Deformations of Conically Singular Cayley Submanifolds

In this article, we study the deformation theory of conically singular Cayley submanifolds. In particular, we prove a result on the expected dimension of a moduli space of Cayley deformations of a conically singular Cayley submanifold. Moreover, when the Cayley submanifold is a two-dimensional complex submanifold of a Calabi–Yau four-fold, we show by comparing Cayley and complex deformations that in this special case the moduli space is a smooth manifold. We also perform calculations of some of the quantities discussed for some examples.


Introduction
Cayley submanifolds are calibrated submanifolds that arise naturally in manifolds with exceptional holonomy Spin (7). Introduced by Harvey and Lawson [6], calibrated submanifolds are by construction volume minimising, and hence minimal submanifolds. Cayley submanifolds exist in abundance, with the simplest examples being any twodimensional complex submanifold of a Calabi-Yau four-fold.
The deformation theory of compact calibrated submanifolds in manifolds with special holonomy was studied by McLean [22]. A major obstruction to generalising these results to noncompact submanifolds is the failure of an elliptic operator on a noncompact manifold to be Fredholm. However, by introducing a weighted norm on spaces of sections of a given vector bundle on a particular type of noncompact manifold, it is possible to overcome this difficulty, as long as one is careful about the choice of weight. It is therefore possible to study certain moduli spaces of noncompact calibrated submanifolds.
In this article, the noncompact submanifolds that we study are conically singular. Motivated by the SYZ conjecture, an interesting problem is whether a Spin(7)manifold can be fibred by Cayley submanifolds with some singular fibres. Conically singular Cayley submanifolds are natural candidates for these singular fibres. Deformations of conically singular special Lagrangian submanifolds in Calabi-Yau manifolds and coassociative submanifolds of G 2 -manifolds have been studied by Joyce [9] and Lotay [18], respectively. Deformations of compact Cayley submanifolds with boundary and asymptotically cylindrical Cayley submanifolds have been studied by Ohst [26,27].
We say that a manifold with a singular point is conically singular if a neighbourhood of the singular point is diffeomorphic to a cone C ∼ = L × (0, ), and moreover the metric approaches the cone metric like r μ−1 as r → 0, for some μ > 1. A submanifold with a singular point is conically singular if, in a neighbourhood of the singular point, we can identify the submanifold with a normal graph over a cone which decays with rate r μ for μ > 1. We will prove a series of results on the moduli space of Cayley deformations of a Cayley submanifold, conically singular with cone C and rate μ, that also have a conical singularity at the same point with cone C and rate μ.
In Theorem 3.8, we prove that the expected dimension of this moduli space is given by the index of a first-order linear elliptic operator acting on smooth normal vector fields that decay like r μ close to the singular point. Motivated by other work of the author [23], we pay special attention to Cayley deformations of a conically singular complex surface N inside a Calabi-Yau four-fold M. In Theorem 3.12, we will show that the elliptic operator in Theorem 3.8 is We will then study the moduli space of complex deformations of N in M that are conically singular at the same point with the same rate and cone as N . We will show in Theorem 3.14 that this moduli space is a smooth manifold, and moreover that there are no infinitesimal Cayley deformations of N that are not infinitesimal complex deformations of N in Corollary 3.16. Note that a calibration argument of Harvey and Lawson [6,II.4 Thm 4.2] shows that the complex and Cayley deformations of any compactly supported two-dimensional complex current in a Calabi-Yau four-fold are the same. Corollary 3.16 gives some geometric intuition for this result by relating the moduli spaces of complex and Cayley deformations to the operator (1.1).
In the later sections of this article, we will focus on the elliptic operators whose indices we are interested in. In particular, we will characterise the exceptional weights for which these operators are not Fredholm. We will also apply the Atiyah-Patodi-Singer Index Theorem [2] to write down an expression for the index of these operators in terms of topological and spectral invariants of the manifold.
We will conclude this article by performing a series of calculations, including the dimension of the space of infinitesimal Cayley and complex deformations of three complex cones in C 4 motivated by the work of Kawai [13] on deformations of associative submanifolds of the seven sphere.
Layout In Sect. 2, we will discuss some background results which the reader may find useful on Cayley submanifolds, conically singular manifolds and Fredholm theory on noncompact manifolds. Section 3 contains the results on the deformation theory of conically singular Cayley and complex submanifolds. In Sect. 4, we characterise the set D of exceptional weights for which the operators we discuss in this article, before deducing a version of the Atiyah-Patodi-Singer theorem for these operators. In Sect. 5, we perform calculations of some of the quantities discussed in this article for some examples.
Notation When M is a complex manifold, we denote by p,q M the bundle of ( p, q)-forms p T * 1,0 M ⊗ q T * 0,1 M. A Calabi-Yau manifold M will have Kähler form ω, complex structure J and holomorphic volume form . If N is a submanifold of M, we denote the normal bundle of N in M by ν M (N ). Moreover, if N is a complex submanifold of M then we denote by ν 1,0 M (N ) and ν 0,1 M (N ) the holomorphic and antiholomorphic normal bundles of N in M, respectively. Submanifolds will be taken to be embedded unless otherwise stated.

Definition 2.1
Let (x 1 , . . . , x 8 ) be coordinates on R 8 with the Euclidean metric g 0 = dx 2 1 + · · · + dx 2 8 . Define a four-form on R 8 by 3) The choice of constant in (2.2) was to ensure that Re is a calibration, which in turn ensures that is a calibration. Writing down an expression for (2.3) in local coordinates at any point of M and comparing to expression (2.1) we see that we can view (M, , g) as a Spin (7)-manifold, where g is the Riemannian metric defined using ω and J . Examining expression (2.3) we see that complex surfaces and special Lagrangians in a Calabi-Yau four-fold are Cayley submanifolds.
We can decompose bundles of forms on Spin (7)-manifolds into irreducible representations of Spin (7). The following proposition is taken from [11,Prop 11.4.4].
If x, u, v, w are orthogonal, then where π 7 (x ∧ y ) = 1 2 x ∧ y + (x, y, ·, ·) and denotes the musical isomorphism T X → T * X . Moreover, if e 1 , . . . , e 8 is an orthonormal frame for T X so that and 0 are identified then (2.5)

Deformation Theory of Compact Cayley Submanifolds
We begin by studying a compact Cayley submanifold Y of a Spin (7)-manifold X .
The results here are due to McLean [22, §6], although are taken in this form from a paper of the author [23]. We first use the tubular neighbourhood theorem to identify the moduli space of Cayley deformations of Y in X with the kernel of a partial differential operator.
we find that ∇ v e 1 ∧ (e i ) (e 1  Finally, note that since the metric g on X is parallel with respect to the Levi-Civita connection, Now suppose that M is a four-dimensional Calabi-Yau manifold and N is a twodimensional complex submanifold of M. We can apply the above results to study the Cayley deformations of N in M, but we will exploit the complex structure of N and M to give these results nicer forms. The following results are due to the author, and proofs can be found in [23].
To begin with, we identify the normal bundle and E with natural vector bundles on N .
We can find the linear part of G. Proposition 2.8 [23,Prop 4.4] Let N be a compact complex surface in a Calabi-Yau four-fold M. Let G be the partial differential operator defined in Proposition 2.6. Then the linearisation of G at zero is equal to the operator Therefore v is an infinitesimal complex deformation of N if, and only if, So we can see from this result (in combination with the explicit isomorphism given in Proposition 2.4) that an infinitesimal ) such that∂v +∂ * w = 0 is a complex deformation of N if and only if∂v = 0 =∂ * w. Therefore an infinitesimal Cayley deformation of N in M that is not complex would satisfy∂v = −∂ * w. The following theorem follows from the above results in combination with a local argument reproduced in Lemma A.1. Theorem 2.9 [23,Thm 4.9] Let N be a compact complex surface inside a fourdimensional Calabi-Yau manifold M. Then the moduli space of Cayley deformations of N in M near N is isomorphic to the moduli space of complex deformations of N in M, which near N is a smooth manifold of dimension Remark Comparing this to Kodaira's theorem [14, Thm 1] on the deformation theory of compact complex submanifolds, we see that we agree with the infinitesimal deformation space, but in this special case where the ambient manifold is Calabi-Yau we can integrate all infinitesimal complex deformations to true complex deformations.

