Atiyah–Jones conjecture for blown-up surfaces

E. Gasparim and R. J. Milgram

Abstract

We show that if the Atiyah–Jones conjecture holds for a surface X, then it also holds for the blow-up of X at a point. Since the conjecture is known to hold for ℙ^{2} and for ruled surfaces, it follows that the conjecture is true for all rational surfaces.

Given a 4-manifold X, let MI_{k}(X) denote the moduli space of rank 2 instantons on X with charge k and let C_{k}(X) denote the space of all connections on X with charge k. In 1978, Atiyah and Jones [AJ] conjectured that the inclusion MI_{k}(X) → C_{k}(X) induces an isomorphism in homology and homotopy through a range that grows with k. The original statement of the conjecture was for the case when X is a sphere, but the question readily generalizes for other 4-manifolds.

The stable topology of these moduli spaces was understood in 1984, when Taubes [Ta] constructed instanton patching maps t_{k}: MI_{k}(X) → MI_{k+1}(X) and showed that the stable limit lim_{k→∞} MI_{k} indeed has the homotopy type of C_{k}(X). However, understanding the behavior of the maps t_{k} at finite stages is a finer question. Using Taubes’ results, to prove the Atiyah–Jones conjecture it would now suffice to show that the maps t_{k} induce isomorphism in homology and homotopy through a range.

In 1993, Boyer, Hurtubise, Milgram and Mann [BHMM] proved that the Atiyah–Jones conjecture holds for the sphere S^{4} and in 1995, Hurtubise and Mann [HM] proved that the conjecture is true for ruled surfaces.

In this paper we show that if the Atiyah–Jones conjecture holds true for a complex surface X then it also holds for the surface X̃ obtained by blowingup X at a point. In particular, it follows that the conjecture holds true for all rational surfaces.

Kobayashi and Hitchin gave a one-to-one correspondence between instantons on a topological bundle E over X and holomorphic structures on E, see [LT]. Using this correspondence we translate the Atiyah–Jones conjecture into the language of holomorphic bundles and compare the topologies of the moduli spaces 𝔐_{k}(X) and 𝔐_{k}(X̃) of stable holomorphic bundles on X, resp. X̃, having c_{1} = 0 and c_{2} = k.

The structure of the proof is the following. First we give a concrete description of instantons on ℂ̃^{2} (section 1) and compute its numerical invariants (section 2). We then study moduli spaces 𝔐_{k}(X̃) of instantons or equivalently stable bundles on X̃, with respect to a polarization L̃ = NL − ℓ where L is a polarization on X. We show that removing the singular points of this moduli space does not affect our homology calculations and thereafter work only with its smooth points. Moreover, the direct image of a stable bundle might yield an unstable bundle, which does not correspond to an instanton on X. We show that removing such unstable bundles from 𝔐_{k}(X) does not affect our homology calculations either (section 3).

We show that any instanton on X̃ is uniquely determined by holomorphic patching of an instanton on X and an instanton on ℂ̃^{2} (section 4). We then prove that the moduli space of (framed) instantons on ℂ̃^{2} with charge i is smooth and of complex dimension not greater than r_{i}(2r_{i}+1) where r_{i} = ⌈\sqrt{i}⌉ (section 5). As a consequence we have a stratifications

𝔐_{k}^{f}(X̃) ≃ ⋃_{i=0}^{k} 𝔐_{k−i}^{f}(X) × N_{i}^{f}

and we show that Taubes’ map 𝔐_{k} → 𝔐_{k+1} is homotopically equivalent to a map that preserves the stratifications (section 6). Using the Leray spectral sequences for these stratifications, we then show that Atiyah–Jones conjecture for X̃ follows from the corresponding statement for X.

1. Moduli of bundles on ℂ̃^{2} with fixed splitting type Let π: X̃ → X be the blow-up of a point x on a compact surface X and

let ℓ be the exceptional divisor. In this paper all bundles have rank 2 and c_{1} = 0. Given a bundle E over X we pull it back to X̃ and then modify it by gluing in new data near the exceptional divisor to construct a bundle Ẽ satisfying Ẽ|_{X̃−ℓ} ≃ E|_{X−x}. The difference of Chern classes c_{2}(Ẽ) − c_{2}(E) depends only on the restriction of Ẽ to a small neighborhood N(ℓ) of ℓ. To begin with, suppose that N(ℓ) is isomorphic to the blow-up of ℂ^{2} at the origin, denoted ℂ̃^{2}. Given a rank two bundle V on ℂ̃^{2} with vanishing first Chern class, there exists an integer j ≥ 0 determined by the restriction of V to the exceptional divisor, such that V |_{ℓ} ≃ O_{ℙ^{1}}(j) ⊕ O_{ℙ^{1}}(−j). This j is called the splitting type of the bundle.

Theorem [G3] Every holomorphic rank two bundle V over ℂ̃^{2} with vanishing first Chern class and splitting type j is an algebraic extension of the form

0 → Oℂ̃^{2}(−j) → V → Oℂ̃^{2}(j) → 0

and the extension class can be represented as a polynomial of degree at most 2j − 2.

It follows that the local (= over N(ℓ)) moduli problem can be studied by considering extensions of line bundles modulo bundle isomorphism even though this is definitely not the case for bundles on the compact surface X̃. The fact that V is determined by an extension of degree 2j − 2 implies that V is determined over N(ℓ) by its restriction to the 2j − 2nd formal neighborhood of the exceptional divisor. Hence, our initial assumption that N(ℓ) ≃ ℂ̃^{2} is not restrictive. Consider the quotient space

M_{j}: = {bundles on ℂ̃^{2} with splitting type j}/ ∼,

where ∼ denotes bundle isomorphism. M_{j} is topologized as follows: We first choose canonical coordinates for N(ℓ) = U ∪ V, where U ≃ ℂ^{2} has coordinates (z, u) and V ≃ ℂ^{2} has coordinates (ξ, v) with (ξ, v) = (z^{−1}, zu) in U ∩ V ≃ ℂ − 0 × ℂ. Then, in these coordinates, we write the extension class as a complex polynomial with n = 2j(j + 1) coordinates (see [G3]). In such canonical coordinates, elements of M_{j} are represented by transition matrices of the form

(\begin{matrix}z^{j} & p \\ 0 & z^{−j} \end{matrix}) .  (∗)

We set p ∼ p′ if the corresponding bundles are holomorphically equivalent. This construction identifies M_{j} to a quotient of ℂ^{n} under a group action and endows M_{j} with the structure of a quotient stack. For more on the structure of M_{j} see [G3] and [G5]. For a decomposition of M_{j} into Hausdorff strata see [BG2]. To study the moduli problem we calculate numerical invariants.

