Quaternionic Structures in Mathematics and Physics SISSA - Trieste, 5-9 September 1994

Instantons on Quaternion-Kähler Manifolds

Yasuyuki Nagatomo

Institute of Mathematics, University of Tsukuba

Abstract. On 4n-dimensional quaternion-Kähler manifolds, self-dual connections are defined which is the same as self-dual connections in 4-dimensional Riemannian geometry in the case n = 1. Every simply connected quaternion-Kähler manifold with non-zero scalar curvature admits only one standard self-dual connection in the case n ≧ 2. As for anti-self-dual connections, we have vanishing theorems on the Salamon twistor spaces. ADHM-construction are generalized on the quaternionic projective spaces and 2-plane complex Grassmannians. A few recent results are stated in the last section.

0. Introduction On quaternion-Kähler manifolds, Self-Dual connections are defined by Nitta [Ni] and

Mamone Capria-Salamon [M-S]. These correspond to Self-Dual connections in the 4dimensional case.

The purpose of this note is to give classification theorems for self-dual and anti-self-dual bundles and to describe the moduli spaces. This note consists of two parts. In the former part, the rigidity theorem for self-dual connections is proved. In the case of the quaternionic projective space, the extended BPST-instanton is defined in a similar way, because the standard metric of S^{4n+3} gives the canonical connection to the Hopf fibration S^{4n+3} ⟶ ℍP^{n}. However, any deformation preserving self-duality does not exist except n = 1. It is essential that the curvature form of an arbitrary self-dual connection is parallel in higher dimentional case. Consequently, it is easy to understand the holonomy algebra, which is a Lie algebra 𝔰𝔭(1). Then holonomy theorem yields our theorem.

In the latter part, we treat anti-self-dual connections and so the Salamon twistor space Z. In ADHM-construction on the 4-dimensional sphere and Buchdahl’s construction on the complex projective space, the following vanishing theorem plays an essential role.

H^{1}(Z, F̃(−2)) = 0,

where F̃ is the pull back bundle of an anti-self-dual bundle and O(−2) is a line bundle which is associated with the complex contact structure of the Penrose twistor space Z. In §3, the above vanishing theorem is generalized to an arbitrary compact quaternion-Kähler manifold with positive scalar curvature. We establish an analogue of the Penrose transform between the cohomology group H^{1}(Z, F̃(−2)) and the solution space of quaternionic operator which is a second-order operator. The positivity of this operator implies our vanishing theorem.

In §4, we use the above vanishing theorem to generalize ADHM-construction to an arbitrary quaternionic projective space. A monad method is available for our purpose as in the 4-dimensional case. To obtain a monad, we use Beilinson’s spectral sequence on the complex projective spaces. Then, further vanishing theorems are needeed. Our vanishing theorem combined with the Hirzebruch-Riemann-Roch formula gives sufficient conditions for a monad construction. The main idea is to intertwine even dimensional complex projective space between twistor spaces of quaternionic projective spaces.

In §5, a vanishing theorem H^{1}(Z, F̃(−2)) is extended. Mamone Capria and Salamon showed that there are elliptic complexes on quaternion-Kähler manifolds, and related them to Dolbeault complexes on the twistor space [M-S]. Baston extended these quaternionic complexes to include second order operators [B]. In this way, we have that there are onoto-one correspondence between the Dolbeault cohomology groups on the twistor space and the cohomology of the corresponding elliptic complex on a quaternion-Kähler manifold. This elliptic complex is composed of quaternionic operators. We define new operators on a quaternion-Kähler manifold to obtain relations between quaternionic operators. The Weitzenböck formula for our operators yields vanishing theorems.

In §6, a complex Grassmannian manifold of 2-planes are the base space. We generalize Buchdahl’s classification along the almost same line as in §4. Main difference are the following two points. Beilinson type spectral sequence is not known explicitly, and so we make spectral sequence for holomorphic vector bundles on the twistor space. On the other hand, the ring structure of the cohomology of the twistor space is complicated, compared with complex projective space. Consequently, to get further vanishing theorems, we prepare two divisors on the twistor space.

