Abstract
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.
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1 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 two-dimensional 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 \(C\cong L\times (0,\epsilon )\), and moreover the metric approaches the cone metric like \(r^{\mu -1}\) as \(r\rightarrow 0\), for some \(\mu >1\). 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 \(r^\mu \) for \(\mu >1\). 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 \(\mu \), that also have a conical singularity at the same point with cone C and rate \(\mu \).
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 \(r^\mu \) 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 \(\mathbb {C}^4\) 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 \(\mathcal {D}\) 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 \(\Lambda ^{p,q}M\) the bundle of (p, q)-forms \(\Lambda ^pT^{*1,0}M\otimes \Lambda ^qT^{*0,1}M\). A Calabi–Yau manifold M will have Kähler form \(\omega \), complex structure J and holomorphic volume form \(\Omega \). If N is a submanifold of M, we denote the normal bundle of N in M by \(\nu _M(N)\). Moreover, if N is a complex submanifold of M then we denote by \(\nu ^{1,0}_M(N)\) and \(\nu ^{0,1}_M(N)\) the holomorphic and antiholomorphic normal bundles of N in M, respectively. Submanifolds will be taken to be embedded unless otherwise stated.
2 Preliminaries
2.1 Cayley Submanifolds
We follow Joyce [11, Defn 11.4.2] to define Spin(7)-manifolds.
Definition 2.1
Let \((x_1,\dots ,x_8)\) be coordinates on \(\mathbb {R}^8\) with the Euclidean metric \(g_0=dx_1^2+\dots +dx_8^2\). Define a four-form on \(\mathbb {R}^8\) by
where \(dx_{ijkl}:=dx_i\wedge dx_j\wedge dx_k\wedge dx_l\).
Let X be an eight-dimensional oriented manifold. For each \(p\in X\) define the subset \(\mathcal {A}_pX\subseteq \Lambda ^4T^*_pX\) to be the set of four-forms \(\Phi \) for which there exists an oriented isomorphism \(T_pX\rightarrow \mathbb {R}^8\) identifying \(\Phi \) and \(\Phi _0\) given in (2.1), and define \(\mathcal {A}X\) to be the vector bundle with fibre \(\mathcal {A}_pX\).
A four-form \(\Phi \) on X that satisfies \(\Phi |_p\in \mathcal {A}_pX\) for all \(p\in X\) defines a metric g on X. We call \((\Phi ,g)\) a Spin(7)-structure on X. If \(\nabla \) denotes the Levi-Civita connection of g then say \((\Phi ,g)\) is a torsion-freeSpin(7)-structure on X if \(\nabla \Phi =0\).
Then \((X,\Phi ,g)\) is a Spin(7)-manifold if X is an eight-dimensional oriented manifold and \((\Phi ,g)\) is a torsion-free Spin(7)-structure on X.
Given a Spin(7)-manifold \((X,\Phi ,g)\) then \(\Phi \) is a calibration on X, known as the Cayley calibration. An oriented, four-dimensional submanifold Y of X is said to be Cayley if
i.e. Y is calibrated by \(\Phi \).
Definition 2.2
Let \((M^m,J,\omega ')\) be a compact Kähler manifold with trivial canonical bundle \(K_M:= \Lambda ^{m,0}M\), i.e. with nowhere vanishing section \(\alpha \) with \(\bar{\partial }\alpha =0\). Then by Yau’s proof of the Calabi conjecture, there exists a Ricci-flat Kähler form \(\omega \in [\omega ']\). Choose a holomorphic, nowhere vanishing section \(\Omega \in \Omega ^{m,0}(M)\) so that
Say that \((M,J,\omega ,\Omega )\) is a Calabi–Yau manifold.
Given a Calabi–Yau four-fold \((M,J,\omega ,\Omega )\), we can define a Cayley form on M by
The choice of constant in (2.2) was to ensure that Re \(\Omega \) is a calibration, which in turn ensures that \(\Phi \) 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 \((M,\Phi ,g)\) as a Spin(7)-manifold, where g is the Riemannian metric defined using \(\omega \) 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].
Proposition 2.1
Let X be a Spin(7)-manifold. Then the bundle of two-forms M admits the following decomposition into irreducible representations of Spin(7):
where \(\Lambda ^k_l\) denotes the irreducible representation of Spin(7) on k-forms of dimension l.
Remark
If Y is a Cayley submanifold, then we can view \(\Lambda ^2_+Y\) as a subbundle of \(\Lambda ^2_7|_Y\) via the map \(\alpha \mapsto \pi _7(\alpha )\) [22, p. 741], where \(\pi _7:\Lambda ^2M\rightarrow \Lambda ^2_7\) is the projection map which will be described explicitly in Proposition 2.2. We will denote by E the orthogonal complement of \(\Lambda ^2_+Y\) in \(\Lambda ^2_7|_Y\), so that
The following result allows us to characterise Cayley submanifolds of a Spin(7)-manifold \((X,\Phi ,g)\) in terms of a four-form that vanishes exactly when restricted to a Cayley submanifold of X.
Proposition 2.2
[30, Lem 10.15] Let X be a eight-dimensional manifold with Spin(7)-structure \((\Phi ,g)\) and let Y be an oriented four-dimensional submanifold of X. Then there exists \(\tau \in C^\infty \left( \Lambda ^4 X\otimes \Lambda ^2_7\right) \) so that Y (endowed with the correct choice of orientation) is a Cayley submanifold of X if, and only if, \(\tau |_Y\equiv 0\).
If x, u, v, w are orthogonal, then
where \(\pi _7(x^\flat \wedge y^\flat )=\frac{1}{2}\left( x^\flat \wedge y^\flat +\Phi (x,y,\cdot ,\cdot )\right) \) and \(\flat \) denotes the musical isomorphism \(TX\rightarrow T^*X\). Moreover, if \(e_1,\dots , e_8\) is an orthonormal frame for TX so that \(\Phi \) and \(\Phi _0\) are identified then
2.2 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.
Proposition 2.3
[22, Thm 6.3] Let \((X,g,\Phi )\) be a Spin(7)-manifold with compact Cayley submanifold Y. Let \(\exp \) denote the exponential map and for a normal vector field v define \(Y_v:=\exp _v(Y)\). The moduli space of Cayley deformations of Y in X is isomorphic near Y to the kernel of the following partial differential operator
where \(\tau \) is defined in Proposition 2.2, V is an open neighbourhood of the zero section in \(\nu _X(Y)\) and E was defined in (2.4), with \(\pi :\Lambda ^2_7|_Y\rightarrow E\) the projection map.
Moreover, we have that the linearisation of F at zero is the operator
where \(\{e_1,e_2,e_3,e_4\}\) is a frame for TY with dual coframe \(\big \{e^1,e^2,e^3,e^4\big \}\), \(\nabla ^\perp :TY\otimes \nu _X(Y)\rightarrow \nu _X(Y)\) denotes the connection on \(\nu _X(Y)\) induced by the Levi-Civita connection of X and \(\pi _7\) denotes the projection of two-forms onto \(\Lambda ^2_7\) as in Proposition 2.2.
Remark
Normal vector fields in the kernel of (2.7) are called infinitesimal Cayley deformations of Y in X.
Proof
(partial) We will prove only that the linearisation of F takes the form (2.7), since this expression is different to that of McLean.
We have that
Take a local orthonormal frame \(\{e_1,\dots , e_8\}\) for TX with \(TY = \{e_1,\dots ,e_4\}\), and in this frame
where \(e^i = g(e_i,\, \cdot \,)\) and \(e^{ijkl}:=e^i\wedge e^j\wedge e^k\wedge e^l\). We have that
Using a formula such as [8, Eqn (4.3.26)], we find that
where we have used that \(\tau \) vanishes on four tangent vectors to a Cayley submanifold. By definition of \(\tau \) given in Proposition 2.2, we have that
and so it remains to show that if \(\Phi \) is parallel then so is \(\tau \). From Eq. (2.5), we see that
We can see that the second sum in the above expression will vanish when evaluated on \(e_1,e_2,e_3,e_4\), so it remains to compute
for \(i=2,\dots ,8\). Since
we find that
Similarly, since \(i\ne 1\),
Using the explicit expression for \(\Phi \), we have that
Finally, note that since the metric g on X is parallel with respect to the Levi-Civita connection,
and so we find that
which vanishes since \(\Phi \) is parallel. \(\square \)
Now suppose that M is a four-dimensional Calabi–Yau manifold and N is a two-dimensional 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.
Proposition 2.4
[23, Prop 3.2 and 3.3] Let N be a two-dimensional submanifold of a Calabi–Yau four-fold \((M,J,\omega ,\Omega )\). Then
with \(\nu ^{0,1}_M(N)\cong \Lambda ^{0,2}N\otimes \nu ^{1,0}_M(N)\) via the map
where \(\sharp :\nu ^{*0,1}_M(N)\rightarrow \nu ^{1,0}_M(N)\), and
With these isomorphisms in place, we can modify Proposition 2.3.
Proposition 2.5
[23, Prop 3.5] Let N be a complex surface in a Calabi–Yau manifold M. Then the infinitesimal Cayley deformations of N in M can be identified with the kernel of the operator
We will now apply McLean’s method to study the complex deformations of N in M. We begin by finding a form which vanishes exactly when restricted to a two-dimensional complex submanifold.
Proposition 2.6
[23, Prop 4.2] Let Y an oriented four-dimensional submanifold of a four-dimensional Calabi–Yau manifold \((M,J,\omega ,\Omega )\). Then Y is a complex submanifold of M if, and only if, for all vector fields u, v, w on Y,
where \(\sigma (u,v,w):= Re \Omega (u,v,w,\cdot )\).
We can now define a partial differential operator whose kernel can be identified with the moduli space of complex deformations of N in M.
Proposition 2.7
[23, Prop 4.3] Let N be a compact complex surface inside a four-dimensional Calabi–Yau manifold M. Then the moduli space of complex deformations of N is isomorphic near N to the kernel of
where \(\sigma \) was defined in Proposition 2.6 and V is an open neighbourhood of the zero section in \(\nu _M(N)\).
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
where \(v\in C^\infty (\nu _M(N)\otimes \mathbb {C})\). Therefore v is an infinitesimal complex deformation of N if, and only if,
Moreover, we have that, if \(v=v_1\oplus v_2\) where \(v_1\in \nu ^{1,0}_M(N)\) and \(v_2\in \nu ^{0,1}_M(N)\)
and
So we can see from this result (in combination with the explicit isomorphism given in Proposition 2.4) that an infinitesimal Cayley deformation of N, \(v\oplus w \in C^\infty (\nu ^{1,0}_M(N)\oplus \Lambda ^{0,2}N\otimes \nu ^{1,0}_M(N))\) such that \(\bar{\partial }v +\bar{\partial }^* w=0\) is a complex deformation of N if and only if \(\bar{\partial }v=0=\bar{\partial }^*w\). Therefore an infinitesimal Cayley deformation of N in M that is not complex would satisfy \(\bar{\partial }v=-\bar{\partial }^* w\). 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 four-dimensional 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
where
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.
2.3 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.
2.3.1 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 \(L\times (0,\epsilon )\), and the metric on the manifold is close to the cone metric on \(L\times (0,\epsilon )\), 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 let \(\hat{z}\in Z\). Suppose that \(\hat{Z}:=Z\backslash \{\hat{z}\}\) is a smooth Riemannian manifold with metric g. Then we say that Z is conically singular at \(\hat{z}\) with cone C and rate \(\lambda \) if there exist \(\epsilon >0\), \(\lambda >1\), a closed Riemannian manifold \((L,g_L)\) of dimension one less than Z, an open set \(\hat{z}\in U\subseteq Z\) and a diffeomorphism
such that
where r is the coordinate on \((0,\infty )\) on the cone \(C=(0,\infty )\times L\), \(g_C=dr^2+r^2g_L\) is the cone metric on C and \(\nabla _C\) is the Levi-Civita connection of \(g_C\).
Definition 2.4
Let Z be a conically singular manifold at \(\hat{z}\) with cone \((0,\infty )\times L\). Use the notation of Definition 2.3. We say that a smooth function \(\rho :\hat{Z}\rightarrow (0,1]\) is a radius function for Z if \(\rho \equiv 1\) on \(Z\backslash U\), while on \(U\backslash \{\hat{z}\}\) there exist constants \(0<c<1\) and \(C>1\) such that
on \((0,\epsilon )\times L\).
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 at \(\hat{x}\) with metric g on \(\hat{Z}:=Z\backslash \{\hat{z}\}\). Let \(\rho \) be a radius function for Z. For a vector bundle F define the weighted Sobolev space \(L^p_{k,\mu }(F)\) to be the set of sections \(\sigma \in L^p_{k,\text {loc}}(F)\) such that
is finite.
2.3.2 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 \((\hat{Z},g)\) is a Riemannian manifold. Then we say that \(\hat{Z}\) is asymptotically cylindrical if there exist \(\delta >0\), a closed Riemannian manifold \((L,g_L)\) of dimension one less than Z, an open set \(U\subseteq Z\) and a diffeomorphism
such that for all \(j\in \mathbb {N}\cup \{0\}\)
where t is the coordinate on \((0,\infty )\) on the cylinder \(C=(0,\infty )\times L\), \(g_\infty =dt^2+g_L\) is the cylindrical metric on C and \(\nabla _\infty \) is the Levi-Civita connection of \(g_\infty \).
Notice that if \((\hat{Z},g)\) is a conically singular manifold with radius function \(\rho \) then \((\hat{Z},\rho ^{-2}g)\) is asymptotically cylindrical.
We have the following weighted spaces on an asymptotically cylindrical manifold.
Definition 2.7
Let \((\hat{Z},g)\) be an asymptotically cylindrical manifold. For a vector bundle F over \(\hat{Z}\), define the weighted Sobolev spaces \(W^p_{k,\delta }(F)\) to be the space of sections \(\sigma \in L^p_{k,\text {loc}}(F)\) so that
where \(\rho :\hat{Z}\rightarrow (0,1]\) is a smooth function satisfying \(c e^{-t}\le \rho (t)\le Ce^{-t}\) on the cylindrical end of \(\hat{Z}\) and is equal to one elsewhere.
We have the following relationship between the weighted spaces \(W^p_{k,\delta }\) and \(L^p_{k,\mu }\).
