j. differential geometry

63 (2003) 279-347

SPECIAL LAGRANGIAN SUBMANIFOLDS WITH ISOLATED CONICAL SINGULARITIES.

V. SURVEY AND APPLICATIONS

DOMINIC JOYCE

Abstract

This is the last in a series of five papers studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_{1}, …, x_{n} locally modelled on special Lagrangian cones C_{1}, …, C_{n} in ℂ^{m} with isolated singularities at 0. Readers are advised to begin with this paper.

We survey the major results of the previous four papers, giving brief explanations of the proofs. We apply the results to describe the boundary of a moduli space of compact, nonsingular SL m-folds N in M. We prove the existence of special Lagrangian connected sums N_{1}# ⋯ #N_{k} of SL mfolds N_{1}, …, N_{k} in M. We also study SL 3-folds with T^{2}-cone singularities, proving results related to ideas of the author on invariants of Calabi-Yau 3-folds, and the SYZ Conjecture.

Let X be a compact SL m-fold with isolated conical singularities x_{i} and cones C_{i} for i = 1, …, n. The first paper studied the regularity of X near its singular points, and the the second the moduli space of deformations of X. The third and fourth papers construct desingularizations of X, realizing X as a limit of a family of compact, nonsingular SL m-folds N^{t} in M for small t > 0. Let L_{i} be an asymptotically conical SL m-fold in ℂ^{m} asymptotic to C_{i} at infinity. We make N^{t} by gluing tL_{i} into X at x_{i} for i = 1, …, n.

1. Introduction

Special Lagrangian m-folds (SL m-folds) are a distinguished class of real m-dimensional minimal submanifolds which may be defined in ℂ^{m}, or in Calabi-Yau m-folds, or more generally in almost Calabi-Yau mfolds (compact Kähler m-folds with trivial canonical bundle). We write an almost Calabi-Yau m-fold as M or (M, J, ω, Ω), where the manifold

Received 04/07/2003.

M has complex structure J, Kähler form ω and holomorphic volume form Ω.

This is the fifth in a series of five papers [17, 18, 19, 20] studying SL m-folds with isolated conical singularities. That is, we consider an SL m-fold X in an almost Calabi-Yau m-fold M for m > 2 with singularities at x_{1}, …, x_{n} in M, such that for some special Lagrangian cones C_{i} in T_{x_{i}}M ≅ ℂ^{m} with C_{i} \ {0} nonsingular, X approaches C_{i} near x_{i}, in an asymptotic C^{1} sense.

New readers of the series are advised to begin with this paper. We shall survey the major results of [17, 18, 19, 20], giving explanations, but avoiding the long, technical analytic proofs of previous papers. We also integrate the results to give an (incomplete) description of the boundary of a moduli space of compact SL m-folds, and apply them to prove some conjectures in [9, 13] on connected sums of SL m-folds, and T^{2}- cone singularities of SL 3-folds.

Having a good understanding of the singularities of special Lagrangian submanifolds will be essential in clarifying the Strominger-Yau– Zaslow conjecture on the mirror symmetry of Calabi-Yau 3-folds [29], and also in resolving conjectures made by the author [9] on defining new invariants of Calabi-Yau 3-folds by counting special Lagrangian homology 3-spheres with weights. The series aims to develop such an understanding for simple singularities of SL m-folds.

We begin in §2 with an introduction to almost Calabi-Yau and special Lagrangian geometry, and the deformation theory of compact SL m-folds. Section 3 defines SL m-folds with conical singularities, our subject, gives examples of special Lagrangian cones, and some basics on homology and cohomology.

Section 4 describes the first paper [17] on the regularity of SL mfolds X with conical singularities x_{1}, …, x_{n}. We study the asymptotic behaviour of X and its derivatives near x_{i}, how quickly it converges to the cone C_{i}.

In §5 we discuss the second paper [18] on the deformation theory of compact SL m-folds X with conical singularities in an almost Calabi-Yau m-fold M. We find that the moduli space M_{X} of deformations of X in M is locally homeomorphic to the zeroes of a smooth map Φ : I_{X′} → O_{X′} between finite-dimensional vector spaces, and if the obstruction space O_{X′} is zero then M_{X} is a smooth manifold. We also study deformations in smooth families of almost Calabi-Yau m-folds (M, J^{s}, ω^{s}, Ω^{s}) for s ∈ F ⊂ ℝ^{d}.

Section 6 is an aside on asymptotically conical SL m-folds (AC SL m-folds) in ℂ^{m}, that is, nonsingular, noncompact SL m-folds L in ℂ^{m} which are asymptotic at infinity to an SL cone C at a prescribed rate λ. Our main sources are [17, §7] and Marshall [23]. The theories of AC SL m-folds and SL m-folds with conical singularities are similar in many respects.

Section 7 explains the third and fourth papers [19, 20] on desingularizations of a compact SL m-fold X with conical singularities x_{i} with cones C_{i} for i = 1, …, n in an almost Calabi-Yau m-fold M. We take AC SL m-folds L_{i} in ℂ^{m} asymptotic to C_{i} at infinity, and glue tL_{i} into X at x_{i} for small t > 0 to get a smooth family of compact, nonsingular SL m-folds Ñ^{t} in M, with Ñ^{t} → X as t → 0. We also study desingularizations in families of almost Calabi-Yau m-folds (M, J^{s}, ω^{s}, Ω^{s}) for s ∈ F.

The new material of the paper is §8–§10. We study the moduli space M_{N} of compact, nonsingular SL m-folds N in §2, the moduli space M_{X} of compact SL m-folds X with conical singularities in §5, and the moduli space Mλ_{L} of AC SL m-folds in ℂ^{m} with rate λ in §6. Section 8 explains how these three kinds of moduli space fit together.

The idea is that M_{N} has a compactification \overline{M}_{N} with boundary ∂M_{N} = \overline{M}_{N} \ M_{N} consisting of singular SL m-folds. Suppose N is constructed as in §7 by desingularizing X with conical singularities x_{1}, …, x_{n} by gluing in AC SL m-folds L_{1}, …, L_{n}. Then in good cases we expect M_{X} ⊆ ∂M_{N}, and M_{N} may be locally modelled on a subset

of M_{X} × M0_{L_{1}} × ⋯ × M0_{Ln} near X.

Section 9 considers connected sums of SL m-folds. Suppose X is a compact, immersed SL m-fold in M with transverse self-intersection points x_{1}, …, x_{n}. This includes the case where X is a union of q > 1 embedded SL m-folds X_{1}, …, X_{q}, and the x_{i} are intersections between X_{j} and X_{k}. Then we can consider X to be an SL m-fold with conical singularities, with each cone C_{i} the union of two transverse SL m-planes Π_{i}^{+}, Π_{i}^{−} ≅ ℝ^{m} in ℂ^{m}.

When Π_{i}^{±} satisfy an angle criterion, Lawlor [21] constructed a family of AC SL m-folds L_{i}^{±,A} for A > 0 with cone Π_{i}^{+} ∪ Π_{i}^{−}, diffeomorphic to S^{m−1} × ℝ. We apply the results of §7 to construct SL m-folds Ñ^{t} by gluing tL_{i}^{±,A} into X at x_{i}. These Ñ^{t} are multiple connected sums of X with itself. In this way we re-prove and extend a result of Lee [8].

Finally, §10 studies SL 3-folds X with conical singularities with cone C the U(1)^{2}-invariant SL T^{2}-cone due to Harvey and Lawson [5, §III.3.A]. These have particularly nice properties. For instance, the moduli space M_{X} is always smooth, and under topological conditions the compactified moduli space \overline{M}_{N} of desingularizations N is near X a nonsingular manifold with boundary M_{X}. We prove several conjectures from [9, 13].

Acknowledgements. I would like to thank Stephen Marshall, Mark Haskins, Tadashi Tokieda, Ivan Smith, Sema Salur, and Adrian Butscher for useful conversations. I was supported by an EPSRC Advanced Research Fellowship whilst writing this paper.

2. Special Lagrangian geometry

We begin with some background from symplectic geometry. Then special Lagrangian submanifolds (SL m-folds) are introduced both in ℂ^{m} and in almost Calabi-Yau m-folds. We also describe the deformation theory of compact SL m-folds. Some references for this section are McDuff and Salamon [24], Harvey and Lawson [5], McLean [26], and the author [16].

2.1 Background from symplectic geometry

We start by recalling some elementary symplectic geometry, which can be found in McDuff and Salamon [24]. Here are the basic definitions.

Definition 2.1. Let M be a smooth manifold of even dimension 2m. A closed 2-form ω on M is called a symplectic form if the 2m-form ω^{m} is nonzero at every point of M. Then (M, ω) is called a symplectic manifold. A submanifold N in M is called Lagrangian if dim N = m = \frac{1}{2} dim M and ω|_{N} ≡ 0.

The simplest example of a symplectic manifold is ℝ^{2m}.

Definition 2.2. Let ℝ^{2m} have coordinates (x_{1}, …, x_{m}, y_{1}, …, y_{m}), and define the standard metric g′ and symplectic form ω′ on ℝ^{2m} by

g′ = ∑_{j=1}^{m}(dx_{j}^{2} + dy_{j}^{2}) and ω′ = ∑_{j=1}^{m} dx_{j} ∧ dy_{j}. (1)

Then (ℝ^{2m}, ω′) is a symplectic manifold. When we wish to identify ℝ^{2m} with ℂ^{m}, we take the complex coordinates (z_{1}, …, z_{m}) on ℂ^{m} to be z_{j} = x_{j} + iy_{j}. For R > 0, define B_{R} to be the open ball of radius R about 0 in ℝ^{2m}.