In §7, it is showed that the moduli space of “1-instantons” on Gr_{2}(ℂ^{n+2}) is identified with an open cone over ℙ(∧^{2}ℂ^{n+2}). This result is a generalization of Buchdahl and Donaldson. The monad which we obtain in §6 translates our problem into linear algebraic terms. The vertex of this moduli space represents a reducible connection. A point in the boundary manifold ℙ(∧^{2}ℂ^{n+2}) represents an ideal instanton whose support is a quaternionic submanifold ℍP^{i} (i = 0, ⋯, [\frac{n}{2}]) of Gr_{2}(ℂ^{n+2}).

In §8, recent results about instanton bundles are stated.

The results in §5,§6 and §7 are due to joint works with Professor Takashi Nitta.

1. Preliminaries Let (M, g) be a compact quaternion-Kähler manifold such that dim_{ℝ}M = 4n. The vector bundle ∧^{2}T^{∗}M has the following holonomy invariant decomposition:

∧^{2}T^{∗}M = S^{2}ℍ ⊕ S^{2}𝔼 ⊕ (S^{2}ℍ ⊕ S^{2}𝔼)^{⊥},

where ℍ and 𝔼 are (local) vector bundles associated with the standard representations of Sp(1) and Sp(n), respectively. (This decomposition can also be explained in terms of the Hodge ∗-operator as in the 4-dimensional case. See [G-P].)

Definition 1.1. [Ni, M-S, G-P] An ω ∈ Γ(∧^{2}T^{∗}M) = Ω^{2}(M) is called a Self-Dual (SD) form or an Anti-Self-Dual (ASD) form if

ω ∈ Γ(S^{2}ℍ) or Γ(S^{2}𝔼),

respectively.

We shall treat metric connections on a complex vector bundle E equipped with a hermitian metric h over M.

Definition 1.2. [Ni, M-S, G-P] A connection ∇ is called SD or ASD if its curvature 2-form R^{∇} is a SD form or an ASD form, respectively..

Theorem 1.3. [Ni, M-S, G-P] Every SD (or ASD ) connection is a Yang-Mills connection.

Remark. Moreover, if M is compact, then an arbitrary SD or ASD connection minimizes the Yang-Mills functional [M-S, G-P].

2. Rigidity of Self-Dual connections First of all, we state a result about Self-Dual connections. Here, the dimension of M

is assumed to be greater than or equal to 8 (i.e n > 1.) We give a few examples of SD connections.

Example 2.1. A quaternion-Kähler manifold M is an Einstein manifold. Using this fact, we can show that the induced connection D on S^{2}ℍ is a SD connection in the same way as 4-dimensional case.

Example 2.2. The quaternionic projective space ℍP^{n} is a typical example of a compact quaternion-Kähler manifold. The Hopf fibration S^{4n+3} → ℍP^{n} is a principal Sp(1)- bundle. The canonical connection induced by the standard metric on S^{4n+3} is a SD connection.

A key property of SD connections is as follows.

Lemma 2.3. Let E be a vector bundle over M with a connection ∇. If S is a section of End(E) ⊗ S^{2}ℍ which satisfies the Bianchi’s identity d^{∇}S = 0, then S is parallel with respect to the induced connection ∇.

From the definition of SD connection, we get

Corollaly 2.4. Any Self-Dual connection is a parallel Yang-Mills connection.

Let P be a reduced principal Sp(n) · Sp(1)-bundle associated to the tangent bundle T M. If we define a SO(3)-bundle P′ = P/Sp(n), then P′ × 𝔰𝔬(3) is nothing but S^{2}ℍ. A principal bundle P′ has a connection induced by the Riemannian connection D of M, which is denoted by φ. The next theorem asserts that φ is the only one SD connection.

Theorem 2.5. Let M be a simply-connected quaternion-Kähler manifold of dimension 4n > 4 which has a non-zero scalar curvature. Assume that (E, h) is a rank r vector bundle with a metric h. If (E, h) admits a metric non-flat Self-Dual connection ∇, then (E, ∇) can be expressed as follows.