Conically Singular and Asymptotically Cylindrical Manifolds
We will now give some facts about closely related conically singular and asymptotically cylindrical manifolds that we will require later.

Conically Singular Manifolds
Heuristically speaking, a conically singular manifold can be thought of as a compact topological space that is a smooth Riemannian manifold away from a point. If the manifold near this point is diffeomorphic to a product L × (0, ), and the metric on the manifold is close to the cone metric on L × (0, ), then we call the manifold conically singular. This idea is made formal in the following definition, taken from [18, Defn 3.1]. Definition 2.3 Let Z be a connected Hausdorff topological space and letẑ ∈ Z . Suppose thatẐ := Z \{ẑ} is a smooth Riemannian manifold with metric g. Then we say that Z is conically singular atẑ with cone C and rate λ if there exist > 0, λ > 1, a closed Riemannian manifold (L, g L ) of dimension one less than Z , an open setẑ ∈ U ⊆ Z and a diffeomorphism where r is the coordinate on (0, ∞) on the cone C = (0, ∞) × L, g C = dr 2 + r 2 g L is the cone metric on C and ∇ C is the Levi-Civita connection of g C .

Definition 2.4
Let Z be a conically singular manifold atẑ with cone (0, ∞) × L. Use the notation of Definition 2.3. We say that a smooth function ρ :Ẑ → (0, 1] is a radius function for Z if ρ ≡ 1 on Z \U , while on U \{ẑ} there exist constants 0 < c < 1 and C > 1 such that We will now define weighted Sobolev spaces for conically singular manifolds. The definition given here may be deduced from [15,Defn 4.1].

Definition 2.5
Let Z be an m-dimensional conically singular manifold atx with metric g onẐ := Z \{ẑ}. Let ρ be a radius function for Z . For a vector bundle F define the weighted Sobolev space L p k,μ (F) to be the set of sections σ ∈ L p k,loc (F) such that is finite.

Asymptotically Cylindrical Manifolds
An asymptotically cylindrical manifold is topologically the same as a conically singular manifold, but metrically they are conformally equivalent. Compare the following definition to Definition 2.3.

Definition 2.6
Suppose that (Ẑ , g) is a Riemannian manifold. Then we say thatẐ is asymptotically cylindrical if there exist δ > 0, a closed Riemannian manifold (L, g L ) of dimension one less than Z , an open set U ⊆ Z and a diffeomorphism where t is the coordinate on (0, ∞) on the cylinder C = (0, ∞) × L, g ∞ = dt 2 + g L is the cylindrical metric on C and ∇ ∞ is the Levi-Civita connection of g ∞ .
Notice that if (Ẑ , g) is a conically singular manifold with radius function ρ then (Ẑ , ρ −2 g) is asymptotically cylindrical. We have the following weighted spaces on an asymptotically cylindrical manifold.
where ρ :Ẑ → (0, 1] is a smooth function satisfying ce −t ≤ ρ(t) ≤ Ce −t on the cylindrical end ofẐ and is equal to one elsewhere.
We have the following relationship between the weighted spaces W

Fredholm Theory on Noncompact Manifolds
A key part in the argument for proving a result on the moduli space of Cayley deformations of compact manifolds is the observation that an elliptic operator on a compact manifold is Fredholm. Unfortunately, this result fails in general when the underlying manifold is not compact, even in the simplest of settings. However, when the noncompact manifold is topologically a compact manifold with a cylindrical end, a theory was developed for certain types of elliptic operators. Definition 2.8 LetẐ be a manifold with a cylindrical end L × (0, ∞). Let be a differential operator on compactly supported smooth sections of vector bundles. We say that A is translation invariant if it is invariant under the natural R + -action on the cylindrical end L × (0, ∞) ofẐ . IfẐ has an asymptotically cylindrical metric g, then we say that an operator

is asymptotically translation invariant if there exists a translation invariant operator
and δ > 0 such that for all j = 0, . . . , m and k ∈ N ∪ {0} where ∇ is the Levi-Civita connection of g. Here a j , a ∞ j ∈ C ∞ (F * 1 ⊗ F 2 ⊗ (T Z) ⊗ j ) and '·' denotes tensor product followed by contraction.
The following result may be deduced from the work of Lockhart and McOwen [16,Thm 6.2] in combination with Lemma 2.10.

Proposition 2.11
Let Z be a conically singular manifold atẑ, ρ a radius function for Z and T q sẐ be the vector bundle of (s, q)-tensors onẐ := Z \{ẑ}. Let be a linear mth-order elliptic differential operator with smooth coefficients such that there exists λ ∈ R so thatÃ Similarly, the isomorphism preserves both the images of (2.16) and (2.17) and their cokernels. Therefore, (2.16) and (2.17) are Fredholm for exactly the same values of μ ∈ R and moreover have the same Fredholm index. The result follows from applying [16, Thm 6.2] to the asymptotically translation invariant operatorÃ.
We will characterise the set D A for the operators that feature in this article in Sect. 4.

Conically Singular Cayley Submanifolds
The following definition gives a preferred choice of coordinates around any given point of X . This definition is analogous to [10,Defn 3.6] and [18,Defn 3.3], which are coordinate systems for almost Calabi-Yau manifolds and G 2 -manifolds, respectively. We note here that the only difference between the definition of conically singular in these works is the type of coordinate system chosen near the singular point. In a general Riemannian manifold, it suffices to choose coordinates around the singular point that identify the metric at this point with the Euclidean metric on R d .

Definition 3.1
Let (X , g, ) be a Spin(7)-manifold. Then given x ∈ X , there exist η > 0, an open set x ∈ V ⊆ X , η > 0 and a diffeomorphism where B η (0) denotes the ball of radius η around zero in R 8 , with χ(0) = x and so that dχ | 0 : Call χ a Spin(7) coordinate system for X around x.
Call two Spin(7)-coordinate systems χ,χ for X around x equivalent if In particular, when the Spin (7)-manifold X is a four-dimensional Calabi-Yau manifold, we can choose a holomorphic volume form for X so that χ is a biholomorphism and dχ | 0 identifies the Ricci-flat Kähler form ω with ω 0 and with 0 , the Euclidean Kähler form and holomorphic volume form, respectively.
We may now define conically singular submanifolds of Spin(7)-manifolds. This definition is again analogous to [10, Defn 3.6] and [18,Defn 3.4]. Definition 3.2 Let (X , g, ) be a Spin(7)-manifold and Y ⊆ X compact and connected such that there existsx ∈ Y such thatŶ := Y \{x} is a smooth submanifold of X . Choose a Spin(7)-coordinate system χ for X aroundx. We say that Y is conically singular (CS) atx with rate μ and cone C if there exist 1 < μ < 2, 0 < < η, a compact Riemannian submanifold (L, g L ) of S 7 of dimension one less than Y , an open setx ∈ U ⊂ X and a smooth map φ : where ι : (0, ∞) × L → R 8 is the inclusion map given by ι(r , l) = rl, ∇ is the Levi-Civita connection of the cone metric g C = dr 2 + r 2 g L on C, and | · | is computed using g C .

Remark
If the smooth, noncompact submanifoldŶ is a Cayley (complex) submanifold of the Spin(7)-manifold (Calabi-Yau four-fold) X then we say that Y is a CS Cayley (complex) submanifold of X .
Conically singular submanifolds come with a rate 1 < μ < 2. We must have that μ > 1 to guarantee that a conically singular submanifold is a conically singular manifold (in the sense of Definition 2.3). The reason for asking that μ < 2 is so that μ does not depend on the choice of equivalent Spin(7)-coordinate system around the singular point of the conically singular submanifold. Lemma 3.1 Let Y be a conically singular submanifold atx with rate μ and cone C of a Spin(7)-manifold (X , g, ) with Spin(7)-coordinate system χ aroundx. Then Definition 3.2 is independent of choice of equivalent Spin(7)-coordinate system.
Proof Letχ be another Spin(7)-coordinate system for X aroundx equivalent to χ . Then χ andχ and their differentials agree at zero. Let φ : (0, ) × L → B η (0) be the map from Definition 3.2. We will show that Y is conically singular in X with Spin(7)-coordinate systemχ aroundx. Takingφ :=χ −1 • χ • φ, we have that . . , and φ(r , l) = rl + O(r μ ). So we see that Y is conically singular atx with cone C in (X , g, ) with Spin(7)-coordinate system χ, but in order for Y to be CS with rate μ in this case, Eq. (3.3) tells us that we must have that μ < 2.
The following definition is independent of choice of equivalent Spin(7)-coordinate system. It is analogous to [18,Defn 3.5].

