ij} ∈ Λ_{ℂ}^{1}(M).

Then for k = 1 the claim is that the 1-forms β_{ij} are of type (1, 0). If this is true, then for any k ≥ 1,

∇ (θ_{i_{1}} ∧ … ∧ θ_{i_{k}}) = = (∇θ_{i_{1}}) ∧ θ_{i_{2}} ∧ … ∧ θ_{i_{k}} + θ_{i_{1}} ∧ (∇θ_{i_{2}}) ∧ θ_{i_{3}} ∧ … ∧ θ_{i_{k}} + … = Σ_{j}β_{i_{1},j} ⊗ (θ̄_{j} ∧ θ_{i_{2}} ∧ … ∧ θ_{i_{k}}) + β_{i_{2},j} ⊗ (θ_{i_{1}} ∧ θ̄_{j} ∧ … ∧ θ_{i_{k}}) + …

+(something in Λ_{ℂ}^{1} ⊗_{ℂ} Λ^{k,0}),

so that the second fundamental form of Λ^{k,0}(M) ⊂ Λ_{ℂ}^{k}(M) is of type (1, 0).

Now for the case k = 1, i.e. to see that the 1-forms β_{ij} above are of type (1, 0), we argue as follows. First, we pick the frame θ_{1}, …, θ_{n} to be a unitary frame, i.e. (θ_{i}, θ̄_{j}) = δ_{ij}, It then follows that

0 = d(θ_{i}, θ_{j}) = (∇θ_{i}, θ_{j}) + (θ_{i}, ∇θ_{j}) = β_{ij} + β_{ji},

i.e. β_{ij} = −β_{ji}.

Next, by the torsion-freeness of ∇, we have

dθ_{i} = anti-symmetrization of ∇θ_{i} = Σ_{j} (α_{ij} ∧ θ_{j} + β_{ij} ∧ θ̄_{j}) .

Now, for an integrable almost-complex structure, we have the vanishing of the (0, 2)-component of d : Λ^{1,0} → Λ^{2}, hence, taking (0, 2)-components of the last equation we have

0 = Σ_{j}β′′_{ij} ∧ θ̄_{j},

where β′′_{ij} denotes the (0, 1)-component of β_{ij}. If we write β′′_{ij} = Σ_{k}β′′_{ijk}θ̄_{k}, then the last equation reads

0 = β′′_{ijk} − β′′_{ikj},

i.e. β′′_{ijk} = β′′_{ikj}, and this, combined with the previous β′′_{ijk} = −β′′_{jik} yields β′′_{ij} = 0 (we use here “the S_{3}-lemma”: any tensor T_{ijk} which is symmetric in one pair of indices and anti-symmetric in another pair is identically zero). QED

In summary, we have obtained the following:

Proposition 1 Let M^{2n} be a hermitian manifold (a riemannian manifold with an orthogonal complex structure). If we equip its canonical bundle Λ^{n,0}(M) with the connection induced by the Levi-Civita connection of M, then its curvature Ω is given by the formula

Ω = iR̃(ω) + Φ^{∗} ∧ Φ,

where ω is the Kähler 2-form associated with the hermitian structure, R̃ is the curvature operator associated to the riemannian structure (see Definition 4 in Section 2), and

Φ^{∗} ∧ Φ ≤ 0.

This last inequality means, recalling our sign conventions of Definition 2 in Section 2, that i(Φ^{∗} ∧ Φ) is a non-positive real (1, 1)-form.

Remark. There is an analogous statement for an almost-Kähler manifold, i.e. when the almost-complex structure is not necessarily integrable but the Kähler form is closed (so we have a symplectic manifold). The difference is that in this case the second fundamental form is of type (0, 1), hence the correction term Φ^{∗} ∧ Φ in the curvature formula is non-negative.

4 Some applications

As an immediate corollary to Proposition 1 we obtain the following result of LeBrun [2]:

Corollary 2 There is no complex structure on S^{6} which is orthogonal with respect to the round metric.

Proof. For the round metric on a sphere, R̃ is the identity operator. Therefore, for any orthogonal complex structure, the formula of Proposition 1 for the curvature of the canonical line bundle gives

Ω = iω + Φ^{∗} ∧ Φ ≤ iω < 0.

It follows that the closed 2-form Ω is non-degenerate, i.e. symplectic, which is impossible since H^{2}(S^{6}) = 0. QED

The next corollary extends the above conclusion to a C^{2}-neighbourhood of the round metric on S^{6}.

Corollary 3 Let g be a riemannian metric on S^{6} satisfying the following conditions:

• The curvature operator R̃ is positive (ie all its eigen values are positive).

• At each point x ∈ S^{6}, the ratio of the largest eigen-value λ_{max} of R̃ to the lowest eigen-value λ_{min} satisfies λ_{max}/λ_{min} < 7/5 = 1.4.

Then (S^{6}, g) does not admit an orthogonal complex structure.

The proof of the last corollary is based on the following

Lemma 4 Let V be a 2n-dimensional euclidean vector space with an orthogonal complex structure J, and let ω = (J·, ·) be the associated Kähler form. Let Ω_{0} be an imaginary (1, 1)-form satisfying Ω_{0} ≤ iω. (See Definition 2 in Section 2 for the sign convention. Note that in particular, since iω < 0, Ω_{0} is also negative, hence non-degenerate).

Then, if Ω is any imaginary 2-form satisfying

||Ω − Ω_{0}|| < \frac{1}{2\sqrt{n}},

Ω is non-degenerate.

Proof of Lemma 4. First, a brief reminder about norms. We use the euclidean norm on V to embed Λ^{2}(V^{∗}) ⊂ End(V) as antisymmetric endomorphisms: α ↦ A, where A is given by (v, Aw) = α(v, w). In fact, this is the inverse of our map of Definition 3 in Section 2, A ↦ Â = α.