(i) Suppose that the structure group of the bundle S^{2}ℍ can be lifted to Sp(1). In other words, there exists a principal Sp(1)-bundle P̃′ whose adjoint bundle is the bundle S^{2}ℍ. (See Example 2.2.) In this case, we have a representation ρ of Sp(1) such that

E ≅ P̃′ ×_{ρ} ℂ^{r} and ∇ = ρ_{∗}φ̃,

where φ̃ denotes the connection in P̃′ induced from the connection φ in P′, (ii) Otherwise, there exists a representation ρ of SO(3) such that

E ≅ P′ ×_{ρ} ℂ^{r} and ∇ = ρ_{∗}φ.

Remark 1. Salamon showed that a compact quaternion-Kähler manifold which has a positive scalar curvature is simply connected [S].

Remark 2. Combined with a result of Salamon [S], a compact quaternion-Kähler manifold with a positive scalar curvature can not admit any Sp(1)-Self-Dual connections except the quaternionic projective space.

For more details, see [Na1].

3. Vanishing Theorem Next, we focus attention on ASD connections. Salamon showed that there is a twistor

fibration π : Z → M over an arbitrary quaternion-Kähler manifold M [S]. The total space Z with compact fibre ℙ^{1} has naturally a complex structure in the same way as 4-dimensional case. For example, a complex projective space ℙ^{2n+1} is the twistor space of ℍP^{n}. To construct and classify ASD bundles E on ℍP^{1} ≅ S^{4}, Atiyah, Drinfeld, Hitchin and Manin make use of the twistor space ℙ^{3}. The crucial step in this work is the vanishing of the sheaf cohomology group

H^{1}(ℙ^{3}, Ẽ(−2)) = 0,

where Ẽ is the pull-back bundle on ℙ^{3}. (We do not distinguish between a vector bundle and the associated locally free sheaf.) First, the above vanishing theorem was proved by Drinfeld-Manin [D-M] and Rawnsley [R] in the case of S^{4}. Hitchin gave a generalization to a compact half-conformally-flat manifold with positive scalar curvature [H].

In higher dimensional case, Nitta and Mamone Capria-Salamon showed the following.

Theorem 3.1 [Ni, M-S]. Any ASD bundle on M pulls back to a holomorphic bundle on Z.

So, if we attempt to generalize ADHM-construction, it is natural to ask whether certain cohomology groups vanish. Here the author introduces a summary of [Na2], in which the following theorem is proved.

Theorem 3.2. Let M be a compact quaternion-Kähler manifold with positive scalar curvature and E be an ASD bundle on M with a unitary structure. If the twistor space of M is denoted by Z and the pull-back bundle of E on Z is denoted by Ẽ, then we have

H^{1}(Z, Ẽ ⊗ L^{−2}) = 0,

where the line bundle L, which is introduced by Salamon [S], restricted to an arbitrary fiber (or real line ) is the hyperplane bundle on ℙ^{1}.

outline of proof. We identify H^{1}(Z, Ẽ⊗L^{−2}) with the corresponding Dolbeault cohomology group. If we fix a base point x ∈ M, we get a transformation T : H^{0,1}(Z, Ẽ ⊗L^{−2}) → Γ(E) in the following way.

H^{0,1}(Z, Ẽ ⊗ L^{−2}) ⟶_{restriction}H^{0,1}(ℙ_{x}, O(−2)) ⊗ E_{x}
∼

Serre duality

H^{1,0}(ℙ_{x}, O(2))^{∗} ⊗ E_{x}
≅H^{0}(ℙ_{x}, O)^{∗} ⊗ E_{x}
≅E_{x}.

It can be showed that the horizontal part of the element of the above Dolbeault cohomology group is completely determined in terms of vertical part, from the complex structure of the twistor space Z. Hence, a transformation T is injective.

An element of H^{0,1}(Z, Ẽ ⊗ L^{−2}) is denoted by ω. We put

T (ω) = s ∈ Γ(E).