Definition 3.3
Let Y be a conically singular submanifold atx with rate μ and cone C of a Spin(7)-manifold (X , g, ) with Spin(7)-coordinate system χ . Denote by ζ := dχ | 0 : T 0 R 8 → Tx X . Define the tangent cone of Y atx to bê where ι : C → R 8 is the inclusion map given in Definition 3.2.
On a Calabi-Yau manifold M we are given a Ricci-flat metric ω that we often have no explicit expression for. The following lemma tells us that Definition 3.2 is independent of choice of Kähler metric on M. Proof Suppose that N is a CS submanifold of M with respect to ω atx. Choose a Spin(7)-coordinate system for M aroundx, with χ (0) = x and dχ | 0 : C 4 → Tx M is an isomorphism identifying ω 0 with ω |x . Then as noted before Definition 3.1, since dχ | 0 identifies the Euclidean metric g 0 with the metric g |x defined by ω and the complex structure, it suffices to check that Definition 3.2 is satisfied with χ instead of a Spin(7)-coordinate system.
Firstly note that since A is a linear map, C = Aι(C) = {Av | v ∈ ι(C)} is also a cone. Denote by L the link of C (considered as a Riemannian submanifold of S 7 ), and for any > 0 write ι : L × (0, ) → C 4 for the inclusion map (r , l ) → r l .
Define φ : Then this map is well defined (taking smaller if necessary) and moreover χ • φ is a diffeomorphism onto its image. Moreover, by a similar argument to Lemma 3.1 we have that since μ < 2, where g C = dr 2 + (r ) 2 g L is the cone metric on C and ∇ C is the Levi-Civita connection of C .
Finally, we have that and so the tangent cone to N atx is the same in each case.
Remark Note that the proof Lemma 3.2 also shows that if N is conically singular with respect to one Spin(7)-coordinate system, it is conically singular with respect to any other Spin(7)-coordinate system, although with a different cone in general, but the same tangent cone.
We can now construct an example of a conically singular complex surface inside a Calabi-Yau four-fold.

Example
We will model our conically singular complex surface on the following complex cone in C 4 . Define C to be the set of (z 1 , z 2 , z 3 , z 4 ) ∈ C 4 satisfying Clearly, if z ∈ C, then also λz ∈ C for any λ ∈ R\{0}, and so C is a cone. Checking the rank of the matrix 4z 3 1 4z 3 2 4z 3 3 4z 3 4 3z 2 1 3z 2 2 3z 2 3 3z 2 As we will discuss in more detail in Sect. 4, a complex cone C in C 4 has both a real link L := S 7 ∩ C, and a complex link := π(L), where π : S 7 → CP 3 is the Hopf fibration. We can view the real link of a complex cone as a circle bundle over the complex link of the cone.
Consider the singular submanifold N of M defined to be the set of all [z 0 : z 1 : z 2 : z 3 : z 4 : z 5 ] ∈ CP 5 satisfying The complex Jacobian matrix of the defining equations of N is given by It can be calculated that there are six singular points on N of the form [ω : 0 : 0 : 0 : where ω is a 6th root of −1.
We will now prove that N satisfies Definition 3.2. We will exploit Lemma 3.2 and check the definition using the metric on M induced from the Fubini-Study metric on CP 5 , denoted by ω.

Tubular Neighbourhood Theorems
In this section, we will prove a tubular neighbourhood theorem for conically singular submanifolds so that we can identify deformations of conically singular submanifolds with normal vector fields. We will do this in two steps. Firstly, in Proposition 3.3 we will construct a tubular neighbourhood of a cone in R n using the well-known tubular neighbourhood theorem for compact submanifolds. We will use this to construct a tubular neighbourhood of a conically singular submanifold in Proposition 3.
We can construct open sets V C ⊆ ν R n (C), invariant under (3.5), containing the zero section and T C ⊆ R n , invariant under multiplication by positive scalars, containing C that grow like r and a dilation equivariant diffeomorphism Proof We will first address the claim that there exists an R + -action on ν R n (C) so that (3.5) holds. First note that points in ν R n (C) take the form where r ∈ R + , l ∈ L and v(r , l) ∈ ν r ,l (C). Notice that for any r , r ∈ R + the spaces ν r ,l (C) and ν r ,l (C) are naturally isometric. Define an action of R + on ν R (C) by is a group action in the usual sense.
To prove the tubular neighbourhood part of this proposition, we first apply the usual tubular neighbourhood theorem to the compact submanifold L of S n−1 . (Recall that we need a metric on S n−1 to define the exponential map. We take this to be the standard round metric on S n−1 .) This gives us an open set V L ⊆ ν S n−1 (L) containing the zero section and an open set T L ⊆ S 7 containing L and a diffeomorphism L : V L → T L , so that L maps the zero section of ν S n−1 (L) to L. Again write points in ν R n (C) as (r , l, v(r , l)), where v ∈ ν r ,l (C), and similarly points in It is clear that V C is invariant under the R + -action (3.6) by construction of V C and the R + -action. We see that where |V L | is the diameter of the set V L . Now define Then it is clear that T C is dilation invariant, in the sense that it is clearly invariant under multiplication by positive scalars, and that C ⊆ T C . We see that T C grows like r in the sense that if t ∈ T , l ∈ L and r ∈ R + then It is clear that C is well-defined, bijective and smooth. It is also clear that

Finally we have that
by definition of L and so C maps the zero section of ν R n (C) to C.
We can use this result to prove a tubular neighbourhood theorem for a conically singular submanifold. We will construct a tubular neighbourhood forŴ nearx. Denote C := C ∩ B (0). Use the notation of Definition 3.2. Fix a coordinate system χ : B η (0) ⊆ R n → V ⊆ Z with χ(0) =x and dχ | 0 identifying the Euclidean metric with the metric on Tx Z . Choose φ : C → R n uniquely by asking that Then since for 1 < μ < 2 as r → 0, making smaller if necessarily, we can guarantee that φ(r , l) lies in the tubular neighbourhood of C given by Proposition 3.3. We can therefore identify φ(C ) with a normal vector field v φ on C.
Applying Proposition 3.3 gives us V C ⊆ ν R n (C), T C ⊆ R n and a diffeomorphism Denote by V C the restriction of V C to C , and define for v ∈ V φ and .

Deformation Problem
The moduli space that we will consider will be defined in Definition 3.5 below, and this moduli space will be identified with the kernel of a nonlinear partial differential operator in Proposition 3.5. First, we will define a weighted norm on spaces of differentiable sections of a vector bundle.

Weighted Norms on Spaces of Differentiable Sections
Let Z be an n-dimensional CS manifold with a radius function ρ, F a vector bundle overẐ (the nonsingular part of Z ) with a metric and connection.

Moduli Space
We will now formally define the moduli space of conically singular Cayley deformations of a Cayley submanifold that we will be studying in this article.

Definition 3.5
Let Y be a conically singular Cayley submanifold atx with cone C and rate μ of a Spin (7)-manifold (X , g, ) with respect to some Spin(7)-coordinate system χ , and denote the tangent cone of Y atx byĈ. WriteŶ := Y \{x}. Define the moduli space of conically singular (CS) Cayley deformations of Y in X ,M μ (Y ), to be the set of CS Cayley submanifolds Y atx with cone C, rate μ and tangent coneĈ of X so that there exists a continuous family of topological embeddings ι t : We will now end this section by identifying the moduli space of Cayley CS deformations of a CS Cayley submanifold of a Spin (7)-manifold with the kernel of a nonlinear partial differential operator.