2. Computation of local numerical invariants In §1 we saw that an element V ∈ M_{j} is determined by its splitting type

j and extension class p. We write V = V (j, p). It is very important to note that the bundle associated to V is trivial on the complement of ℓ.

Lemma [G2] Every holomorphic bundle on ℂ̃^{2} with vanishing first Chern class is trivial on ℂ̃^{2} minus the exceptional divisor.

It follows that the local bundle V (j, p) can be glued to any bundle E pulled back from X to form a new bundle Ẽ on X̃. We also note that the topology of Ẽ does not depend on the attaching map. In fact, the attaching is given by a holomorphic map ψ: ℂ^{2} − {0} → Sl(2, ℂ). By Hartog’s theorem ψ extends to the origin, and therefore has degree zero, thus not contributing to the Chern numbers of Ẽ. Hence 2 pairs of data (E, V, p, ψ) and (E, V, p, ψ′) determine the same topological bundle, although in general they determine distinct holomorphic bundles.

Lemma [G4] Every holomorphic rank two bundle Ẽ on X̃ with c_{1}(Ẽ) = 0 is topologically determined by a triple (E, j, p) where E is a bundle over X, j is a non-negative integer and p is a polynomial.

Here E = π_{∗}(Ẽ)^{∨∨} and the Chern class difference c = c_{2}(Ẽ) − c_{2}(E) depends only on the local data V (j, p). We call c the charge of the bundle near the exceptional divisor. The terminology comes from instanton moduli spaces, where c represents the change in topological charge of the instanton on X̃ given by the patching of V (j, p). Friedman and Morgan [FM] give the bounds j ≤ c_{2}(Ẽ) − c_{2}(E) ≤ j^{2}.

Theorem [G1] The bounds j ≤ c_{2}(Ẽ) − c_{2}(E) ≤ j^{2} are sharp.

Using elementary transformations, we obtain that all intermediate values occur. The following result implies non-emptiness of the strata appearing in the stratification on §6.

Theorem [BG1] For every integer k satisfying j ≤ k ≤ j^{2} there exists a holomorphic bundle on X̃ with splitting type j and such that c_{2}(Ẽ)−c_{2}(E) = k.

For the global moduli problem on X̃ we need to identify the spaces N_{i} of isomorphism classes of bundles on a neighborhood of the exceptional divisor that have fixed charge i. We define

N_{i} = {bundles on ℂ̃^{2} having charge i}/ ∼ .

Proposition 2.1 N_{i} has complex dimension 2i − 3.

Proof. When j = i the generic sets of M_{j} and N_{i} are the same, and are identified with ℙ^{2i−3} minus a complex subvariety of codimension at least two [BG3,Thm. 3.5]. The set of all additional points on N_{i} has strictly lower dimension [BG1, Thm. 4.1].

Remark: Although being quite distinct, M_{j} and N_{j} share the same generic set. On the former j denotes the splitting type, whereas on the latter j denotes the charge addition. If one desires to have a concept of stability for bundles on N(ℓ), a good definition would be to call a bundle on N(ℓ) stable, if its charge equals its splitting type. This certainly gives moduli spaces. The definition proposed earlier in [Kn] turned out to be satisfied by every bundle on N(ℓ).

3. Moduli of bundles on X̃ For the moduli problem on X̃ we need stability conditions. If L is an

ample divisor on X then, for large N, the divisor L̃ = NL − ℓ is ample on X̃. We fix, once and for all, polarizations L and L̃ on X and X̃ respectively. For a polarized surface Y, the notation 𝔐_{k}(Y) stands for moduli of rank 2 bundles on Y, slope stable with respect to the fixed polarization, with vanishing first Chern class and second Chern class k. If E is L-stable on X, then π^{∗}(E) is L̃- stable on X̃ [FM, Thm. 5.5]. Hence, the pull back map induces an inclusion of moduli spaces 𝔐_{k}(X) ↪ 𝔐_{k}(X̃).

Our objective is to show that given a map 𝔐_{k}(X) → 𝔐_{k+1}(X) inducing a homology equivalence through a range, there is a map 𝔐_{k}(X̃) → 𝔐_{k+1}(X̃) inducing a homology equivalence through a comparable range, and that both ranges go to ∞ with k. Given that the homology equivalence for the case of X = ℙ^{2} holds up to dimension ⌊k/2⌋ − 1, we expect to obtain an equivalence range at best equal to this one. From here on, we will feel free to modify 𝔐_{k}(X̃) in any way that does not change topology up to dimension k/2 = c_{2}/2. Firstly, we remove singularities of the moduli spaces; secondly, we remove semistable bundles on 𝔐_{i}(X) that are not stable and thirdly we add frames. We then obtain a stratification of the framed moduli space 𝔐_{k}^{f}(X̃) by smooth submanifolds.

3.1 Removing singularities

We remove the singular points of the moduli space, in order to work only with smooth manifolds. We use the following results of Kirwan and Donaldson.

Theorem ([Ki], Cor. 6,.4) Let X be a quasi-projective variety and m a nonnegative integer such that every x_{0} ∈ X has a neighborhood in X isomorphic to

{x ∈ ℂ^{N}|f_{1}(x) = ⋯ = f_{M}(x) = 0}

for some integers N, M, and holomorphic functions f_{i} depending on x_{0} with M ≤ m. If Y is a closed subvariety of codimension k in X, then for q < k−m,

H_{q}(X − Y, ℤ) ≃ H_{q}(X, ℤ).