Darboux’s Theorem [24, Th. 3.15] says that every symplectic manifold is locally isomorphic to (ℝ^{2m}, ω′). Our version easily follows.

Theorem 2.3. Let (M, ω) be a symplectic 2m-manifold and x ∈ M. Then there exist R > 0 and an embedding Υ : B_{R} → M with Υ(0) = x such that Υ^{∗}(ω) = ω′, where ω′ is the standard symplectic form on ℝ^{2m} ⊃ B_{R}. Given an isomorphism υ : ℝ^{2m} → T_{x}M with υ^{∗}(ω|_{x})=ω′, we can choose Υ with dΥ|_{0} =υ.

Let N be a real m-manifold. Then its tangent bundle T^{∗}N has a canonical symplectic form ω̂, defined as follows. Let (x_{1}, …, x_{m}) be local coordinates on N. Extend them to local coordinates (x_{1}, …, x_{m}, y_{1}, …, y_{m}) on T^{∗}N such that (x_{1}, ..., y_{m}) represents the 1-form y_{1}dx_{1} + ⋯ + y_{m}dx_{m} in T_{(x_{1},...,x_{m})}^{∗}N. Then ω̂ = dx_{1} ∧ dy_{1} + ⋯ + dx_{m} ∧ dy_{m}.

Identify N with the zero section in T^{∗}N. Then N is a Lagrangian submanifold of T^{∗}N. The Lagrangian Neighbourhood Theorem [24, Th. 3.33] shows that any compact Lagrangian submanifold N in a symplectic manifold looks locally like the zero section in T^{∗}N.

Theorem 2.4. Let (M, ω) be a symplectic manifold and N ⊂ M a compact Lagrangian submanifold. Then there exists an open tubular neighbourhood U of the zero section N in T^{∗}N, and an embedding Φ : U → M with Φ|_{N} = id : N → N and Φ^{∗}(ω) = ω̂, where ω̂ is the canonical symplectic structure on T^{∗}N.

We shall call U, Φ a Lagrangian neighbourhood of N. Such neighbourhoods are useful for parametrizing nearby Lagrangian submanifolds of M. Suppose that Ñ is a Lagrangian submanifold of M which is C^{1}-close to N. Then Ñ lies in Φ(U), and is the image Φ(Γ(α)) of the graph Γ(α) of a unique C^{1}-small 1-form α on N.

As Ñ is Lagrangian and Φ^{∗}(ω) = ω̂ we see that ω̂|_{Γ(α)} ≡ 0. But one can easily show that ω̂|_{Γ(α)} = −π^{∗}(dα), where π : Γ(α) → N is the natural projection. Hence dα = 0, and α is a closed 1-form. This establishes a 1-1 correspondence between small closed 1-forms on N and Lagrangian submanifolds Ñ close to N in M, which is an essential tool in proving the results of §5 and §7.

2.2 Special Lagrangian submanifolds in ℂ^{m}

We define calibrations and calibrated submanifolds, following [5].

Definition 2.5. Let (M, g) be a Riemannian manifold. An oriented tangent k-plane V on M is a vector subspace V of some tangent space T_{x}M to M with dim V = k, equipped with an orientation. If V is an oriented tangent k-plane on M then g|_{V} is a Euclidean metric on V, so combining g|_{V} with the orientation on V gives a natural volume form vol_{V} on V, which is a k-form on V.

Now let φ be a closed k-form on M. We say that φ is a calibration on M if for every oriented k-plane V on M we have φ|_{V} ⩽ vol_{V}. Here φ|_{V} = α · vol_{V} for some α ∈ ℝ, and φ|_{V} ⩽ vol_{V} if α ⩽ 1. Let N be an oriented submanifold of M with dimension k. Then each tangent space T_{x}N for x ∈ N is an oriented tangent k-plane. We say that N is a calibrated submanifold if φ|_{T_{x}N} = vol_{T_{x}N} for all x ∈ N.

It is easy to show that calibrated submanifolds are automatically minimal submanifolds [5, Th. II.4.2]. Here is the definition of special Lagrangian submanifolds in ℂ^{m}, taken from [5, §III].

Definition 2.6. Let ℂ^{m} have complex coordinates (z_{1}, …, z_{m}), and define a metric g′, a real 2-form ω′ and a complex m-form Ω′ on ℂ^{m} by

g′ = |dz_{1}|^{2} + ⋯ + |dz_{m}|^{2},
(2)  ω′ = \frac{i}{2}(dz_{1} ∧ dz_{1} + ⋯ + dz_{m} ∧ dz_{m}),  and
Ω′ = dz_{1} ∧ ⋯ ∧ dz_{m}.

Then g′, ω′ are as in Definition 2.2, and Re Ω′ and Im Ω′ are real m-forms on ℂ^{m}. Let L be an oriented real submanifold of ℂ^{m} of real dimension m. We say that L is a special Lagrangian submanifold of ℂ^{m}, or SL m-fold for short, if L is calibrated with respect to Re Ω′, in the sense of Definition 2.5.

Harvey and Lawson [5, Cor. III.1.11] give the following alternative characterization of special Lagrangian submanifolds:

Proposition 2.7. Let L be a real m-dimensional submanifold of ℂ^{m}. Then L admits an orientation making it into an SL submanifold of ℂ^{m} if and only if ω′|_{L} ≡ 0 and Im Ω′|_{L} ≡ 0.

Thus special Lagrangian submanifolds are Lagrangian submanifolds satisfying the extra condition that Im Ω′|_{L} ≡ 0, which is how they get their name.

2.3 Almost Calabi-Yau m-folds and SL m-folds

We shall define special Lagrangian submanifolds not just in Calabi-Yau manifolds, as usual, but in the much larger class of almost Calabi-Yau manifolds.

Definition 2.8. Let m ⩾ 2. An almost Calabi-Yau m-fold is a quadruple (M, J, ω, Ω) such that (M, J) is a compact m-dimensional complex manifold, ω is the Kähler form of a Kähler metric g on M, and Ω is a non-vanishing holomorphic (m, 0)-form on M.

We call (M, J, ω, Ω) a Calabi-Yau m-fold if in addition ω and Ω satisfy

ω^{m}/m! = (−1)^{m(m−1)/2}(i/2)^{m}Ω ∧ Ω. (3)

Then for each x ∈ M there exists an isomorphism T_{x}M ≅ ℂ^{m} that identifies g_{x}, ω_{x} and Ω_{x} with the flat versions g′, ω′, Ω′ on ℂ^{m} in (2). Furthermore, g is Ricci-flat and its holonomy group is a subgroup of SU(m).

This is not the usual definition of a Calabi-Yau manifold, but is essentially equivalent to it.

Definition 2.9. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold, and N a real m-dimensional submanifold of M. We call N a special Lagrangian submanifold, or SL m-fold for short, if ω|_{N} ≡ Im Ω|_{N} ≡ 0. It easily follows that Re Ω|_{N} is a nonvanishing m-form on N. Thus N is orientable, with a unique orientation in which Re Ω|_{N} is positive.

Again, this is not the usual definition of SL m-fold, but is essentially equivalent to it. In Definition 9.9 we give a more general definition of SL m-fold involving a phase e^{iθ}. Suppose (M, J, ω, Ω) is an almost Calabi-Yau m-fold, with metric g. Let ψ : M → (0, ∞) be the unique smooth function such that

ψ^{2m}ω^{m}/m! = (−1)^{m(m−1)/2}(i/2)^{m}Ω ∧ Ω, (4)

and define g̃ to be the conformally equivalent metric ψ^{2}g on M. Then Re Ω is a calibration on the Riemannian manifold (M, g̃), and SL mfolds N in (M, J, ω, Ω) are calibrated with respect to it, so that they are minimal with respect to g̃.

If M is a Calabi-Yau m-fold then ψ ≡ 1 by (3), so g̃ = g, and an m-submanifold N in M is special Lagrangian if and only if it is calibrated w.r.t. Re Ω on (M, g), as in Definition 2.6. This recovers the usual definition of special Lagrangian m-folds in Calabi-Yau m-folds.

2.4 Deformations of compact SL m-folds

The deformation theory of special Lagrangian submanifolds was studied by McLean [26, §3], who proved the following result in the Calabi-Yau case. The extension to the almost Calabi-Yau case is described in [16, §9.5].

Theorem 2.10. Let N be a compact SL m-fold in an almost Calabi-Yau m-fold (M, J, ω, Ω). Then the moduli space M_{N} of special Lagrangian deformations of N is a smooth manifold of dimension b^{1}(N), the first Betti number of N.

Here is a sketch of the proof of Theorem 2.10. Let g be the Kähler metric on M, and define ψ : M → (0, ∞) by (4). Applying Theorem 2.4 gives an open neighbourhood U of N in T^{∗}N and an embedding Φ : U → M. Let π : U → N be the natural projection. Define an m-form β on U by β = Φ^{∗}(Im Ω). If α is a 1-form on N let Γ(α) be the graph of α in T^{∗}N, and write C^{∞}(U) ⊂ C^{∞}(T^{∗}N) for the subset of 1-forms whose graphs lie in U.

Then each submanifold Ñ of M which is C^{1}-close to N is Φ(Γ(α)) for some small α ∈ C^{∞}(U). Here is the condition for Ñ to be special Lagrangian.

Lemma 2.11. In the situation above, if α ∈ C^{∞}(U) then Ñ = Φ(Γ(α)) is an SL m-fold in M if and only if dα = 0 and π_{∗}(β|_{Γ(α)}) = 0.