Lemma 2.10
[15, Prop and Defn 4.4] Let Z be a conically singular manifold at \(\hat{z}\) of dimension m with metric g on \(\hat{Z}:=Z\backslash \{\hat{z}\}\). Let \(\rho \) be a radius function for Z. Let \(T^q_s\hat{Z}\) be the vector bundle of (s, q)-tensors on \(\hat{Z}\). Denote by \(W^p_{k,\delta }(T^q_s\hat{Z})\) the weighted space of Definition 2.7 with metric \(\rho ^{-2}g\) and denote by \(L^p_{k,\mu }(T^q_s\hat{Z})\) the weighted space of Definition 2.5. Then these spaces are isomorphic, with isomorphism given by
2.4 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
Let \(\hat{Z}\) be a manifold with a cylindrical end \(L\times (0,\infty )\). 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 \(\mathbb {R}_+\)-action on the cylindrical end \(L\times (0,\infty )\) of \(\hat{Z}\). If \(\hat{Z}\) has an asymptotically cylindrical metric g, then we say that an operator
is asymptotically translation invariant if there exists a translation invariant operator
and \(\delta >0\) such that for all \(j=0,\dots , m\) and \(k\in \mathbb {N}\cup \{0\}\)
where \(\nabla \) is the Levi-Civita connection of g. Here \(a_j,a_j^\infty \in C^\infty (F_1^*\otimes F_2\otimes (TZ)^{\otimes j})\) and ‘\(\cdot \)’ 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 at \(\hat{z}\), \(\rho \) a radius function for Z and \(T^q_s\hat{Z}\) be the vector bundle of (s, q)-tensors on \(\hat{Z}:=Z\backslash \{\hat{z}\}\). Let
be a linear \(m\text {th}\)-order elliptic differential operator with smooth coefficients such that there exists \(\lambda \in \mathbb {R}\) so that
is asymptotically translation invariant to some translation invariant operator
Then
is a bounded map and there exists a discrete set \(\mathcal {D}_{A}\subseteq \mathbb {R}\) such that (2.16) is Fredholm if, and only if, \(\mu \in \mathbb {R}\backslash \mathcal {D}_{A}\). In this case (2.15) and (2.16) are Fredholm for the same set of weights, and moreover their Fredholm indices differ by a constant independent of \(\mu \).
Proof
By Lemma 2.10, the map
is an isomorphism which restricts to an isomorphism between the kernel of (2.16) and the kernel of
since by definition
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 \(\mu \in \mathbb {R}\) and moreover have the same Fredholm index. The result follows from applying [16, Thm 6.2] to the asymptotically translation invariant operator \(\tilde{A}\). \(\square \)
We will characterise the set \(\mathcal {D}_A\) for the operators that feature in this article in Sect. 4.
3 Deformations of Conically Singular Cayley Submanifolds
3.1 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 \(\mathbb {R}^d\).
Definition 3.1
Let \((X,g,\Phi )\) be a Spin(7)-manifold. Then given \(x\in X\), there exist \(\eta >0\), an open set \(x\in V\subseteq X\), \(\eta >0\) and a diffeomorphism
where \(B_\eta (0)\) denotes the ball of radius \(\eta \) around zero in \(\mathbb {R}^8\), with \(\chi (0)=x\) and so that \(d\chi |_0:\mathbb {R}^8\rightarrow T_xX\) is an isomorphism identifying \((\Phi |_x,g|_x)\) with \((\Phi _0,g_0)\). Call \(\chi \) a Spin(7) coordinate system for X around x.
Call two Spin(7)-coordinate systems \(\chi ,\tilde{\chi }\) for X around xequivalent if
as maps \(\mathbb {R}^8\rightarrow T_xX\).
In particular, when the Spin(7)-manifold X is a four-dimensional Calabi–Yau manifold, we can choose a holomorphic volume form \(\Omega \) for X so that \(\chi \) is a biholomorphism and \(d\chi |_0\) identifies the Ricci-flat Kähler form \(\omega \) with \(\omega _0\) and \(\Omega \) with \(\Omega _0\), 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 \((X,g,\Phi )\) be a Spin(7)-manifold and \(Y\subseteq X\) compact and connected such that there exists \(\hat{x}\in Y\) such that \(\hat{Y}:=Y\backslash \{\hat{x}\}\) is a smooth submanifold of X. Choose a Spin(7)-coordinate system \(\chi \) for X around \(\hat{x}\). We say that Y is conically singular (CS) at \(\hat{x}\) with rate \(\mu \) and cone C if there exist \(1<\mu <2\), \(0<\epsilon <\eta \), a compact Riemannian submanifold \((L,g_L)\) of \(S^7\) of dimension one less than Y, an open set \(\hat{x}\in U\subset X\) and a smooth map \(\phi :(0,\epsilon )\times L\rightarrow B_\eta (0)\subseteq \mathbb {R}^8\) such that \(\Psi =\chi \circ \phi :(0,\epsilon )\times L\rightarrow U\backslash \{\hat{x}\}\) is a diffeomorphism and \(\phi \) satisfies
where \(\iota :(0,\infty )\times L\rightarrow \mathbb {R}^8\) is the inclusion map given by \(\iota (r,l)=rl\), \(\nabla \) is the Levi-Civita connection of the cone metric \(g_C=dr^2+r^2g_L\) on C, and \(|\cdot |\) is computed using \(g_C\).
Remark
If the smooth, noncompact submanifold \(\hat{Y}\) 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 \(1<\mu <2\). We must have that \(\mu >1\) to guarantee that a conically singular submanifold is a conically singular manifold (in the sense of Definition 2.3). The reason for asking that \(\mu <2\) is so that \(\mu \) 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 at \(\hat{x}\) with rate \(\mu \) and cone C of a Spin(7)-manifold \((X,g,\Phi )\) with Spin(7)-coordinate system \(\chi \) around \(\hat{x}\). Then Definition 3.2 is independent of choice of equivalent Spin(7)-coordinate system.
Proof
Let \(\tilde{\chi }\) be another Spin(7)-coordinate system for X around \(\hat{x}\) equivalent to \(\chi \). Then \(\chi \) and \(\tilde{\chi }\) and their differentials agree at zero. Let \(\phi :(0,\epsilon )\times L\rightarrow B_\eta (0)\) be the map from Definition 3.2. We will show that Y is conically singular in X with Spin(7)-coordinate system \(\tilde{\chi }\) around \(\hat{x}\). Taking \(\tilde{\phi }:=\tilde{\chi }^{-1}\circ \chi \circ \phi \), we have that
since \(\tilde{\chi }^{-1}\circ \chi (x)=x+x^TAx+\dots \), and \(\phi (r,l)=rl+O(r^{\mu })\). So we see that Y is conically singular at \(\hat{x}\) with cone C in \((X,g,\Phi )\) with Spin(7)-coordinate system \(\tilde{\chi }\), but in order for Y to be CS with rate \(\mu \) in this case, Eq. (3.3) tells us that we must have that \(\mu <2\). \(\square \)
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 at \(\hat{x}\) with rate \(\mu \) and cone C of a Spin(7)-manifold \((X,g,\Phi )\) with Spin(7)-coordinate system \(\chi \). Denote by \(\zeta :=d\chi |_0:T_0\mathbb {R}^8\rightarrow T_{\hat{x}}X\). Define the tangent cone of Y at \(\hat{x}\) to be
where \(\iota :C\rightarrow \mathbb {R}^8\) is the inclusion map given in Definition 3.2.
On a Calabi–Yau manifold M we are given a Ricci-flat metric \(\omega \) 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.
Lemma 3.2
Let M be a Calabi–Yau four-fold with Ricci-flat Kähler form \(\omega \) and let N be a CS submanifold of M as in Definition 3.2. Then if \(\omega '\) is any other Kähler form on M then N is still a conically singular submanifold of M with the same rate \(\mu \in (1,2)\) and tangent cone.
Proof
Suppose that N is a CS submanifold of M with respect to \(\omega \) at \(\hat{x}\). Choose a Spin(7)-coordinate system for M around \(\hat{x}\),
for some \(\eta >0\) and open \(V\subseteq M\) containing \(\hat{x}\), so that \(\chi (0)=\hat{x}\) and \(d\chi |_0:\mathbb {C}^4\rightarrow T_{\hat{x}}M\) is an isomorphism identifying the standard Euclidean Kähler form and holomorphic volume form \((\omega _0,\Omega _0)\) with \((\omega |_{\hat{x}},\Omega |_{\hat{x}})\). Let \(\phi \), \(\epsilon \), \(C=(0,\infty )\times L\), \(\iota \) and \(\mu \) be as in Definition 3.2.
Now given any other Kähler form \(\omega '\) on M, we can find by [5, p. 107] \(\eta '>0\), an open set \(x\in V'\subseteq M\) and a biholomorphism
with \(\chi '(0)=x\) and \(d\chi '|_0:\mathbb {C}^4\rightarrow T_{\hat{x}}M\) is an isomorphism identifying \(\omega _0\) with \(\omega '|_{\hat{x}}\). Then as noted before Definition 3.1, since \(d\chi '|_0\) identifies the Euclidean metric \(g_0\) with the metric \(g'|_{\hat{x}}\) defined by \(\omega '\) and the complex structure, it suffices to check that Definition 3.2 is satisfied with \(\chi '\) instead of a Spin(7)-coordinate system.
Since \(\chi \) and \(\chi '\) are diffeomorphisms, \(d\chi |_0\) and \(d\chi '|_0\) are isomorphisms \(\mathbb {C}^4\rightarrow T_{\hat{x}}M\). Then \(A:=(d\chi '|_0)^{-1}\circ d\chi |_0\) is an invertible linear map \(\mathbb {C}^4\rightarrow \mathbb {C}^4\). We will show that N is conically singular in \((M,\omega ')\) with cone \(C'=A\iota (C)\) and rate \(\mu \).
Firstly note that since A is a linear map, \(C'=A\iota (C)=\{Av \,|\, v\in \iota (C)\}\) is also a cone. Denote by \(L'\) the link of \(C'\) (considered as a Riemannian submanifold of \(S^7\)), and for any \(\epsilon '>0\) write \(\iota ':L'\times (0,\epsilon ')\rightarrow \mathbb {C}^4\) for the inclusion map \((r',l')\mapsto r'l'\).
Define \(\phi ':(0,\epsilon ')\times L'\rightarrow \mathbb {C}^4\) by \(\phi '=\chi '^{-1}\circ \chi \circ \phi \circ A^{-1}\), where \(\epsilon '=\epsilon \Vert A\Vert \). Then this map is well defined (taking \(\epsilon '\) smaller if necessary) and moreover \(\chi '\circ \phi '\) is a diffeomorphism onto its image. Moreover, by a similar argument to Lemma 3.1 we have that
since \(\mu <2\), where \(g_{C'}=d{r'}^2+(r')^2g_{L'}\) is the cone metric on \(C'\) and \(\nabla _{C'}\) is the Levi-Civita connection of \(C'\).
Finally, we have that
and so the tangent cone to N at \(\hat{x}\) is the same in each case. \(\square \)
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 \(\mathbb {C}^4\). Define C to be the set of \((z_1,z_2,z_3,z_4)\in \mathbb {C}^4\) satisfying
Clearly, if \(z\in C\), then also \(\lambda z\in C\) for any \(\lambda \in \mathbb {R}\backslash \{0\}\), and so C is a cone.
Checking the rank of the matrix
at each point of C, we see that the only singular point of C is zero.
As we will discuss in more detail in Sect. 4, a complex cone C in \(\mathbb {C}^4\) has both a real link \(L:=S^7\cap C\), and a complex link \(\Sigma :=\pi (L)\), where \(\pi :S^7\rightarrow \mathbb {C}P^3\) 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.
In this case, the complex link \(\Sigma \) of C is the complex curve in \(\mathbb {C}P^3\) is given by \([z_0:z_1:z_2:z_3]\in \mathbb {C}P^3\) satisfying
We can apply the adjunction formula [7, Prop 2.2.17] to find that the canonical bundle of \(\Sigma \) is given by
where \(\mathcal {O}_{\mathbb {C}P^3}(k)\) denotes the \(-k\text {th}\) (tensor) power of the tautological line bundle over \(\mathbb {C}P^3\) if k is a negative integer, the \(k\text {th}\) power of the dual of the tautological line bundle if k is a positive integer, and the trivial line bundle if \(k=0\). Then it follows from the Hirzebruch–Riemann–Roch theorem [7, Thm 5.1.1] that the genus of \(\Sigma \) is
Now consider the Calabi–Yau four-fold M defined by
Consider the singular submanifold N of M defined to be the set of all \([z_0:z_1:z_2:z_3:z_4:z_5]\in \mathbb {C}P^5\) 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 \([\omega :0:0:0:0:1]\), where \(\omega \) is a \(6\text {th}\) 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 \(\mathbb {C}P^5\), denoted by \(\omega \).
Denote the singular points of N by \(\{p_1,\dots ,p_6\}\), where \(p_k=[\omega _k:0:0:0:0:1]\) for \(\omega _k:= e^{i(2k-1)\pi /6}\). We must construct maps \(\chi _k\) so that there exist \(\eta _k>0\) and open sets \(p_k\in V_k\subseteq M\) and diffeomorphisms
with \(\chi _k(0)=p_k\) and so that
for \(k=1,\dots , 6\).
For \(k=1,\dots 6\), define \(\chi _k:B_{\eta _k}(0)\rightarrow M\) by
where if \(a=re^{i\theta }\) for \(r>0\) and \(-\pi <\theta \le \pi \), we define \(a^{1/6}:=r^{1/6}e^{i\theta /6}\). It is clear that (3.4) is a diffeomorphism onto its image. The induced Fubini–Study metric on M pulls back under \(\chi _k\) to the Euclidean metric on \(\mathbb {C}^4\) at each \(p_k=[\omega _k:0:0:0:0:1]\). Taking \(\phi =\iota \), where \(\iota :C\rightarrow \mathbb {C}^4\) is the inclusion map, we see that \(\phi \circ \chi \) is a diffeomorphism C to N, and so the definition of conically singular is trivially satisfied.
3.2 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 \(\mathbb {R}^n\) 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 3.4. Propositions 3.3 and 3.4 use ideas of similar results proved by Joyce [10, Thm 4.6] for special Lagrangian cones and Lotay [18, Prop 6.4] for CS coassociative submanifolds.
Proposition 3.3
(Tubular neighbourhood theorem for cones) Let C be a cone in \(\mathbb {R}^n\) with link L and let g be a Riemannian metric on \(\mathbb {R}^n\) (not necessarily the Euclidean metric). There exists an action of \(\mathbb {R}_+\) on \(\nu _{\mathbb {R}^n}(C)\) (defined by g)
so that
We can construct open sets \(V_C\subseteq \nu _{\mathbb {R}^n}(C)\), invariant under (3.5), containing the zero section and \(T_C\subseteq \mathbb {R}^n\), invariant under multiplication by positive scalars, containing C that grow like r and a dilation equivariant diffeomorphism
in the sense that \(\Xi _C(t\cdot v)=t\,\Xi _C(v)\) for all \(v\in \nu _{\mathbb {R}^n}(C)\). Moreover, \(\Xi _C\) maps the zero section of \(\nu _{\mathbb {R}^n}(C)\) to C.