Proposition 3.5 Let Y be a CS Cayley submanifold atx with cone C and rate μ ∈
(1, 2) of a Spin (7)-manifold (X , g, ). Let τ be the 2 7 -valued four-form defined in Proposition 2.2, π : 2 7 → E be the projection map for the splitting given in (2.4) and V ⊆ ν X (Ŷ ),T ⊆ X andˆ be the open sets and diffeomorphism from the CS tubular Then we can identify the moduli space of CS Cayley deformations of Y in X near Y with the kernel of the following differential operator Let v be a smooth normal vector field onŶ , and letŶ v :=ˆ v (Ŷ ). Use the notation of Definition 3.2. Choose φ : (0, ) × L → B (0) uniquely by requiring that Making and U smaller if necessary, by the definition of the tubular neighbourhood map in Proposition 3.4, we can define a map φ v : for all j ∈ N as r → 0. Now we can write and so (3.8) holds if, and only if, for j ∈ N as r → 0. But examining the definition of φ v , we see that we can identify φ v − φ with the graph of v C , and so (3.8) holds if, and only if, for j ∈ N as r → 0. But then by definition of v C this is equivalent to So we see that the moduli space of CS Cayley deformations of Y in X can be identified with the kernel of (3.7). K. Moore

Cayley Deformations of a CS Cayley Submanifold
In this section, we prove Theorem 3.8 on the expected dimension of the moduli space of CS Cayley deformations of a conically singular Cayley submanifold Y in a Spin (7)manifold X .
The following lemma is similar to [ (3.11) Moreover, we may deduce that is a smooth map of Banach spaces for any 1 < p < ∞ and k ∈ N with k > 1 + 4/ p. where we use the notation of Proposition 3.4 and Definition 3.2. The estimates follow by fixing some r 0 > 0, performing the estimate on the compact manifold {r 0 } × L, and extending these estimates to (0, ) × L using the equivariance properties of F C . Now that we have described the behaviour of the operatorF close to the singular point of the conically singular manifoldŶ , we will prove a weighted elliptic regularity result for normal vector fields in the kernel ofF. (7)manifold X . LetF be the map defined in Proposition 3.5. Then

Proposition 3.7 Let Y be a conically singular Cayley submanifold of a Spin
for any μ ∈ (1, 2)\D, 1 < p < ∞ and k ∈ N satisfying k > 1 + 4/ p. Here D is the set of exceptional weights given by applying Proposition 2.11 to the linear part ofF.
This is a little trickier than it seems, since we have that for any > 0, Recall that in Lemma 3.6, we saw that we could writê where D was defined in Proposition 2.3, andQ is nonlinear. By Proposition 2.11 there exists a discrete set D so that is Fredholm as long as λ / ∈ D. Take 0 < < (μ − 1)/2 small enough so that where Coker λ D denotes the cokernel of (3.13). Since [μ − , μ] ∩ D = ∅, we know that (see [16,Lem 7 and by our choice of . Therefore we have that Dv =Q(v) ∈ L p k,μ−1 (V ), and it is orthogonal to Coker μ D by (3.14). Therefore there existsv ∈ L p k+1,μ (V ) with Dv = Dv. But then we must have that Here we perform a trick similar to that in [12,Prop 4.6]. Taylor expandingF(v) around zero we can writeF(v) as a polynomial in v and ∇v. Differentiating and gathering terms we can write Consider the second-order elliptic linear operator By Sobolev embedding, we know that v ∈ C l μ (V ), for l ≥ 2 by choice of p and k, and therefore the coefficients of the linear operator L v lie in C l−1 loc (V ). Local regularity for linear elliptic operators with coefficients in Hölder spaces (a nice statement is given in [11,Thm 1.4.2], taken from [25, Thm 6.2.5]) tells us that v ∈ C l+1 loc (V ) which is an improvement on the regularity of v, and so bootstrapping we may deduce that v ∈ C ∞ loc (V ). (This is why we must differentiateF(v), to ensure that the coefficients of the linear operator have enough regularity to improve the regularity of v.) Therefore the coefficients of the operator L v are smooth and so we may apply an estimate of Lockhart .
We may finally deduce the main theorem of this section, on the expected dimension of the moduli space of Cayley CS deformations of a CS Cayley submanifold of a Spin (7)-manifold X . Moreover, the expected dimension ofM μ (Y ) is given by the index of the linear elliptic operator D : The linearisation ofF at zero is the operator whereK is the kernel of (3.17) andX is closed, and whereÔ μ is the finite-dimensional obstruction space and Then the mapF which is surjective. WriteK =K × {0} for the kernel of (3.18). We then have that Now we may apply the Banach space implicit function theorem to findK 0 ⊆K containing zero,X 0 ⊆X ,Ô 0 ⊆Ô μ and a smooth mapĝ = (ĝ 1 ,ĝ 2 ) :K 0 →X 0 ×Ô 0 so thatF So we may identify the kernel ofF, and thereforeM μ (Y ) with the kernel ofĝ 2 :K 0 → O 0 , a smooth map between finite-dimensional spaces (since (3.17) is Fredholm). Sard's theorem tells us that the expected dimension of the kernel ofĝ 2 is given by the index of the operator (3.17).

Cayley Deformations of a CS Complex Surface
In this section, we prove Theorem 3.12 which gives the expected dimension of the moduli space of CS Cayley deformations of a two-dimensional conically singular complex submanifold N of a Calabi-Yau four-fold M in terms of the index of the operator∂ +∂ * acting on weighted sections of a vector bundle overN (the nonsingular part of N ).

Deformation Problem
We would like to study the moduli space given in Definition 3.5 for the CS Cayley submanifold N that is a complex submanifold of a Calabi-Yau four-fold M. We will now identify this moduli space with the kernel of a nonlinear partial differential operator.
is the image ofV ⊗ C from the tubular neighbourhood theorem under the isomorphism given in Proposition 2.4, andF cx is defined so that the following diagram commutes whereF is the operator defined in Proposition 3.5 and we use the isomorphisms given in Proposition 2.4.

Moreover, the linearisation ofF cx at zero is the operator
Proof By Proposition 3.5 we can identify the moduli space of CS Cayley deformations of N in M with the kernel ofF, which is the same as the kernel ofF cx .
Since the linearisation of the operator ofF is given by the operator D defined in Proposition 2.3, the local argument of Proposition 2.5 still holds, and so we see that the linearisation ofF cx at zero is given by the operator (3.19) as claimed.

Cayley Deformations of a CS Complex Surface
In this section, we will give analogies of the results of Sect. 3.4, which were on analytic properties of the operatorF defined in Proposition 3.5, for the operatorF cx defined in Proposition 3.9. Due to the relation between the operatorsF andF cx noted in the proof of Proposition 3.9, these results follow immediately from their counterparts.
sufficiently small, there exist constants C k > 0 so that Moreover, we may deduce that

23)
is a smooth map of Banach spaces for any 1 < p < ∞ and k ∈ N with k > 1 + 4/ p.
Proof SinceF cx is defined by composing the operatorF defined in Proposition 3.5 with isomorphisms of vector bundles, the estimates (3.21) and (3.22) follow from the estimates (3.10) and (3.11), respectively, since the isomorphisms defined in Proposition 2.4 are isometries. Moreover, since these isomorphisms are smooth, the claim that (3.23) is a smooth map of Banach spaces follows from the corresponding fact forF from Lemma 3.6.
We may now give a weighted elliptic regularity result forF cx .
for any μ ∈ (1, 2)\D, 1 < p < ∞ and k ∈ N. Here D is the set of exceptional weights given by applying Proposition 2.11 to the linear part ofF cx .
Proof This follows from Proposition 3.7 in combination with the fact that the kernels ofF, defined in Proposition 3.5, andF cx are isomorphic by definition, and the isomorphism given in Proposition 2.4 is an isometry.
We deduce the following theorem on the moduli space of CS Cayley deformations of a CS complex surface inside a Calabi-Yau four-fold. This theorem can be proved by an identical argument to the proof of Theorem 3.8, but we will deduce it as a corollary of Theorem 3.8.

Complex Deformations of a CS Complex Surface
In this section, we will compare the CS complex and Cayley deformations of a CS complex surface inside a four-dimensional Calabi-Yau manifold. We will now identify the moduli space of CS complex deformations of a CS complex surface in a Calabi-Yau manifold M with the kernel of a nonlinear partial differential operator.

25)
where σ was defined in Proposition 2.6. Moreover, the kernel ofĜ is isomorphic to the kernel of its linear part given by the map (3.26) The kernel of (3.26) is isomorphic to Proof By definition of σ we see that normal vector fields in the kernel ofĜ correspond to complex deformations ofN , and a similar argument to Proposition 3.5 shows that weighted smooth sections of ν M (N ) ⊗ C give conically singular deformations ofN as required. The linear part ofĜ follows from Proposition 2.8, which was a local argument, and similarly that the kernel ofĜ is equal to the kernel of its linear part follows from the local argument reproduce in Lemma A.1. Finally, that the kernel of (3.26) is equal to (3.27) follows from Proposition 2.8, where we proved that where π 1,0 : ν M (N ) ⊗ C → ν 1,0 M (N ) and the isomorphism of Proposition 2.4.
This proposition allows us to prove that the CS complex deformations of a conically singular complex surface are unobstructed. This theorem is a generalisation of Theorem 2.9 to conically singular submanifolds.  To compare CS complex and Cayley deformations of a CS complex surface, we require the following result.