Proof. By Definition 2.9, Ñ is an SL m-fold if and only if ω|Ñ ≡ Im Ω|Ñ ≡ 0. Pulling back by Φ and pushing forward by π : Γ(α) → N, we see that Ñ is special Lagrangian if and only if π_{∗}(ω̂|_{Γ(α)}) ≡ π_{∗}(β|_{Γ(α)}) ≡ 0, since Φ^{∗}(ω) = ω̂ and Φ^{∗}(Im Ω) = β. But π_{∗}(ω̂|_{Γ(α)}) = −dα, and the lemma follows. q.e.d.

We rewrite the condition π_{∗}(β|_{Γ(α)}) = 0 in terms of a function F.

Definition 2.12. Define F : C^{∞}(U) → C^{∞}(N) by π_{∗}(β|_{Γ(α)}) = F (α) dV_{g}, where dV_{g} is the volume form of g|_{N} on N. Then Lemma 2.11 shows that if α∈C^{∞}(U) then Φ(Γ(α)) is special Lagrangian if and only if dα = F (α) = 0.

In [18, Prop. 2.10] we compute the expansion of F up to first order in α.

Proposition 2.13. This function F may be written

F (α)[x] = −d^{∗}(ψ^{m}α) + Q(x, α(x), ∇α(x)) for x ∈ N, (5)

where Q : {(x, y, z) : x ∈ N, y ∈ T_{x}^{∗}N ∩ U, z ∈ ⊗^{2}T_{x}^{∗}N} → ℝ is smooth and Q(x, y, z) = O(|y|^{2} + |z|^{2}) for small y, z.

From Definition 2.12 and Proposition 2.13 we see that the moduli space M_{N} of special Lagrangian deformations of N is locally approximately isomorphic to the vector space of 1-forms α with dα = d^{∗}(ψ^{m}α) = 0. But by Hodge theory, this is isomorphic to the de Rham cohomology group H^{1}(N, ℝ), and is a manifold with dimension b^{1}(N).

To carry out this last step rigorously requires some technical machinery: one must work with certain Banach spaces of sections of Λ^{k}T^{∗}N for k = 0, 1, 2, use elliptic regularity results to prove that the map α ↦ (dα, dF |_{0}(α)) is surjective upon the appropriate Banach spaces, and then use the Implicit Mapping Theorem for Banach spaces to show that the kernel of the map is what we expect. This concludes our sketch of the proof of Theorem 2.10.

We extend Theorem 2.10 to families of almost Calabi-Yau m-folds.

Definition 2.14. Let (M, J, ω, Ω) be an almost Calabi-Yau mfold. A smooth family of deformations of (M, J, ω, Ω) is a connected open set F ⊂ ℝ^{d} for d ⩾ 0 with 0 ∈ F called the base space, and a smooth family {(M, J^{s}, ω^{s}, Ω^{s}) : s ∈ F} of almost Calabi-Yau structures with (J^{0}, ω^{0}, Ω^{0}) = (J, ω, Ω).

If N is a compact SL m-fold in (M, J, ω, Ω), the moduli of deformations of N in each (M, J^{s}, ω^{s}, Ω^{s}) for s ∈ F make up a big moduli

space M\begin{matrix}F \\ N \end{matrix}.

Definition 2.15. Let {(M, J^{s}, ω^{s}, Ω^{s}) : s ∈ F} be a smooth family of deformations of an almost Calabi-Yau m-fold (M, J, ω, Ω), and N be a compact SL m-fold in (M, J, ω, Ω). Define the moduli space M\begin{matrix}F \\ N \end{matrix} of deformations of N in the family F to be the set of pairs (s, N̂) for which s ∈ F and N̂ is a compact SL m-fold in (M, J^{s}, ω^{s}, Ω^{s}) which is diffeomorphic to N and isotopic to N in M. Define a projection π^{F} : M\begin{matrix}F \\ N \end{matrix} → F by π^{F}(s, N̂) = s. Then M\begin{matrix}F \\ N \end{matrix} has a natural topology, and π^{F} is continuous.

The following result is proved by Marshall [23, Th. 3.2.9], using similar methods to Theorem 2.10.

Theorem 2.16. Let {(M, J^{s}, ω^{s}, Ω^{s}) : s ∈ F} be a smooth family of deformations of an almost Calabi-Yau m-fold (M, J, ω, Ω), with base space F ⊂ ℝ^{d}. Suppose N is a compact SL m-fold in (M, J, ω, Ω) with [ω^{s}|_{N}] = 0 in H^{2}(N, ℝ) and [Im Ω^{s}|_{N}] = 0 in H^{m}(N, ℝ) for all s ∈ F. Let M\begin{matrix}F \\ N \end{matrix} be the moduli space of deformations of N in F, and π^{F} : M\begin{matrix}F \\ N \end{matrix} → F the natural projection.

Then M\begin{matrix}F \\ N \end{matrix} is a smooth manifold of dimension d + b^{1}(N), and π^{F} :

M\begin{matrix}F \\ N \end{matrix} → F a smooth submersion. For small s ∈ F the moduli space Ms_{N} = (π^{F})^{−1}(s) of deformations of N in (M, J^{s}, ω^{s}, Ω^{s}) is a nonempty smooth manifold of dimension b^{1}(N), with M0_{N} = M_{N}.

Here a necessary condition for the existence of an SL m-fold N̂ isotopic to N in (M, J^{s}, ω^{s}, Ω^{s}) is that [ω^{s}|_{N}] = [Im Ω^{s}|_{N}] = 0 in H^{∗}(N, ℝ), since [ω^{s}|_{N}] and [ω^{s}|N̂] are identified under the natural isomorphism between H^{2}(N, ℝ) and H^{2}(N̂, ℝ), and similarly for Im Ω^{s}.

The point of the theorem is that these conditions [ω^{s}|_{N}] = [Im Ω^{s}|_{N}] = 0 are also sufficient for the existence of N̂ when s is close to 0 in F. That is, the only obstructions to existence of compact SL m-folds when we deform the underlying almost Calabi-Yau m-fold are the obvious cohomological ones.

3. SL cones and conical singularities

We begin in §3.1 with some definitions on special Lagrangian cones. Section 3.2 gives examples of SL cones, and §3.3 defines SL m-folds with conical singularities, the subject of the paper. Section 3.4 discusses homology and cohomology of SL m-folds with conical singularities.

3.1 Preliminaries on special Lagrangian cones

We define special Lagrangian cones, and some notation.

Definition 3.1. A (singular) SL m-fold C in ℂ^{m} is called a cone if C = tC for all t > 0, where tC = {t x : x ∈ C}. Let C be a closed SL cone in ℂ^{m} with an isolated singularity at 0. Then Σ = C ∩ S^{2m−1} is a compact, nonsingular (m−1)-submanifold of S^{2m−1}, not necessarily connected. Let g_{Σ} be the restriction of g′ to Σ, where g′ is as in (2).

Set C′ = C \ {0}. Define ι : Σ × (0, ∞) → ℂ^{m} by ι(σ, r) = rσ. Then ι has image C′. By an abuse of notation, identify C′ with Σ × (0, ∞) using ι. The cone metric on C′ ≅ Σ × (0, ∞) is g′ = ι^{∗}(g′) = dr^{2} + r^{2}g_{Σ}.

For α ∈ ℝ, we say that a function u : C′ → ℝ is homogeneous of order α if u ∘ t ≡ t^{α}u for all t > 0. Equivalently, u is homogeneous of order α if u(σ, r) ≡ r^{α}v(σ) for some function v : Σ → ℝ.

In [17, Lem. 2.3] we study homogeneous harmonic functions on C′.

Lemma 3.2. In the situation of Definition 3.1, let u(σ, r) ≡ r^{α}v(σ) be a homogeneous function of order α on C′ = Σ×(0, ∞), for v ∈ C^{2}(Σ).

Then

Δu(σ, r) = r^{α−2}(Δ_{Σ}v − α(α + m − 2)v), (6)

where Δ, Δ_{Σ} are the Laplacians on (C′, g′) and (Σ, g_{Σ}). Hence, u is harmonic on C′ if and only if Δ_{Σ}v = α(α + m − 2)v.

Following [17, Def. 2.5], we define:

Definition 3.3. In the situation of Definition 3.1, suppose m > 2 and define

(7) D_{Σ} = {α ∈ ℝ : α(α + m − 2) is an eigenvalue of Δ_{Σ}}.

Then D_{Σ} is a countable, discrete subset of ℝ. By Lemma 3.2, an equivalent definition is that D_{Σ} is the set of α ∈ ℝ for which there exists a nonzero homogeneous harmonic function u of order α on C′.

Define m_{Σ} : D_{Σ} → ℕ by taking m_{Σ}(α) to be the multiplicity of the eigenvalue α(α + m − 2) of Δ_{Σ}, or equivalently the dimension of the vector space of homogeneous harmonic functions u of order α on C′. Define N_{Σ} : ℝ → ℤ by

N_{Σ}(δ) = − ∑_{α∈D_{Σ}∩(δ,0)} m_{Σ}(α) if δ < 0, and
N_{Σ}(δ) = ∑_{α∈D_{Σ}∩[0,δ]} m_{Σ}(α) if δ ⩾ 0.

Then N_{Σ} is monotone increasing and upper semicontinuous, and is discontinuous exactly on D_{Σ}, increasing by m_{Σ}(α) at each α ∈ D_{Σ}. As the eigenvalues of Δ_{Σ} are nonnegative, we see that D_{Σ} ∩ (2 − m, 0) = ∅ and N_{Σ} ≡ 0 on (2 − m, 0).