Then \overline{∂}ω = 0 implies that

(∇^{∗}∇ + 4nct)s = 0,

where ∇^{∗}∇ is the Laplace operator, c is a positive constant depending only on n ([S]) and t is the scalar curvature. From the hypothesis of theorem 3.1, ∇^{∗}∇ + 4nct is positive as an operator. So, the equation (∇^{∗}∇ + 4nct)s = 0 has only a trivial solution. □

Remark. The above operator is explained in terms of a Dirac operator and a twistor operator in 4-dimensional case [H]. In higher dimensional case, we can define a Dirac-like operator and a twistor operator (see also [S-2] and [B]). Moreover, the above operator ∇^{∗}∇ + 4nct is also closely related to these operator via the Weitzenböck formula. Using these operators, we can show various vanishing theorems which will be explained in the section 5.

Since Ẽ restricted to an arbitrary fibre ℙ^{1} is holomorphically trivial, we obtain the following.

Corollary 3.3. If we put M = ℍP^{n} in Theorem 3.2, then

H^{1}(ℙ^{2n+1}, Ẽ(k)) = 0, k ≦ −2,

where E(k) means E ⊗ O(k).

4. k-instantons on ℍP^{n} In this section, we shall confine ourselves exclusively to a quaternionic projective space

ℍP^{n}. We treat a bundle E on ℍP^{n} satisfying the following.

(1) E has an ASD Sp(r)-connection (r ≧ 2), and

(2) its total Chern clss c(E) = (1 − x)^{−k}, where k ∈ ℤ_{+} and x is the standard positive generator of H^{4}(ℍP^{n}, ℤ).

Remark. When r = n and k = 1 , the pull-back bundle Ẽ on ℙ^{2n+1} is the null correlation bundle. In general, if r = n, Ẽ is called a mathematical instanton bundle [O-S, S-T, A-O]. On the other hand, in the case n = 1, E is a k-instanton bundle on ℍP^{1} ≅ S^{4}.

Then, the following vanishing theorem holds (See [K-N]).

Theorem 4.1.

H^{i}(ℙ^{2n+1}, Ẽ(p)) = 0 for 2 ≦ i ≦ n and p ∈ ℤ,

Remark 1. Applying Serre duality, from a symplectic structure of E, we know

H^{i}(ℙ^{2n+1}, Ẽ(p)) = 0 for 2 ≦ i ≦ 2n − 1 and p ∈ ℤ,

Remark 2. Theorem 4.1 is proved under a weaker condition with respect to its total Chern class such as

c_{2}(E) = k, c_{4}(E) = \frac{1}{2}k(k + 1),

because we make use of the Hirzebruch-Riemann-Roch theorem only on ℙ^{4}.

Combined with Beilinson’s spectral sequence (see [O-S-S p.240]), our vanishing theorem (Corollary 3.3 and Theorem 4.1) yields the next classification theorem [K-N].

Theorem 4.2. Ẽ is the cohomology bundle of a monad of the type:

W (−1) ⟶ V ⟶ W^{∗}(1),

where W^{∗} = H^{1}(ℙ^{2n+1}, Ẽ(−1)) , V = H^{1}(ℙ^{2n+1}, Ẽ ⊗ Ω^{1}) (see [C-S],or[O-S]).

and dim_{ℂ}H^{1}(ℙ^{2n+1}, Ẽ(−1)) = k.

Remark 1. A ”monad” is a complex of vector bundles

A ⟶^{a} B ⟶^{b} C,

with homomorphisms a and b between them, such that a is injective and b is surjective. The quotient bundle

E = Kerb/Ima

is called the cohomology of the monad.

Remark 2. If the base space is ℍP^{1} ≅ S^{4}, the monad in Theorem 4.2 corresponds to one used in ADHM-construction (See [A] or [B-H]).