Proof
We will first address the claim that there exists an \(\mathbb {R}_+\)-action on \(\nu _{\mathbb {R}^n}(C)\) so that (3.5) holds. First note that points in \(\nu _{\mathbb {R}^n}(C)\) take the form
where \(r\in \mathbb {R}_+\), \(l\in L\) and \(v(r,l)\in \nu _{r,l}(C)\). Notice that for any \(r,r'\in \mathbb {R}_+\) the spaces \(\nu _{r,l}(C)\) and \(\nu _{r',l}(C)\) are naturally isometric. Define an action of \(\mathbb {R}_+\) on \(\nu _{\mathbb {R}}(C)\) by
Then \(|t\cdot v(r,l)|_{tr,l}=|t v(r,l)|_{r,l}=t|v(r,l)|_{r,l}\) as claimed. Notice that \(t\cdot (t'\cdot v)=(tt')\cdot v\) and so (3.6) 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 \(V_L\subseteq \nu _{S^{n-1}}(L)\) containing the zero section and an open set \(T_L\subseteq S^7\) containing L and a diffeomorphism
so that \(\Xi _L\) maps the zero section of \(\nu _{S^{n-1}}(L)\) to L. Again write points in \(\nu _{\mathbb {R}^n}(C)\) as (r, l, v(r, l)), where \(v\in \nu _{r,l}(C)\), and similarly points in \(\nu _{S^{n-1}}(L)\) as (l, v(l)) where \(v\in \nu _{l}(L)\cong \nu _{r,l}(C)\). Then define
It is clear that \(V_C\) is invariant under the \(\mathbb {R}_+\)-action (3.6) by construction of \(V_C\) and the \(\mathbb {R}_+\)-action. We see that \(V_C\) grows like r in the sense that if \(v=(r,l,v(r,l))\in V_C\) then
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 \(C\subseteq T_C\). We see that \(T_C\) grows like r in the sense that if \(t\in T\), \(l\in L\) and \(r\in \mathbb {R}_+\) then
where \(|T_L|\) is the diameter of the set \(T_L\). Define
It is clear that \(\Xi _C\) is well-defined, bijective and smooth. It is also clear that
Finally we have that
by definition of \(\Xi _L\) and so \(\Xi _C\) maps the zero section of \(\nu _{\mathbb {R}^n}(C)\) to C. \(\square \)
We can use this result to prove a tubular neighbourhood theorem for a conically singular submanifold.
Proposition 3.4
Let W be a conically singular submanifold of Z at \(\hat{x}\) with cone C and rate \(\mu \). Write \(\hat{W}:=W\backslash \{\hat{x}\}\). Then there exist open sets \(\hat{V}\subseteq \nu _Z(\hat{W})\) containing the zero section and \(\hat{T}\subseteq Z\) containing \(\hat{W}\) and a diffeomorphism
that takes the zero section of \(\nu _Z(\hat{W})\) to \(\hat{W}\). Moreover, we can choose \(\hat{V}\) and \(\hat{T}\) to grow like \(\rho \) as \(\rho \rightarrow 0\).
Proof
Notice that \(K:=W\backslash U\) is a compact submanifold of Z. So by the compact tubular neighbourhood theorem we can find open sets \(\hat{V}_1\subseteq \nu _Z(K)\) containing the zero section and \(\hat{T}_1\subseteq Z\) containing K and a diffeomorphism
We will construct a tubular neighbourhood for \(\hat{W}\) near \(\hat{x}\). Denote \(C_\epsilon :=C\cap B_\epsilon (0)\). Use the notation of Definition 3.2. Fix a coordinate system \(\chi :B_\eta (0)\subseteq \mathbb {R}^n\rightarrow V\subseteq Z\) with \(\chi (0)=\hat{x}\) and \(d\chi |_0\) identifying the Euclidean metric with the metric on \(T_{\hat{x}}Z\). Choose \(\phi :C_\epsilon \rightarrow \mathbb {R}^n\) uniquely by asking that
Then since
for \(1<\mu <2\) as \(r\rightarrow 0\), making \(\epsilon \) smaller if necessarily, we can guarantee that \(\phi (r,l)\) lies in the tubular neighbourhood of C given by Proposition 3.3. We can therefore identify \(\phi (C_\epsilon )\) with a normal vector field \(v_\phi \) on C.
Applying Proposition 3.3 gives us \(V_C\subseteq \nu _{\mathbb {R}^n}(C)\), \(T_C\subseteq \mathbb {R}^n\) and a diffeomorphism
Denote by \(V_{C_\epsilon }\) the restriction of \(V_C\) to \(C_\epsilon \), and define
with
for \(v\in V_\phi \) and
Then \(\Xi _C:V_\phi \rightarrow T_\phi \) is a diffeomorphism by construction.
Write \(\hat{U}:=U\backslash \{\hat{x}\}\). Define \(\hat{V}_2:=F(V_\phi )\subseteq \nu _Z(\hat{U})\), where F is the isomorphism \(\nu _{B_\epsilon (0)}(C_\epsilon )\rightarrow \nu _Z(\hat{U})\) induced from \(\Psi \) and \(\iota \) and \(\hat{T}_2:=\chi (T_\phi )\). By definition, these sets grow with order \(\rho \) as \(\rho \rightarrow 0\). Then
is a diffeomorphism taking the zero section of \(\nu _Z(\hat{U})\) to \(\hat{U}\). Define \(\hat{V}\), \(\hat{T}\) and \(\hat{\Xi }\) by interpolating smoothly between \(\hat{V}_1\) and \(\hat{V}_2\), \(\hat{T}_1\) and \(\hat{T}_2\) and \(\hat{\Xi }_1\) and \(\hat{\Xi }_2\). \(\square \)
3.3 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.
3.3.1 Weighted Norms on Spaces of Differentiable Sections
Let Z be an n-dimensional CS manifold with a radius function \(\rho \), F a vector bundle over \(\hat{Z}\) (the nonsingular part of Z) with a metric and connection.
Definition 3.4
Let \(\lambda \in \mathbb {R}\) and \(k\in \mathbb {N}\). Define the space \(C^k_\lambda (F)\) to be the space of sections \(\sigma \in C^k_\text {loc}(F)\) satisfying
We say that \(\sigma \in C^\infty _\lambda (F)\) if \(\sigma \in C^k_\lambda (F)\) for all \(k\in \mathbb {N}\).
The space \(C^k_\lambda (F)\) is a Banach space, but \(C^\infty _\lambda (F)\) is not in general.
3.3.2 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 at \(\hat{x}\) with cone C and rate \(\mu \) of a Spin(7)-manifold \((X,g,\Phi )\) with respect to some Spin(7)-coordinate system \(\chi \), and denote the tangent cone of Y at \(\hat{x}\) by \(\hat{C}\). Write \(\hat{Y}:=Y\backslash \{\hat{x}\}\). Define the moduli space of conically singular (CS) Cayley deformations of Y in X, \(\hat{\mathcal {M}}_{\mu }(Y)\), to be the set of CS Cayley submanifolds \(Y'\) at \(\hat{x}\) with cone C, rate \(\mu \) and tangent cone \(\hat{C}\) of X so that there exists a continuous family of topological embeddings \(\iota _t:Y\rightarrow X\) with \(\iota _0(Y)=Y\) and \(\iota _1(Y)=Y'\), so that \(\iota _t(\hat{x})=\hat{x}\) for all \(t\in [0,1]\) and so that \(\hat{\iota }_t:=\iota _t|_{\hat{Y}}\) is a smooth family of embeddings \(\hat{Y}\rightarrow X\) with \(\hat{\iota }_0(\hat{Y})=\hat{Y}\) and \(\hat{\iota }_1({\hat{Y}})=\hat{Y}':=Y'\backslash \{\hat{x}\}\).
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 at \(\hat{x}\) with cone C and rate \(\mu \in (1,2)\) of a Spin(7)-manifold \((X,g,\Phi )\). Let \(\tau \) be the \(\Lambda ^2_7\)-valued four-form defined in Proposition 2.2, \(\pi :\Lambda ^2_7\rightarrow E\) be the projection map for the splitting given in (2.4) and \(\hat{V}\subseteq \nu _X(\hat{Y})\), \(\hat{T}\subseteq {X}\) and \(\hat{\Xi }\) be the open sets and diffeomorphism from the CS tubular neighbourhood theorem 3.4. For \(v\in C^\infty (\nu _X(\hat{Y}))\) taking values in \(\hat{V}\) write \(\hat{\Xi }_v\) for the diffeomorphism \(\hat{\Xi }\circ v: \hat{Y}\rightarrow \hat{Y}_v:=\hat{\Xi }_v(\hat{Y})\).
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
Proof
The deformation \(\hat{Y}_v\) is Cayley if, and only if, \(\tau |_{\hat{Y}_v}\equiv 0\), which since \(\hat{\Xi }_v\) is a diffeomorphism is equivalent to \(*_{\hat{Y}}\hat{\Xi }^*_v(\tau |_{\hat{Y}_v})=0\). By a local argument in [23, Prop 3.4] based on a similar argument of Harvey and Lawson [6, IV.2.C Thm 2.20] this is equivalent to \(\hat{F}(v)=0\). Since \(v,\tau , \hat{\Xi }_v\) are all smooth, we see that \(\hat{F}\) takes values in \(C^\infty _\text {loc}(E)\) at claimed.
It remains to show that \(Y_v:=\hat{Y}_v\cup \{\hat{x}\}\) is a CS submanifold of X at \(\hat{x}\) with cone C and rate \(\mu \) (with respect to the same Spin(7)-coordinate system as Y) if, and only if, \(v\in C^\infty _\mu (\hat{V})\).
Let v be a smooth normal vector field on \(\hat{Y}\), and let \(\hat{Y}_v:=\hat{\Xi }_v(\hat{Y})\). Use the notation of Definition 3.2. Choose \(\phi :(0,\epsilon )\times L\rightarrow B_\epsilon (0)\) uniquely by requiring that
Now we can use \(\Psi \) and \(\iota \) to identify \(\nu _{X}(\hat{U})\) with \(\nu _{B_\epsilon (0)}(\iota (C_\epsilon ))\), where \(\hat{U}:=U\backslash \{\hat{x}\}\) and \(C_\epsilon :=(0,\epsilon )\times L\). Write \(v_C\) for the section of \(\nu _{B_\epsilon (0)}(\iota (C_\epsilon ))\) corresponding to v under this identification.
Making \(\epsilon \) and U smaller if necessary, by the definition of the tubular neighbourhood map in Proposition 3.4, we can define a map \(\phi _v:C_\epsilon \rightarrow B_\epsilon (0)\) by
where \(\Xi _\phi \) was defined in the proof of Proposition 3.4, so that \(\chi \circ \phi _v:C_\epsilon \rightarrow \Xi _v(\hat{U})\subseteq \hat{Y}_v\) is a diffeomorphism. So we see that for \(Y_v\) to be a CS submanifold of X with rate \(\mu \) and cone C we must have that
for all \(j\in \mathbb {N}\) as \(r\rightarrow 0\). Now we can write
and so (3.8) holds if, and only if,
for \(j\in \mathbb {N}\) as \(r\rightarrow 0\). But examining the definition of \(\phi _v\), we see that we can identify \(\phi _v-\phi \) with the graph of \(v_C\), and so (3.8) holds if, and only if,
for \(j\in \mathbb {N}\) as \(r\rightarrow 0\). But then by definition of \(v_C\) this is equivalent to
for \(j \in \mathbb {N}\) as \(\rho \rightarrow 0\), that is, \(v\in C^j_\mu (\hat{V})\) for all \(j\in \mathbb {N}\). So we see that the moduli space of CS Cayley deformations of Y in X can be identified with the kernel of (3.7). \(\square \)
3.4 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 [10, Thm 5.1] and [18, Prop 6.9].
Lemma 3.6
Let Y be a conically singular Cayley submanifold of a Spin(7)-manifold X. Let \(\hat{F}\) be the operator defined in Proposition 3.5. Then we can write
for \(x\in \hat{Y}\), where
is smooth, D was defined in Proposition 2.3 and
is a section of E. Let \(\mu >1\). Then for each \(k\in \mathbb {N}\), for \(v\in C^{k+1}_\mu (\hat{V})\) with \(\Vert v\Vert _{C^1_1}\) sufficiently small, there exist constants \(C_k>0\) so that
and if \(v\in L^p_{k+1,\mu }(\hat{V})\) with \(\Vert v\Vert _{C^1_1}\) sufficiently small, with \(k>1+4/p\), there exist constants \(D_k>0\) such that
Moreover, we may deduce that
is a smooth map of Banach spaces for any \(1<p<\infty \) and \(k\in \mathbb {N}\) with \(k>1+4/p\).
Proof
We omit the details of the proof of this result because of similarities to the above-cited works. For the full proof the reader may consult the author’s PhD thesis [24, Lem 5.3.1]. We briefly describe how we find the estimate near the singular point of N, elsewhere it suffices to argue as for compact Cayley submanifolds (see, for example [23, Lem 3.4]). Close to the singular point, it suffices to estimate the following operator on the cone C,
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 \(\{r_0\}\times L\), and extending these estimates to \((0,\epsilon )\times L\) using the equivariance properties of \(\hat{F}_C\). \(\square \)
Now that we have described the behaviour of the operator \(\hat{F}\) close to the singular point of the conically singular manifold \(\hat{Y}\), we will prove a weighted elliptic regularity result for normal vector fields in the kernel of \(\hat{F}\).
Proposition 3.7
Let Y be a conically singular Cayley submanifold of a Spin(7)-manifold X. Let \(\hat{F}\) be the map defined in Proposition 3.5. Then
for any \(\mu \in (1,2)\backslash \mathcal {D}\), \(1<p<\infty \) and \(k\in \mathbb {N}\) satisfying \(k>1+4/p\). Here \(\mathcal {D}\) is the set of exceptional weights given by applying Proposition 2.11 to the linear part of \(\hat{F}\).
Proof
We will first show that if \(v\in C^\infty _\mu (\hat{V})\) satisfying \(\hat{F}(v)=0\), then \(v\in L^p_{k+1,\mu }\). This is a little trickier than it seems, since we have that for any \(\epsilon >0\), \(C^\infty _\mu (\hat{V})\subseteq L^p_{k,\mu -\epsilon }(\hat{V})\), which is weaker than what we require. We will show that if \(v\in L^p_{k+1,\mu -\epsilon }(\hat{V})\), for \(\epsilon >0\) sufficiently small, satisfies \(\hat{F}(v)=0\), then we may deduce that \(v\in L^p_{k+1,\mu }(\hat{V})\). Recall that in Lemma 3.6, we saw that we could write
where D was defined in Proposition 2.3, and \(\hat{Q}\) is nonlinear. By Proposition 2.11 there exists a discrete set \(\mathcal {D}\) so that
is Fredholm as long as \(\lambda \notin \mathcal {D}\). Take \(0<\epsilon <(\mu -1)/2\) small enough so that \([\mu -\epsilon ,\mu ]\cap \mathcal {D}=\emptyset \). Let \(v\in L^p_{k+1,\mu -\epsilon }(\hat{V})\) and suppose that \(\hat{F}(v)=0\). Since (3.13) is Fredholm when \(\lambda =\mu -\epsilon \), we can write
where \(\hat{\mathcal {O}}_{\mu -\epsilon }\) is finite-dimensional and
where \(\text {Coker}_{\lambda }D\) denotes the cokernel of (3.13). Since \([\mu -\epsilon ,\mu ]\cap \mathcal {D}=\emptyset \), we know that (see [16, Lem 7.1])
Now since \(\hat{F}(v)=0\), we have that \(Dv=-\hat{Q}(v)\), and so \(\hat{Q}(v)\) is orthogonal to \(\text {Coker}_{\mu -\epsilon } D\). Also \(\hat{Q}(v)\in L^p_{k,2\mu -2-2\epsilon }(E)\subseteq L^p_{k,\mu -1}(E)\) by Lemma 3.6 since \(v\in L^p_{k+1,\mu -\epsilon }(\hat{V})\) and by our choice of \(\epsilon \). Therefore we have that \(Dv=\hat{Q}(v)\in L^p_{k,\mu -1}(\hat{V})\), and it is orthogonal to \(\text {Coker}_{\mu }D\) by (3.14). Therefore there exists \(\bar{v}\in L^p_{k+1,\mu }(\hat{V})\) with \(Dv=D\bar{v}\). But then we must have that \(v-\bar{v}\in \text {Ker}_{\mu -\epsilon }D=\text {Ker}_\mu D\), since \([\mu -\epsilon ,\mu ]\cap \mathcal {D}=\emptyset \), and so \(v\in L^p_{k+1,\mu }(\hat{V})\), as required.