Given a complex surface S with polarization H, let Σ_{k} ⊂ 𝔐_{k}(S) denote the algebraic subvariety representing bundles E with H^{2}(End_{0}E) ≠ 0. A local model of the moduli space is given by the kernel of the Kuranishi map Sym^{2}H^{1}(EndE) → H^{2}(End_{0}E) parametrizing small deformations of E. In case H^{2}(End_{0}E) = 0, small deformations of E are unobstructed and E is a smooth point of the moduli space. Therefore, singular points satisfy H^{2}(End_{0}E) ≠ 0 and the singular part of 𝔐_{k} is contained in Σ_{k}.

Theorem ([Do], Thm. 5.8) There are constants a, b depending only on S and the ray spanned by H in H_{2}(S) such that:

dim_{ℂ}Σ_{k} ≤ a + b\sqrt{k} + 3k.

Proposition 3.1 For large k, removing the singularity set Sing of 𝔐_{k}(S) does not change homology in dimension less than k. That is, for q < k

H_{q}(𝔐_{k}(S)) = H_{q}(𝔐_{k}(S) − Sing).

Proof. By Kuranishi theory, points E ∈ 𝔐_{k}(S) satisfying H^{2}(End_{0}E) = 0 are smooth points. Therefore, the singularity set of 𝔐_{k}(S) is contained in Σ_{k}. Moreover, the moduli space is defined on a neighborhood of a singular point by dimH^{2}(End_{0}E) equations. In fact, in the proof of his theorem cited above, Donaldson shows that dimH^{2}(End_{0}E) ≤ a + b\sqrt{k} + 3k. Therefore, using Kirwan’s result we have that H_{q}(𝔐_{k}(S)) = H_{q}(𝔐_{k}(S) − Sing) for q < dim𝔐_{k}(S)−2(a+b\sqrt{k}+3k) < dim𝔐_{k}(S)−7k = 8k−3(1−b_{1} +b^{+})−7k < k for large k.

We now consider only the smooth part of 𝔐_{k}(X̃), which in what follows stands for just the set of its smooth points. Similarly, when considering 𝔐_{k}(X) we will consider only its set of smooth points, keeping in mind that this does not change the homology up to dimension k.

3.2 Removing unstable bundles

Given an L̃-stable bundle Ẽ, the bundle E = (π_{∗}Ẽ)^{∨∨} is L-semistable [FM, Thm. 5.5]. We want to remove the set of strictly semistable bundles that appear in this process, leaving only stable bundles over X, as these are the ones corresponding to instantons.

Proposition 3.2 There is a bound, depending only on X, on the dimension of the set of semistable bundles on X that are not stable.

Proof. Let E be a semistable bundle on X that is not stable. Since we are only considering bundles with c_{1} = 0 we must have that there is a destabilizing line bundle L with deg(L) = 0. Hence, E is an extension

0 → L → E → F → 0,  (1)

where F is a rank 1 sheaf satisfying F^{∨∨} = L^{−1} and fitting into a short exact sequence

0 → F → F^{∨∨} → Q → 0,

with Q supported only at points. It follows that the dimension of the set 𝔐^{su}: = 𝔐^{ss} − 𝔐^{s} of semistable bundles on X that are not stable is given by

dim𝔐^{su} = dim(Ext^{1}(F, L) × Jac(X)).

It is a basic fact that dimJac(X) = h^{1}(X, O_{X}) is the irregularity of X. Claim: dimExt^{1}(F, L) = h^{1}(X, O_{X}) − h^{2}(X, O_{X}) + ϵ, where ϵ = 0 or 1. In fact, taking Hom(., L) on the short exact sequence (1) gives

0 → Hom(F, L) → Hom(F^{∨∨}, L) → Hom(Q, L) →
→ Ext^{1}(F, L) → Ext^{1}(F^{∨∨}, L) → Ext^{1}(Q, L) → .

But, because Q is supported at points, we have Hom(Q, L) = Ext^{1}(Q, L) = 0, and consequently

Ext^{1}(F, L) = Ext^{1}(F^{∨∨}, L) = H^{1}(L^{2}).

An application of Hirzebruch–Riemann–Roch gives

h^{1}(L^{2}) = h^{0}(L^{2}) − (c_{1}^{2} + c_{2})

and since X is a surface c_{1} = −K, we have

h^{1}(L^{2}) = h^{0}(L^{2}) − (K^{2} + c_{2}) = h^{0}(L^{2}) − (1 + p_{a}),

where p_{a} = h^{2}(X, O_{X})−h^{1}(X, O_{X}) is the arithmetic genus of X. To calculate h^{0}(L^{2}) there are two possibilities, either L^{2} has no holomorphic sections, in which case, h^{0}(L^{2}) = 0; or else, suppose s is a holomorphic section of L^{2} and let S its divisor of zeros. Since the divisor L defining the polarization is ample, it has positive intersection with nonzero divisors. But, deg(L) = 0, hence S · L = 0 forcing S ≡ 0. Therefore, L^{2} is trivial and in this case h^{0}(L^{2}) = 1. It follows that

dim𝔐^{su} = dimExt^{1}(F, L) + dimJac(X) = 2h^{1}(X, O_{X}) − h^{2}(X, O_{X}) + ϵ

where ϵ = 0 or 1.

Based on the results of this section, we assume in what follows that the moduli spaces 𝔐_{i} are smooth and contain only stable points.

4. The gluing construction We show that a framed bundle on X̃ is uniquely determined by a pair of

framed bundles on X and ℂ̃^{2}. We fix a neighborhood N(ℓ) of the exceptional divisor inside of X̃ and write X̃ = (X̃ − {ℓ}) ∪ N(ℓ). Put N_{0} : = (X̃ − {ℓ})∩N(ℓ). The blow-up map gives an isomorphism i_{1} : X̃ −{ℓ} → X −{x}. Based on the results of §1, we can assume that there is an isomorphism i_{2} : N(ℓ) → ℂ̃^{2}. Over the intersection N_{0} we have isomorphisms N(x)−{x} ←^{i_{1}} N_{0} →^{i_{2}} ℂ̃^{2} − {0}, which we still denote by the same letters i_{1} and i_{2}. Using this isomorphisms we write framed bundles on X̃ as pairs of framed bundles on X and ℂ̃^{2}.

Lemma 4.1 There is a one-to-one correspondence between holomorphic vector bundles on X̃ − {ℓ} and holomorphic vector bundles on X.