N )) is an infinitesimal CS Cayley deformation ofN if, and only if, it is an infinitesimal complex deformation ofN . That is, (∂ +∂ * )w = 0 if, and only if,
, that is, whether the integrals on both sides converge. Let ρ be a radius function for N . We have that by Hölder's inequality. This is finite since which again is finite since for μ ∈ (1, 2). Therefore  (4.2) We will first characterise the set of exceptional weights D for which (4.1) and (4.2) are not Fredholm. We will then explain how we can apply the Atiyah-Patodi-Singer index theorem to operators on conically singular manifolds, before applying this result to the operator (4.2).

Finding the Exceptional Weights for the Operators D and@ +@ *
In this section, we will find the set D of exceptional weights for which the linear elliptic operators (4.1) and (4.2) that appeared in Sect. 3 are not Fredholm. To do this, we will study these operators acting on Cayley and complex cones in R 8 . We will see that the exceptional weights are actually eigenvalues for differential operators on the links of these cones.

Nearly Parallel G 2 Structure on S 7
We can consider R 8 as a cone with link S 7 . Let ( 0 , g 0 ) be the Euclidean Spin (7)structure (as given in Definition 2.1). Define a three-form ϕ on S 7 by the following relation: Then (ϕ, g) is a G 2 -structure on S 7 (here g is the standard round metric on S 7 ). Notice that this G 2 -structure is not torsion-free, however, since 0 is closed we have that dϕ = 4 * ϕ.

Exceptional Weights for the Operator D
Let Y be a CS Cayley submanifold atx with rate μ and cone C of a Spin(7)-manifold X and writeŶ := Y \{x}. Recall the linear elliptic operator defined in Proposition 2.3.
We will now describe the set of exceptional weights for D in terms of an eigenvalue problem on the link of C.

Proposition 4.1 Let Y be a CS Cayley submanifold atx with cone C = L × (0, ∞) and rate μ of a Spin(7)-manifold X . Let D D denote the set of λ ∈ R for which
is not Fredholm.
Then λ ∈ D D if, and only if, there exists 0 = v ∈ C ∞ (ν S 7 (L)) so that

5)
where for {e 1 , e 2 , e 3 } an orthonormal frame for T L and ∇ ⊥ the connection on the normal bundle of L in S 7 induced by the Levi-Civita connection of the round metric on S 7 ,

6)
where × is the cross product on S 7 induced from the nearly parallel G 2 -structure (ϕ, g) defined by for any vector fields u, v, w on S 7 .

Remark
The operator D L can be defined on any associative submanifold of a G 2manifold, that is, a manifold with torsion-free G 2 -structure. Normal vector fields in its kernel correspond to infinitesimal associative deformations of the associative submanifold. This can be deduced from the work of McLean [22, Thm 5-2], however, the operator first appears in this form in [1,Eqn 14]. Infinitesimal associative deformations of an associative submanifold of S 7 with its nearly parallel G 2 -structure, however, satisfy (4.5) with λ = 1 as shown by Kawai [13, Lem 3.5]. Proposition 4.1 can be considered as a different proof of this fact.
Proof We can apply Proposition 2.11 to the operator D. Suppose that ρ is a radius function for Y . Then since the given Spin(7)-structure on X approaches the Euclidean Spin(7)-structure as we move close to the singular point of Y , is asymptotic to the differential operator where D 0 is defined similarly to D but using the Euclidean Spin(7)-structure pulled back to X by a Spin(7)-coordinate system χ for X aroundx (see Definition 3.1). We will verify in the subsequent calculation thatD ∞ is indeed translation invariant. By Proposition 2.11 in combination with the discussion in [16, p. 416], we see that λ ∈ D D if, and only if, there exists a normal vector field v ∈ C ∞ (ν L (S 7 )) satisfying where since ν rl,R 8 (C) ∼ = ν l,S 7 (L) for all r > 0 we can consider (r , l) → (r , r λ−1 v(l)) as a normal vector field on the cone. Note also that the induced Euclidean metric on the normal bundle of C in R 8 takes the form r 2 h, where h is the metric on the normal bundle of L in S 7 induced from the round metric on S 7 .
Let {e 1 , e 2 , e 3 } denote a local orthonormal frame for T L with dual coframe {e 1 , e 2 , e 3 }, and denote by 0 the Euclidean Cayley form on R 8 and ϕ the nearly parallel G 2 -structure on S 7 defined in (4.3). Let ∇ denote the Levi-Civita connection of the cone metric and ∇ denote the Levi-Civita connection of g L (and induced by g S 7 on normal vector fields). We compute that Now we wish to replace the musical isomorphism : ν R 8 (C) → ν * R 8 (C) with the musical isomorphism L : ν S 7 (L) → ν * S 7 (L). Since the metric on ν R 8 (C) is of the form r 2 h, where h is a metric on ν S 7 (L), we find that At this point, we may verify that the operatorD ∞ = r −1 D 0 r −1 is translation invariant. Writing r = e −λt , we see that the expression above implies that where∇ is the Levi-Civita connection of the product metricg = dt 2 + g L . This expression makes it clear thatD ∞ is a translation invariant operator on the cylinder L × (0, ∞).
Therefore we see that We find that Since by definition, we see that λ ∈ D D if, and only if, there exists 0 = v ∈ C ∞ (ν S 7 (L)) such that

Exceptional Weights for the Operator@ +@ *
Let N be a CS complex surface with rate μ and cone C inside a Calabi-Yau fourfold M, and writeN for its nonsingular part. In order to prove an analogous result to Proposition 4.1 for the operator we will need some preliminary facts about complex cones. The real link of a complex cone C is a circle bundle over the complex link of C.

Definition 4.2 Let
C be a complex cone in C n+1 , and denote by J the standard complex structure on C n+1 . The Reeb vector field is defined to be ξ := J r ∂ ∂r .
If p| L : L → is the restriction of the Hopf projection to L, then at each l ∈ L, ξ l spans the kernel of dp| l : T l L → T p(l) . With these definitions, we may characterise the set of exceptional weights for the operator (4.7) in terms of an eigenproblem on the link of a cone. is not Fredholm. Let L denote the real link of C. Then λ ∈ D if, and only if, there exists a nontrivial pair v ∈ C ∞ (ν 1,0 S 7 (L)) and w ∈ C ∞ ( 0,1 h L ⊗ ν 1,0 S 7 (L)) so that where ξ is the Reeb vector field on L. Here ∇ acts on 0,1 h L as the Levi-Civita connection of the metric on L and on ν 1,0 S 7 (L) as the normal part of the Levi-Civita connection on S 7 .
Proof Similarly to the proof of Proposition 4.1, if ρ is a radius function for N then we can see that onN is asymptotic to the operator on the cone C, which we will see in the calculation below is translation invariant.
If v ∈ C ∞ (ν S 7 (L) ⊗ C) we can think of r μ v as a complexified normal vector field on C, and moreover the complex structure J on C 4 induces a splitting of the complexified normal bundle of L in S 7 into holomorphic and antiholomorphic parts (the i and −i eigenspaces of J , respectively). Also, by definition of the Reeb vector field, if we take θ ∈ C ∞ ( 1 L) to be the dual one-form to ξ we have that dr − irθ is a (0, 1)-form on C. By definition, 0,2 C = 2 T * 0,1 C, and T * 0,1 C = dr − irθ ⊕ 0,1 h L so we can see that a (0, 2)-form on C must be of the form where w ∈ C ∞ ( 0,1 h L). By Proposition 2.11 in combination with the discussion in [16, p. 416], we deduce that λ ∈ D if, and only if, there exists v ∈ C ∞ (ν S 7 (L)) and Denote by ∇ the Levi-Civita connection of the cone metric and ∇ the Levi-Civita connection of g L (induced from the Levi-Civita connection of g S 7 on normal vector fields). We can calculate that On (0, 2)-forms, the operators∂ * and d * coincide. Therefore, for a local orthonormal frame { ∂ ∂r , ξ/r , e 1 /r , e 2 /r } for T C with J (r ∂ ∂r ) = ξ and J e 1 = e 2 we have that∂ * C We calculate that since for the Levi-Civita connection of the cone metric on a one-form α on the link ∇ ∂ ∂r α = −α/r . Moreover, again using properties of the Levi-Civita connection of the cone metric and recalling that the complex structure on a Kähler manifold is parallel. Finally, notice that for c 1 , c 2 = ±1 or ±i since ∇ X dr = r X L and the complex structure J is parallel. Therefore since 0,1 L is a rank one vector bundle. We deduce that (4.12) Equating (4.11) and minus (4.12), we find that λ ∈ D if, and only if, as claimed. Finally, we verify thatÃ ∞ is translation invariant. Using the above calculations, with a coordinate transformation of the form r = e −t we find that where∇ denotes the Levi-Civita of the product metricg = dt 2 + g L . We can see from this expression that the operatorÃ ∞ is translation invariant, as claimed.