To prove Theorem 4.1, we employ an algebraic geometrical technique and induction with respect to the dimension of the base space. The main idea is to intertwine the twistor spaces with even dimensional complex projective space. Explicitly, we consider the following diagram,

ℙ^{2n−1} ⟶ ℙ^{2n+1}
π↓  ↓^{π}
ℍP^{n−1} ⟶ ℍP^{n}

and the sequence ℙ^{2n−1} ⊂ ℙ^{2n} ⊂ ℙ^{2n+1}, where ℙ^{2n−1} and ℙ^{2n+1} are the corresponding twistor spaces, respectively. Making use of the standard sequence of sheaves

0 ⟶ O(−1)|_{ℙ^{2n}} ⟶ O|_{ℙ^{2n}} ⟶ O|_{ℙ^{2n−1}} ⟶ 0,

0 ⟶ O(−1)|_{ℙ^{2n+1}} ⟶ O|_{ℙ^{2n+1}} ⟶ O|_{ℙ^{2n}} ⟶ 0,

we get imformation on ℍP^{n} from one on ℍP^{n−1}.

5. Vanishing Theorem II Baston showed that there are elliptic complexes on compact quaternion-Kähler man-

ifolds with positive scalar curvature ([B] and see also [S-2]). The elliptic complexes are related to the Dolbeault complexes on the twistor spaces. Here we state vanishing theorems for these elliptic complexes, which are generalizations of Theorem 3.2.

We denote by S^{k}ℍ and ∧^{i}𝔼 kth-symmetric product of ℍ and i-th skew-symmetric product of 𝔼, respectively. Recall the Clebsch-Gordan formula

S^{k}ℍ ⊗ ℍ ≅ S^{k−1}ℍ ⊕ S^{k+1}ℍ.

The tensor product S^{k}ℍ ⊗ ∧^{i}𝔼 ⊗ ℍ ⊗ 𝔼 involves the two components S^{k−1}ℍ ⊗ ∧^{i+1}𝔼 and S^{k+1}ℍ ⊗ ∧^{i+1}𝔼. Using the contraction Cont : S^{k}ℍ ⊗ ∧^{i}𝔼 ⊗ ℍ ⊗ 𝔼 → S^{k−1}ℍ ⊗ ∧^{i+1}𝔼, and the symmetrization Sym : S^{k}ℍ ⊗ ∧^{i}𝔼 ⊗ ℍ ⊗ 𝔼 → S^{k+1}ℍ ⊗ ∧^{i+1}𝔼, We define two types of differential operators on M (cf. [S-2] and [B]).

Definition 5.1. Quaternionic operators D and D are defined by

D_{k} =Cont∇ : Γ(S^{k}ℍ ⊗ ∧^{i}𝔼) → Γ(S^{k−1}ℍ ⊗ ∧^{i+1}𝔼),
D_{k} =Sym∇ : Γ(S^{k}ℍ ⊗ ∧^{i}𝔼) → Γ(S^{k+1}ℍ ⊗ ∧^{i+1}𝔼).

Remark 1. When no confusion can arise, we will omit the subscript k.

Remark 2. In case of n = 1, the above operators D and D are nothing but the Dirac operator and the twistor operator, respectively.

As quaternionic analogues of (complex) Kähler geometry, we define

L : S^{k}ℍ ⊗ ∧^{i}𝔼 → S^{k}ℍ ⊗ ∧^{i+2}𝔼

as L(ϕ) =_{def.} ϕ ∧ ω_{𝔼}, where ω_{𝔼} is the section of ∧^{2}𝔼 determined by an invariant symplectic

form [S]. The operator Λ : S^{k}ℍ⊗∧^{i+2}𝔼 → S^{k}ℍ⊗∧^{i}𝔼 denotes the formal adjoint operator of L.

Baston proved the following theorem.

Theorem 5.2. ([B] and see also [S-2]) There exist elliptic complexes on a compact quaternion-Kähler manifold M such that

B_{k} : 0 → S^{k−2}ℍ →_{D} S^{k−3}ℍ ⊗ 𝔼 →_{D} ⋯ →_{D} ∧^{k−2}𝔼

→_{△} ∧^{k}𝔼 →_{D} ℍ ⊗ ∧^{k+1}𝔼 →_{D} ⋯ →_{D} S^{2n−k}ℍ → 0,

where

△ = −DD + 4ctL.