We define the stability index of C, and stable and rigid cones [18, Def. 3.6].

Definition 3.4. Let C be an SL cone in ℂ^{m} for m > 2 with an isolated singularity at 0, let G be the Lie subgroup of SU(m) preserving C, and use the notation of Definitions 3.1 and 3.3. Then [18, eq. (8)] shows that

(8) m_{Σ}(0) = b^{0}(Σ), m_{Σ}(1) ⩾ 2m and m_{Σ}(2) ⩾ m^{2} − 1 − dim G.

Define the stability index s-ind(C) to be

(9) s-ind(C) = N_{Σ}(2) − b^{0}(Σ) − m^{2} − 2m + 1 + dim G.

Then s-ind(C) ⩾ 0 by (8), as N_{Σ}(2) ⩾ m_{Σ}(0) + m_{Σ}(1) + m_{Σ}(2) by definition. We call C stable if s-ind(C) = 0.

Following [17, Def. 6.7], we call C rigid if m_{Σ}(2) = m^{2} − 1 − dim G. As

s-ind(C) ⩾ m_{Σ}(2) − (m^{2} − 1 − dim G) ⩾ 0,

we see that if C is stable, then C is rigid.

We shall see in §5 that s-ind(C) is the dimension of an obstruction space to deforming an SL m-fold X with a conical singularity with cone C, and that if C is stable then the deformation theory of X simplifies. An SL cone C is rigid if all infinitesimal deformations of C as an SL cone come from SU(m) rotations of C. This will be useful in the geometric measure theory material of §4.2.

3.2 Examples of special Lagrangian cones

In our first example we can compute the data of §3.1 very explicitly.

Example 3.5. Here is a family of special Lagrangian cones constructed by Harvey and Lawson [5, §III.3.A]. For m ⩾ 3, define

Cm_{HL} = {(z_{1}, …, z_{m}) ∈ ℂ^{m} : i^{m+1}z_{1} ⋯ z_{m} ∈ [0, ∞), |z_{1}| = ⋯ = |z_{m}|}.

Then Cm_{HL} is a special Lagrangian cone in ℂ^{m} with an isolated singularity at 0, and Σm_{HL} = Cm_{HL} ∩ S^{2m−1} is an (m−1)-torus T^{m−1}. Both Cm_{HL} and Σm_{HL} are invariant under the U(1)^{m−1} subgroup of SU(m) acting by

(z_{1}, …, z_{m})↦(e^{iθ_{1}}z_{1}, …, e^{iθ_{m}}z_{m}) for θ_{j} ∈ ℝ with θ_{1}+⋯+θ_{m} =0. (10)

In fact ± Cm_{HL} for m odd, and Cm_{HL}, iCm_{HL} for m even, are the unique SL cones in ℂ^{m} invariant under (10), which is how Harvey and Lawson constructed them.

The metric on Σm_{HL} ≅ T^{m−1} is flat, so it is easy to compute the eigenvalues of Δ_{Σ{m \atop HL}}. This was done by Marshall [23, §6.3.4]. There is a 1-1 correspondence between (n_{1}, …, n_{m−1}) ∈ ℤ^{m−1} and eigenvectors of Δ_{Σ{m \atop HL}} with eigenvalue

m ∑_{i=1}^{m−1} n_{i}^{2} − ∑_{i,j=1}^{m−1} n_{i}n_{j}. (11)

Using (11) and a computer we can find the eigenvalues of Δ_{Σ{m \atop HL}} and their multiplicities. The Lie subgroup Gm_{HL} of SU(m) preserving Cm_{HL} has identity component the U(1)^{m−1} of (10), so that dim Gm_{HL} = m−1. Thus we can calculate D_{Σ{m \atop HL}}, m_{Σ{m \atop HL}}, N_{Σ{m \atop HL}}, and the stability index s-ind(Cm_{HL}). This was done by Marshall [23, Table 6.1] and the author [18, §3.2]. Table 1 gives the data m, N_{Σ{m \atop HL}}(2), m_{Σ{m \atop HL}}(2) and s-ind(Cm_{HL}) for 3 ⩽ m ⩽ 12.

m  3 4 5 6  7  8  9 10 11 12
N_{Σ{m \atop HL}}(2)  13 27 51 93 169 311 331 201 243 289
m_{Σ{m \atop HL}}(2)  6 12 20 30 42 126 240 90 110 132
s-ind(Cm_{HL}) 0 6 20 50 112 238 240 90 110 132

Table 1: Data for U(1)^{m−1}-invariant SL cones Cm_{HL} in ℂ^{m}.

One can also prove that

N_{Σ{m \atop HL}}(2) = 2m^{2} + 1 and m_{Σ{m \atop HL}}(2) = s-ind(Cm_{HL}) = m^{2} − m for m ⩾ 10. (12)

As Cm_{HL} is stable when s-ind(Cm_{HL}) = 0 we see from Table 1 and (12) that C3_{HL} is a stable cone in ℂ^{3}, but Cm_{HL} is unstable for m ⩾ 4. Also Cm_{HL} is rigid when m_{Σ{m \atop HL}}(2) = m^{2} − m, as dim Gm_{HL} = m − 1. Thus Cm_{HL} is rigid if and only if m ≠ 8, 9, by Table 1 and (12).

Here is an example taken from [10, Ex. 9.4], chosen as it is easy to write down.

Example 3.6. Let a_{1}, …, a_{m} ∈ ℤ with a_{1} + ⋯ + a_{m} = 0 and highest common factor 1, such that a_{1}, …, a_{k} > 0 and a_{k+1}, …, a_{m} < 0 for 0 < k < m. Define

\begin{matrix}L_{0}^{a_{1},...,a_{m}} = \{(ie^{ia_{1}θ}x_{1}, e^{ia_{2}θ}x_{2}, …, e^{ia_{m}θ}x_{m}) : θ ∈ [0, 2π), \\ x_{1}, …, x_{m} ∈ ℝ, a_{1}x_{1}^{2} + ⋯ + a_{m}x_{m}^{2} = 0\}. \end{matrix} (13)

Then L_{0}^{a_{1},...,a_{m}} is an immersed SL cone in ℂ^{m}, with an isolated singularity at 0.

Define C^{a_{1},...,a_{m}} = {(x_{1}, …, x_{m}) ∈ ℝ^{m} : a_{1}x_{1}^{2} + ⋯ + a_{m}x_{m}^{2} = 0}. Then C^{a_{1},...,a_{m}} is a quadric cone on S^{k−1}×S^{m−k−1} in ℝ^{m}, and L_{0}^{a_{1},...,a_{m}} is the image of an immersion Φ : C^{a_{1},...,a_{m}} ×S^{1} → ℂ^{m}, which is generically 2:1. Therefore L_{0}^{a_{1},...,a_{m}} is an immersed SL cone on (S^{k−1} × S^{m−k−1} × S^{1})/ℤ_{2}.

Further examples of SL cones are constructed by Harvey and Lawson [5, §III.3], Haskins [6], the author [10, 11], and others. Special Lagrangian cones in ℂ^{3} are a special case, which may be treated using the theory of integrable systems. In principle this should yield a classification of all SL cones on T^{2} in ℂ^{3}. For more information see McIntosh [25] or the author [15].

3.3 Special Lagrangian m-folds with conical singularities

Now we can define conical singularities of SL m-folds, following [17, Def. 3.6].

Definition 3.7. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold for m > 2, and define ψ : M → (0, ∞) as in (4). Suppose X is a compact singular SL m-fold in M with singularities at distinct points x_{1}, …, x_{n} ∈ X, and no other singularities.

Fix isomorphisms υ_{i} : ℂ^{m} → T_{x_{i}}M for i = 1, …, n such that υ_{i}^{∗}(ω) = ω′ and υ_{i}^{∗}(Ω) = ψ(x_{i})^{m}Ω′, where ω′, Ω′ are as in (2). Let C_{1}, …, C_{n} be SL cones in ℂ^{m} with isolated singularities at 0. For i = 1, …, n let Σ_{i} = C_{i} ∩ S^{2m−1}, and let μ_{i} ∈ (2, 3) with

(14) (2, μ_{i}] ∩ D_{Σ_{i}} = ∅, where D_{Σ_{i}} is defined in (7).

Then we say that X has a conical singularity or conical singular point at x_{i}, with rate μ_{i} and cone C_{i} for i = 1, …, n, if the following holds.

By Theorem 2.3 there exist embeddings Υ_{i} : B_{R} → M for i = 1, …, n satisfying Υ_{i}(0) = x_{i}, dΥ_{i}|_{0} = υ_{i} and Υ_{i}^{∗}(ω) = ω′, where B_{R} is the open ball of radius R about 0 in ℂ^{m} for some small R > 0. Define ι_{i} : Σ_{i} × (0, R) → B_{R} by ι_{i}(σ, r) = rσ for i = 1, …, n.

Define X′ = X \ {x_{1}, …, x_{n}}. Then there should exist a compact subset K ⊂ X′ such that X′ \ K is a union of open sets S_{1}, …, S_{n} with S_{i} ⊂ Υ_{i}(B_{R}), whose closures S_{1}, …, S_{n} are disjoint in X. For i = 1, …, n and some R′ ∈ (0, R] there should exist a smooth ϕ_{i} : Σ_{i}×(0, R′) → B_{R} such that Υ_{i}∘ϕ_{i} : Σ_{i}×(0, R′) → M is a diffeomorphism Σ_{i} × (0, R′) → S_{i}, and

|∇^{k}(ϕ_{i} − ι_{i})| = O(r^{μ_{i}−1−k}) as r → 0 for k = 0, 1. (15)

Here ∇ is the Levi-Civita connection of the cone metric ι_{i}^{∗}(g′) on Σ_{i} × (0, R′), | . | is computed using ι_{i}^{∗}(g′). If the cones C_{1}, …, C_{n} are stable in the sense of Definition 3.4, then we say that X has stable conical singularities.