Proof. It is trivial that there is a one-to-one correspondence between bundles on X̃ − {ℓ} and bundles on X − x, because these are isomorphic surfaces. If E is a holomorphic bundle on X − x it extends uniquely to a bundle on X by Hartog’s because X is 2-dimensional.

Definitions

• A framed bundle πẼ: Ẽ^{f} → X̃ is a bundle Ẽ on X̃ together with a pair of pointwise linearly independent sections (f_{1}, f_{2}): N_{0} → π−1 ; Ẽ(N_{0}).

• A framed bundle π_{V} : V^{f} → ℂ̃^{2} is a bundle V on ℂ̃^{2} together with a pair of pointwise linearly independent sections (t_{1}, t_{2}): ℂ̃^{2} − {0} → π_{V}^{−1}(ℂ̃^{2}) − {0}.

• A framed bundle π_{E}: E^{f} → X is a bundle E on X together with a pair of pointwise linearly independent sections (s_{1}, s_{2}): N(x) → π_{E}^{−1}(N(x)), where N(x) is a small neighborhood of x. We will always consider N(x) = π(N(ℓ)).

Proposition 4.2 An isomorphism class [Ẽ] of a framed bundle on X̃ is uniquely determined by a pair of isomorphism classes of framed bundles [E^{f}] on X and [V^{f}] on ℂ̃^{2}. We write Ẽ^{f} = (E^{f}, V^{f}).

Proof. By construction Ẽ = E ∐_{(s_{1},s_{2})=(t_{1},t_{2})} V is made by identifying the

bundles as well as the sections over N_{0}, so that the bundles satisfy Ẽ^{f}|_{N_{0}} = i_{1}^{∗}(E|_{N(x)}) = i_{2}^{∗}(V |_{ℂ̃^{2}−{0}}) and the framing (f_{1}, f_{2}) of Ẽ satisfies (f_{1}, f_{2}) = (s_{1}, s_{2}) ∘ i_{1} = (t_{1}, t_{2}) ∘ i_{2}.

Let ϕ: E^{f} → E′^{f} be an isomorphism such that ϕ ∘ (s_{1}, s_{2}) = (s′_{1}, s′_{2}) and let ξ: V^{f} → V′^{f} be an isomorphism such that ξ ∘ (t_{1}, t_{2}) = (t′_{1}, t′_{2}). We have the following diagrams of bundle maps:

Ẽ^{f}|_{X̃−{ℓ}} →  E^{f}  →^{ϕ} E′^{f}  ← Ẽ′^{f}|_{X̃−{ℓ}}
↓ πẼ  ↓ π_{E}  ↓ π_{E′}  ↓ πẼ′
X̃ − {ℓ} →^{i_{1}} X − {x} = X − {x} ←^{i_{1}}  X̃ − {ℓ}

Hence,

Ẽ′^{f}|_{X̃−{ℓ}} = i_{1}^{∗}(E′^{f}|_{X−{x}}) = i_{1}^{∗} ∘ ϕ(E^{f}|_{X−{x}}) = i_{1}^{∗} ∘ ϕ ∘ i_{1∗}(Ẽ^{f}|_{X̃−{ℓ}}) (1)

showing that i_{1}^{∗} ∘ ϕ ∘ i_{1∗} is an isomorphism of Ẽ and Ẽ′ over X̃ − {ℓ} such that

ϕ ∘ (f_{1}, f_{2}) = ϕ ∘ (s_{1}, s_{2}) ∘ i_{1} = (s′_{1}, s′_{2}) ∘ i_{1} = (f′_{1}, f′_{2}).  (2)

On the other hand we have the second diagram of bundle maps:

\begin{aligned}Ẽ^{f}|_{N(ℓ)} & → & V^{f} & →^{ξ} & V′^{f} & ← & Ẽ′^{f}|_{N(ℓ)} \\ ↓ πẼ & & ↓ π_{V} & & ↓ π_{V′} & & ↓ πṼ′ \\ N(ℓ) & →^{i_{2}} & ℂ̃^{2} & = & ℂ̃^{2} & ←^{i_{2}} & N(ℓ) \end{aligned}.

Therefore,

Ẽ′^{f}|_{N(ℓ)} = i_{2}^{∗}(V′^{f}|ℂ̃^{2}) = i_{2}^{∗} ∘ ξ(V^{f}|ℂ̃^{2}) = i_{2}^{∗} ∘ ξ ∘ i_{2∗}(Ẽ^{f}|_{N(ℓ)})  (3),

showing that i_{2}^{∗} ∘ ξ ∘ i_{2∗} is an isomorphism of Ẽ and Ẽ′ over N(ℓ) such that

ξ ∘ (f_{1}, f_{2}) = ξ ∘ (t_{1}, t_{2}) ∘ i_{2} = (t′_{1}, t′_{2}) ∘ i_{2} = (f′_{1}, f′_{2}).  (4)

These isomorphisms agree over the intersection N_{0}, in fact, by (1) and (3)

i_{1}^{∗} ∘ ϕ ∘ i_{1∗}(Ẽ^{f}|_{N_{0}}) = i_{1}^{∗} ∘ ϕ(E^{f}|_{N(x)}) = i_{1}^{∗}(E′^{f}|_{N(x)}) = i_{2}^{∗}(V′^{f}|_{ℂ̃^{2}−{0}}) =
= i_{2}^{∗} ∘ ξ(V^{f}|_{ℂ̃^{2}−{0}}) = i_{2}^{∗} ∘ ξ ∘ i_{2∗}(Ẽ^{f}|_{N_{0}})

and moreover they also preserve the framings over the intersection, since over N_{0}, we have, by (2) and (4)

ϕ ∘ (f_{1}, f_{2}) = (f′_{1}, f′_{2}) = ξ ∘ (f_{1}, f_{2}).

By the gluing lemma this gives an isomorphism over the entire X̃ and we get Ẽ′ ≃ Ẽ.

5. Framed moduli spaces Notation: With the added superscript 𝔐^{f}(X̃), 𝔐^{f}(X), M^{f} and N^{f} denote the framed versions of 𝔐(X̃), 𝔐(X), M and N respectively, framed according to the definitions of section 4.

This section contains a series of lemmas, which prove the following result.