Next, the euclidean structure on V induces a euclidean norm || · ||_{E} on End(V) by ||A||_{E}^{2} = Σ|A_{ij}|^{2}, where A_{ij} are the components of an element A ∈ End(V) with respect to an orthonormal basis of V. This norm is multiplicative, i.e. ||AB||_{E} ≤ ||A||_{E}||B||_{E}. Using this multiplicativity property, one can show that if A ∈ End(V) satisfies ||A||_{E} < 1, then I + A + A^{2} + ... is convergent, thus giving an inverse to I − A.

Unfortunately, the norm || · ||_{E} induces on Λ^{2}(V^{∗}) a norm which differs by a constant from the standard norm on Λ^{2}(V^{∗}): for any 2-form Ω, ||Ω||_{E} = \sqrt{2}||Ω||. For example, the Kähler form ω has (standard) norm \sqrt{n}, whereas the corresponding endomorphism, namely J, has norm \sqrt{2n}. In what follows, we will work with the || · ||_{E} norm on 2-forms.

Now, we can diagonalize ω and Ω_{0} simultaneously (over ℂ), obtaining

ω = iΣθ_{j} ∧ θ̄_{j}, Ω_{0} = Σλ_{j}θ_{j} ∧ θ̄_{j},

with {θ_{j}} a unitary frame, and the condition Ω_{0} ≤ iω implies λ_{j} ≤ −1. Then

Ω_{0}^{−1} = Σ\frac{1}{λ_{j}} θ_{j} ∧ θ̄_{j},

thus

||Ω_{0}^{−1}||_{E}^{2} = 2Σ|\frac{1}{λ_{j}} |^{2} ≤ 2n.

Now,

Ω = Ω_{0} + (Ω − Ω_{0}) = Ω_{0} (I + Ω_{0}^{−1}(Ω − Ω_{0})) ,

and our condition of ||Ω − Ω_{0}|| < 1/(2\sqrt{n}) translates to

||(Ω − Ω_{0})||_{E} < \frac{1}{\sqrt{2n}},

hence

||Ω_{0}^{−1}(Ω − Ω_{0})||_{E} ≤ ||Ω_{0}^{−1}||_{E} ||(Ω − Ω_{0})||_{E} < \sqrt{2n} · \frac{1}{\sqrt{2n}} = 1,

and so Ω is non-degenerate. QED

Proof of Corollary 3. Let us suppose there is a complex structure on S^{6} which is orthogonal with respect to a metric g whose curvature operator satisfies the said conditions. From Proposition 1, the curvature of the associated canonical line bundle is given by

Ω = iR̃(ω) + Φ^{∗} ∧ Φ = i(R̃ω − ω) + Ω_{0},

where Ω_{0} = iω + Φ^{∗} ∧ Φ ≤ iω. Now we apply the the previous lemma. We conclude that Ω is non-degenerate provided ||R̃ω−ω|| < 1/(2\sqrt{3}) (pointwise).

Now, by rescaling the metric if necessary (this does not affect of course the orthogonality of the complex structure), we can bring the eigen-values of R̃ to the range (5/6, 7/6), so that the eigen-values of R̃ − I are in the range (−1/6, 1/6). This implies that ||(R̃ − I)α|| < (1/6)||α|| (pointwise) for any α ∈ Λ^{2}(M), so in particular

||R̃ω − ω|| < \frac{1}{6}||ω|| = \frac{1}{6}\sqrt{3} = \frac{1}{2\sqrt{3}}.

And so, according to the lemma above, the closed 2-form Ω is non-degenerate, i.e. symplectic, which is impossible since H^{2}(S^{6}) = 0. QED

Remarks.

1. It is tempting to generalize Corollary 3 to the case of a hermitian structure on a 2n-dimensional manifold with a positive curvature operator satisfying λ_{max}/λ_{min} < (2n+1)/(2n−1). Unfortunately, such a generalization is useless, because of the well-known “sphere-theorem”, which implies that the universal cover of a complete riemannian manifold satisfying our curvature bound is a sphere, on which a complex structure is in question only in dimension 6 (because in all dimensions except 2 and 6 the n-sphere does not admit even an almost-complex structure), so we are back to our case.

2. However, we believe that one should be able to use Proposition 1 beyond what we have done here, because of the following argument. The condition of orthogonality of a complex structure with respect to a riemannian metric is obviously conformaly invariant. On the other hand, the curvature restriction in Corollary 3 is not conformaly invariant. Thus, Corollary 3 can be improved by including any metric on S^{6} which is conformal to a metric satisfying the given curvature condition, but one hopes for a more explicit condition, say in terms of the Weyl tensor. So far, we were not able to derive such a condition.

3. Another direction in which one could possibly use Proposition 1 is by applying it to some specific classes of hermitian structures. In such cases one may be able to give a more delicate estimate of the terms in the formula of Proposition 1, especially the R̃ω term.

References

[1] P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons, 1994.

[2] C. LeBrun, Orthogonal complex structures on S^{6}, Proc. Amer. Math. Soc. 101 (1987), 136-138.

[3] S. Salamon, Orthogonal complex structures, Diff. Geom. & App., Proc. Conf., Aug. 28 – Sept. 1 , 1995, Brno, Czech Republic, Massaryk University, Brno 1996, 103-117.

[4] K. Sekigawa and L. Vanhecke, Almost-hermitian manifolds with vanishing first Chern classes or Chern numbers, Rend. Sem. Mat. Univ. Pol. Torino 50, 2 (1992), 195-208.

[5] C. M. Wood, Harmonic almost-complex structures, Compositio Mathematica 99 (1995), 183-212.