We will see in Theorem 4.4 that if (15) holds for k = 0, 1 and some μ_{i} satisfying (14), then we can choose a natural ϕ_{i} for which (15) holds for all k ⩾ 0, and for all rates μ_{i} satisfying (14). Thus the number of derivatives required in (15) and the choice of μ_{i} both make little difference. We choose k = 0, 1 in (15), and some μ_{i} in (14), to make the definition as weak as possible.

We suppose m > 2 for two reasons. Firstly, the only SL cones in ℂ^{2} are finite unions of SL planes ℝ^{2} in ℂ^{2} intersecting only at 0. Thus any SL 2-fold with conical singularities is actually nonsingular as an immersed 2-fold, so there is really no point in studying them. Secondly, m = 2 is a special case in the analysis of [17, §2], and it is simpler to exclude it. Therefore we will suppose m > 2 throughout the paper.

Here are the reasons for the conditions on μ_{i} in Definition 3.7:

• We need μ_{i} > 2, or else (15) does not force X to approach C_{i} near x_{i}.

• The definition involves a choice of Υ_{i} : B_{R} → M. If we replace Υ_{i} by a different choice Υ̃_{i} then we should replace ϕ_{i} by ϕ̃_{i} = (Υ̃_{i}^{−1} ∘ Υ_{i}) ∘ ϕ_{i} near 0 in B_{R}. Calculation shows that as Υ_{i}, Υ̃_{i} agree up to second order, we have |∇^{k}(ϕ̃_{i} − ϕ_{i})| = O(r^{2−k}).

Therefore we choose μ_{i} < 3 so that these O(r^{2−k}) terms are absorbed into the O(r^{μ_{i}−1−k}) in (15). This makes the definition independent of the choice of Υ_{i}, which it would not be if μ_{i} > 3.

• Condition (14) is needed to prove the regularity result Theorem 4.4, and also to reduce to a minimum the obstructions to deforming compact SL m-folds with conical singularities studied in §5.

3.4 Homology and cohomology

Next we discuss homology and cohomology of SL m-folds with conical singularities, following [17, §2.4]. For a general reference, see for instance Bredon [2]. When Y is a manifold, write H^{k}(Y, ℝ) for the k^{th} de Rham cohomology group and H_{cs}^{k}(Y, ℝ) for the k^{th} compactly-supported de Rham cohomology group of Y. If Y is compact then H^{k}(Y, ℝ) = H_{cs}^{k}(Y, ℝ). The Betti numbers of Y are b^{k}(Y) = dim H^{k}(Y, ℝ) and b_{cs}^{k}(Y) = dim H_{cs}^{k}(Y, ℝ).

Let Y be a topological space, and Z ⊂ Y a subspace. Write H_{k}(Y, ℝ) for the k^{th} real singular homology group of Y, and H_{k}(Y; Z, ℝ) for the k^{th} real singular relative homology group of (Y; Z). When Y is a manifold and Z a submanifold we define H_{k}(Y, ℝ) and H_{k}(Y; Z, ℝ) using smooth simplices, as in [2, §V.5]. Then the pairing between (singular) homology and (de Rham) cohomology is defined at the chain level by integrating k-forms over k-simplices.

Suppose X is a compact SL m-fold in M with conical singularities x_{1}, …, x_{n} and cones C_{1}, …, C_{n}, and set X′ = X \ {x_{1}, …, x_{n}} and Σ_{i} = C_{i} ∩ S^{2m−1}, as in §3.3. Then X′ is the interior of a compact manifold X′ with boundary ∐_{i=1}^{n} Σ_{i}. Using this we show in [17, §2.4] that there is a natural long exact sequence

⋯→H_{cs}^{k}(X′, ℝ)→H^{k}(X′, ℝ)→⨁_{i=1}^{n}H^{k}(Σ_{i}, ℝ)→H_{cs}^{k+1}(X′, ℝ)→⋯, (16)

and natural isomorphisms

H_{k}(X; {x_{1}, …, x_{n}}, ℝ)^{∗} ≅ H_{cs}^{k}(X′, ℝ) ≅ H_{m−k}(X′, ℝ)
≅ H^{m−k}(X′, ℝ)^{∗} for all k ⩾ 0

(17)

(18) and H_{cs}^{k}(X′, ℝ) ≅ H_{k}(X, ℝ)^{∗} for all k > 1.

The inclusion ι : X → M induces homomorphisms ι_{∗} : H_{k}(X, ℝ) → H_{k}(M, ℝ).

4. The asymptotic behaviour of X near x_{i}

We now review the work of [17] on the regularity of SL m-folds with conical singularities. Let M be an almost Calabi-Yau m-fold and X an SL m-fold in M with conical singularities at x_{1}, …, x_{n}, with identifications υ_{i} and cones C_{i}. We study how quickly X converges to the cone υ(C_{i}) in T_{x_{i}}M near x_{i}.

We start in §4.1 by writing X in a special coordinate system near x_{i}, as the graph of an exact 1-form η_{i} = dA_{i} on C′_{i} = C_{i} \ {0}. The special Lagrangian condition reduces to a nonlinear elliptic p.d.e. on the function A_{i}. In §4.2 we explain how elliptic regularity of this p.d.e. implies that A_{i} and its derivatives decay quickly near x_{i}.

4.1 Lagrangian neighbourhood theorems

In [17, Th. 4.3] we extend the Lagrangian Neighbourhood Theorem, Theorem 2.4, to special Lagrangian cones.

Theorem 4.1. Let C be an SL cone in ℂ^{m} with isolated singularity at 0, and set Σ = C∩S^{2m−1}. Define ι : Σ×(0, ∞) → ℂ^{m} by ι(σ, r) = rσ, with image C \ {0}. For σ ∈ Σ, τ ∈ T_{σ}^{∗}Σ, r ∈ (0, ∞) and u ∈ ℝ, let (σ, r, τ, u) represent the point τ + u dr in T_{(σ,r)}^{∗}(Σ×(0, ∞)). Identify Σ × (0, ∞) with the zero section τ =u=0 in T^{∗}(Σ × (0, ∞)). Define an action of (0, ∞) on T^{∗}(Σ×(0, ∞)) by

t : (σ, r, τ, u) ⟼ (σ, tr, t^{2}τ, tu) for t ∈ (0, ∞), (19)

so t^{∗}(ω̂) = t^{2}ω̂ for ω̂ the canonical symplectic structure on T^{∗}(Σ × (0, ∞)).

Then there exists an open neighbourhood U_{C} of Σ×(0, ∞) in T^{∗}(Σ× (0, ∞)) invariant under (19) given by

U_{C} = {(σ, r, τ, u) ∈ T^{∗}(Σ × (0, ∞)) : |(τ, u)| < 2ζr} for some ζ > 0, (20)

where | . | is calculated using the cone metric ι^{∗}(g′) on Σ × (0, ∞), and an embedding Φ_{C} : U_{C} → ℂ^{m} with Φ_{C}|_{Σ×(0,∞)} = ι, Φ∗_{C}(ω′) = ω̂ and Φ_{C} ∘ t = t Φ_{C} for all t > 0, where t acts on U_{C} as in (19) and on ℂ^{m} by multiplication.

These U_{C}, Φ_{C} are a Lagrangian neighbourhood of C′ in ℂ^{m} which is equivariant under the action of dilations. Effectively they are a special coordinate system on ℂ^{m} near C′, in which ω′ assumes a simple form. In [17, Th. 4.4] we use U_{C_{i}}, Φ_{C_{i}} to construct a particular choice of ϕ_{i} in Definition 3.7.

Theorem 4.2. Let (M, J, ω, Ω), ψ, X, n, x_{i}, υ_{i}, C_{i}, Σ_{i}, μ_{i}, R, Υ_{i} and ι_{i} be as in Definition 3.7. Theorem 4.1 gives ζ > 0, neighbourhoods U_{C_{i}} of Σ_{i} × (0, ∞) in T^{∗}(Σ_{i} × (0, ∞)) and embeddings Φ_{C_{i}} : U_{C_{i}} → ℂ^{m} for i = 1, …, n.

Then for sufficiently small R′ ∈ (0, R] there exist unique closed 1forms η_{i} on Σ_{i} × (0, R′) for i = 1, …, n written η_{i}(σ, r) = η_{i}^{1}(σ, r) + η_{i}^{2}(σ, r)dr for η_{i}^{1}(σ, r) ∈ T_{σ}^{∗}Σ_{i} and η_{i}^{2}(σ, r) ∈ ℝ, and satisfying |η_{i}(σ, r)| < ζr and

|∇^{k}η_{i}| = O(r^{μ_{i}−1−k}) as r → 0 for k = 0, 1, (21)

computing ∇, | . | using the cone metric ι_{i}^{∗}(g′), such that the following holds.