An Eigenproblem on the Complex Link
In Proposition 4.2 we characterised the set of exceptional weights D for which the operator (4.8) is not Fredholm in terms of an eigenproblem on the real link of a complex cone C. In this section, we will introduce a trick used by Lotay [20, §6] to study an eigenvalue problem on the link of a coassociative cone which is a circle bundle over a complex curve in CP 2 . This will allow us to give an equivalent eigenvalue problem to (4.9)-(4.10) on the real link of C completely in terms of operators and vector bundles on the complex link of C.
Let C be a complex cone in C 4 with real link L ⊆ S 7 and complex link ⊆ CP 3 . Suppose we have a problem of the following form: Find all of the functions f on L that satisfy for some m ∈ Z, where ξ is the Reeb vector field on C.
We would like to understand the relationship between the operator∂ h on the real link of C and∂ on the complex link C.
Basic functions, forms and vector fields are special because they are in one-one correspondence with functions, forms and vector fields on . It follows from [28, Lem 1] that∂ h acting on basic functions, forms or vector fields on L is equivalent to∂ acting on functions, forms or vector fields on . In Problem (4.13), when m = 0, f is not basic. However, a simple trick allows us to pretend that f is basic.
By the definition of the complex link, we may identify the cone C with the vector bundle O CP 3 (−1)| , that is, the tautological line bundle over CP 3 restricted to . This is then a trivial (real) line bundle over L and therefore has a global section given by the map x → s(x) = x for x ∈ L. It is easy to see that L ξ s = is, and therefore

Theorem 4.3 (Hirzebruch-Riemann-Roch) Let be a Riemann surface and let F be a holomorphic vector bundle over . Denote by h 0 ( , F) the dimension of the space of holomorphic sections of F. Let K denote the canonical bundle of . Then
where deg(F) is the degree of the vector bundle F, rk(F) is the rank of the vector bundle and g is the genus of .
We will now apply the trick that we described above to rephrase the eigenvalue problem (4.9)-(4.10) on the real link of a cone as an eigenvalue problem on the complex link of the cone.

Proposition 4.4 Let C be a complex cone in C 4 with real link L and complex link
. Then given λ ∈ R and m ∈ Z, pairs v ∈ C ∞ ν 1,0 14) are in a one-one correspondence with pairsṽ ∈ C ∞ ν 1,0 where ξ is the Reeb vector field, and the eigenvalue problem (4.9)-(4.10).
Proof We can pull back v and w to basic sections of ν 1,0 S 7 (L)⊗O CP 3 (m)| and 0,1 h L ⊗ ν 1,0 S 7 (L) ⊗ O CP 3 (m)| over L, respectively. As mentioned above, these sections are in one-one correspondence with sectionsṽ andw of ν 1,0 S 7 (L) and 0,1 h L ⊗ ν 1,0 S 7 (L), respectively, satisfying So we see that v and w are in one-one correspondence withṽ andw satisfying (4.16), andṽ andw satisfy∂ Let ∇ denote the Levi-Civita connection of g L (induced from g S 7 on normal vector fields). By [28, Lemma 3, §5], we see that any horizontal vector field X on S 7 viewed as a circle bundle over CP 3 satisfies horizontal part(∇ X ξ) = J X. and so for any vector field of type (1, 0), we have that Moreover, if α is a (0, 1)-form, then for any (0, 1)-vector field X we have that Therefore (4.16) implies that and therefore∂ as required.

Dimension of the Moduli Space of Complex Deformations of a CS Complex Surface
In this section, we will deduce a version of the Atiyah-Patodi-Singer index theorem for operators on conically singular manifolds. We will then apply this result to prove Theorem 4.8, an index formula for the operator (4.2), which allows us to compare the dimension of the moduli space of CS complex deformations of a conically singular complex surface to what we will think of as the dimension of the moduli space of all complex deformations of a CS complex surface in a Calabi-Yau four-fold based on Kodaira's theorem [14, Thm 1] on deformations of complex submanifolds of complex varieties.

The Atiyah-Patodi-Singer Index Theorem for Conically Singular Manifolds
The Atiyah-Patodi-Singer index theorem is predominantly for a certain type of elliptic operator on a manifold with boundary. However, as a corollary to the main theorem, an index theorem for translation invariant operators on a manifold with a cylindrical end is also proved, which we quote here.

Theorem 4.5 [2, Thm 3.10 & Cor 3.14] Let
be a linear elliptic first-order translation invariant differential operator on a manifold Z with a cylindrical end L × (0, ∞) that takes the special form on L ×(0, ∞), where u is the inward normal coordinate, σ : F 1 | L → F 2 | L is a bundle isomorphism and B is a self-adjoint elliptic operator on L. Then where h, η, α 0 and h ∞ (F 2 ) are defined as follows: We will now explain how we can apply the Atiyah-Patodi-Singer index Theorem 4.5 to elliptic operators on conically singular manifolds.
We first give a technical result that relates the adjoint of a differential operator on a conically singular manifold to the adjoint of the related asymptotically translation invariant operator acting on the conformally equivalent manifold with cylindrical end.

Lemma 4.6
Let Z be an m-dimensional conically singular manifold atẑ and let ρ be a radius function for Z . WriteẐ := Z \{ẑ}, and g for the metric onẐ . Let be a linear first-order differential operator onẐ and suppose there exists λ ∈ R so thatÃ is an asymptotically translation invariant operator. Then the formal adjoint of the operatorÃ (with respect to the metric ρ −2 g) is the formal adjoint of A with respect to g. Moreover, using the notation of Definitions 2.7 and 2.5, the kernel of is isomorphic to the kernel of for any μ ∈ R, k ∈ N and 1 < p < ∞.
where we have used that A * is the formal adjoint of A with respect to the metric g, which shows thatÃ * := ρ s−q+m A * ρ λ−m−s +q , is the formal adjoint ofÃ with respect to the metric ρ −2 g. By Lemma 2.10 is an isomorphism and so by definition ofÃ * and A * the kernels of (4. 19) and (4.20) are isomorphic.
We may now deduce the following proposition from Theorem 4.5 and Lemma 4.6 to give an index theorem for operators on conically singular submanifolds.