Theorem 5.1 N_{i}^{f} is a smooth complex manifold of dimension not greater than r_{i}(2r_{i} + 1).

We identify the space N_{i}^{f} with the moduli space M_{i}(Σ) of bundles on the first Hirzebruch surface Σ = ℙ(O(1) ⊕ O(−1)) having second Chern class i, and that are trivial (and framed) over the line at infinity ℓ_{∞}. We then show that M_{i}(Σ) is smooth. Recall N_{0} is the deleted neighborhood of the exceptional divisor, that is, N_{0} = N(ℓ) − ℓ.

Lemma 5.2 For all E, E′ in N_{i}^{f}, the map

H^{0}(Hom(E, E′)) → H^{0}(Hom(E|_{N_{0}}, E′|_{N_{0}})

is injective.

Proof. N_{0} is open in ℂ̃^{2}, hence two holomorphic functions that coincide in N_{0} and are globally defined must be equal.

Lemma 5.3 In order to study endomorphisms of a bundle V ∈ N_{i} it suffices to chose a fixed transition matrix T for V and then to consider endomorphisms fixing T.

Proof. In fact, suppose there is an automorphism ϕ of V taking T to T̄. Then, ϕ = (X, Y) is given by a pair of transition matrices X ∈ Γ(U) and Y ∈ Γ(V) such that Y T X = T̄. But, since T and T̄ represent the same bundle V, there are change of coordinates A ∈ Γ(U) and B ∈ Γ(V) such that T = BT̄A. Now define a new endomorphism of V by ϕ̄ = (XA, BY), then ϕ̄(T) = BY T XA = BT̄A = T.

Lemma 5.4 An automorphism of a framed bundle (V, f) ∈ N_{i}^{f} over N_{0} can be written in the form T ↦ X^{−1}T X with X ∈ Γ(U_{0})∩Γ(V_{0}) and X = (\begin{matrix}a & b \\ 0 & d \end{matrix}).

Proof. Let ϕ(V) = (X, Y) given by Y T X = T be an automorphism of V so that ϕ(V, f) = (V, f̄) where f and f̄ are equivalent framings of V. Let f = (A, B) and f̄ = (Ā, B̄) be the expressions for these framings in coordinate charts, with A, Ā ∈ Γ(U_{0}) and B, B̄ ∈ Γ(V_{0}), where U_{0} = U ∩ N_{0} and V_{0} = V ∩ N_{0}. Hence ϕ(A) = XA = Ā and ϕ(B) = Y B = B̄.

On the other hand, since f and f̄ are trivializations of the trivial bundle on N_{0}, there exists G ∈ Γ(sl(2, ℂ), N_{0}) such that Ā = GA and B̄ = G^{−1}B. Therefore on U_{0} we have Ā = GA = XA and on V_{0} we have B̄ = G^{−1}B = Y B. Now A and B are invertible matrices, hence X^{−1} = G^{−1} = Y on U_{0} ∩V_{0}.

Suppose X = (\begin{matrix}a & b \\ c & d \end{matrix}). From the equality XT = T X we get −z^{−j}c = z^{j}c, which immediately implies that c = 0.

Lemma 5.5An automorphism of a framed bundle (V, f) ∈ N_{i}^{f} over N(ℓ) ≃ ℂ̃^{2} can be written in the form T ↦ X^{−1}T X with X = (\begin{matrix}a_{0} + a & b \\ 0 & d_{0} + d \end{matrix}) where a, b, d ∈ Γ(U_{0}) ∩ Γ(V_{0}) and a, b and d vanish over the exceptional divisor and a_{0}, d_{0} are constants..

Proof. By lemma 0.3 X = (\begin{matrix}a & b \\ 0 & d \end{matrix}) over N_{0} and we must check what are the possible extensions of X to the full coordinate charts U and V. Now consider any possible holomorphic extension (\begin{matrix}ā & b̄ \\ c̄ & d̄ \end{matrix}) to the entire U− chart. We claim the extension is also of the form X = (\begin{matrix}ā & b̄ \\ 0 & d̄ \end{matrix}). In fact, U ≃ ℂ^{2} and c̄ = c = 0 on U_{0} which is an open subset of U hence c̄ = 0 everywhere on U. Similarly on V we have the form X = (\begin{matrix}ᾱ & β̄ \\ 0 & δ̄ \end{matrix}). Write x̄ = x_{o}(z)+x(z, u) for x ∈ \{a, b, d\} and write ȳ = ȳ_{0}(z^{−1}) + y(z^{−1}, zu) for y = {α, β, δ}. Over the exceptional divisor one has p = 0 = u and the equality (∗) becomes

(\begin{matrix}z^{j}a_{0} & z^{j}b_{0} \\ 0 & z^{−j}d_{0} \end{matrix}) = (\begin{matrix}z^{j}α_{0} & z^{−j}β_{0} \\ 0 & z^{−j}δ_{0} \end{matrix})

where a_{0}, b_{0}, d_{0} are holomorphic in z whereas α_{0}, β_{0}, δ_{0} are holomorphic in z^{−1}. It immediately follows that b_{0} = β_{0} = 0 and that a_{0} = α_{0} and d_{0} = δ_{0} are constants.

Lemma 5.6 A framed bundle (V, f) ∈ N_{i}^{f} has no traceless automorphisms, unless V splits.

Proof. The equality trace(λX) = λ^{2}trace(X) holds for any X ∈ GL(2, ℂ) and constant λ. Therefore, the bundle V has traceless GL(2, ℂ) automorphisms if and only if it has traceless SL(2, ℂ) automorphims. So, we assume det(X) = 1. Over the exceptional divisor X|_{ℓ} = (\begin{matrix}a_{0} & 0 \\ 0 & d_{0} \end{matrix}). Hence a_{o}d_{0} = 1. If trace(X) is zero, then we also have a_{0} + d_{0} = 0. It follows that a_{0} = ±i. Suppose a_{0} = i, the other case is analogous. Since X is traceless, it then follows that X = (\begin{matrix}i + a & b \\ 0 & −i − a \end{matrix}). On the other hand det(X) = −(i + a)^{2} = 1 implies that either a = 0 or a = −2i. Therefore X = ±(\begin{matrix}i & b \\ 0 & −i \end{matrix}). Now the equality XT = T X gives

(\begin{matrix}iz^{j} & ip + z^{−j}b \\ 0 & −iz^{−j} \end{matrix}) = (\begin{matrix}iz^{j} & −ip + z^{j}b \\ 0 & −iz^{−j} \end{matrix})

implying

2ip = (z^{j} − z^{−j})b.

