The anti-self-dual deformation complex and a conjecture of Singer

1.1 The ASD deformation complex

Let M ⁢ ( M 4 ) be the space of smooth Riemannian metrics on M 4 , and R ⁢ ( M 4 ) the bundle of algebraic curvature tensors. We can view W + as a mapping

Let g ∈ M ⁢ ( M 4 ) be an ASD metric. We can identify the formal tangent space of ℳ at 𝑔 with sections of the bundle of symmetric two-tensors, S 2 ⁢ ( T ∗ ⁢ M 4 ) . Let

denote the linearisation of W + at 𝑔, i.e., for h ∈ Γ ⁢ ( S 2 ⁢ ( T ∗ ⁢ M 4 ) ) ,

The choice of a conformal class of metrics [ g ] determines the bundle of algebraic Weyl tensors, W = W ⁢ ( M 4 , [ g ] ) , and the subbundle W + ⊂ W ⊂ S 0 2 ⁢ ( Λ + 2 ) , where S 0 2 ⁢ ( Λ + 2 ) is the bundle of symmetric, trace-free endomorphisms of Λ + 2 . Note that

In fact, since 𝑔 is ASD, by conformal invariance, D g ⁢ ( f ⁢ g ) = 0 for any f ∈ C ∞ ⁢ ( M 4 ) ; hence

We also let

denote the L 2 -formal adjoint of D g . Although the formula for D g is somewhat involved, the formula for D ∗ is much more compact (see [24, Proposition A.4]):

where 𝑃 denotes the Schouten tensor (see Section 2).

Let K g : Γ ⁢ ( T ∗ ⁢ M 4 ) → S 0 2 ⁢ ( T ∗ ⁢ M 4 ) denote the Killing operator,

where δ g ⁢ ω = ∇ k ω k is the divergence of 𝜔. The kernel of 𝒦 consists of those one-forms whose dual vector fields are conformal Killing. Moreover, by diffeomorphism and conformal invariance, Im ⁡ K ⊂ ker ⁡ D .

The ASD deformation complex is given by

This complex is elliptic; see [29, Section 2]. The associated cohomology groups are given by

where, for h ∈ Γ ⁢ ( S 2 ⁢ ( T ∗ ⁢ M 4 ) ) , ( δ g ⁢ h ) j = g i ⁢ k ⁢ ∇ k h i ⁢ j is the divergence. The Atiyah–Singer index theorem can be used to calculate the index of (1.2):

Vanishing of the cohomology groups also provides information on the local structure of the moduli space of ASD conformal structures.

This leads to the following definition.

By the work of Floer [17] and Donaldson–Friedman [13], if ( M 1 , g 1 ) and ( M 2 , g 2 ) are unobstructed ASD manifolds, then the connected sum M 1 ⁢ # ⁢ M 2 admits an ASD metric. Thus, we are lead to the question: under what condition is an ASD manifold unobstructed? The following conjecture is often attributed to Singer^[1].

For ASD Einstein manifolds, the operators D ⁢ D ∗ and D ∗ ⁢ D were explicitly computed in [25] and [30]. It follows from these calculations (and can also be seen by more or less direct calculation) that ( S 4 , g c ) and ( − C ⁢ P 2 , g FS ) are unobstructed. We remark that, for non-Einstein ASD manifolds, the formulas for these operators are fairly intractable.

The vanishing of H ASD 2 ⁢ ( M 4 , [ g ] ) can sometimes be verified when the twistor space is explicitly known. For example, the LCF metrics k ⁢ # ⁢ S 3 × S 1 (see [34, Theorem 8.2], [14]), and the ASD metrics on m ⁢ # ⁢ ( − C ⁢ P 2 ) constructed by LeBrun [33], are unobstructed. Our goal in this paper is to provide a criterion for the vanishing of H ASD 2 that only involves conformal invariants of the ASD manifold (and in particular does not depend on verifying any properties of the twistor space). Our main result is the following.

The proof of Theorem 1.5 relies on two key ideas. The first is that any element U ∈ ker ⁡ D ∗ can be associated to a self-dual harmonic two-form z = z ⁢ ( U ) ∈ Λ + 2 ⁢ ( A ) taking its values in the adjoint tractor bundle (see Section 2). Therefore, 𝑧 satisfies a twisted version of the usual Weitzenböck formula for self-dual (real-valued) two-forms. This twisted version provides us with two identities for 𝑈 (see Theorem 3.6).