Moreover, if E is an ASD bundle on M, we can couple these complexes with E.

From now on, the author present a joint work with T.Nitta [N-N1]. To investigate these complexes, we denote by D^{∗} and D^{∗} the formal adjoint operator of D and D respectively. Then we have the following formulae.

Proposition 5.3.

D_{m+1}D_{m} + \frac{m + 2}{m + 1}D_{m−1}D_{m} = 2ct(m + 2)L,
[Λ, L] = (n − i)id.,
[L, D^{∗}] = −D,
[D, Λ] = −D^{∗},
[D, Λ] = D^{∗},
[L, D^{∗}] = D.

Using these formulae, we get the following.

Theorem 5.4. Let M be a compact quaternion-Kähler manifold (dimM = 4n) with positive scalar curvature and F be a unitary anti-self-dual bundle on M. Then

H^{i}(Γ(M, F ⊗ B_{k})) = 0 for i < k − 2 or (i, k) = (0, 2), (1, 3)
and i ≦ n − 1.

Combined with the Penrose transform by Baston ([B]) which yields H^{i}(Γ(M, E⊗B_{k})) = H^{i+1}(Z, Ẽ ⊗ L^{−k}), Theorem 5.4 implies vanishing theorems for the Dolbeault cohomology groups.

Corollaly 5.5. Let M be a 4n-dimensional compact quaternion-Kähler manifold with positive scalar curvature and Z be the twistor space of M. If Ẽ is the pull back bundle of E on M which has a unitary structure and an anti-self-dual connection, then we have

H^{i}(Z, Ẽ ⊗ L^{−k}) = 0, for 1 ≦ i ≦ n and i + k + 1 < 0.

and

H^{2}(Z, Ẽ ⊗ L^{−3}) = 0 (if n ≧ 2).

Applying Serre duality, we obtain

Corollary 5.6. Under the same assumption in Corollary 5.5, the following vanishing theorems hold:

H^{i}(Z, Ẽ ⊗ L^{−k}) = 0, for n + 1 ≦ i ≦ 2n and i + k > 0

and

H^{2n−1}(Z, Ẽ ⊗ L^{−2n+1}) = 0 (if n ≧ 2).

Remark. Under the same argument, vanishing theorems can be obtained for M with negative scalar curvature. Combined with the Penrose transform due to Mamone Capria-Salamon [M-S], these results generalize M.Thornber’s vanishing theorem [T].

Theorem 5.7. If M is a compact quaternion-Kähler manifold with negative scalar curvature (dim M = 4n) and Ẽ is the pull-back of a unitary anti-self-dual bundle on M, then

H^{i}(Z, Ẽ ⊗ L^{−k}) = 0 for k + i ≧ 1 and i ≦ n − 1.

Remark. Horan extends independently Thornber’s theorem [Ho].

6. k-instantons on Gr_{2}(ℂ^{n+2})

The twistor space of Gr_{2}(ℂ^{n+2}) is a flag manifold

F_{n} =_{def.} {(l, V)|0 ∈ l ⊂ V ⊂ ℂ^{n+2}, dim_{ℂ}l = 1, dim_{ℂ}V = n + 1},

(see [S]). Here we work on Gr_{2}(ℂ^{4}), because no big change arises in higher dimensional case and see [N-N2] for more details.

An F_{2} has double (holomorphic) fibrations to ℙ^{3} and ℙ^{3}^{∗} such that

p_{1} : (l, V) ⟶ [l]
p_{2} : (l, V) ⟶ [V].

We denote by x and y pull-back elements of H^{2}(F_{2}, ℚ) of the standard positive generators of H^{2}(ℙ^{3}, ℤ) and H^{2}(ℙ^{3}^{∗}, ℤ) respectively. Then the cohomology ring H^{∗}(F_{2}, ℚ) is generated by x and y. On the other hand, the cohomology ring H^{2}(G_{2}(ℂ^{4}), ℚ) is regarded as a subring of H^{2}(F_{2}, ℚ) by the twistor fibration. Hence, an element of H^{2}(Gr_{2}(ℂ^{4}), ℚ) may be written with x and y.