Conversely, let \(v\in L^p_{k+1,\mu }(\hat{V})\) satisfy \(\hat{F}(v)=0\). Here we perform a trick similar to that in [12, Prop 4.6]. Taylor expanding \(\hat{F}(v)\) around zero we can write \(\hat{F}(v)\) as a polynomial in v and \(\nabla v\). Differentiating and gathering terms we can write
Consider the second-order elliptic linear operator
By Sobolev embedding, we know that \(v\in C^l_\mu (\hat{V})\), for \(l\ge 2\) by choice of p and k, and therefore the coefficients of the linear operator \(L_v\) lie in \(C^{l-1}_\text {loc}(\hat{V})\). 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 \(v\in C^{l+1}_\text {loc}(\hat{V})\) which is an improvement on the regularity of v, and so bootstrapping we may deduce that \(v\in C^\infty _\text {loc}(\hat{V})\). (This is why we must differentiate \(\hat{F}(v)\), 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 and McOwen [16, Eq. 2.4] in combination with a change of coordinates which tells us that
Since \(\hat{F}(v)=0=\nabla \hat{F}(v)\), we have that
Since \(E(x,v(x),\nabla v(x))\) is a polynomial in v and \(\nabla v\) with coefficients that depend on the \(C^1_1\)-norm of v, and \(v\in C^1_\mu (\hat{V})\) and \(L^p_{k+1,\mu }(\hat{V})\), we have that \(E(x,v(x),\nabla v(x))\in L^p_{k,\mu -1}(E)\subseteq L^p_{k,\mu -2}(E)\). Therefore Eq. (3.15) tells us that \(v\in L^p_{k+2,\mu }(\hat{V})\), from which we may deduce that v is in fact in \(C^\infty _\mu (\hat{V})\). \(\square \)
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.
Theorem 3.8
Let Y be a CS Cayley submanifold at \(\hat{x}\) with cone C and rate \(\mu \in (1,2)\backslash \mathcal {D}\) of a Spin(7)-manifold X. Let D denote the first-order elliptic differential operator defined in (2.7). Then there exist a smooth manifold \(\hat{K}_0\), which is an open neighbourhood of 0 in the kernel of (3.16), and a smooth map \(\hat{g}_2\) from \(\hat{K}_0\) into the cokernel of (3.16) with \(\hat{g}_2(0)=0\) so that an open neighbourhood of Y in the moduli space of CS Cayley deformations of Y in X, \(\hat{\mathcal {M}}_\mu (Y)\) from Definition 3.5, is homeomorphic to an open neighbourhood of 0 in \( Ker \hat{g}_2\).
Moreover, the expected dimension of \(\hat{\mathcal {M}}_\mu (Y)\) is given by the index of the linear elliptic operator
If the cokernel of (3.16) is \(\{0\}\) then \(\hat{\mathcal {M}}_\mu (Y)\) is a smooth manifold near Y of the same dimension as the kernel of (3.16). Here \(\mathcal {D}\) is the set of weights \(\mu \in \mathbb {R}\) for which (3.16) is not Fredholm from Proposition 2.11.
Proof
By Propositions 3.5 and 3.7, we can identify \(\hat{\mathcal {M}}_\mu (Y)\) near Y with the kernel of the operator
The linearisation of \(\hat{F}\) at zero is the operator
which is elliptic. Since \(\mu \notin \mathcal {D}\), (3.17) is Fredholm. Therefore we may decompose
where \(\hat{K}'\) is the kernel of (3.17) and \(\hat{X}'\) is closed, and
where \(\hat{\mathcal {O}}_\mu \) is the finite-dimensional obstruction space and
Then the map
has
which is surjective. Write \(\hat{K}=\hat{K}\times \{0\}\) for the kernel of (3.18). We then have that
Now we may apply the Banach space implicit function theorem to find \(\hat{K}_0\subseteq \hat{K}\) containing zero, \(\hat{X}'_0\subseteq \hat{X}'\), \(\hat{\mathcal {O}}_0\subseteq \hat{\mathcal {O}}_\mu \) and a smooth map \(\hat{g}=(\hat{g}_1,\hat{g_2}):\hat{K}_0\rightarrow \hat{X}'_0\times \hat{\mathcal {O}}_0\) so that
So we may identify the kernel of \(\hat{F}\), and therefore \(\hat{M}_\mu (Y)\) with the kernel of \(\hat{g}_2:\hat{K}_0\rightarrow \hat{\mathcal {O}}_0\), a smooth map between finite-dimensional spaces (since (3.17) is Fredholm). Sard’s theorem tells us that the expected dimension of the kernel of \(\hat{g}_2\) is given by the index of the operator (3.17). \(\square \)
3.5 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 \(\bar{\partial }+\bar{\partial }^*\) acting on weighted sections of a vector bundle over \(\hat{N}\) (the nonsingular part of N).
3.5.1 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.
Proposition 3.9
Let N be a CS complex surface at \(\hat{x}\) with cone C and rate \(\mu \in (1,2)\) inside a Calabi–Yau four-fold M. Write \(\hat{N}:=N\backslash \{\hat{x}\}\). Then the moduli space of CS Cayley deformations of N in M, \(\hat{\mathcal {M}}_\mu (N)\), can be identified with the kernel of the operator
where \(\hat{U}\subseteq \nu ^{1,0}_M(\hat{N})\oplus \Lambda ^{0,2}\hat{N}\otimes \nu ^{1,0}_M(\hat{N})\) is the image of \(\hat{V}\otimes \mathbb {C}\) from the tubular neighbourhood theorem under the isomorphism given in Proposition 2.4, and \(\hat{F}^ cx \) is defined so that the following diagram commutes
where \(\hat{F}\) is the operator defined in Proposition 3.5 and we use the isomorphisms given in Proposition 2.4.
Moreover, the linearisation of \(\hat{F}^ cx \) 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 of \(\hat{F}\), which is the same as the kernel of \(\hat{F}^\text {cx}\).
Since the linearisation of the operator of \(\hat{F}\) 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 of \(\hat{F}^\text {cx}\) at zero is given by the operator (3.19) as claimed. \(\square \)
3.5.2 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 operator \(\hat{F}\) defined in Proposition 3.5, for the operator \(\hat{F}^\text {cx}\) defined in Proposition 3.9. Due to the relation between the operators \(\hat{F}\) and \(\hat{F}^\text {cx}\) noted in the proof of Proposition 3.9, these results follow immediately from their counterparts.
Lemma 3.10
Let N be a conically singular complex surface inside a Calabi–Yau four-fold M. Let \(\hat{F}^ cx \) be the operator defined in Proposition 3.9. Then we can write
for \(x\in \hat{N}\), where
is smooth and \(\hat{Q}^ cx (w)(x):=\hat{Q}^ cx (x,w(x),\nabla w(x))\) is a section of \(\Lambda ^{0,1}\hat{N}\otimes \nu _M^{1,0}(\hat{N})\). Let \(\mu >1\). Then for each \(k\in \mathbb {N}\), for \(w\in C^{k+1}_\mu (\hat{U})\) with \(\Vert w\Vert _{C^1_1}\) sufficiently small, there exist constants \(C_k>0\) so that
and if \(w\in L^p_{k+1,\mu }(\hat{U})\) with \(\Vert w\Vert _{C^1_1}\) sufficiently small, there exist constants \(D_k>0\) such that
Moreover, we may deduce that
is a smooth map of Banach spaces for any \(1<p<\infty \) and \(k\in \mathbb {N}\) with \(k>1+4/p\).
Proof
Since \(\hat{F}^\text {cx}\) is defined by composing the operator \(\hat{F}\) 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 for \(\hat{F}\) from Lemma 3.6. \(\square \)
We may now give a weighted elliptic regularity result for \(\hat{F}^ cx \).
Proposition 3.11
Let N be a conically singular complex surface inside a Calabi–Yau four-fold M. Let \(\hat{F}^ cx \) be the map defined in Proposition 3.9. Then
for any \(\mu \in (1,2)\backslash \mathcal {D}\), \(1<p<\infty \) and \(k\in \mathbb {N}\). Here \(\mathcal {D}\) is the set of exceptional weights given by applying Proposition 2.11 to the linear part of \(\hat{F}^ cx \).
Proof
This follows from Proposition 3.7 in combination with the fact that the kernels of \(\hat{F}\), defined in Proposition 3.5, and \(\hat{F}^\text {cx}\) are isomorphic by definition, and the isomorphism given in Proposition 2.4 is an isometry. \(\square \)
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.
Theorem 3.12
Let N be a CS complex surface at \(\hat{x}\) with cone C and rate \(\mu \in (1,2)\backslash \mathcal {D}\) of a Calabi–Yau four-fold M. Then the expected dimension of \(\hat{\mathcal {M}}_\mu (N)\) is given by the index of the linear elliptic operator
Moreover if the cokernel of (3.24) is \(\{0\}\) then \(\hat{\mathcal {M}}_\mu (N)\) is a smooth manifold near N of the same dimension as the (complex) dimension of the kernel of (3.24). Here \(\mathcal {D}\) is the set of weights \(\mu \in \mathbb {R}\) for which (3.16) is not Fredholm from Proposition 2.11.
Proof
By Theorem 3.8, the expected dimension of \(\hat{\mathcal {M}}_\mu (N)\) is given by the index of the operator (3.16). Since, by Proposition 2.5 we can consider the operator (3.24) as the composition of the operator (3.16) with the isomorphisms from Proposition 2.4, which are isometries, we may deduce that the index of (3.16) and (3.24) are equal.
\(\square \)
3.6 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.
Definition 3.6
Let N be a CS complex surface at \(\hat{x}\) with rate \(\mu \) and cone C inside a Calabi–Yau manifold M with respect to some Spin(7)-coordinate system \(\chi \), and denote by \(\hat{C}\) the tangent cone of N. Write \(\hat{N}:=N\backslash \{\hat{x}\}\). Define the moduli space of conically singular (CS) complex deformations of N in M, \(\hat{\mathcal {M}}^\text {cx}_\mu (N)\), to be the set of CS complex surfaces \(N'\) at \(\hat{x}\) with cone C, rate \(\mu \) and tangent cone \(\hat{C}\) of M so that there exists a continuous family of topological embeddings \(\iota _t:N\rightarrow M\) with \(\iota _0(N)=N\) and \(\iota _1(N)=N'\), so that \(\iota _t(\hat{x})=\hat{x}\) for all \(t\in [0,1]\) and so that \(\hat{\iota }_t:=\iota _t|_{\hat{N}}\) is a smooth family of embeddings \(\hat{N}\rightarrow M\) with \(\hat{\iota }_0(\hat{N})=\hat{N}\) and \(\hat{\iota }_1({\hat{N}})=\hat{N}':=N'\backslash \{\hat{x}\}\).
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.
Proposition 3.13
Let N be a conically singular complex surface at \(\hat{x}\) with rate \(\mu \) and cone C inside a Calabi–Yau four-fold M. Write \(\hat{N}:=N\backslash \{\hat{x}\}\). Let \(\hat{V}\subseteq \nu _M(\hat{N})\otimes \mathbb {C}\) be the open set and \(\hat{\Xi }:\hat{V}\rightarrow \hat{T}\) the diffeomorphism defined in the tubular neighbourhood theorem 3.4. For \(v\in C^\infty _ loc (\hat{V})\) write \(\Xi _v:=\Xi \circ v\), and define \(\hat{N}_v:=\Xi _v(\hat{N})\). Then the moduli space of CS complex deformations of N in M, \(\hat{\mathcal {M}}_\mu ^ cx (N)\), is isomorphic near N to the kernel of
where \(\sigma \) was defined in Proposition 2.6. Moreover, the kernel of \(\hat{G}\) is isomorphic to the kernel of its linear part given by the map
The kernel of (3.26) is isomorphic to
Proof
By definition of \(\sigma \) we see that normal vector fields in the kernel of \(\hat{G}\) correspond to complex deformations of \(\hat{N}\), and a similar argument to Proposition 3.5 shows that weighted smooth sections of \(\nu _M(\hat{N})\otimes \mathbb {C}\) give conically singular deformations of \(\hat{N}\) as required. The linear part of \(\hat{G}\) follows from Proposition 2.8, which was a local argument, and similarly that the kernel of \(\hat{G}\) 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 \(\pi _{1,0}:\nu _M(\hat{N})\otimes \mathbb {C}\rightarrow \nu ^{1,0}_M(\hat{N})\) and the isomorphism of Proposition 2.4.
\(\square \)
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.
Theorem 3.14
Let N be a conically singular complex surface at \(\hat{x}\) with rate \(\mu \in (1,2)\) and cone C inside a Calabi–Yau four-fold M. The moduli space of CS complex deformations of N in M, \(\hat{\mathcal {M}}_\mu ^ cx (N)\) given in Definition 3.6, is a smooth manifold of dimension
where
Proof
By Proposition 3.13 the moduli space of CS complex deformations of N in M can be identified with the kernels of the operators (3.29) and (3.30). Equation (3.28) follows since the kernels of the operators (3.29) and (3.30) are isomorphic [23, Cor 4.6]. \(\square \)
To compare CS complex and Cayley deformations of a CS complex surface, we require the following result.
Proposition 3.15
Let N be a CS complex surface at \(\hat{x}\) with cone C and rate \(\mu \in (1,2)\) in a Calabi–Yau four-fold M. Write \(\hat{N}:=N\backslash \{\hat{x}\}\). Then \(w\in L^2_{k+1,\mu }(\nu ^{1,0}_M(\hat{N})\oplus \Lambda ^{0,2}\hat{N}\otimes \nu ^{1,0}_M(\hat{N}))\) is an infinitesimal CS Cayley deformation of \(\hat{N}\) if, and only if, it is an infinitesimal complex deformation of \(\hat{N}\). That is, \((\bar{\partial }+\bar{\partial }^*)w=0\) if, and only if, \(\bar{\partial }w=0=\bar{\partial }^*w\).