But b has only positive powers of z, hence the r.h.s. contains powers of z greater of equal to j whereas the l.h.s. only has powers of z strictly smaller than j. Hence b = 0 = p. We conclude that (V, f) has no traceless automorphisms unless V splits, in which case the automorphisms are multiples of

(\begin{matrix}i & 0 \\ 0 & −i. \end{matrix}) .

Proposition 5.7M_{i}(Σ) is smooth.

Proof. By [L] Theorem 1.1, given the conditions of lemma 0.1 above, M_{i}(Σ) is smooth at E provided that H^{2}(sl(E) ⊗ O_{Σ}(−ℓ_{∞})) = 0, where sl(E) is the bundle of traceless endomorphisms of E. But lemma 5.6 implies that there are no traceless endomorphisms of E, unless E splits, in which case sl(E) is trivial. The same conclusions then holds also for bundles on M(Σ). Therefore H^{2}(sl(E) ⊗ O_{Σ}(−ℓ_{∞})) = 0.

Notation:

• S_{i,j} denotes the space of framed bundles having charge i and splitting type j. Hence S_{i,j} = N_{i}^{f} ∩ M_{j}^{f}.

• For any integer i, let r_{i}^{2} be the smallest square not lower than i. Hence r_{i} = ⌈\sqrt{i}⌉.

Proposition 5.8 dimN_{i}^{f} is not greater than r_{i}(2r_{i} + 1).

Proof. The dimension of N_{i}^{f} is calculated in the following sequence of 3 lemmas, showing that

dimN_{i}^{f} ≤ dimM_{r_{i}}^{f} = dimS_{r_{i}^{2},r_{i}} = r_{i}(2r_{i} + 1).

Lemma 5.9 dimN_{i}^{f} ≤ dimM_{r_{i}}^{f}. In particular, dimS_{i,j} ≤ dimS_{i,r_{i}} for all j ≥ r_{i}.

Proof. By [G1] the inequality j ≤ i ≤ j^{2} between the charge i and the splitting type j is sharp. Therefore, bundles on N_{i}^{f} have splitting type j varying as r_{i} ≤ j ≤ i.

Let R be a bundle on N_{i}^{f} with splitting type r_{i}. The dimension of N_{i}^{f} at R is given by dimDef_{N i}(R) + dimF ram(R), where Def_{N_{i}}(R) is the space of local deformations of R inside N_{i} and F ram(R) is the space of inequivalent reframings of R.

We claim that Def_{N i}(R) ⊂ Def_{M_{r_{i}}}(R). In fact, by [G3] R can be written as an extension of O(r_{i}) by O(−r_{i}), and it follows that any small deformation of R has splitting type smaller or equal to r_{i}. But deformations with splitting type lower than r_{i} fall outside of N_{i}. Hence Def_{N i}(R) ⊂ Def_{M_{r_{i}}}(R) and consequently dimN_{i}^{f} ≤ dimM_{r_{i}}^{f}.

Lemma 5.10 dimM_{j}^{f} = dimS_{j^{2},j}. In particular, dimS_{k,j} ≤ dimS_{j^{2},j}, for all k ≤ j^{2}.

Proof. By ]G3] elements of M_{j} are determined by pairs (j, p) where p has the form p = ∑_{i=1}^{2j−2} ∑_{l=i−j+1}^{j−1} p_{il}z^{l}u^{i}. Consequently, M_{j} is parametrized by the N coefficients of the polynomial and is identified with a quotient ℂ^{N}/ ∼ where ∼ is the equivalence determined by bundle isomorphism. The relation p ∼ λp holds for all λ ≠ 0 and for all p and this implies that any neighborhood of 0 ∈ ℂ^{N}/ ∼ contains the entire space. By [BG2, thm. 4.1] the space of framed isomorphism classes of bundles having underlying data (j, 0) equals S_{j^{2},j.} It follows that dimS_{k,j} ≤ dimS_{j^{2},j} = dimM_{j}.

Lemma 5.11 dimS_{j^{2},j} = j(2j + 1).

Proof. S_{j^{2},j} is the space of inequivalent framings of the split bundle O(j) ⊕ O(−j). In canonical coordinates as in (∗) the split bundle is represented by the transition matrix T = (\begin{matrix}z^{j} & 0 \\ 0 & z^{−j} \end{matrix} ) and has trivializing section on N_{0} given by s_{1} = (u^{j}, 0) and s_{2} = (0, u^{−j}). To calculate the dimension of the space of inequivalent framings, we use the method of [Hu, p. 116]. The space of all possible framings is obtained from the canonical one by the action of the group Sl(2, ℂ)_{N_{0}} of holomorphic maps N_{0} → Sl(2, ℂ). The space of inequivalent framings will be the quotient of Sl(2, ℂ)_{N_{0}} by the subgroup H_{N_{0}} of maps that extend to an isomorphism also over the exceptional divisor. We set A = (\begin{matrix}u^{j} & 0 \\ 0 & u^{−j} \end{matrix} ) and B = (\begin{matrix}zu^{j} & 0 \\ 0 & zu^{−j} \end{matrix} ) . We have the following diagram of transition matrices

U × ℂ ⟶^{T} V × ℂ
A ↓  ↓ B
U × ℂ ⟶^{I} V × ℂ
G ↓  ↓ G
U × ℂ ⟶^{I} V × ℂ
A^{−1} ↓  ↓ B^{−1}
U × ℂ ⟶^{T} V × ℂ

The condition that G ∈ HN_{0} is then that A^{−1}GA be holomorphic nonsingular in U ∩ N_{0} and that B^{−1}GB be holomorphic nonsingular in V ∩ N_{0}.

A^{−1}GA = (\begin{matrix}G_{11} & u^{2}G_{12} \\ u^{−2j}G_{21} & G_{22} \end{matrix}) .