Define ϕ_{i} :Σ_{i}×(0, R′) →B_{R} by ϕ_{i}(σ, r) =Φ_{C_{i}}(σ, r, η_{i}^{1}(σ, r), η_{i}^{2}(σ, r)). Then Υ_{i} ∘ ϕ_{i} : Σ_{i} × (0, R′) → M is a diffeomorphism Σ_{i} × (0, R′) → S_{i} for open sets S_{1}, …, S_{n} in X′ with S_{1}, …, S_{n} disjoint, and K = X′ \ (S_{1} ∪ ⋯ ∪ S_{n}) is compact. Also ϕ_{i} satisfies (15), so that R′, ϕ_{i}, S_{i}, K satisfy Definition 3.7.

We explained in §2.1 that in a Lagrangian neighbourhood U, Φ of a Lagrangian m-fold N gives a 1-1 correspondence between nearby Lagrangian m-folds Ñ and closed 1-forms on N. Theorem 4.2 uses this correspondence for the Lagrangian neighbourhoods U_{C_{i}}, Φ_{C_{i}} of Theorem 4.1. This is why the 1-forms η_{i} are closed. We can extend Theorem 2.4 to SL m-folds with conical singularities [17, Th. 4.6], in a way compatible with Theorems 4.1 and 4.2.

Theorem 4.3. Suppose (M, J, ω, Ω) is an almost Calabi-Yau mfold and X a compact SL m-fold in M with conical singularities at x_{1}, …, x_{n}. Let the notation ψ, υ_{i}, C_{i}, Σ_{i}, μ_{i}, R, Υ_{i} and ι_{i} be as in Definition 3.7, and let ζ, U_{C_{i}}, Φ_{C_{i}}, R′, η_{i}, η_{i}^{1}, η_{i}^{2}, ϕ_{i}, S_{i} and K be as in Theorem 4.2.

Then making R′ smaller if necessary, there exists an open tubular neighbourhood U_{X′} ⊂ T^{∗}X′ of the zero section X′ in T^{∗}X′, such that under d(Υ_{i} ∘ ϕ_{i}) : T^{∗}(Σ_{i} × (0, R′)) → T^{∗}X′ for i = 1, …, n we have

(d(Υ_{i} ∘ ϕ_{i}))^{∗}(U_{X′}) = {(σ, r, τ, u) ∈ T^{∗}(Σ_{i} × (0, R′)) : |(τ, u)| < ζr}, (22)

and there exists an embedding Φ_{X′} : U_{X′} → M with Φ_{X′}|_{X′} = id : X′ → X′ and Φ∗_{X′}(ω) = ω̂, where ω̂ is the canonical symplectic structure on T^{∗}X′, such that

Φ_{X′} ∘ d(Υ_{i} ∘ ϕ_{i})(σ, r, τ, u) ≡ Υ_{i} ∘ Φ_{C_{i}}(σ, r, τ + η_{i}^{1}(σ, r), u + η_{i}^{2}(σ, r)) (23)

for all i = 1, …, n and (σ, r, τ, u) ∈ T^{∗}(Σ_{i} × (0, R′)) with |(τ, u)| < ζr. Here |(τ, u)| is computed using the cone metric ι_{i}^{∗}(g′) on Σ_{i} × (0, R′).

This is an essential tool in the deformation theory of §5 and desingularization results of §7, as it gives a special coordinate system on M near X′ in which ω assumes a simple form. In these coordinate, deformations or desingularizations of X become graphs of closed 1-forms on X′ away from x_{i}, as in §2.1.

4.2 Regularity of X near x_{i}

The results of §4.1 used only the fact that X′ is Lagrangian in (M, ω). Our next theorems make essential use of the special Lagrangian condition. In [17, §5] we study the asymptotic behaviour of the maps ϕ_{i} of Theorem 4.2. Combining [17, Th. 5.1], [17, Lem. 4.5] and [17, Th. 5.5] proves:

Theorem 4.4. In the situation of Theorem 4.2 we have η_{i} = dA_{i} for i = 1, …, n, where A_{i} : Σ_{i} × (0, R′) → ℝ is given by A_{i}(σ, r) = ∫_{0}^{r} η_{i}^{2}(σ, s)ds. Suppose μ′_{i} ∈ (2, 3) with (2, μ′_{i}] ∩ D_{Σ_{i}} = ∅ for i = 1, …, n. Then

|∇^{k}(ϕ_{i} − ι_{i})| = O(r^{μ′_{i}−1−k}), |∇^{k}η_{i}| = O(r^{μ′_{i}−1−k}) and

|∇^{k}A_{i}| = O(r^{μ′_{i}−k}) as r → 0 for all k ⩾ 0 and i = 1, …, n.

(24)

Hence X has conical singularities at x_{i} with cone C_{i} and rate μ′_{i}, for all possible rates μ′_{i} allowed by Definition 3.7. Therefore, the definition of conical singularities is essentially independent of the choice of rate μ_{i}.

Theorem 4.4 in effect strengthens the definition of SL m-folds with conical singularities, Definition 3.7, as it shows that (15) actually implies the much stronger condition (24) on all derivatives.

The proof works by treating X′ near x_{i} as a deformation of the SL cone C_{i} in ℂ^{m}. Thus we can apply Proposition 2.13 with N replaced by Σ_{i} × (0, R′) and α = η_{i} = dA_{i}, and we find that A_{i} satisfies the second-order nonlinear p.d.e.

d^{∗}(ψ^{m}dA_{i})(σ, r) = Q(σ, r, dA_{i}(σ, r), ∇^{2}A_{i}(σ, r)) (25)

for (σ, r) ∈ Σ_{i} × (0, R′), where Q is a smooth nonlinear function.

When r is small the Q term in (25) is also small and (25) approximates Δ_{i}A_{i} = 0, where Δ_{i} is the Laplacian on the cone C_{i}. Therefore (25) is elliptic for small r. Using known results on the regularity of solutions of nonlinear second-order elliptic p.d.e.s, and a theory of analysis on weighted Sobolev spaces on manifolds with cylindrical ends developed by Lockhart and McOwen [22], we can then prove (24).

Our next result [17, Th. 6.8] is an application of geometric measure theory. For an introduction to the subject, see Morgan [27]. Geometric measure theory studies measure-theoretic generalizations of submanifolds called integral currents, which may be very singular, and is particularly powerful for minimal submanifolds. As from §2 SL m-folds are minimal submanifolds w.r.t. an appropriate metric, many major results of geometric measure theory immediately apply to special Lagrangian integral currents, a very general class of singular SL m-folds with strong compactness properties.

Theorem 4.5. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold and define ψ : M → (0, ∞) as in (4). Let x ∈ M and fix an isomorphism υ : ℂ^{m} → T_{x}M with υ^{∗}(ω) = ω′ and υ^{∗}(Ω) = ψ(x)^{m}Ω′, where ω′, Ω′ are as in (2).

Suppose that T is a special Lagrangian integral current in M with x ∈ T^{∘}, where T^{∘} = supp T \ supp ∂T, and that υ_{∗}(C) is a multiplicity 1 tangent cone to T at x, where C is a rigid special Lagrangian cone in ℂ^{m}, in the sense of Definition 3.4. Then T has a conical singularity at x, in the sense of Definition 3.7.

This is a weakening of Definition 3.7 for rigid cones C. Theorem 4.5 also holds for the larger class of Jacobi integrable SL cones C, [17, Def. 6.7].

Basically, Theorem 4.5 shows that if a singular SL m-fold T in M is locally modelled on a rigid SL cone C in only a very weak sense, then it necessarily satisfies Definition 3.7. One moral of Theorems 4.4 and 4.5 is that, at least for rigid SL cones C, more-or-less any sensible definition of SL m-folds with conical singularities is equivalent to Definition 3.7.

Theorem 4.5 is proved by applying regularity results of Allard and Almgren, and Adams and Simon, mildly adapted to the special Lagrangian situation, which roughly say that if a tangent cone C_{i} to X at x_{i} has an isolated singularity at 0, is multiplicity 1, and rigid, then X has a parametrization ϕ_{i} near x_{i} which satisfies (15) for some μ_{i} > 2. It then quickly follows that X has a conical singularity at x_{i}, in the sense of Definition 3.7.

As discussed in [17, §6.3], one can use other results from geometric measure theory to argue that for tangent cones C which are not Jacobi integrable, Definition 3.7 may be too strong, in that there could exist examples of singular SL m-folds with tangent cone C which are not covered by Definition 3.7, as the decay conditions (15) are too strict.

5. Moduli of SL m-folds with conical singularities

Next we review the work of [18] on deformation theory for compact SL m-folds with conical singularities. Following [18, Def. 5.4], we define the space M_{X} of compact SL m-folds X̂ in M with conical singularities deforming a fixed SL m-fold X with conical singularities.

Definition 5.1. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold and X a compact SL m-fold in M with conical singularities at x_{1}, …, x_{n} with identifications υ_{i} : ℂ^{m} → T_{x_{i}}M and cones C_{1}, …, C_{n}. Define the moduli space M_{X} of deformations of X to be the set of X̂ such that:

(i) X̂ is a compact SL m-fold in M with conical singularities at x̂_{1}, …, x̂_{n} with cones C_{1}, …, C_{n}, for some x̂_{i} and identifications υ̂_{i} : ℂ^{m} → T_{x̂_{i}}M.

(ii) There exists a homeomorphism ι̂ : X → X̂ with ι̂(x_{i}) = x̂_{i} for i = 1, …, n such that ι̂|_{X′} : X′ → X̂′ is a diffeomorphism and ι̂ and ι are isotopic as continuous maps X → M, where ι : X → M is the inclusion.

In [18, Def. 5.6] we define a topology on M_{X}, and explain why it is the natural choice. We will not repeat the complicated definition here; readers are referred to [18, §5] for details.