Proof
Suppose that \(w\in L^2_{k+1,\mu }(\nu ^{1,0}_M(\hat{N})\oplus \Lambda ^{0,2}\hat{N}\otimes \nu ^{1,0}_M(\hat{N}))\) satisfies \(\bar{\partial }w=-\bar{\partial }^* w\) for \(\mu \in (1,2)\). Then \(\bar{\partial }^*\bar{\partial }w=0\). We will check whether
holds for \(u\in L^2_{1,\mu }(\nu ^{1,0}_M(\hat{N})\oplus \Lambda ^{0,2}\hat{N}\otimes \nu ^{1,0}_M(\hat{N}))\) and \(v\in L^2_{1,\mu -1}(\nu ^{1,0}_M(\hat{N})\oplus \Lambda ^{0,2}\hat{N}\otimes \nu ^{1,0}_M(\hat{N}))\), that is, whether the integrals on both sides converge. Let \(\rho \) be a radius function for N. We have that
by Hölder’s inequality. This is finite since
since \(\mu \in (1,2)\). Similarly,
which again is finite since
for \(\mu \in (1,2)\). Therefore
and so \(\bar{\partial }w=0\). \(\square \)
This allows us to find that CS complex and Cayley deformations of a CS complex surface in a Calabi–Yau four-fold have the same expected dimension, and therefore with an application of a result of Harvey and Lawson we may deduce that the moduli space of Cayley deformations of a CS complex surface is a smooth manifold.
Corollary 3.16
Let N be a CS complex surface inside a Calabi–Yau four-fold M. Then the moduli space of CS Cayley deformations of N in M is the same as the moduli space of CS complex deformations of N in M, and is therefore a smooth manifold.
Proof
There are no infinitesimal CS Cayley deformations of N by Proposition 3.15, i.e. no \(w\in C^\infty _\mu (\nu ^{1,0}_M(\hat{N})\oplus \Lambda ^{0,2}\hat{N}\otimes \nu ^{1,0}_M(\hat{N}))\) satisfying
where \(\bar{\partial }w \ne 0\). Comparing the expected dimension of the moduli space of CS Cayley deformations of N in M, computed in Theorem 3.12, to the dimension of the moduli space of CS complex deformations of N in M, computed in Theorem 3.14, we see that these spaces must have the same expected dimension. To deduce that the moduli space of Cayley deformations is a smooth manifold, we apply [6, II.4 Thm 4.2] which says that complex and Cayley deformations of a compactly supported complex current are the same. \(\square \)
4 Index Theory
Let Y be a CS Cayley submanifold of a Spin(7)-manifold X with nonsingular part \(\hat{Y}\) and let N be a CS complex surface inside a four-dimensional Calabi–Yau manifold M with nonsingular part \(\hat{N}\). In this section, we will be interested in the index of the operators
from Proposition 2.3 on sections with compact support and extended by density to the above spaces, and
We will first characterise the set of exceptional weights \(\mathcal {D}\) 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).
4.1 Finding the Exceptional Weights for the Operators D and \(\bar{\partial }+\bar{\partial }^*\)
In this section, we will find the set \(\mathcal {D}\) 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 \(\mathbb {R}^8\). We will see that the exceptional weights are actually eigenvalues for differential operators on the links of these cones.
4.1.1 Nearly Parallel \(G_2\) Structure on \(S^7\)
We can consider \(\mathbb {R}^8\) as a cone with link \(S^7\). Let \((\Phi _0,g_0)\) be the Euclidean Spin(7)-structure (as given in Definition 2.1). Define a three-form \(\varphi \) on \(S^7\) by the following relation:
Then \((\varphi ,g)\) 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 \(\Phi _0\) is closed we have that
\(G_2\)-structures \((\varphi ,g)\) satisfying (4.4) are called nearly parallel.
4.1.2 Exceptional Weights for the Operator D
Let Y be a CS Cayley submanifold at \(\hat{x}\) with rate \(\mu \) and cone C of a Spin(7)-manifold X and write \(\hat{Y}:=Y\backslash \{\hat{x}\}\). 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 at \(\hat{x}\) with cone \(C=L\times (0,\infty )\) and rate \(\mu \) of a Spin(7)-manifold X. Let \(\mathcal {D}_D\) denote the set of \(\lambda \in \mathbb {R}\) for which
is not Fredholm.
Then \(\lambda \in \mathcal {D}_D\) if, and only if, there exists \(0\ne v\in C^\infty (\nu _{S^7}(L))\) so that
where for \(\{e_1,e_2,e_3\}\) an orthonormal frame for TL and \(\nabla ^\perp \) the connection on the normal bundle of L in \(S^7\) induced by the Levi-Civita connection of the round metric on \(S^7\),
where \(\times \) is the cross product on \(S^7\) induced from the nearly parallel \(G_2\)-structure \((\varphi ,g)\) 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 \(G_2\)-manifold, 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 \(\lambda =1\) 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 \(\rho \) 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 \(\chi \) for X around \(\hat{x}\) (see Definition 3.1). We will verify in the subsequent calculation that \(\tilde{D}_\infty \) is indeed translation invariant.
By Proposition 2.11 in combination with the discussion in [16, p. 416], we see that \(\lambda \in \mathcal {D}_D\) if, and only if, there exists a normal vector field \(v\in C^\infty (\nu _L(S^7))\) satisfying
where since \(\nu _{rl,\mathbb {R}^8}(C)\cong \nu _{l,S^7}(L)\) for all \(r>0\) we can consider \((r,l)\mapsto (r,r^{\lambda -1} v(l))\) as a normal vector field on the cone. Note also that the induced Euclidean metric on the normal bundle of C in \(\mathbb {R}^8\) takes the form \(r^2h\), 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 TL with dual coframe \(\{e^1,e^2,e^3\}\), and denote by \(\Phi _0\) the Euclidean Cayley form on \(\mathbb {R}^8\) and \(\varphi \) the nearly parallel \(G_2\)-structure on \(S^7\) defined in (4.3). Let \(\nabla \) denote the Levi-Civita connection of the cone metric and \(\overline{\nabla }\) denote the Levi-Civita connection of \(g_L\) (and induced by \(g_{S^7}\) on normal vector fields). We compute that
since \(\nabla _{\frac{\partial }{\partial r}}^\perp v=r^{-1}v\) as the metric on the normal bundle is of the form \(r^2h\). Using the definition of \(\varphi \) in (4.3), we find that
Now we wish to replace the musical isomorphism \(\flat :\nu _{\mathbb {R}^8}(C)\rightarrow \nu ^*_{\mathbb {R}^8}(C)\) with the musical isomorphism \(\flat _L:\nu _{S^7}(L)\rightarrow \nu ^*_{S^7}(L)\). Since the metric on \(\nu _{\mathbb {R}^8}(C)\) is of the form \(r^2h\), where h is a metric on \(\nu _{S^7}(L)\), we find that
At this point, we may verify that the operator \(\tilde{D}_\infty =r^{-1}D_0 r^{-1}\) is translation invariant. Writing \(r=e^{-\lambda t}\), we see that the expression above implies that
where \(\tilde{\nabla }\) is the Levi-Civita connection of the product metric \(\tilde{g}=dt^2+g_L\). This expression makes it clear that \(\tilde{D}_\infty \) is a translation invariant operator on the cylinder \(L\times (0,\infty )\).
Notice that \(E\cong \nu _{S^7}(L)\) via the map
where \(\sharp _L:\nu ^*_{S^7}(L)\rightarrow \nu _{S^7}(L)\) is the musical isomorphism, with inverse map
Therefore we see that
We find that
Since by definition,
we see that \(\lambda \in \mathcal {D}_D\) if, and only if, there exists \(0\ne v\in C^\infty (\nu _{S^7}(L))\) such that
\(\square \)
4.1.3 Exceptional Weights for the Operator \(\bar{\partial }+\bar{\partial }^*\)
Let N be a CS complex surface with rate \(\mu \) and cone C inside a Calabi–Yau four-fold M, and write \(\hat{N}\) 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.
Definition 4.1
Let C be a complex cone in \(\mathbb {C}^{n+1}\), with real link \(L:=C\cap S^{2n+1}\). Consider the Hopf projection \(p:S^{2n+1}\rightarrow \mathbb {C}P^n\). Define the complex link\(\Sigma \) of C to be the image of L under the Hopf projection, i.e. \(\Sigma :=p(L)\subseteq \mathbb {C}P^n\).
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 \(\mathbb {C}^{n+1}\), and denote by J the standard complex structure on \(\mathbb {C}^{n+1}\). The Reeb vector field is defined to be
Notice that \(|\xi |_L=1\).
If \(p|_L:L\rightarrow \Sigma \) is the restriction of the Hopf projection to L, then at each \(l\in L\), \(\xi _l\) spans the kernel of \(dp|_l:T_lL\rightarrow T_{p(l)}\Sigma \).
Definition 4.3
Let C be a complex cone in \(\mathbb {C}^{n+1}\) with real link L. Let \(\alpha \) be a p-form on L. We say that \(\alpha \) is horizontal if \(\xi \lrcorner \,\alpha =0\), where \(\xi \) is the Reeb vector field. Denote by \(\Lambda ^p_hL\) the vector bundle of horizontal p-forms on L. Denote by \(d_h\) the projection of the exterior derivative onto horizontal forms.
By definition of the Reeb vector field, we see if J is the complex structure on \(\mathbb {C}^{n+1}\) then \(J(\Lambda ^1_hL)\subseteq \Lambda ^1_hL\). So we have a well-defined splitting \(\Lambda ^1_hL=\Lambda ^{1,0}_hL\oplus \Lambda ^{0,1}_hL\) of one-forms into the \(\pm i\) eigenspaces of J. Define the operator \(\bar{\partial }_h\) on functions to be the projection of \(d_h\) onto horizontal (0, 1)-forms.
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.
Proposition 4.2
Let N be a CS complex surface at \(\hat{x}\) with rate \(\mu \) and cone C inside a Calabi–Yau four-fold M. Write \(\hat{N}:=N\backslash \{\hat{x}\}\). Let \(\mathcal {D}\) denote the set of \(\lambda \in \mathbb {R}\) for which
is not Fredholm. Let L denote the real link of C. Then \(\lambda \in \mathcal {D}\) if, and only if, there exists a nontrivial pair \(v\in C^\infty (\nu ^{1,0}_{S^7}(L))\) and \(w\in C^\infty (\Lambda ^{0,1}_hL\otimes \nu ^{1,0}_{S^7}(L))\) so that
where \(\xi \) is the Reeb vector field on L. Here \(\overline{\nabla }\) acts on \(\Lambda ^{0,1}_hL\) as the Levi-Civita connection of the metric on L and on \(\nu ^{1,0}_{S^7}(L)\) as the normal part of the Levi-Civita connection on \(S^7\).
Proof
Similarly to the proof of Proposition 4.1, if \(\rho \) is a radius function for N then we can see that
on \(\hat{N}\) is asymptotic to the operator
on the cone C, which we will see in the calculation below is translation invariant. If \(v\in C^\infty (\nu _{S^7}(L)\otimes \mathbb {C})\) we can think of \(r^\mu v\) as a complexified normal vector field on C, and moreover the complex structure J on \(\mathbb {C}^4\) 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 \(\theta \in C^\infty (\Lambda ^1L)\) to be the dual one-form to \(\xi \) we have that \(dr-ir\theta \) is a (0, 1)-form on C. By definition, \(\Lambda ^{0,2}C= \Lambda ^2T^{*0,1}C\), and \(T^{*0,1}C=\langle dr-ir\theta \rangle \oplus \Lambda ^{0,1}_hL\) so we can see that a (0, 2)-form on C must be of the form
where \(w\in C^\infty (\Lambda ^{0,1}_hL)\). By Proposition 2.11 in combination with the discussion in [16, p. 416], we deduce that \(\lambda \in \mathcal {D}\) if, and only if, there exists \(v\in C^\infty (\nu _{S^7}(L))\) and \(w\in C^\infty (\Lambda ^{0,1}_hL\otimes \nu ^{1,0}_{S^7}(L))\) so that
Denote by \(\nabla \) the Levi-Civita connection of the cone metric and \(\overline{\nabla }\) 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
and therefore, since \(\nabla _{\frac{\partial }{\partial r}}v=r^{-1}v\),
On (0, 2)-forms, the operators \(\bar{\partial }^*\) and \(d^*\) coincide. Therefore, for a local orthonormal frame \(\{\frac{\partial }{\partial r},\xi /r,e_1/r,e_2/r\}\) for TC with \(J(r\frac{\partial }{\partial r})=\xi \) and \(Je_1=e_2\) we have that
We calculate that
since for the Levi-Civita connection of the cone metric on a one-form \(\alpha \) on the link \(\nabla _{\frac{\partial }{\partial r}}\alpha =-\alpha /r\). 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 \(c_1,c_2=\pm 1\) or \(\pm i\) since \(\nabla _Xdr=rX^{\flat _L}\) and the complex structure J is parallel. Therefore
since \(\Lambda ^{0,1}L\) is a rank one vector bundle. We deduce that
Equating (4.11) and minus (4.12), we find that \(\lambda \in \mathcal {D}\) if, and only if, there exist \(v\in C^\infty (\nu _{S^7}^{1,0}(L))\) and \(w\in C^\infty (\Lambda ^{0,1}_hL\otimes \nu ^{1,0}_{S^7}(L))\) satisfying
as claimed.
Finally, we verify that \(\tilde{A}_\infty \) is translation invariant. Using the above calculations, with a coordinate transformation of the form \(r=e^{-t}\) we find that
where \(\tilde{\nabla }\) denotes the Levi-Civita of the product metric \(\tilde{g}=dt^2+g_L\). We can see from this expression that the operator \(\tilde{A}_\infty \) is translation invariant, as claimed. \(\square \)
4.1.4 An Eigenproblem on the Complex Link
In Proposition 4.2 we characterised the set of exceptional weights \(\mathcal {D}\) 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 \(\mathbb {C}P^2\). 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 \(\mathbb {C}^4\) with real link \(L\subseteq S^7\) and complex link \(\Sigma \subseteq \mathbb {C}P^3\). Suppose we have a problem of the following form: Find all of the functions f on L that satisfy
for some \(m\in \mathbb {Z}\), where \(\xi \) is the Reeb vector field on C.
We would like to understand the relationship between the operator \(\bar{\partial }_h\) on the real link of C and \(\bar{\partial }_\Sigma \) on the complex link C.