We consider a bundle E over Gr_{2}(ℂ^{4}) satisfying the following.

(1) E has an ASD Sp(r)-connection (r ≧ 1), and

(2) c_{2}(E) = k(x^{2} + y^{2}) and c_{4}(E) = k^{2}x^{2}y^{2}, where k ∈ ℤ.

Remark. When n = 1, the bundle E is a k-instanton bundle on ℙ^{2} which Buchdahl classified by using a monad on F_{1} [Bu].

In the case of ℍP^{n}, a monad can be obtained via vanishing theorems and Beilinson’s spectral sequence on ℙ^{2n+1} in section 4. Instead of Beilinson’s spectral sequence, we have a Beilinson-type spectral sequence on F_{n}. To represent this spectral sequence, we define vector bundles on F_{n}.

Definition 6.1. A vector bundle Q_{1} denotes the quotient bundle of p_{1}^{∗}Ω_{ℙ^{n+1}} by O(−1, −1), where Ω_{ℙ^{n+1}} is the holomorphic cotangent bundle on ℙ^{n+1} and in general, O(p, q) is the line bundle p_{1}^{∗}O(p) ⊗ p_{2}^{∗}O(q). The quotient bundle Q_{2} is defined in a similar way.

0 ⟶ O(−1, −1) ⟶p_{1}^{∗}Ω_{ℙ^{n+1}} ⟶ Q_{1} ⟶ 0
0 ⟶ O(−1, −1) ⟶p_{2}^{∗}Ω_{ℙ^{n+1}^{∗}} ⟶ Q_{2} ⟶ 0.

Proposition 6.2. For an arbitrary holomorphic vector bundle S on F_{n}, there exists a spectral sequence E_{r}^{p,q} converging to

E_{∞}^{p,q} = \{ \begin{matrix}∑_{p=0}^{2n+1} E_{∞}^{−p,p} = S, & if p + q = 0 \\ 0, & otherwise. \end{matrix}

The E_{1}-terms satisfy exact sequences

⋯ → ∑_{r=0}^{p−1} H^{q}(F_{n}, ∧^{r}Q_{1}^{∗} ⊗ S(−p, 0)) ⊗ ∧^{p−1−r}Q_{2}^{∗}(0, −p)
→E_{1}^{−p,q}
→ ∑_{r=0}^{p} H^{q}(F_{n}, ∧^{r}Q_{1}^{∗} ⊗ S(−p, 0)) ⊗ ∧^{p−r}Q_{2}^{∗}(0, −p) → · · · .

where, for example, S(p, q) means S ⊗ O(p, q) as usual.

Using a similar (but more complicated) argument as Theorem 4.2, we have the classification theorem.

Theorem 6.3. Ẽ is the cohomology bundle of the following monad,

W_{1}(−1, 0) ⊕ W_{2}(0, −1) → V → W_{1}^{∗}(1, 0) ⊕ W_{2}^{∗}(0, 1),

whrere W_{1}^{∗} = H^{1}(F_{2}, Ẽ(−1, 0)), W_{2}^{∗} = H^{1}(F_{2}, Ẽ(0, −1)) and V is a trivial bundle of rank 4k + 2r. (dim_{ℂ}H^{1}(F_{2}, Ẽ(−1, 0)) = dim_{ℂ}H^{1}(F_{2}, Ẽ(0, −1)) = k).

Remark. If the base space is Gr_{2}(ℂ^{3}) ≅ ℙ^{2}^{∗}, the monad in Theorem 6.3 is what is used in Buchdahl’s construction (See [Bu]).

7. Moduli of 1-instantons on Gr_{2}(ℂ^{n+2})

For simplicity, we will restrict our attention to 1-instantons on G_{2}(ℂ^{4}). Then, the monad given in section 6 is

O(−1, 0) ⊕ O(0, −1) → V → O(1, 0) ⊕ O(0, 1).