The second key idea is to make a judicious choice of conformal representative in order to show that (1.4) implies the vanishing of 𝑈. As explained in Section 4, we choose a conformal metric whose scalar curvature satisfies a differential inequality involving the Schouten tensor (see Theorem 4.1), and is adapted to the curvature terms appearing in the Weitzenböck formula(s) for 𝑈. Curiously, to prove the existence of this metric, we consider a modification of the functional determinant of a conformally covariant elliptic operator first computed by Branson and Ørsted [4]. A robust theory for the existence of critical points of this functional was developed by Chang and Yang [10], and we are able to show that their ideas also give existence for our modified functional.

If Y ⁢ ( M 4 , [ g ] ) > 0 , then by the estimate of Aubin,

Therefore, (1.4) will fail (for topological reasons) if ( 2 ⁢ χ + 2 ⁢ τ ) is sufficiently negative. On the other hand, using Kobayashi’s estimate [31] of the Yamabe invariant of connected sums, it is easy to construct examples of ASD metrics on connected sums of − C ⁢ P 2 or S 3 × S 1 satisfying (1.4).

1.2 Organisation

In Section 2, we provide the necessary background on the tractor bundle and associated connection and metric. The main result (for our purposes) is Proposition 2.2, in which we associate to U ∈ ker ⁡ D ∗ a twisted SD harmonic two-form

In Section 3, we compute two Weitzenböck formulas for 𝑈 that follow from this correspondence. In Section 4, we give the proof of Theorem 1.5, and relegate the PDE aspects of our work to the appendix.

2.1 Some conventions for Riemannian geometry

For index calculations, we will use Penrose’s abstract index notation, unless otherwise indicated. In this, we write E a and E a as alternative notations for, respectively, the cotangent and tangent bundles and the contraction ω ⁢ ( v ) of a one-form 𝜔 with a tangent vector v a is written with a repeated index ω a ⁢ v a (mimicking the Einstein summation convention). Tensor bundles are denoted then by adorning the symbol ℰ with appropriate indices and sometimes also indicating symmetries. For example, E ( a ⁢ b ) is the notation for S 2 ⁢ T ∗ ⁢ M , the subbundle of symmetric tensors in T ∗ ⁢ M ⊗ T ∗ ⁢ M .

Then our convention for the Riemann tensor R a ⁢ b c d is such that

where ∇ a is the Levi-Civita connection of a metric g a ⁢ b , 𝑣 any tangent vector field, and 𝜔 is any one-form. Using the metric to raise and lower indices, we have, for example,

This may be decomposed as

where the completely trace-free part W a ⁢ b ⁢ c ⁢ d is the Weyl tensor and P a ⁢ b is the Schouten tensor. It follows that

where R b ⁢ c = R a ⁢ b a c is the Ricci tensor and its metric trace R = g a ⁢ b ⁢ R a ⁢ b is the scalar curvature. We will use 𝐽 to denote the metric trace of Schouten, i.e., J : = g a ⁢ b P a ⁢ b . Lastly, we have

where C a ⁢ b ⁢ c and B a ⁢ b are Cotton and Bach tensors, respectively. It should be noted that the Bianchi identities imply ( n − 3 ) ⁢ C a ⁢ b ⁢ c = ∇ d W d ⁢ a ⁢ b ⁢ c .

2.2 The tractor bundle and connection

To treat and work with objects that are conformally invariant, it is natural work, at least partly, in the setting of conformal manifolds. Here, by a conformal manifold ( M , c ) , we mean a smooth manifold of dimension n ≥ 3 equipped with an equivalence class 𝒄 of Riemannian metrics, where g a ⁢ b , g ̂ a ⁢ b ∈ c means that g ̂ a ⁢ b = Ω 2 ⁢ g a ⁢ b for some smooth positive function Ω. On a general conformal manifold ( M , c ) , there is no distinguished connection on T ⁢ M . But there is an invariant and canonical connection on a closely related bundle, namely the conformal tractor connection on the standard tractor bundle; see [2, 7].