Definition 4.4
Call a function, horizontal vector field or horizontal differential form f on Lbasic if
Basic functions, forms and vector fields are special because they are in one-one correspondence with functions, forms and vector fields on \(\Sigma \). It follows from [28, Lem 1] that \(\bar{\partial }_h\) acting on basic functions, forms or vector fields on L is equivalent to \(\bar{\partial }_\Sigma \) acting on functions, forms or vector fields on \(\Sigma \). In Problem (4.13), when \(m\ne 0\), 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 \(\mathcal {O}_{\mathbb {C}P^3}(-1)|_\Sigma \), that is, the tautological line bundle over \(\mathbb {C}P^3\) restricted to \(\Sigma \). This is then a trivial (real) line bundle over L and therefore has a global section given by the map \(x\mapsto s(x)=x\) for \(x\in L\). It is easy to see that \(\mathcal {L}_{\xi }s=is\), and therefore
is a section of the vector bundle \(\mathcal {O}_{\mathbb {C}P^3}(m)|_\Sigma \) satisfying
and therefore pushes down to a well-defined section of the vector bundle \(\mathcal {O}_{\mathbb {C}P^3}(m)|_\Sigma \). Since \(\mathcal {O}_{\mathbb {C}P^3}(m)|_{\Sigma }\) is a trivial line bundle over L, we can still consider \(f\otimes s^{-m}\) as a function on L. Therefore we can rephrase Problem (4.13) as follows: Find all basic sections \(\tilde{f}\) of \(\mathcal {O}_{\mathbb {C}P^3}(m)|_\Sigma \rightarrow L\) satisfying
This is now equivalent to finding the sections \(\tilde{f}\) of \(\mathcal {O}_{\mathbb {C}P^3}(m)|_\Sigma \rightarrow \Sigma \) that satisfy
Therefore we have reduced Problem (4.13) to asking: How many holomorphic sections of the line bundle \(\mathcal {O}_{\mathbb {C}P^3}(m)|_\Sigma \) are there?
This problem is easily solved using the Hirzebruch–Riemann–Roch Theorem [7, Thm 5.1.1].
Theorem 4.3
(Hirzebruch–Riemann–Roch) Let \(\Sigma \) be a Riemann surface and let F be a holomorphic vector bundle over \(\Sigma \). Denote by \(h^0(\Sigma ,F)\) the dimension of the space of holomorphic sections of F. Let \(K_\Sigma \) denote the canonical bundle of \(\Sigma \). 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 \(\Sigma \).
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 \(\mathbb {C}^4\) with real link L and complex link \(\Sigma \). Then given \(\lambda \in \mathbb {R}\) and \(m\in \mathbb {Z}\), pairs \(v\in C^\infty \left( \nu _{\mathbb {C}P^3}^{1,0}(\Sigma )\otimes \mathcal {O}_{\mathbb {C}P^3}(m)|_\Sigma \right) \) and \(w\in C^\infty \left( \Lambda ^{0,1}\Sigma \otimes \nu _{\mathbb {C}P^3}^{1,0}(\Sigma )\otimes \mathcal {O}_{\mathbb {C}P^3}(m)|_{\Sigma }\right) \) so that
are in a one-one correspondence with pairs \(\tilde{v}\in C^\infty \left( \nu ^{1,0}_{S^7}(L)\right) \) and \(\tilde{w}\in C^\infty \left( \Lambda ^{0,1}_hL\otimes \nu ^{1,0}_{S^7}(L)\right) \) satisfying
where \(\xi \) 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 \(\nu ^{1,0}_{S^7}(L)\otimes \mathcal {O}_{\mathbb {C}P^3}(m)|_\Sigma \) and \(\Lambda ^{0,1}_hL\otimes \nu ^{1,0}_{S^7}(L)\otimes \mathcal {O}_{\mathbb {C}P^3}(m)|_\Sigma \) over L, respectively. As mentioned above, these sections are in one-one correspondence with sections \(\tilde{v}\) and \(\tilde{w}\) of \(\nu ^{1,0}_{S^7}(L)\) and \(\Lambda ^{0,1}_hL\otimes \nu ^{1,0}_{S^7}(L)\), respectively, satisfying
So we see that v and w are in one-one correspondence with \(\tilde{v}\) and \(\tilde{w}\) satisfying (4.16), and \(\tilde{v}\) and \(\tilde{w}\) satisfy
Let \(\nabla \) 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 \(\mathbb {C}P^3\) satisfies
and so for any vector field of type (1, 0), we have that
Moreover, if \(\alpha \) 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. \(\square \)
4.2 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.
4.2.1 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 \(\hat{Z}\) with a cylindrical end \(L\times (0,\infty )\) that takes the special form
on \(L\times (0,\infty )\), where u is the inward normal coordinate, \(\sigma :F_1|_L\rightarrow F_2|_L\) is a bundle isomorphism and B is a self-adjoint elliptic operator on L. Then
where \(h,\eta , \alpha _0\) and \(h_\infty (F_2)\) are defined as follows:
-
(i)
\(\alpha _0(x)\) is the constant term in the asymptotic expansion (as \(t\rightarrow 0\)) of
$$\begin{aligned} \sum e^{-t\mu '}|\phi '_\mu (x)|^2-\sum e^{-t\mu ''}|\phi ''_\mu (x)|^2, \end{aligned}$$where \(\mu ',\phi '_\mu \) denote the eigenvalues and eigenfunctions of \(A^*A\) on the double of Z (where Z is the compact manifold with boundary L obtained by removing the cylindrical end of \(\hat{Z}\)), and \(\mu '',\phi ''_\mu \) are the corresponding objects for \(AA^*\).
-
(ii)
\(h= dim Ker B=\) multiplicity of the 0-eigenvalue of B.
-
(iii)
\(\eta (s)=\sum _{\lambda \ne 0}( sign \lambda ) |\lambda |^{-s}\), where \(\lambda \) runs over the eigenvalues of B.
-
(iv)
\(h_\infty (F_2)\) is the dimension of the subspace of \( Ker B\) consisting of limiting values of extended \(L^2\) sections f of \(F_2\) satisfying \(A^*f=0\).
Here we call f an extended\(L^2\)-section of F if \(f\in L^2_\text {loc}(F)\) and on the cylindrical end of \(\hat{Z}\), for large t, f takes the form
for \(g\in L^2(F)\) and \(f_\infty \in Ker B\).
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 at \(\hat{z}\) and let \(\rho \) be a radius function for Z. Write \(\hat{Z}:=Z\backslash \{\hat{z}\}\), and g for the metric on \(\hat{Z}\). Let
be a linear first-order differential operator on \(\hat{Z}\) and suppose there exists \(\lambda \in \mathbb {R}\) so that
is an asymptotically translation invariant operator. Then the formal adjoint of the operator \(\tilde{A}\) (with respect to the metric \(\rho ^{-2}g\))
is of the form
where
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 \(\mu \in \mathbb {R}\), \(k\in \mathbb {N}\) and \(1<p<\infty \).
Proof
Let \(v\in C^\infty _0(T^{q}_s\hat{Z})\) and \(w\in C^\infty _0(T^{q'}_{s'}\hat{Z})\). Notice that if \(a,b\in C^\infty _0(T^q_s\hat{Z})\) then
Then
where we have used that \(A^*\) is the formal adjoint of A with respect to the metric g, which shows that
is the formal adjoint of \(\tilde{A}\) with respect to the metric \(\rho ^{-2}g\). By Lemma 2.10
is an isomorphism and so by definition of \(\tilde{A}^*\) and \(A^*\) the kernels of (4.19) and (4.20) are isomorphic. \(\square \)
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 at \(\hat{z}\) with radius function \(\rho \). Let \(T^q_s\hat{Z}\) be the vector bundle of (s, q)-tensors on \(\hat{Z}:=Z\backslash \{\hat{z}\}\). Let
be a first-order linear elliptic differential operator so that for some \(\lambda \in \mathbb {R}\)
is asymptotically translation invariant to a translation invariant operator \(\tilde{A}_\infty \) acting on sections of \(T^q_s\hat{Z}\), which takes the special form (4.17) on the end of \(\hat{Z}\). Then for \(\mu \in \mathbb {R}\backslash \mathcal {D}\), given in Proposition 2.11, the index of
differs by a constant independent of \(\mu \) from the index \( ind _\mu A_\infty \) of
which satisfies
for \(\epsilon >0\) chosen so that \((0,\epsilon ]\cap \mathcal {D}=\emptyset \) 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 operator \(\tilde{A}_\infty \)).
Proof
As we saw in the proof of Proposition 2.11, A and \(\tilde{A}\) have isomorphic kernel and cokernel when acting on weighted Sobolev spaces \(L^p_{k+1,\mu }(T^q_s\hat{Z})\) and \(W^p_{k+1,\mu }(T^q_s\hat{Z})\), respectively, and moreover the index of these operators differ from the index of \(\tilde{A}_\infty \) by a constant independent of the weight \(\mu \).
Note that the definition of asymptotically translation invariant only determines the behaviour of \(\tilde{A}_\infty \) on the cylindrical end of \((\hat{Z},\rho ^{-2}g)\). We may choose a preferred operator \(\tilde{A}_\infty \) by interpolating between \(\tilde{A}\) on the compact piece of \(\hat{Z}\) and any such operator \(\tilde{A}_\infty \) on the cylindrical end of \(\hat{Z}\). Since \(\tilde{A}_\infty \) is translation invariant, we can apply Theorem 4.5 to \(\tilde{A}_\infty \). Let \(\text {Ker}_\mu \, \tilde{A}_\infty \) and \(\text {Ker}_\mu \, \tilde{A}_\infty ^*\) denote the kernels of
respectively, where \(\tilde{A}^*_\infty \) is the formal adjoint of \(\tilde{A}_\infty \) with respect to the metric \(\rho ^{-2}g\), where g is the metric on \(\hat{Z}\). Then Theorem 4.5 yields that
By definition of \(\tilde{A}_\infty \), \(\text {Ker}_0\, \tilde{A}_\infty \cong \text {Ker}_0\, A_\infty \), where \(\text {Ker}_\mu \, A_\infty \) denotes the kernel of (4.22), and by Lemma 4.6, \(\text {Ker}_0 \, \tilde{A}_\infty ^*\cong \text {Ker}_{\lambda -m}\, A_\infty ^*\), where \(A^*_\infty \) is the formal adjoint of \(A_\infty \) with respect to the metric g and \(\text {Ker}_\mu \, A_\infty ^*\) denotes the kernel of
So we see that
Denote by \(\mathcal {D}\) the subset of \(\mathbb {R}\) for which \(\mu \in \mathcal {D}\) if, and only if, (4.22) is not Fredholm. Then since we expect that \(0\in \mathcal {D}\) in general, the index of \(A_\infty \) for the weight 0 may not be defined. Take \(\epsilon >0\) so that
Then \(\text {ind}_\epsilon \, A_\infty \) is well-defined. Since \(\epsilon >0\), we have that
where \(\text {Ker}_\mu \, A_\infty \) denotes the kernel of (4.22). It is claimed that
To see this, suppose that \(\alpha \in \text {Ker}_0 A_\infty \). Then by elliptic regularity, \(\alpha \) is smooth, and by definition of weighted norm on \(L^2_{k+1,0}(T^q_s\hat{Z})\)\(\alpha \) must decay to zero as \(r\rightarrow 0\) and so we must have that \(\alpha =\mathcal {O}(r^{\epsilon '})\) for some \(\epsilon '>0\). Taking \(\epsilon '\) smaller if necessary we can guarantee that \(\mathcal {D}\cap (0,\epsilon ']=\emptyset \). The rate of decay of \(\alpha \) allows us to deduce that \(\alpha \in L^2_{k+1,\epsilon ''}(T^q_s\hat{Z})\) where \(0<\epsilon ''<\epsilon '\). But then we are done, since there is no exceptional weight between \(\epsilon \) and \(\epsilon ''\), and so [16, Lem 7.1] says that \(\text {Ker}_\epsilon A_\infty =\text {Ker}_{\epsilon '}A_\infty \). Notice that this tells us that the function \(\mu \mapsto \text {dim Ker}_\mu A_\infty \) is upper semi-continuous at zero.
Since \(\epsilon >0\)
The above argument also shows that the function \(\mu \mapsto \text {dim Ker}_\mu A^*_\infty \) is upper semi-continuous (in particular at \(\mu =\lambda -m\)) and so the set
is nonempty, but its elements are exactly the limiting values of extended \(L^2\)-sections f of \(T^{q'}_{s'}\hat{Z}\) satisfying \(\tilde{A}^*_\infty f=0\). To see this, recall that \(\text {Ker}_{\lambda -m}A^*_\infty \cong \text {Ker}_0\tilde{A}^*_\infty \) and \(\text {Ker}_{-\epsilon +\lambda -m}A^*_\infty \cong \text {Ker}_{-\epsilon }\tilde{A}^*_\infty \). Following Atiyah, Patodi and Singer [2, Prop 3.11], we can describe any \(v\in \text {Ker}_{\delta }\tilde{A}_\infty ^*\) on the end \(L\times (0,\infty )\) of \(\hat{Z}\) as
Setting \(\delta =0,-\epsilon \) in (4.26) we see that the sections in \(\text {Ker}_{-\epsilon }\tilde{A}^*_\infty \backslash \text {Ker}_0\tilde{A}^*_\infty \) are exactly the terms in the above expression corresponding to \(\lambda =0\), which are exactly the limiting values of extended \(L^2\) sections in the kernel of \(\tilde{A}^*_\infty \) as claimed.
Therefore
i.e. exactly the dimension of the space of limiting values of extended \(L^2\)-sections f of \(T^{q'}_{s'}\hat{Z}\) satisfying \(A^*_{\infty }f=0\). This allows us to deduce that
Applying this to (4.25) we find that
as claimed. \(\square \)
4.2.2 An Application of the APS Index Theorem
Having discussed in the previous section the set of exceptional weights \(\mathcal {D}\) for the operator (4.7) in more detail, we will apply the Atiyah–Patodi–Singer index theorem to the operator \(\bar{\partial }+\bar{\partial }^*\) to compare the dimension of the space of CS complex deformations of a CS complex surface in a Calabi–Yau four-fold to what we might expect to be the dimension of the space of all complex deformations of the complex surface from Kodaira’s theorem [14, Theorem 1].
Theorem 4.8
Let N be a CS complex surface at \(\hat{x}\) with cone C and rate \(\mu \in (1,2)\backslash \mathcal {D}\), where \(\mathcal {D}\) is the set of exceptional weights defined in Proposition 2.11, inside a Calabi–Yau four-fold M. Write \(\hat{N}:=N\backslash \{\hat{x}\}\). Let, for \(k>4/p+1\),
and denote the index of this operator by
Then
where \(\chi (N,\nu ^{1,0}_M(N))\) is the holomorphic Euler characteristic of \(\nu ^{1,0}_M(N)\), \(\mathcal {D}\) is the set of \(\lambda \in \mathbb {R}\) for which (4.14)–(4.15) has a nontrivial solution and then \(d(\lambda )\) is the dimension of the solution space, \(\eta \) is the \(\eta \)-invariant which we can now define to be
Remark
We interpret this as follows. The term \(\chi \left( N,\nu ^{1,0}_M(N)\right) \) is interpreted as the dimension of the space of all complex deformations of N in M, since this is what we can expect if Kodaira’s theorem [14, Theorem 1] remains valid for complex varieties. Theorem 3.12 tells us that \(\text {ind}_\mu (\bar{\partial }+\bar{\partial }^*)\) is the expected dimension of the space of CS Cayley deformations of N in M (which by Proposition 3.15 we can interpret as the expected dimension of the space of CS complex deformations of N in M, although Theorem 3.14 tells us that in fact this should be equal to just the dimension of the kernel of (4.28), which is what we expect to happen generically anyway). The term d(1) represents deformations of N that have a different tangent cone to N at \(\hat{x}\).