In [18, Th. 6.10] we describe M_{X} near X, in terms of a smooth map Φ between the infinitesimal deformation space I_{X′} and the obstruction space O_{X′}.

Theorem 5.2. Suppose (M, J, ω, Ω) is an almost Calabi-Yau m-fold and X a compact SL m-fold in M with conical singularities at x_{1}, …, x_{n} and cones C_{1}, …, C_{n}. Let M_{X} be the moduli space of deformations of X as an SL m-fold with conical singularities in M, as in Definition 5.1. Set X′ = X \ {x_{1}, …, x_{n}}.

Then there exist natural finite-dimensional vector spaces I_{X′}, O_{X′} with I_{X′} isomorphic to the image of H_{cs}^{1} (X′, ℝ) in H^{1} (X′, ℝ) and dim O_{X′} = ∑_{i=1}^{n} s-ind(C_{i}), where s-ind(C_{i}) is the stability index of Definition 3.4. There exists an open neighbourhood U of 0 in I_{X′}, a smooth map Φ : U → O_{X′} with Φ(0) = 0, and a map Ξ : {u ∈ U : Φ(u) = 0} → M_{X} with Ξ(0) = X which is a homeomorphism with an open neighbourhood of X in M_{X}.

Here is a sketch of the proof. For simplicity, consider first the subset of X̂ ∈ M_{X} which have the same singular points x_{1}, …, x_{n} and identifications υ_{1}, …, υ_{n} as X. If X̂ is C^{1} close to X in an appropriate sense then X̂′ = Φ_{X′}(Γ(α)), where U_{X′}, Φ_{X′} is the Lagrangian neighbourhood map of Theorem 4.3, and Γ(α) ⊂ U_{X′} is the graph of a small 1-form α on X′.

Since X̂′ is Lagrangian, α is closed, as in §2.1. Also, if ϕ_{i}, η_{i} and ϕ̂_{i}, η̂_{i} are as in Theorem 4.2 for X, X̂ then (Υ_{i} ∘ ϕ_{i})^{∗}(α) = η̂_{i} − η_{i} on Σ_{i} × (0, R′), so applying Theorem 4.4 to X, X̂ shows that if i = 1, …, n and μ′_{i} ∈ (2, 3) with (2, μ′_{i}] ∩ D_{Σ_{i}} = ∅ then

(26)  |∇^{k}α(x)| = O(d(x, x_{i})^{μ′_{i}−1−k}) near x_{i} for all k ⩾ 0.

As α is closed it has a cohomology class [α] ∈ H^{1}(X′, ℝ). Since (26) implies that α decays quickly near x_{i}, it turns out that α must be exact near x_{i}. Therefore [α] can be represented by a compactly-supported form on X′, and lies in the image of H_{cs}^{1}(X′, ℝ) in H^{1}(X′, ℝ).

Choose a vector space I_{X′} of compactly-supported 1-forms on X′ representing the image of H_{cs}^{1}(X′, ℝ) in H^{1}(X′, ℝ). Then we can write α = β + df, where β ∈ I_{X′} with [α] = [β] is unique, and f ∈ C^{∞}(X′) is unique up to addition of constants. As X̂′ is special Lagrangian we find that f satisfies a second-order nonlinear elliptic p.d.e. similar to (25):

d^{∗}(ψ^{m}(β + df))(x) = Q(x, (β + df)(x), (∇β + ∇^{2}f)(x)) (27)

for x ∈ X′. The linearization of (27) at β = f = 0 is d^{∗}(ψ^{m}(β + df)) = 0.

We study the family of small solutions β, f of (27) for which f has the decay near x_{i} required by (26). There is a ready-made theory of analysis on manifolds with cylindrical ends developed by Lockhart and McOwen [22], which is well-suited to this task. We work on certain weighted Sobolev spaces L_{k,μ}^{p}(X′) of functions on X′.

By results from [22] it turns out that the operator f ↦ d^{∗}(ψ^{m}df) is a Fredholm map L_{k,μ}^{p}(X′) → L_{k−2,μ−2}^{p}(X′), with cokernel of dimension ∑_{i=1}^{n} N_{Σ_{i}}(2). This cokernel is in effect the obstruction space to deforming X with x_{i}, υ_{i} fixed, as it is the obstruction space to solving the linearization of (27) in f at β = f = 0.

By varying x_{i}, υ_{i}, and allowing f to converge to different constants on the ends of X′ rather than zero, we overcome many of these obstructions. This reduces the dimension of the obstruction space O_{X′} from ∑_{i=1}^{n} N_{Σ_{i}}(2) to ∑_{i=1}^{n} s-ind(C_{i}). The obstruction map Φ is constructed using the Implicit Mapping Theorem for Banach spaces. This concludes our sketch.

If the C_{i} are stable then O_{X′} = {0} and we deduce [18, Cor. 6.11]:

Corollary 5.3. Suppose (M, J, ω, Ω) is an almost Calabi-Yau mfold and X a compact SL m-fold in M with stable conical singularities, and let M_{X} and I_{X′} be as in Theorem 5.2. Then M_{X} is a smooth manifold of dimension dim I_{X′}.

This has clear similarities with Theorem 2.10. Here is another simple condition for M_{X} to be a manifold near X, [18, Def. 6.12].

Definition 5.4. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold and X a compact SL m-fold in M with conical singularities, and let I_{X′}, O_{X′}, U and Φ be as in Theorem 5.2. We call X transverse if the linear map dΦ|_{0} : I_{X′} → O_{X′} is surjective.

If X is transverse then {u ∈ U : Φ(u) = 0} is a manifold near 0, so Theorem 5.2 yields [18, Cor. 6.13]:

Corollary 5.5. Suppose (M, J, ω, Ω) is an almost Calabi-Yau mfold and X a transverse compact SL m-fold in M with conical singularities, and let M_{X}, I_{X′} and O_{X′} be as in Theorem 5.2. Then M_{X} is near X a smooth manifold of dimension dim I_{X′} − dim O_{X′}.

In [18, §7] we extend all this to families of almost Calabi-Yau mfolds. Combining Definitions 2.15 and 5.1, we define moduli spaces in families:

Definition 5.6. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold and X a compact SL m-fold in M with conical singularities at x_{1}, ..., x_{n}. Suppose {(M, J^{s}, ω^{s}, Ω^{s}) : s ∈ F} is a smooth family of deformations of (M, J, ω, Ω). Define the moduli space M\begin{matrix}F \\ X \end{matrix} of deformations of X in the family F to be the set of pairs (s, X̂) such that:

(i) s ∈ F and X̂ is a compact SL m-fold in (M, J^{s}, ω^{s}, Ω^{s}) with conical singularities at x̂_{1}, …, x̂_{n} with cones C_{1}, …, C_{n}, for some x̂_{i}.

(ii) There exists a homeomorphism ι̂ : X → X̂ with ι̂(x_{i}) = x̂_{i} for i = 1, …, n such that ι̂|_{X′} : X′ → X̂′ is a diffeomorphism and ι̂ and ι are isotopic as continuous maps X → M, where ι : X → M is the inclusion.

Define a projection π^{F} : M\begin{matrix}F \\ X \end{matrix} → F by π^{F}(s, X̂) = s. In [18, Def. 7.5] we define a natural topology on M\begin{matrix}F \\ X \end{matrix}, for which π^{F} is continuous.

Here [18, Th. 7.9] is the families analogue of Theorem 5.2.

Theorem 5.7. Suppose (M, J, ω, Ω) is an almost Calabi-Yau mfold and X a compact SL m-fold in M with conical singularities at x_{1}, …, x_{n}. Let M_{X}, X′, I_{X′}, O_{X′}, U, Φ and Ξ be as in Theorem 5.2.

Suppose {(M, J^{s}, ω^{s}, Ω^{s}) : s ∈ F} is a smooth family of deformations of (M, J, ω, Ω), in the sense of Definition 2.14, such that ι_{∗}(γ) · [ω^{s}] = 0 for all γ ∈ H_{2}(X, ℝ) and s ∈ F, where ι : X → M is the inclusion, and [X] · [Im Ω^{s}] = 0 for all s ∈ F, where [X] ∈ H_{m}(M, ℝ) and [Im Ω^{s}] ∈ H^{m}(M, ℝ). Let M\begin{matrix}F \\ X \end{matrix} and π^{F} : M\begin{matrix}F \\ X \end{matrix} → F be as in Definition 5.6.

Then there exists an open neighbourhood U^{F} of (0, 0) in F × U, a smooth map Φ^{F} : U^{F} → O_{X′} with Φ^{F}(0, u) ≡ Φ(u), and a map Ξ^{F} :

\{(s, u) ∈ U^{F} : Φ^{F}(s, u) = 0\} → M\begin{matrix}F \\ X \end{matrix} with Ξ^{F}(0, u) ≡ (0, Ξ(u)) and π^{F} ∘

Ξ^{F}(s, u) ≡ s, which is a homeomorphism with an open neighbourhood of

(0, X) in M\begin{matrix}F \\ X \end{matrix}.

The conditions ι_{∗}(γ) · [ω^{s}] = 0 for all γ and [X] · [Im Ω^{s}] = 0 are necessary conditions for the existence of any SL m-fold X̂ with conical singularities isotopic to X in (M, J^{s}, ω^{s}, Ω^{s}). Here are the families analogues of Definition 5.4 and Corollaries 5.3 and 5.5, taken from [18, Def. 7.11 & Cor.s 7.10 & 7.12].