Here we review the basic conformal tractor calculus on Riemannian and conformal manifolds. See [2, 11, 19] for more details. Unless stated otherwise, calculations will be done with the use of generic g ∈ c .

On an 𝑛-manifold 𝑀, the top exterior power of the tangent bundle Λ n ⁢ T ⁢ M is a line bundle. Thus its square K : = ( Λ n T M ) ⊗ 2 is canonically oriented, and so one can take oriented roots of it: given w ∈ R , we set

and refer to this as the bundle of conformal densities. For any vector bundle 𝒱, we write V ⁢ [ w ] to mean V [ w ] : = V ⊗ E [ w ] . For example, E ( a ⁢ b ) ⁢ [ w ] denotes the symmetric second tensor power of the cotangent bundle tensored with E ⁢ [ w ] , i.e., S 2 ⁢ T ∗ ⁢ M ⊗ E ⁢ [ w ] on 𝑀. On a fixed Riemannian manifold, 𝒦 is canonically trivialised by the square of the volume form, and so 𝒦 and its roots are not usually needed explicitly. However, if we wish to change the metric conformally, or work on a conformal structure, then these objects become important.

Since each metric in a conformal class determines a trivialisation of 𝒦, it follows easily that, on a conformal structure, there is a canonical section g a ⁢ b ∈ Γ ⁢ ( E ( a ⁢ b ) ) ⁢ [ 2 ] . This has the property that, for each positive section σ ∈ Γ ⁢ ( E + ⁢ [ 1 ] ) (called a scale), g a ⁢ b : = σ − 2 g a ⁢ b is a metric in 𝒄. Moreover, the Levi-Civita connection of g a ⁢ b preserves 𝜎 and therefore g a ⁢ b . Thus it makes sense to use the conformal metric to raise and lower indices, even when we are choosing a particular metric g a ⁢ b ∈ c and its Levi-Civita connection for calculations. It turns out that this simplifies many computations, and so, in this section, we will do that without further mention. (In the subsequent sections, we will work with a fixed metric and use that to trivialise density bundles – so indices will be raised and lowered using the metric.)

Considering Taylor series for sections of E ⁢ [ 1 ] , one recovers the jet exact sequence at 2-jets,

Note that J 2 ⁢ E ⁢ [ 1 ] and its sequence (2.2) are canonical objects on any smooth manifold. But, with a conformal structure 𝒄, we have the orthogonal decomposition of E a ⁢ b ⁢ [ 1 ] into trace-free and trace parts

Thus we can canonically quotient J 2 ⁢ E ⁢ [ 1 ] by the image of E ( a ⁢ b ) 0 ⁢ [ 1 ] under 𝜄 (in (2.2)). The resulting quotient bundle is denoted T ∗ (or E A in abstract indices) and called the conformal cotractor bundle. Observing that the jet exact sequence at 1-jets (of E ⁢ [ 1 ] ),

we see at once that T ∗ has a composition series (or filtration structure)

(2.3)

What this notation means is that E ⁢ [ − 1 ] is a subbundle of T ∗ , and the quotient of T ∗ by E ⁢ [ − 1 ] (which is J 1 ⁢ E ⁢ [ 1 ] ) has E a ⁢ [ 1 ] as a subbundle, whereas there is a canonical projection X : T ∗ → E ⁢ [ 1 ] . In abstract indices, we write X A for this map and call it the canonical tractor.

Given a choice of metric g ∈ c , the formula

(where Δ is the Laplacian ∇ a ∇ a ) gives a second-order differential operator on E ⁢ [ 1 ] which is a linear map J 2 ⁢ E ⁢ [ 1 ] → E ⁢ [ 1 ] ⊕ E a ⁢ [ 1 ] ⊕ E ⁢ [ − 1 ] that clearly factors through T ∗ and so determines an isomorphism

In subsequent discussions, we will use (2.5) to split the tractor bundles without further comment. Thus, given g ∈ c , an element V A of E A may be represented by a triple ( σ , μ a , ρ ) , or equivalently by

The last display defines the algebraic splitting operators Y : E ⁢ [ 1 ] → T ∗ and Z : T ∗ ⁢ M ⁢ [ 1 ] → T ∗ (determined by the choice g a ⁢ b ∈ c ) which may be viewed as sections Y A ∈ Γ ⁢ ( E A ⁢ [ − 1 ] ) and Z A a ∈ Γ ⁢ ( E A a ⁢ [ − 1 ] ) . We call the sections X A , Y A and Z A a tractor projectors.