The condition u^{−2j}G_{21} ∈ Γ(U) implies that the part of G_{21} that does not extend to an isomorphism over the divisor has the form ∑_{i=0}^{2j−1} ∑_{l≤i} g_{il}z^{l}u^{i}. There are j(2j + 1) such coefficients. It follows that dimS_{j^{2},j} = j(2j + 1).

6. Topology of 𝔐_{k}(X̃)

We assume that the Atiyah–Jones conjecture holds true for X, that is, we assume that there exist maps r_{k}: 𝔐_{k}^{f}(X) → 𝔐_{k+1}^{f}(X) inducing isomorphisms in homology r_{k}_{∗}: H_{q}(𝔐_{k}^{f}(X)) → H_{q}(𝔐_{k+1}^{f}(X)) for q ≤ ⌊k/2⌋ − c where c is a constant.

6.1 Statement of the conjecture

For a 4-manifold X, let MI_{k}(X) denote the moduli space of rank 2 instantons on E with charge k and let B_{k}^{ϵ}(X) denote the space of connections on X whose anti-self-dual part of the curvature has norm less than ϵ. Given a point x_{o} ∈ X, Taubes [Ta] constructed a map

t_{k}^{x_{0}} : MI_{k}(X) → B_{k+1}^{ϵ}(X)

by patching a small instanton on a neighborhood of x_{0}. The map t_{k}^{x_{0}} takes an instanton on X with charge k to a connection on X with charge k + 1 whose anti-self-dual part of the L^{2} norm of its curvature is less than ϵ. Taubes also constructed a strong deformation retract

τ_{k+1} : B_{k+1}^{ϵ}(X) → MI_{k+1}(X).

Based on Taubes’ results, the Atiyah–Jones conjecture in homology is equivalent to the statement that the maps i_{k} = τ_{k+1} ∘ t_{k}^{x_{0}} induce isomorphisms in homology

i_{k}_{∗}: H_{q}(MI_{k}(X)) → H_{q}(MI_{k+1}(X))

through a range q(k) increasing with k.

For a Kähler surface X, Kobayashi and Hitchin gave a one-to-one correspondence between instantons on a topological bundle E on X and holomorphic structures on E, see [LT]. The correspondence is made by taking the holomorphic part of an anti-self-dual connection. Let

K_{X}: MI_{k}(X) → 𝔐_{k}(X)

denote the Kobayashi–Hitchin map on X given by K(∇ = ∂ +\overline{∂}) = \overline{∂}, and let H_{X}: = D_{X}^{−1} be the inverse map. For the noncompact surface X = ℂ̃^{2} this correspondence takes an instanton to a holomorphic bundle with an added trivialization at infinity [Ki]. In terms of instantons proposition 4.2 can be restated as:

Proposition 6.1 Every instanton on X̃ is obtained by holomorphic patching and instanton on X to an instanton on ℂ̃^{2}.

6.2 Stratifications Definition A smooth manifold M is L-stratified if there is a decomposition of M into disjoint submanifolds M(K) such that

(1) The index set K = {K} is finite with a given fixed well ordering ≤ .

(2) If K_{0} is the smallest element in (K, ≤), then M(K_{0}) is an open-dense subset of M.

(3) For all K ∈ K the union of the submanifolds of the same or smaller order

Z(K) = ∪_{K′≤K}M(K′)

is an open-dense submanifold of M.

(4) For all K ∈ K the normal bundle, ν(K), of M(K) in M is orientable.

Proposition 6.1 There is an L-stratification of the moduli space of framed bundles on X̃ as

𝔐_{k}^{f}(X̃) ≃ ⋃_{i=0}^{k} 𝔐_{k−i}^{f}(X) × N_{i}^{f}.  (∗)

Proof. The existence of the decomposition follows directly from the definition of the N_{i}. The first stratum K_{0} = 𝔐_{k}^{f}(X) × N_{0}^{f} equals the set of pull-back bundles and is dense in 𝔐_{k}^{f}(X̃). To show this gives an L-stratification we need to prove that the normal bundle of each stratum K_{i} = 𝔐_{k−i}^{f}(X) × N_{i}^{f} is orientable. Using the results from sections 3.1 and 3.2, we may assume that 𝔐_{k}^{f}(X̃) is smooth and that 𝔐_{i}^{f}(X) is smooth and contains only stable bundles. By theorem 5.1 N_{i}^{f} is a complex manifold. It then follows that each stratum K_{i} is a complex submanifold of 𝔐_{k}^{f}(X̃) and therefore has orientable normal bundle.

6.3 Maps between spectral sequences

Whereas it is not strictly true that Taubes map on X̃ preserves the stratification (∗), we show that the map on 𝔐_{k}(X̃) induced by Taubes map is homotopy equivalent to a map that preserves the stratification. Recall that we are considering the stratifications;

𝔐_{k}^{f}(X̃) = ⨁ K_{i}, and 𝔐_{k+1}^{f}(X̃) = ⨁ K′_{i}

where K_{i} = 𝔐_{k−i}^{f}(X) × N_{i}^{f} and K′_{i} = 𝔐_{k−i+1}^{f}(X) × N_{i}^{f}.

Let d: 𝔐_{k}^{f}(X̃) → 𝔐_{k}^{f}(X̃) be the map that re-writes a bundle into its two components of the decomposition, d(Ẽ) = (E, V) as in proposition 4.2. We choose a point x_{0} in X that is far from the point x we blew-up, in the sense that x_{0} ∉ π_{∗}(N(ℓ)) and we set x̃_{0} = π^{∗}(x_{0}). Let

t_{k}^{x_{o}}: MI_{k}(X) → B_{k+1}^{ϵ}(X) and tx̃_{o} ; k: MI_{k}(X̃) → B_{k+1}^{ϵ}(X̃)

denote Taubes patching on a small neighborhood of x_{o} in X and of x̃_{0} in X̃ respectively, taking an instanton of charge k to a connection of charge k + 1 whose anti-self-dual part of the L^{2} norm of its curvature is less than ϵ. Taubes also constructed strong deformation retracts

τ_{k+1} : B_{k+1}^{ϵ} → MI_{k+1}.

The idea now is that choosing x to be far from the exceptional divisor, the the patching remains far from the exceptional divisor. From the point of view of vector bundles, this implies that we can choose the map so that it does not alter the bundle near the divisor. We now make this statement precise.