Proof
This follows from Proposition 4.7, since in this case
from [29, Thm 1.6]. \(\square \)
5 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 \(\mathbb {C}^4\), both as a Cayley submanifold and a complex submanifold of \(\mathbb {C}^4\). 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 \(\mathbb {C}^4\), and so deforming the cone as a complex or Cayley cone in \(\mathbb {C}^4\) 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 \(\eta \)-invariant for a complex cone in \(\mathbb {C}^4\).
5.1 Cone Deformations
Let C be a two-dimensional complex cone in \(\mathbb {C}^4\). 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. Write \(v=v_1\oplus v_2\), where \(v_1\in C^\infty (\nu ^{1,0}_{\mathbb {C}^4}(C))\) and \(v_2\in C^\infty (\nu ^{0,1}_{\mathbb {C}^4}(C))\). We know from Proposition 2.5 that v is an infinitesimal Cayley deformation of C if, and only if,
where \(\Omega _0\) is the standard holomorphic volume form on \(\mathbb {C}^4\) and \(\sharp \) denotes the musical isomorphism \(\nu ^{*0,1}_{\mathbb {C}^4}(C)\rightarrow \nu ^{1,0}_{\mathbb {C}^4}(C)\). Moreover, by Proposition 2.6v 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
where \(V_C\subseteq \nu _{\mathbb {R}^8}(C)\) contains the zero section and \(T_C\subseteq \mathbb {C}^4\) contains C. We constructed an action of \(\mathbb {R}_+\) on \(\nu _{\mathbb {C}^4}(C)\) satisfying \(|t\cdot v|=t|v|\), and the map \(\Xi _C\) satisfies
Therefore, to guarantee that \(\Xi _C\circ v\) is a cone in \(\mathbb {C}^4\), we must have that \(v(r,l)=r\cdot \hat{v}(l)\), for some \(\hat{v}\in C^\infty (\nu _{S^7}(L))\). In this case,
for all \(r\in \mathbb {R}_+\). Choosing a metric on \(\nu _{\mathbb {C}^4}(C)\) that is independent of r, we see that \(r\cdot \hat{v}(l)=r\hat{v}(l)\).
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 \(\lambda =1\). 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 \(\lambda =1\). 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.
5.1.1 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 \(\Sigma _1\) is the complex link of \(C_1\).
Proposition 5.1
[13, §6.4.1] The space of infinitesimal associative deformations of \(L_1\) in \(S^7\) has dimension twelve.
5.1.2 Example 2: \(L_2\cong SU(2)/ \mathbb {Z}_2\)
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 \(SU(2)/\mathbb {Z}_2\).
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.
5.1.3 Example 3: \(L_3\cong SU(2)/\mathbb {Z}_3\)
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(2) on \(\mathbb {C}^4\)
where \(a,b\in \mathbb {C}\) 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 \(a,b\in \mathbb {C}\) satisfy \(|a|^2+|b|^2=1\). We see that for
\(L_3\) is invariant under the action of \(\mathbb {Z}_3\), therefore \(L_3\cong SU(2)/\mathbb {Z}_3\). The complex link of the cone \(C_3\) over \(L_3\) is
which is known as the twisted cubic in \(\mathbb {C}P^3\).
This is a particularly interesting example for the following reason [19, Ex 5.8]. Define \(L_3(\theta )\) to be the orbit of the above group action around the point \((\cos \theta ,0,0,\sin \theta )^T\). Then \(L_3(\theta )\) is associative for \(\theta \in [0,\frac{\pi }{4}]\). As noted above, \(L_3(0)=L_3\) is the real link of a complex cone, however, \(L_3(\frac{\pi }{4})\) is the link of a special Lagrangian cone. Therefore there exists a family of Cayley cones in \(\mathbb {C}^4\), including both a complex cone and a special Lagrangian cone, that are related by a group action.
Proposition 5.3
[13, §6.3.2] The space of infinitesimal associative deformations of \(L_3(\frac{\pi }{4})\) in \(S^7\) has dimension thirty.
5.2 Calculations
We will now study the eigenvalue problem (4.9)–(4.10) with \(\lambda =1\) 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 \(\lambda =1\) 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.
Proposition 5.4
Let C be a complex cone in \(\mathbb {C}^4\) with real link L and complex link \(\Sigma \). Infinitesimal complex conical deformations of C in \(\mathbb {C}^4\) are given by holomorphic sections of \(\nu ^{1,0}_{\mathbb {C}P^3}(\Sigma )\). Infinitesimal Cayley conical deformations of C that are not complex are given by \(v\in C^\infty (\nu ^{1,0}_{\mathbb {C}P^3}(\Sigma )\otimes \mathcal {O}_{\mathbb {C}P^3}(m)|_{\Sigma })\) satisfying
where \(-4< m<0\).
Proof
We know that infinitesimal complex deformations C will lie in the kernel of \(\bar{\partial }_C\) or \(\bar{\partial }^*_C\). 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 \(\nu ^{1,0}_{\mathbb {C}P^3}(\Sigma )\otimes \mathcal {O}_{\mathbb {C}P^3}(\lambda -1)|_\Sigma \), and antiholomorphic sections of \(\Lambda ^{0,1}\Sigma \otimes \nu ^{1,0}_{\mathbb {C}P^3}(\Sigma )\otimes \mathcal {O}_{\mathbb {C}P^3}(-3-\lambda )\). Since infinitesimal conical deformations of C will correspond to \(\lambda =1\) here, we see that infinitesimal complex conical deformations of C correspond to holomorphic sections of
and antiholomorphic sections of
by the adjunction formula [7, Prop 2.2.17] since \(K_{\mathbb {C}P^3}|_{\Sigma }=\mathcal {O}_{\mathbb {C}P^3}(-4)|_{\Sigma }\). So we see that infinitesimal conical complex deformations of C arise from holomorphic sections of the holomorphic normal bundle of the complex link in \(\mathbb {C}P^3\). The dimension of the space of infinitesimal conical complex deformations of C is then equal to the real dimension (or twice the complex dimension) of the space of holomorphic sections of the holomorphic normal bundle of the complex link.
Finally, we see that any remaining infinitesimal conical Cayley deformations of C must satisfy the eigenproblem (4.14)–(4.15) with \(\lambda =1\) and \(m\ne 0,-4\). Applying \(\bar{\partial }^*_{\Sigma }\) to (4.14) and using (4.15), we see that the remaining infinitesimal conical Cayley deformations of C are given by \(v\in C^\infty (\nu ^{1,0}_{\mathbb {C}P^3}(\Sigma )\otimes \mathcal {O}_{\mathbb {C}P^3}(m)|_\Sigma )\) satisfying
\(\square \)
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 \(\bar{\partial }_\Sigma \)-Laplacian acting on sections of holomorphic line bundles over \(\mathbb {C}P^1\) equipped with a metric of constant scalar curvature.
Theorem 5.5
[17, Thm 5.1] Let K be a Hermitian line bundle over \(\Sigma \), where \(\Sigma \) is \(\mathbb {C}P^1\) with metric of constant scalar curvature \(\kappa \) equipped with a unitary harmonic connection \(\nabla _K\) of curvature \(F^{\nabla _K}=-iB\omega _\Sigma \) for some \(B\in \mathbb {R}\). Then the spectrum of the operator
is the set
where \(a=0\) if \( deg K\ge 0\), \(a=1\) otherwise.
The space of eigensections of \(2\bar{\partial }^*_\Sigma \bar{\partial }_\Sigma \) with eigenvalue \(\lambda _q\) is identified with the space of holomorphic sections of
when \( deg K\ge 0\), or of holomorphic sections of
when \( deg K <0\). Therefore the multiplicity of \(\lambda _q\) is
5.2.1 Example 1: \(L_1=S^3\)
To calculate the dimension of the space of infinitesimal conical Cayley deformations of the cone \(C_1=\mathbb {C}^2\), which as real link \(L_1=S^3\) and complex link \(\Sigma _1=\mathbb {C}P^1\), 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
for \(v\in C^\infty (\nu ^{1,0}_{\mathbb {C}P^3}(\Sigma _1)\otimes \mathcal {O}_{\mathbb {C}P^3}(m)|_{\Sigma _1})=C^\infty (\mathcal {O}_{\mathbb {C}P^3}(m+1)|_{\Sigma _1}\oplus \mathcal {O}_{\mathbb {C}P^3}(m+1)|_{\Sigma _1})\) and \(-4<m<0\). We can apply Theorem 5.5 to solve (5.2) as long as the connection on \(\mathcal {O}_{\mathbb {C}P^3}(m+1)|_{\Sigma _1}\oplus \mathcal {O}_{\mathbb {C}P^3}(m+1)|_{\Sigma _1}\) takes the form
where \(\nabla _i\) are connections on \(\mathcal {O}_{\mathbb {C}P^3}(m+1)|_{\Sigma _1}\). This is the case here, as can be seen from the relation between the connection on the normal bundle of \(\Sigma _1\) in \(\mathbb {C}P^3\) 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
for \(m\in \mathbb {Z}\) and \(q\in \mathbb {N}\cup \{0\}\) (since the scalar curvature of \(\Sigma _1=\mathbb {C}P^1=S^2(1/2)\) is eight) with \(a=0\) if \(m\ge -1\) and \(a=1\) if \(m\le -2\). It can be checked that this has solution \((q,a,m)=(0,1,-2)\), and so by Theorem 5.5 the dimension of eigensections of (5.2) has dimension \(2\times 2=4\). So we have a total of twelve infinitesimal conical Cayley deformations of C in \(\mathbb {C}^4\).
We sum this up in a proposition.
Proposition 5.6
The real dimension of the space of infinitesimal conical Cayley deformations of \(C_1\) in \(\mathbb {C}^4\) is twelve. The real dimension of the space of infinitesimal conical complex deformations of \(C_1\) in \(\mathbb {C}^4\) is eight.
Remark
Recall that the stabiliser of a Cayley plane in \(\mathbb {R}^8\) is isomorphic to \((SU(2)\times SU(2)\times SU(2))/\mathbb {Z}_2\) and that the dimension of \(Spin(7)/((SU(2)\times SU(2)\times SU(2))/\mathbb {Z}_2)\) is twelve. The stabiliser of a two-dimensional complex plane in \(\mathbb {C}^4\) is isomorphic to \(U(2)\times U(2)\), and the dimension of \(U(4)/(U(2)\times U(2))\) is equal to eight.
5.2.2 Example 2: \(L_2\cong SU(2)/\mathbb {Z}_2\)
We now use Proposition 5.4 to calculate the dimension of the space of infinitesimal conical Cayley deformations of the cone \(C_2\) in \(\mathbb {C}^4\) with link \(L_2\cong SU(2)/\mathbb {Z}_2\) and complex link \(\Sigma _2\) as defined in Sect. 5.1.2. Since \(\Sigma _2\) is a complete intersection of irreducible polynomials of degree 1 and 2 in \(\mathbb {C}P^3\), its normal bundle is given by
The dimension of the space of holomorphic sections of \(\nu ^{1,0}_{\mathbb {C}P^3}(\Sigma _2)\), 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 \(\Sigma _2\subseteq \mathbb {C}P^2\), we see that the Levi-Civita connection on \(\nu ^{1,0}_{\mathbb {C}P^3}(\Sigma _2)\) must be of the form (5.3), so that we may apply Theorem 5.5 to solve the eigenproblem
for \(v\in C^\infty (\mathcal {O}_{\mathbb {C}P^3}(m+1)|_{\Sigma _2}\oplus \mathcal {O}_{\mathbb {C}P^3}(m+2)|_{\Sigma _2})\) with \(-4<m<0\). This reduces again to solving the equations for \(m\in \mathbb {Z}\) and \(q\in \mathbb {N}\cup \{0\}\)
with \(a=0\) for \(m\ge -1\) and \(a=1\) otherwise, which has solution \((q,a,m)=(0,1,-2)\) and
with \(a=0\) for \(m\ge -2\) 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 \(\mathbb {C}^4\) is twenty-two.
Proposition 5.7
The real dimension of the space of infinitesimal conical Cayley deformations of \(C_2\) in \(\mathbb {C}^4\) is twenty-two. The real dimension of the space of infinitesimal conical complex deformations of \(C_2\) in \(\mathbb {C}^4\) 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 \(\mathbb {R}^8\).
5.2.3 Example 3: \(L_3\cong SU(2)/\mathbb {Z}_3\)
Finally, we compute the dimension of the space of infinitesimal conical Cayley deformations of \(C_3\) in \(\mathbb {C}^4\), which has real link \(L_3\cong SU(2)/\mathbb {Z}_3\) and complex link \(\Sigma _3\) as defined in Sect. 5.1.3.
The normal bundle of \(\Sigma _3\) in \(\mathbb {C}P^3\) is [4, Prop 6]
where \(\mathcal {O}_{\Sigma _3}(n)\) denotes the line bundle of degree n over \(\Sigma _3\). By Hirzebruch–Riemann–Roch, Theorem 4.3, the space of holomorphic sections of \(\nu ^{1,0}_{\mathbb {C}P^3}(\Sigma _3)\) has dimension twelve, and so the dimension of the space of infinitesimal conical complex deformations of \(C_3\) in \(\mathbb {C}^4\) has dimension twenty-four.
So it remains to find \(v\in C^\infty (\mathcal {O}_{\Sigma _3}(3m+5)\oplus \mathcal {O}_{\Sigma _3}(3m+5))\) satisfying
Unfortunately, for this example, we cannot directly apply Theorem 5.5 to this problem, so we must find a different way to solve (5.5). We will do this by constructing a moving frame for \(L_3\).
Proposition 5.8
[13, §6.3.2] There exists an orthonormal frame of \(L_3\), denoted \(\{e_1,e_2,e_3\}\), where \(Je_2=e_3\) and \(e_1\) is the Reeb vector field. We have that
We extend this to a frame of \(S^7\) as follows.
Lemma 5.9
There exist orthonormal frames \(\{e_1,e_2,e_3\}\) of \(L_3\) and \(\{f_4,f_5,\)\(f_6,f_7\}\) of \(\nu _{S^7}(L_3)\) such that the structure equations of Proposition B.2 take the following form:
where \(x:L_3\rightarrow S^7\), \(Je_2=e_3, Jf_4=f_5, Jf_6=f_7\), \(\{\omega _1,\omega _2,\omega _3\}\) is an orthonormal coframe of \(L_3\) (\(\omega _i(e_j)=\delta _{ij}\)) and \(\{\eta _4,\eta _5,\eta _6,\eta _7\}\) is an orthonormal coframe of the normal bundle of \(L_3\) in \(S^7\) (\(\eta _a(f_b)=\delta _{ab}\)). Further, the second structure equations of Proposition B.3 are also satisfied.