Proposition 4.7
Let Z be an m-dimensional conically singular manifold atẑ with radius function ρ. Let T q sẐ be the vector bundle of (s, q)-tensors onẐ := Z \{ẑ}. Let be a first-order linear elliptic differential operator so that for some λ ∈ R A := ρ λ+s −q Aρ q−s , is asymptotically translation invariant to a translation invariant operatorÃ ∞ acting on sections of T q sẐ , which takes the special form (4.17) on the end ofẐ . Then for μ ∈ R\D, given in Proposition 2.11, the index of differs by a constant independent of μ from the index ind μ A ∞ of which satisfies for > 0 chosen so that (0, ] ∩ D = ∅ and we use the notation of Theorem 4.5 for the terms on the right hand side of (4.23) (and these terms are defined for the translation invariant operatorÃ ∞ ).
Proof As we saw in the proof of Proposition 2.11, A andÃ have isomorphic kernel and cokernel when acting on weighted Sobolev spaces L p k+1,μ (T q sẐ ) and W p k+1,μ (T q sẐ ), respectively, and moreover the index of these operators differ from the index ofÃ ∞ by a constant independent of the weight μ.
Note that the definition of asymptotically translation invariant only determines the behaviour ofÃ ∞ on the cylindrical end of (Ẑ , ρ −2 g). We may choose a preferred operatorÃ ∞ by interpolating betweenÃ on the compact piece ofẐ and any such operatorÃ ∞ on the cylindrical end ofẐ . SinceÃ ∞ is translation invariant, we can apply Theorem 4.5 toÃ ∞ . Let Ker μÃ∞ and Ker μÃ * ∞ denote the kernels of respectively, whereÃ * ∞ is the formal adjoint ofÃ ∞ with respect to the metric ρ −2 g, where g is the metric onẐ . Then Theorem 4.5 yields that So we see that Denote by D the subset of R for which μ ∈ D if, and only if, (4.22) is not Fredholm. Then since we expect that 0 ∈ D in general, the index of A ∞ for the weight 0 may not be defined. Take > 0 so that Then ind A ∞ is well-defined. Since > 0, we have that where Ker μ A ∞ denotes the kernel of (4.22). It is claimed that To see this, suppose that α ∈ Ker 0 A ∞ . Then by elliptic regularity, α is smooth, and by definition of weighted norm on L 2 k+1,0 (T q sẐ ) α must decay to zero as r → 0 and so we must have that α = O(r ) for some > 0. Taking smaller if necessary we can guarantee that D ∩ (0, ] = ∅. The rate of decay of α allows us to deduce that α ∈ L 2 k+1, (T q sẐ ) where 0 < < . But then we are done, since there is no exceptional weight between and , and so [16,Lem 7.1] says that Ker A ∞ = Ker A ∞ . Notice that this tells us that the function μ → dim Ker μ A ∞ is upper semi-continuous at zero.
Since > 0 The above argument also shows that the function μ → dim Ker μ A * ∞ is upper semicontinuous (in particular at μ = λ − m) and so the set Ẑ satisfying A * ∞ f = 0. This allows us to deduce that Applying this to (4.25) we find that as claimed.

An Application of the APS Index Theorem
Having discussed in the previous section the set of exceptional weights D for the operator (4.7) in more detail, we will apply the Atiyah-Patodi-Singer index theorem to the operator∂ +∂ * to compare the dimension of the space of CS complex deformations  Then

Calculations
In this section, we will calculate some of the quantities studied in this article for some examples. In Sect. 5.1, we will consider deformations of two-dimensional complex cones in C 4 , both as a Cayley submanifold and a complex submanifold of C 4 . In particular, we will consider Cayley deformations of the cone that are themselves cones. The (real) link of such a complex cone is an associative submanifold of S 7 with its nearly parallel G 2 -structure inherited from the Euclidean Spin (7)-structure on C 4 , and so deforming the cone as a complex or Cayley cone in C 4 is equivalent to deforming the link of the cone as an associative submanifold. Homogeneous associative submanifolds of S 7 were classified by Lotay [19], using the classification of homogeneous submanifolds of S 6 of Mashimo [21]. The deformation theory of these submanifolds was studied by Kawai [13], who explicitly calculated the dimension of the space of infinitesimal associative deformations of these explicit examples using techniques from representation theory. Motivated by these calculations, in Sect. 5.2, we will apply the analysis of the earlier sections to compute the dimension of the space of infinitesimal Cayley conical deformations of the complex cones with these links, and check that these calculations match. We will be able to see explicitly which infinitesimal deformations correspond to complex deformations of the cone and which are Cayley but not complex deformations. In particular, we will see that complex infinitesimal deformations and Cayley infinitesimal deformations of a two-dimensional complex submanifold of a Calabi-Yau four-fold are not the same in general. Finally, in Sect. 5.3, we will compute the η-invariant for a complex cone in C 4 .

Cone Deformations
Let C be a two-dimensional complex cone in C 4 . Let v be a normal vector field on C.
If v is sufficiently small, we can apply the tubular neighbourhood theorem for cones, Proposition 3.3, to identify v with a deformation of C.
and v 2 ∈ C ∞ (ν 0,1 C 4 (C)). We know from Proposition 2.5 that v is an infinitesimal Cayley deformation of C if, and only if, where 0 is the standard holomorphic volume form on C 4 and denotes the musical isomorphism ν * 0,1 C 4 (C) → ν 1,0 C 4 (C). Moreover, by Proposition 2.6 v is an infinitesimal complex deformation of C if, and only if, We would like to know what properties v must have in order for the deformation of C corresponding to v to be a cone itself. By Proposition 3.3, in which we constructed the tubular neighbourhood of a cone, we constructed a map 2198 K. Moore where V C ⊆ ν R 8 (C) contains the zero section and T C ⊆ C 4 contains C. We constructed an action of R + on ν C 4 (C) satisfying |t · v| = t|v|, and the map C satisfies C (tr, l, tr · v(r , l)) = t C (r , l, v(r , l)).
Therefore, to guarantee that C •v is a cone in C 4 , we must have that v(r , l) = r ·v(l), for somev ∈ C ∞ (ν S 7 (L)). In this case, for all r ∈ R + . Choosing a metric on ν C 4 (C) that is independent of r , we see that r ·v(l) = rv(l).
Therefore the dimension of the space of infinitesimal conical Cayley deformations of C is equal to the dimensions of the spaces of solutions to the eigenproblems (4.5) and (4.9)-(4.10) with λ = 1. As remarked after the statement of Proposition 4.1, this particular eigenspace can be identified with the space of infinitesimal associative deformations of the link of the cone in S 7 with its nearly parallel G 2 -structure. This problem was studied by Kawai [13], who computed the dimension of these spaces for a range of examples. In terms of the work done here, this is equivalent to solving the eigenproblem (4.5) when λ = 1. We will study the eigenproblem (4.9)-(4.10) for the three examples of complex cones that were studied by Kawai in his paper. Our analysis will allow us to see directly the difference between the infinitesimal conical Cayley and complex deformations of a cone, and we hope that the complex geometry will make these calculations simpler.

Example 1: L 1 = S 3
The first example is the simplest, being just a vector subspace (with the zero vector removed). We take where C 1 is the complex cone, L 1 is the real link of C 1 and 1 is the complex link of C 1 .

Example 2: L 2 ∼ = SU(2)/Z 2
Our second example is a little less trivial. Take Then it can be shown [13,Ex 6.6] that the link of C 2 , L 2 , is isomorphic to the quotient group SU (2)/Z 2 . The complex link of C 2 is Proposition 5.2 ([20, Cor 5.12], [13,Prop 6.26]) The space of infinitesimal associative deformations of L 2 in S 7 has dimension twenty-two.

Example 3: L 3 ∼ = SU(2)/Z 3
Our third example is the most complicated to state, but is certainly the most interesting.
Define the cone C 3 to be the cone over the submanifold L 3 of S 7 which is defined as follows: consider the following action of SU where a, b ∈ C satisfy |a| 2 + |b| 2 = 1. We define L 3 to be the orbit of the above action around the point (1, 0, 0, 0) T , that is, where a, b ∈ C satisfy |a| 2 + |b| 2 = 1. We see that for L 3 is invariant under the action of Z 3 , therefore L 3 ∼ = SU (2)/Z 3 . The complex link of the cone C 3 over L 3 is which is known as the twisted cubic in CP 3 . This is a particularly interesting example for the following reason [19,Ex 5.8]. Define L 3 (θ ) to be the orbit of the above group action around the point (cos θ, 0, 0, sin θ) T . Then L 3 (θ ) is associative for θ ∈ [0, π 4 ]. As noted above, L 3 (0) = L 3 is the real link of a complex cone, however, L 3 ( π 4 ) is the link of a special Lagrangian cone. Therefore there exists a family of Cayley cones in C 4 , including both a complex cone and a special Lagrangian cone, that are related by a group action.