Lemma 6.2 The maps t_{k}: 𝔐_{k}^{f}(X̃) → 𝔐_{k+1}^{f}(X̃) and r_{k}: 𝔐_{k}^{f}(X̃) → 𝔐_{k+1}^{f}(X̃) given by

t_{k}: = KX̃ ∘ τ_{k+1} ∘ tx̃_{0} ; k ∘ HX̃

and

r_{k}: = d^{−1} ∘ KX̃ ∘ τ_{k+1} ∘ (t_{x_{0}}^{k}, 1) ∘ (H_{X}, Hℂ̃^{2}) ∘ d

are homotopically equivalent.

Proof. The following diagram summarizes the situation

\begin{matrix}𝔐_{k}^{f}(X̃) & ⟶^{HX̃} & MI_{k}(X̃) & ⟶^{tx̃0 ; k} & B_{k+1}^{ϵ}(X̃) & ⟶^{τ_{k+1}} & MI_{k+1}(X̃) & ⟶^{KX̃} & 𝔐_{k+1}^{f}(X̃) \\ d ↓ & & || & & || & & || & & d^{−1} ↑ \\ 𝔐_{k}^{f}(X̃) & ⟶^{(H_{X} ,Hℂ̃2 )} & MI_{k}(X̃) & ⟶^{(t_{x0}^{k} ,1)} & B_{k+1}^{ϵ}(X̃) & ⟶^{τ_{k+1}} & MI_{k+1}(X̃) & ⟶^{KX̃} & 𝔐_{k+1}^{f}(X̃) \end{matrix} .

We need only to show that tx̃_{0} ; k ∼ (t_{x_{0}}^{k}, 1). But the space of connections if an affine space, hence the linear homotopy stx̃_{0} ; k + (1 − s)(t_{x_{0}}^{k}, 1) will do (divide by the norm and multiply by ϵ/2 here if necessary).

Proposition 6.3 t_{k}: 𝔐_{k}^{f}(X̃) → 𝔐_{k+1}^{f}(X̃) is homotopy equivalent to a map that preserves the filtrations.

Proof. This is just a restatement of lemma 6.1.

6.4 Proof of AJ for blow-ups Lemma 6.4The real codimension of the stratum K_{i} is at least 12i − 6(r_{i} + 1).

Proof. K_{i} = 𝔐_{k−i}^{f}(X) × N_{i}^{f}. The complex codimension of K_{i} is

codK_{i} = dim𝔐_{k}^{f}(X̃) − (dim𝔐_{k−i}^{f}(X) + dimN_{i}^{f}) =
= 8k − 3(1 − b_{1} + b^{+}) − (8(k − i) − 3(1 − b_{1} + b^{+}) + dimN_{i}^{f}).

According to theorem 5.1, the complex codimension of N_{i}^{f} = r_{i}(2r_{i} + 1). Hence, codK_{i} = 8i − r_{i}(2r_{i} + 1) ≥ 6i − 3(r_{i} + 1).

Let 𝔖_{k} denote the Leray spectral sequence associated to the filtration {K_{i}} of 𝔐_{k}^{f}(X̃) and let 𝔖′_{k+1} denote the Leray spectral sequence associated to the filtration {K′_{i}} of 𝔐_{k+1}^{f}(X̃).

Theorem 6.5 The maps t_{k}: 𝔐_{k}^{f}(X̃) → 𝔐_{k+1}^{f}(X̃) induce an isomorphisms in homology H_{q}(𝔐_{k}^{f}(X̃)) → H_{q}(𝔐_{k+1}^{f}(X̃)) for q ≤ ⌊k/2⌋ − c.

Proof. By proposition 6.3 we may assume that t_{k}: 𝔐_{k}^{f}(X̃) → 𝔐_{k+1}^{f}(X̃) respects the filtrations and therefore induces a map of spectral sequences t_{k}: 𝔖_{k} → 𝔖′_{k+1}. The E^{1} term of 𝔖_{k} is ⨁_{K_{i}} H_{∗}(T ν(K_{i})) and the E^{1} term of 𝔖′_{k+1} is ⨁_{K′_{i}} H_{∗}(T ν(K′_{i})). By hypothesis Atiyah–Jones holds for X, hence we are assuming that for each k and for q ≤ ⌊k/2⌋ − c, H_{q}(𝔐_{k}^{f}(X)) = H_{q}(𝔐_{k+1}^{f}(X)). Consequently

H_{q}(K_{i}) = H_{q}(𝔐_{k−i}^{f}(X) × N_{i}^{f}) = H_{q}(𝔐_{k−i+1}^{f}(X) × N_{i}^{f}) = H_{q}(K′_{i})

for q ≤ ⌊(k−i)/2⌋−c (where we might need to take the largest of the c′s if they vary with i). The Thom isomorphism gives H_{∗}(T ν(K_{i})) = H_{∗}(Σ^{τ(K_{i})}K_{i}), where τ(K_{i}) is the real codimension of K_{i} and similarly for K′_{i}. By lemma 6.4 the real codimension of K_{i} at least 12i − 6(r_{i} + 1). Therefore

H_{q}(T ν(K_{i})) = H_{q}(T ν(K′_{i})) for q ≤ ⌊(k − i)/2⌋ − c + 12i − 6(r_{i} + 1).

Since ⌊(k −i)/2⌋−c+12i−6(r_{i} +1) ≥ ⌊k/2⌋−c, we conclude that t_{k}: 𝔖_{k} → 𝔖′_{k+1} induces an isomorphism of E^{1} terms for all E_{r,s}^{1} with r + s ≤ ⌊k/2⌋ − c.

Corollary 6.4 The Atiyah–Jones conjecture is true for rational surfaces.

Proof. Every rational surface is obtained by blowing up points on ℙ^{2} or on a rational ruled surface, and on these cases the conjecture holds true by [BHMM] and [HM].

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Elizabeth Gasparim, New Mexico State University, Department of Mathematics, Las Cruces NM 88003 USA, gasparim@nmsu.edu

R. James Milgram, Stanford University, Department of Mathematics, Stanford CA 94305 USA, milgram@math.stanford.edu