Proof
Let \(\nabla \) denote the Levi-Civita connection of \(L_3\). We use the notation of Appendix B. Again we take \(\alpha _2=\omega _2\) and \(\alpha _3=\omega _3\) as we may by Proposition B.4. We see that since, using the structure equations given in B.2,
we must have that \(\alpha _1=-\frac{\omega _1}{3}\). We check that
and
Now Eq. (B.3) tells us that we must have that
So we take \(\beta ^4_2=\frac{2}{\sqrt{3}}\omega _2\) and \(\beta ^5_2=\frac{2}{\sqrt{3}}\omega _3\), \(\beta ^6_2=\beta ^7_2=0\) and this is satisfied. To ensure that Eq. (B.4) is satisfied, we seek \(\gamma \) so that
From this we see that we must have that \(\gamma _1=\frac{2}{3}\omega _1\), and \(\gamma _2=a\omega _2\) and \(\gamma _3=a\omega _3\). To determine a, we check Equation (B.5), which tells us that we must have
and therefore we must have \(a=2\). It can be checked that the remaining parts of Eq. (B.5) are satisfied with \(\gamma =(\frac{2}{3}\omega _1,2\omega _2,2\omega _3)\). Therefore we choose \(\{f_4,f_5,f_6,f_7\}\) so that the above choices of \(\gamma ,\beta \) and \(\alpha \) hold, and so the equations claimed hold. \(\square \)
We have that \(\{f_4-if_5,f_6-if_7\}\) is a frame for the holomorphic tangent bundle of \(L_3\) in \(S^7\). We have that
However,
and so we see explicitly that the connection on the normal bundle of \(L_3\) in \(S^7\) is not in a nice diagonal form as we had before. Since we have a moving frame of \(S^7\), we will return to considering the eigenvalue problem (4.9)–(4.10). Writing a section of \(\nu ^{1,0}_{S^7}(L_3)\) as
where \(g_1,g_2\) are functions on \(L_3\) and sections of \(\Lambda ^{0,1}_hL\otimes \nu ^{1,0}_{S^7}(L_3)\) as
where \(\alpha _1,\alpha _2\) are sections of \(\Lambda ^{0,1}_hL\), we seek \(g_1,g_2\in C^\infty (L_3)\) and \(\alpha _1,\alpha _2\in C^\infty (\Lambda ^{0,1}_hL)\) satisfying
and
We must have that
for some \(a\in \mathbb {C}\) (since if \(\alpha _1=0\) then we find infinitesimal conical complex deformations of \(C_3\)), and so we may instead study the eigenvalue problems
and
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 \(g_2(\omega _2-i\omega _3)\) as a \(\Lambda ^{0,1}_hL\)-valued function, which becomes
where now \(\alpha _2\) is a section of \(\Lambda ^{0,1}_hL\otimes \Lambda ^{0,1}_hL\). Supposing that
for \(3m\in \mathbb {Z}\) we see that in order for the eigenproblem (5.12)–(5.13) to make sense we must have
Write \(\mathcal {O}_{\Sigma _3}(d)\) for the degree d line bundle over \(\Sigma _3\). Then as explained in Sect. 4.1.4, we may replace the eigenvalue problems (5.6)–(5.7)–(5.12)–(5.13) with seeking \(g_1\in C^\infty (\mathcal {O}_{\Sigma _3}(3m)),\) and \(\alpha _1\in C^\infty (\mathcal {O}_{\Sigma _3}(3m+2)),\alpha _2\in C^\infty (\mathcal {O}_{\Sigma }(3m+4))\) satisfying
and
We find that \(\alpha _1\) must simultaneously satisfy the following two eigenproblems: applying \(\bar{\partial }_{\Sigma _3}\) to (5.15) and using (5.14) we find that
and applying \(\bar{\partial }_{\Sigma _3}^*\) to (5.16) and using (5.17) we have that
Applying the formula [17, Lem 2.1, 2.2]
where \(\alpha \) is a section of \(\mathcal {O}_{\Sigma _3}(3m+2)\), we see that
for \(\alpha _1\in C^\infty (\mathcal {O}_{\Sigma _3}(3m+2))\). Therefore \(a\in \mathbb {C}\) must satisfy
Solving this equation for a, we find that for \(m\ne 4/3\)
which simplifies to
First considering \(a=a_+\) we apply Theorem 5.5 to see that
is an eigenvalue of \(\bar{\partial }^*_{\Sigma _3}\bar{\partial }_{\Sigma _3}\) acting on sections of \(\mathcal {O}_{\Sigma _3}(3m+2)\) if, and only if, \(m=-2/3\). In this case there are five \(\alpha _1\in C^\infty (\mathcal {O}_{\Sigma _3}(0))\) satisfying
Taking \(g_1=\bar{\partial }^*_{\Sigma _3}\alpha _1\) and \(\alpha _2=\bar{\partial }_{\Sigma _2}\alpha _1\) completes this solution to the eigenproblem (5.14)–(5.15)–(5.16)–(5.17).
Secondly, when \(a=a_-=-2\) Theorem 5.5 tells us that
is an eigenvalue of \(\bar{\partial }^*_{\Sigma _3}\bar{\partial }_{\Sigma _3}\) acting on sections of \(\mathcal {O}_{\Sigma _3}(3m+2)\) if, and only if, \(m=-2/3\), in which case we seek functions \(\alpha _1\) on \(\Sigma _3\) satisfying
Since \(\Sigma _3\) is compact, \(\alpha _1\) must be holomorphic and further constant. Taking \(g_1=\alpha _2=0\) completes our analysis.
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 \(\bar{\partial }^*_{\Sigma _3}\bar{\partial }_{\Sigma _3}\) on sections of \(\mathcal {O}_{\Sigma _3}(6)\).
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 \(\mathbb {C}^4\) is thirty. The real dimension of the space of infinitesimal conical complex deformations of \(C_3\) in \(\mathbb {C}^4\) 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.
5.3 Calculating the \(\eta \)-Invariant for an Example
The final calculation in this article is to compute the \(\eta \)-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 \(\mu \) in a Calabi–Yau manifold M inside the space of all complex deformations of N, for a certain cone C in \(\mathbb {C}^4\), using Theorem 4.8.
We consider our simplest example of a two-dimensional complex cone in \(\mathbb {C}^4\) which is \(C_1=\mathbb {C}^2\). Denote by \(\Sigma _1\) the complex link of \(C_1\), i.e. \(\Sigma _1=\mathbb {C}P^1\). Proposition 4.4 told us that the exceptional weights \(\lambda \in \mathbb {R}\) satisfy an eigenproblem, and to calculate the \(\eta \)-invariant we must first find the dimension of the space of solutions to (4.14)–(4.15) for each \(\lambda \in \mathbb {R}\). Setting \(w=0\) in (4.14)–(4.15), we seek holomorphic sections of \(\nu ^{1,0}_{\mathbb {C}P^3}(\Sigma _1)\otimes \mathcal {O}_{\mathbb {C}P^3}(\lambda -1)|_{\Sigma _1}=\mathcal {O}_{\mathbb {C}P^3}(\lambda )|_{\Sigma _1}\oplus \mathcal {O}_{\mathbb {C}P^3}(\lambda )|_{\Sigma _1}\), for \(\lambda \in \mathbb {N}\cup \{0\}\), which by the Hirzebruch–Riemann–Roch theorem 4.3 has dimension \(2(\lambda +1)\). Similarly, setting \(v=0\) in (4.14)–(4.15), we seek antiholomorphic sections of \(\mathcal {O}_{\mathbb {C}P^3}(-\lambda )|_{\Sigma _1}\oplus \mathcal {O}_{\mathbb {C}P^3}(-\lambda )|_{\Sigma _1}\), which again have dimension \(2(\lambda +1)\).
It remains to compute the multiplicity of \(\lambda \) as an eigenvalue of
where v is a section of \(\mathcal {O}_{\mathbb {C}P^3}(m+1)|_{\Sigma _1}\oplus \mathcal {O}_{\mathbb {C}P^3}(m+1)|_{\Sigma _1}\) and \(\lambda \ne 1+m\) 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 \(\lambda >0\) as an eigenvalue of (5.20) is \(2\lambda (\lambda +1)\) and the multiplicity of integer \(\lambda <-2\) as an eigenvalue of (5.20) is \(2(\lambda +2)(\lambda +1)\). So we have that
and so
where \(\zeta \) is the Riemann zeta function.
We have that the multiplicity of the zero eigenvalue in this case in four. So we have found that
5.4 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.
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Acknowledgements
I would like to thank Jason Lotay for his help, guidance and feedback on this project. I would also like to thank Alexei Kovalev, Yng-Ing Lee and Julius Ross for comments on my PhD thesis, from which this work is taken. This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) Grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis.
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Appendices
Appendix A: Local Argument for Unobstructedness of Complex Deformations
The argument here is taken from the author’s PhD thesis [24, Lem 3.4.6]. A more general version for any complex submanifold of a Calabi–Yau manifold appears in [23, Lem 4.7].
Lemma A.1
Let G denote the operator defined in Proposition 2.7, whose kernel contains exactly those normal vector fields that correspond to complex deformations of a compact complex surface. Then
Proof
We will write the tangent space to a deformation of a complex surface N in a Calabi–Yau four-fold M as a graph over the tangent space of N, identified with a complex subspace of \(\mathbb {C}^4\) and write down the condition equivalent to \(G(v)=0\).
Choose \(p\in N\). Then \(T_pM=T_pN\oplus \nu _p(N)\). Choose an orthonormal basis \(\{e_1,\dots , e_8\}\) for \(T_pM\) with \(Je_i=e_{i+4}\) for \(i=1,\dots 4\) so that
Let \(N'\) be a small deformation of N with diffeomorphism \(f:N\rightarrow N'\). Then there is a natural isometry \(T_pM\rightarrow T_{f(p)}M\) preserving the complex structures J and \(J'\) on these spaces. Denote by \(\{e_1',\dots , e_8'\}\) the orthonormal basis of \(T_{f(p)}M\) where \(e_i\) maps to \(e_i'\) under this isometry, with \(J'e_i'=e'_{i+4}\). Then
where without loss of generality since \(N'\) is a small deformation of N we may take
for \(\lambda ^j_i\in \mathbb {R}\).
We can then evaluate
where \(\{i,j,k\}\subseteq \{1,2,5,6\}\). We have that
where \(e'^{j}(e'_k)=\delta _{jk}\) and \(e'^{ijkl}:=e'^{i}\wedge e'^{j}\wedge e'^{k}\wedge e'^{l}\).
We evaluate \(\sigma (v_i,v_j,v_k)=0\) for \(\{i,j,k\}=\{1,2,5\},\{1,2,6\},\{1,5,6\}\) and \(\{2,5,6\}\). Eliminating duplicate equations, we find that the \(\lambda ^i_j\) must satisfy the following linear equations
and the following nonlinear equations
Since the first four equations may be rewritten as
it is easy to see that if the linear equations are satisfied then all of the equations above are satisfied. \(\square \)
Appendix B: Structure Equations of Spin(7)
We will here give the structure equations of \(S^7\) adapted to an associative submanifold of \(S^7\). To do this, we will consider the sphere \(S^7\) as the group quotient \(Spin(7)/G_2\), that is, we can consider Spin(7) as the \(G_2\) frame bundle over \(S^7\). Bryant [3, Prop 1.1] first wrote down the structure equations of Spin(7), but we will quote them in the following useful form given by Lotay [19, §4].
Proposition B.1
[19, Prop 4.2] We may write the Lie algebra \(\mathfrak {spin}(7)\) of the Lie group \(Spin(7)\subseteq Gl(n,\mathbb {R})\) as
where
and
Now that we have the structure equations for Spin(7), we may construct a moving frame for \(S^7\) adapted to an associative three-fold. If we let \(g:Spin(7)\rightarrow Gl(8,\mathbb {R})\) be the map taking Spin(7) to the identity component of the Lie subgroup of \(Gl(8,\mathbb {R})\) which has Lie algebra \(\mathfrak {spin}(7)\), then we can write \(g=(x\, e \, f)\), where for \(p\in Spin(7)\) we have that \(x(p)\in M_{8\times 1}(\mathbb {R}), e(p)=(e_1(p),e_2(p),e_3(p))\in M_{8\times 3}(\mathbb {R})\) and \(f(p)=(f_4(p),f_5(p),f_6(p),\)\(f_7(p))\in M_{8\times 4}(\mathbb {R})\). We can choose our frame so that x represents a point of our associative three-fold L, e is an orthonormal frame for L and \(\omega \) is an orthonormal coframe for L. Therefore f is an orthonormal frame for the normal bundle of L in \(S^7\), \(\eta \) an orthonormal coframe. Then since the Maurer–Cartan form \(\phi =g^{-1}dg\) takes values in \(\mathfrak {spin}(7)\), we can write
This yields the following results.
Proposition B.2
[19, Prop 4.3] Use the notation above. On the adapted frame bundle of an associative three-fold L in \(S^7\), \(x:L\rightarrow S^7\) and \(\{e_1,e_2,e_3,\)\(f_4,f_5,f_6,f_7\}\) is a local oriented orthonormal basis for \(TA\oplus NA\), so the first structure equations are
Proposition B.3
[19, Prop 4.4] Use the notation above. On the adapted frame bundle of an associative three-fold in \(S^7\), there exists a local tensor of functions \(h=h^a_{jk}=h^a_{kj}\), for \(1\le j,k \le 3\) and \(4\le a \le 7\), such that the second structure equations are
Notice that \([\alpha ]\) is the Levi-Civita connection of L and \(\frac{1}{2}([\alpha -\omega ]_++[\gamma ]_-)\) defines the induced connection on the normal bundle of L in \(S^7\). We have that h defines the second fundamental form \(\mathbf {II}_L\in C^\infty (S^2T^*L;\nu (L))\) of L in \(S^7\), writing
Since the associative submanifolds of \(S^7\) that we are considering are \(S^1\)-bundles over complex curves, we may reduce the structure equations of L.
Proposition B.4
[19, Ex 4.9] Let L be the link of complex cone C in \(\mathbb {C}^4\). Then we can choose a frame of \(TS^7|_L\) such that
This implies that \(\beta ^4_3=-\beta ^5_2,\beta ^5_3=\beta ^4_2, \beta ^6_3=-\beta ^7_2\) and \(\beta ^7_3=\beta ^6_2\). Here \(e_1\) defines the direction of the circle fibres of L over the complex link \(\Sigma \) of C.
Proof
This follows from supposing that the complex structure of \(\mathbb {C}^4\) acts on C as follows:
\(\square \)
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Moore, K. Deformations of Conically Singular Cayley Submanifolds. J Geom Anal 29, 2147–2216 (2019). https://doi.org/10.1007/s12220-018-0074-7
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DOI: https://doi.org/10.1007/s12220-018-0074-7