Note that the rank of V is 4 + 2r. However, it is proved that the condition H^{0}(F_{2}, Ẽ) = 0 implies that 4 + 2r ≦ 8. If Ẽ has a non-trivial section, it is proved that E contains a trivial summand, so E can not have Sp(r)-connection. Moreover, when r = 1, there does not exist a monad of the above type. Consequently, the value of r must be 2. For this argument and a proof of the following theorem, see [N-N3].

Theorem 7.1. The moduli of 1-instanton bundles on G_{2}(ℂ^{4}) is identified with a ( real ) open cone over ℙ^{5} (i.e. not containing the boundary mainfold ℙ^{5} ). Hence, the ( real ) dimension of the moduli equals to 11.

The vertex of this cone represents a direst sum E_{1} ⊕ E_{2} (i.e. a reducible connection). These bundles are the cohomology bundles of monads:

Ẽ_{1} : O(0, −1) →V_{1} → O(1, 0),
Ẽ_{2} : O(−1, 0) →V_{1}^{∗} → O(0, 1).

We also have isomorphisms

Ẽ_{1} ≅ Q_{1}(1, 0) and Ẽ_{2} ≅ Q_{2}(0, 1).

Remark. By Bott-Borel-Weil theorem,

H^{1}(F_{2}, End(Ẽ_{1} ⊕ Ẽ_{2})) = ∧^{2}ℂ^{4} ⊕ ∧^{2}ℂ^{4}^{∗}.

On the other hand, we choose a point p of the boundary of the moduli. The vertex of the cone and p are linked by a straight line l with a parameter 0 ≦ t ≦ 1 and l(0) =the vertex. Then, in the limit (t → 1), two cases occur depending on p.

(1) The curvature density converges to a one point in Gr_{2}(ℂ^{4}).

(2) The curvature density converges to a quaternionic submanifold ℍP^{1} of Gr_{2}(ℂ^{4}).

In higher dimensional cases, a quaternionic submanifold ℍP^{k} of G_{2}(ℂ^{n+2}) appears as the limiting manifold in the above sense.

8. Final Remarks

Remark 1. We can determine irreducible homogeneous bundles with anti-self-dual canonical connections on compact quaternion symmetric spaces G/K in terms of weights, where G is the compact real form of the simply connected complex simple Lie group. Moreover, a monad can be obtained from representation spaces of G. Consequently, ADHMconstruction is generalized from the viewpoint of the representation theory. In other words, one of the way of the deformation of the canonical connection is given on every compact quaternion symmetric space. In the simplest cases including “1-instantons on ℍP^{n} and Gr_{2}(ℂ^{n+2})”, we can describe moduli spaces explicitly using the Bott-Borel-Weil theorem. (For more details, see [Na3].)

Remark 2. On Gr_{2}(ℂ^{n+2}), we have another type of instanton bundles. The structure group of this bundle is SU(n + 1). In the case n = 1, these are the same as bundles in §6, because of the isomorphism SU(2) ≅ Sp(1). The moduli of these SU(n + 1)-instanton is identified with an open cone over ℙ(ℂ^{n+2}), where ℂ^{n+2} is the standard representation space of SU(n + 2) [Na4].

Remark3. Making use of a result of LeBrun-Salamon [L-S] and the Bott-Borel-Weil theorem, we can classify complex line bundles with an anti-self-dual connection over compact quaternion Kähler manifolds. In fact, unless the base manifold is Gr_{2}(ℂ^{n+2}), the only trivial complex line bundle can admit an anti-self-dual connection, which is the product connection. An anti-self-dual connection on a complex line bundle over Gr_{2}(ℂ^{n+2}) is necessaly the canonical connection on some homogeneous line bundle.

Remark4. The vector bundles E_{1} and E_{2} in §7 can admit the only one anti-self-dual connections, respectively. These are also the canonical connections.

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Tsukuba-shi, Ibaraki 305, Japan

   E-mail address: nagatomo@math.tsukuba.ac.jp