Calculations
We will now study the eigenvalue problem (4.9)-(4.10) with λ = 1 for C 1 , C 2 and C 3 defined above. Recall that by Proposition 4.4 we can study the eigenproblem (4.14)-(4.15) with λ = 1 on the complex link instead to make our calculations easier. We first explain how to count infinitesimal conical complex deformations and infinitesimal conical Cayley but noncomplex deformations of a complex cone.
Proof We know that infinitesimal complex deformations C will lie in the kernel of∂ C or∂ * C . Recall that these spaces are isomorphic and so we expect them to have the same dimension. Examining the proof of Proposition 4.2 and comparing to Proposition 4.4, we see that infinitesimal complex deformations of C are given by holomorphic sections of ν 1,0 CP 3 ( ) ⊗ O CP 3 (λ − 1)| , and antiholomorphic sections of 0,1 ⊗ ν 1,0 . Since infinitesimal conical deformations of C will correspond to λ = 1 here, we see that infinitesimal complex conical deformations of C correspond to holomorphic sections of and antiholomorphic sections of Finally, we see that any remaining infinitesimal conical Cayley deformations of C must satisfy the eigenproblem (4.14)-(4.15) with λ = 1 and m = 0, −4. Applying ∂ * to (4.14) and using (4.15), we see that the remaining infinitesimal conical Cayley deformations of C are given by v ∈ C ∞ (ν 1,0 While we can apply the Hirzebruch-Riemann-Roch theorem 4.3 to count holomorphic sections of holomorphic vector bundles, solving eigenproblems for the Laplacian acting sections of vector bundles such as (5.1) is somewhat more difficult, especially since the degree of the line bundle we consider appears in the eigenvalue itself. Such problems have been studied, however, and we will make use of the following result of López Almorox and Tejero Prieto on eigenvalues of the∂ -Laplacian acting on sections of holomorphic line bundles over CP 1 equipped with a metric of constant scalar curvature.
is the set The space of eigensections of 2∂ * ∂ with eigenvalue λ q is identified with the space of holomorphic sections of when deg K < 0. Therefore the multiplicity of λ q is m(λ q ) = 1 + |deg K | + 2q.

Example 1: L 1 = S 3
To calculate the dimension of the space of infinitesimal conical Cayley deformations of the cone C 1 = C 2 , which as real link L 1 = S 3 and complex link 1 = CP 1 , we will apply Proposition 5.4. We first calculate the dimension of the space of holomorphic sections of which by the Hirzebruch-Riemann-Roch theorem 4.3 has dimension four. Therefore, the dimension of the space of infinitesimal conical complex deformations of C 1 is eight. Now we study the eigenproblem and −4 < m < 0. We can apply Theorem 5.5 to solve (5.2) as long as the connection on O CP 3 (m + 1)| 1 ⊕ O CP 3 (m + 1)| 1 takes the form where ∇ i are connections on O CP 3 (m + 1)| 1 . This is the case here, as can be seen from the relation between the connection on the normal bundle of 1 in CP 3 and the connection on the normal bundle of L 1 in S 7 (see [28,Lem 1]) and the fact that the normal bundle of L 1 in S 7 is trivial. Therefore, by Theorem 5.5, solving (5.2) reduces to solving the algebraic equation Remark Recall that the stabiliser of a Cayley plane in R 8 is isomorphic to (SU (2) × SU (2) × SU (2))/Z 2 and that the dimension of Spin(7)/((SU (2) × SU (2) × SU (2))/Z 2 ) is twelve. The stabiliser of a two-dimensional complex plane in C 4 is isomorphic to U (2) × U (2), and the dimension of U (4)/(U (2) × U (2)) is equal to eight.

Example 2: L 2 ∼ = SU(2)/Z 2
We now use Proposition 5.4 to calculate the dimension of the space of infinitesimal conical Cayley deformations of the cone C 2 in C 4 with link L 2 ∼ = SU (2)/Z 2 and complex link 2 as defined in Sect. 5.1.2. Since 2 is a complete intersection of irreducible polynomials of degree 1 and 2 in CP 3 , its normal bundle is given by The dimension of the space of holomorphic sections of ν 1,0 CP 3 ( 2 ), by the Hirzebruch-Riemann-Roch theorem 4.3, has dimension eight, and so we deduce that the space of infinitesimal conical complex deformations of C 2 has dimension sixteen.
Since 2 ⊆ CP 2 , we see that the Levi-Civita connection on ν 1,0 CP 3 ( 2 ) must be of the form (5.3), so that we may apply Theorem 5.5 to solve the eigenproblem with a = 0 for m ≥ −2 and a = 1 otherwise, which has solution (q, a, m) = (1, 0, −2). Therefore by Theorem 5.5 the dimension of the space of solutions to (5.4) has dimension 3+3 = 6. Therefore, the dimension of the space of infinitesimal conical Cayley deformations of C 2 in C 4 is twenty-two.

Proposition 5.7
The real dimension of the space of infinitesimal conical Cayley deformations of C 2 in C 4 is twenty-two. The real dimension of the space of infinitesimal conical complex deformations of C 2 in C 4 is sixteen.
Remark The dimension of Spin(7)/SU (4) is six, which implies that the six Cayley but not complex infinitesimal conical deformations of C 2 are just rigid motions induced by the action of Spin(7) on R 8 .
We must have that for some a ∈ C (since if α 1 = 0 then we find infinitesimal conical complex deformations of C 3 ), and so we may instead study the eigenvalue problems Using the structure equations given in Lemma 5.9, we see that the problem (5.8)-(5.9) is equivalent to the eigenproblem where we consider g 2 (ω 2 − iω 3 ) as a 0,1 h L-valued function, which becomes where now α 2 is a section of 0,1 h L ⊗ 0,1 h L. Supposing that L e 1 g 1 = img 1 , L e 1 α 1 = imα 1 , for 3m ∈ Z we see that in order for the eigenproblem (5.12)-(5.13) to make sense we must have L e 1 α 2 = imα 2 .
Finally, we check the case that m = 4/3. In this case, for the eigenvalues we must have a = −2. However, in this case, the eigenvalue is equal to −4, which is negative and therefore not a possible eigenvalue of∂ * 3∂ 3 on sections of O 3 (6). We have found a total of six infinitesimal conical Cayley deformations of C 3 that are not complex.

Proposition 5.10
The real dimension of the space of infinitesimal conical Cayley deformations of C 3 in C 4 is thirty. The real dimension of the space of infinitesimal conical complex deformations of C 3 in C 4 is twenty-four.
Remark Similarly to Proposition 5.7 we have six infinitesimal conical Cayley deformations of C 3 which are not complex, which again implies that these deformations are just rigid motions.

Calculating the Á-Invariant for an Example
The final calculation in this article is to compute the η-invariant of the Atiyah-Patodi-Singer index theorem 4.5 for one of the examples we considered in Sect. 5.1. This will help us to calculate (what we expect to be) the codimension of the space of conically singular complex CS deformations of a CS complex surface N at C with rate μ in a Calabi-Yau manifold M inside the space of all complex deformations of N , for a certain cone C in C 4 , using Theorem 4.8.
It remains to compute the multiplicity of λ as an eigenvalue of where v is a section of O CP 3 (m + 1)| 1 ⊕ O CP 3 (m + 1)| 1 and λ = 1 + m or −3 − m. Theorem 5.5 tells us that this is equivalent to solving the algebraic equation where q is a positive integer. It can be computed that the multiplicity of integer λ > 0 as an eigenvalue of (5.20) is 2λ(λ + 1) and the multiplicity of integer λ < −2 as an eigenvalue of (5.20) is 2(λ + 2)(λ + 1). So we have that and so where ζ is the Riemann zeta function.
We have that the multiplicity of the zero eigenvalue in this case in four. So we have found that

Concluding Remarks
An ideal result in this area would be to deform a singular calibrated submanifold into a compact nonsingular calibrated submanifold-this would perhaps give new examples of compact calibrated submanifolds. However, this problem seems intractable with the type of analysis applied in this article. One motivation for the complex geometry viewpoint taken in this article is that techniques from algebraic geometry are ideal for this kind of problem. If one could generalise the natural techniques for desingularisation from complex geometry to Cayley submanifolds and thus other calibrated submanifolds this would be a very interesting result. However, whether this is feasible remains to be seen. The author chose to study the Atiyah-Patodi-Singer index theorem in the context of conically singular manifolds, which to the author's knowledge has not been done before, and calculate some of the quantities that appear in the index formula for some examples. In particular, it was hoped that complex geometry would make it easier to calculate some of these quantities, which as one can see from the length of Sect. 5.2.3 is not necessarily the case in practice. Moreover, an explicit calculation of the expected dimension of a moduli space using the Atiyah-Patodi-Singer index theorem will not be accurate since the expression (4.29) will in general differ by a constant from the index of the operator that gives the expected dimension. However, (4.29) and in particular the heuristic interpretation of this expression given in the remark after Theorem 4.8 could be a clue to how one might develop new techniques to study more general moduli spaces of conically singular calibrated submanifolds.