By construction, the tractor bundle is conformally invariant, i.e., determined by ( M , c ) and independent of any choice of g ∈ c . However, the splitting (2.6) is not. Considering the transformation of the operator (2.4) determining the splitting, we see that if g ̂ = Ω 2 ⁢ f , the components of an invariant section of T ∗ should transform according to

where Υ a = Ω − 1 ⁢ ∇ a Ω , and conversely, this transformation of triples is the hallmark of an invariant tractor section. Equivalent to the last display is the rule for how the algebraic splitting operators transform

Given a metric g ∈ c , and the corresponding splittings, as above, the tractor connection is given by the formula

where on the right-hand side the ∇s are the Levi-Civita connection of 𝑔. Using the transformation of components, as in (2.7), and also the conformal transformation of the Schouten tensor,

reveals that the triple on the right-hand side transforms as a one-form taking values in T ∗ , i.e., again by (2.7) except twisted by E a . Thus the right-hand side of (2.9) is the splitting into slots of a conformally invariant connection ∇ T on (a section of) the bundle T ∗ .

There is a nice conceptual origin for the connection (2.9). Using (2.10) and the transformation of the Levi-Civita connection, it is straightforward to verify that the equation

on conformal densities σ ∈ Γ ⁢ ( E ⁢ [ 1 ] ) is conformally invariant. As this is an overdetermined PDE, solutions in general do not exist. Overdetermined linear PDEs are typically studied by prolongation, and it is quickly verified that the tractor parallel transport given by (2.9) is exactly the closed system that arises from prolonging (2.11) (see [2, 11]). From this observation and formula (2.9), it follows that non-trivial solutions of (2.11) are non-vanishing on an open dense set (for 𝑀 connected) on which the metric σ − 2 ⁢ g a ⁢ b is Einstein – an observation that has a number of applications; see [11, 18] and references therein.

The tractor bundle is also equipped with a conformally invariant signature ( n + 1 , 1 ) metric h A ⁢ B ∈ Γ ⁢ ( E ( A ⁢ B ) ) , defined as quadratic form by the mapping

and the polarisation identity. This is important not only by dint of its conformal invariance, but it is easily checked that this tractor metricℎ is preserved by ∇ a T , i.e., ∇ a T h A ⁢ B = 0 . Thus it makes sense to use h A ⁢ B (and its inverse) to raise and lower tractor indices, and we do this henceforth without further comment. In particular, X A = h A ⁢ B ⁢ X B is the canonical tractor (and hence our use of the same kernel symbol). For computations, Table 1 is useful. We see that ℎ may be decomposed into a sum of projections

Finally, for this section, we note that, of course, the curvature of the tractor connection κ a ⁢ b ⁢ C ⁢ D is determined by

and can be written in terms of tractor projectors as

The bundle A : = Λ 2 T is often termed the adjoint tractor bundle, as it is a vector bundle modelled on the Lie algebra of the conformal group SO ⁢ ( n + 1 , 1 ) . So, as expected, the tractor curvature is a two-form taking values in this bundle.

2.3 The differential splitting operator

In dimensions n ≥ 4 , the tractor curvature is the image of a conformally invariant operator 𝑧 acting on the Weyl curvature. The operator is as follows.

The conformal invariance is easily verified directly using (2.8) and the conformal transformation of the Levi-Civita connection. It can also be deduced from the conformal invariance of the tractor curvature κ a ⁢ b ⁢ C ⁢ D . In fact, the map (2.12) is a standard “BGG-splitting operator”, as in the theory [8, 5], and the conformal deformation sequence can be understood as arising from a twisting by Λ 2 ⁢ T of the de Rham complex [6, 20].

We are, in particular, interested in the case of dimension n = 4 . Then it is evident, from formula (2.13), that if 𝑈 is SD (or ASD), then so is z ⁢ ( U ) as a tractor-twisted two-form, as the Hodge-⋆ commutes with the Levi-Civita connection. We obtain more if also U ∈ ker ⁡ D ∗ .

We will give another proof of this result (by direct calculation) in the next section; see Corollary 3.3.