Corollary 5.8. In the situation of Theorem 5.7, suppose X has stable singularities. Then M\begin{matrix}F \\ X \end{matrix} is a smooth manifold of dimension d + dim I_{X′} and π^{F} : M\begin{matrix}F \\ X \end{matrix} → F a smooth submersion. For small s ∈ F the fibre (π^{F})^{−1}(s) is a nonempty smooth manifold of dimension dim I_{X′}, with (π^{F})^{−1}(0) = M_{X}.

Note the similarity of Corollary 5.8 and Theorem 2.16.

Definition 5.9. In the situation of Definition 5.7, we call X transverse in F if the linear map dΦ^{F}|_{(0,0)} : ℝ^{d} × I_{X′} → O_{X′} is surjective. If X is transverse in the sense of Definition 5.4 then it is also transverse in F.

Corollary 5.10. In the situation of Theorem 5.7, suppose X is transverse in F. Then M\begin{matrix}F \\ X \end{matrix} is near (0, X) a smooth manifold of dimension d + dim I_{X′} − dim O_{X′}, and π^{F} : M\begin{matrix}F \\ X \end{matrix} → F is a smooth map near (0, X).

Now there are a number of well-known moduli space problems in geometry where in general moduli spaces are obstructed and singular, but after a generic perturbation they become smooth manifolds. For instance, moduli spaces of instantons on 4-manifolds can be made smooth by choosing a generic metric, and similar things hold for Seiberg–Witten equations, and moduli spaces of pseudo-holomorphic curves in symplectic manifolds.

In [18, §9] we try (but do not quite succeed) to replicate this for moduli spaces of SL m-folds with conical singularities, by choosing a generic Kähler metric in a fixed Kähler class. Our first result is [18, Th. 9.1]:

Theorem 5.11. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold, X a compact SL m-fold in M with conical singularities, and I_{X′}, O_{X′} as in Theorem 5.2. Then there exists a smooth family of deformations {(M, J, ω^{s}, Ω) : s ∈ F} of (M, J, ω, Ω) with [ω^{s}] = [ω] ∈ H^{2}(M, ℝ) for all s ∈ F, such that X is transverse in F in the sense of Definition 5.9, and d=dim F =dim O_{X′}.

Combining this with Corollary 5.10 we see that M\begin{matrix}F \\ X \end{matrix} is a manifold near (0, X) and π^{F} a smooth map near (0, X). It then follows from Sard’s Theorem that for small generic s ∈ F, the moduli space (π^{F})^{−1}(s) of deformations of X in (M, J, ω^{s}, Ω) is a smooth manifold near X.

Thus, given a compact SL m-fold X in (M, J, ω, Ω) with conical singularities, we can perturb ω a little bit in its Kähler class to ω^{s}, and the moduli space Ms_{X} in (M, J, ω^{s}, Ω) will be a smooth manifold near X. More generally [18, Th. 9.3], if W ⊆ M_{X} is a compact subset then we can perturb ω to ω^{s} so Ms_{X} is a smooth manifold near W.

We would like to conclude that by choosing a sufficiently generic perturbation ω^{s} we can make Ms_{X} smooth everywhere. This is the idea of the following conjecture, [18, Conj. 9.5]:

Conjecture 5.12. Let (M, J, ω, Ω) be an almost Calabi-Yau mfold, X a compact SL m-fold in M with conical singularities, and I_{X′}, O_{X′} be as in Theorem 5.2. Then for a second category subset of Kähler forms ω̌ in the Kähler class of ω, the moduli space M̌_{X} of compact SL m-folds X̂ with conical singularities in (M, J, ω̌, Ω) isotopic to X is a manifold of dimension dim I_{X′} − dim O_{X′}.

If we could treat the moduli spaces M_{X} as compact, say if we had a good understanding of the compactification \overline{M}_{X} of M_{X} in §8, then this would follow from [18, Th. 9.3]. However, without knowing M_{X} is compact, the condition that M̌_{X} is smooth everywhere is in effect the intersection of an infinite number of genericity conditions on ω̌, and we do not know that this intersection is dense (or even nonempty) in the Kähler class.

Notice that Conjecture 5.12 constrains the topology and cones of SL m-folds X with conical singularities that can occur in a generic almost Calabi-Yau m-fold, as we must have dim I_{X′} ⩾ dim O_{X′}.

6. Asymptotically conical SL m-folds

We now discuss asymptotically conical SL m-folds L in ℂ^{m}, [17, Def. 7.1].

Definition 6.1. Let C be a closed SL cone in ℂ^{m} with isolated singularity at 0 for m > 2, and let Σ = C ∩ S^{2m−1}, so that Σ is a compact, nonsingular (m − 1)-manifold, not necessarily connected. Let g_{Σ} be the metric on Σ induced by the metric g′ on ℂ^{m} in (2), and r the radius function on ℂ^{m}. Define ι : Σ × (0, ∞) → ℂ^{m} by ι(σ, r) = rσ. Then the image of ι is C \{0}, and ι^{∗}(g′) = r^{2}g_{Σ} +dr^{2} is the cone metric on C \ {0}.

Let L be a closed, nonsingular SL m-fold in ℂ^{m}. We call L asymptotically conical (AC) with rate λ < 2 and cone C if there exists a compact subset K ⊂ L and a diffeomorphism φ : Σ × (T, ∞) → L \ K for some T > 0, such that

|∇^{k}(φ − ι)| = O(r^{λ−1−k}) as r → ∞ for k = 0, 1. (28)

Here ∇, | . | are computed using the cone metric ι^{∗}(g′).

This is very similar to Definition 3.7, and in fact there are strong parallels between the theories of SL m-folds with conical singularities and of asymptotically conical SL m-folds. We continue to assume m > 2 throughout.

In §6.1–§6.2 we review the results of [17, §7] on AC SL m-folds. Section 6.3 covers the deformation theory of AC SL m-folds in ℂ^{m} following Marshall [23] and Pacini [28], and §6.4 discusses examples of AC SL m-folds.

6.1 Cohomological invariants of AC SL m-folds

Let L be an AC SL m-fold in ℂ^{m} with cone C, and set Σ = C ∩ S^{2m−1}. Using the notation of §3.4, as in (16) there is a long exact sequence

⋯ → H_{cs}^{k}(L, ℝ) → H^{k}(L, ℝ) → H^{k}(Σ, ℝ) → H_{cs}^{k+1}(L, ℝ) → · · · . (29)

Following [17, Def. 7.2] we define cohomological invariants Y (L), Z(L) of L.

Definition 6.2. Let L be an AC SL m-fold in ℂ^{m} with cone C, and let Σ = C ∩ S^{2m−1}. As ω′, Im Ω′ in (2) are closed forms with ω′|_{L} ≡ Im Ω′|_{L} ≡ 0, they define classes in the relative de Rham cohomology groups H^{k}(ℂ^{m}; L, ℝ) for k = 2, m. But for k > 1 we have the exact sequence

0 = H^{k−1}(ℂ^{m}, ℝ) → H^{k−1}(L, ℝ)⟶^{≅}H^{k}(ℂ^{m}; L, ℝ) → H^{k}(ℂ^{m}, ℝ) = 0.

Let Y (L) ∈ H^{1}(Σ, ℝ) be the image of [ω′] in H^{2}(ℂ^{m}; L, ℝ) ≅ H^{1}(L, ℝ) under H^{1}(L, ℝ) → H^{1}(Σ, R) in (29), and Z(L) ∈ H^{m−1}(Σ, ℝ) be the image of [Im Ω′] in H^{m}(ℂ^{m}; L, ℝ) ≅ H^{m−1}(L, ℝ) under H^{m−1}(L, ℝ) → H^{m−1}(Σ, R) in (29).

Here are some conditions for Y (L) or Z(L) to be zero, [17, Prop. 7.3]:

Proposition 6.3. Let L be an AC SL m-fold in ℂ^{m} with cone C and rate λ, and let Σ = C ∩ S^{2m−1}. If λ < 0 or b^{1}(L) = 0 then Y (L) = 0. If λ < 2 − m or b^{0}(Σ) = 1 then Z(L) = 0.

6.2 Lagrangian neighbourhood theorems and regularity

Next we give versions of parts of §4.1–§4.2 for AC SL m-folds rather than SL m-folds with conical singularities. Here are the analogues of Theorems 4.2 and 4.3, proved in [17, Th.s 7.4 & 7.5].

Theorem 6.4. Let C be an SL cone in ℂ^{m} with isolated singularity at 0, and set Σ = C∩S^{2m−1}. Define ι : Σ×(0, ∞) → ℂ^{m} by ι(σ, r) = rσ. Let ζ, U_{C} ⊂ T^{∗}(Σ × (0, ∞)) and Φ_{C} : U_{C} → ℂ^{m} be as in Theorem 4.1.

Suppose L is an AC SL m-fold in ℂ^{m} with cone C and rate λ < 2. Then there exists a compact K ⊂ L and a diffeomorphism φ : Σ × (T, ∞) → L \ K for some T > 0 satisfying (28), and a closed 1-form χ on Σ × (T, ∞) written χ(σ, r) = χ^{1}(σ, r) + χ^{2}(σ, r)dr for χ^{1}(σ, r) ∈ T_{σ}^{∗}Σ and χ^{2}(σ, r) ∈ ℝ, satisfying

|χ(σ, r)| < ζr, φ(σ, r) ≡ Φ_{C}(σ, r, χ^{1}(σ, r), χ^{2}(σ, r))

and |∇^{k}χ| = O(r^{λ−1−k}) as r → ∞ for k = 0, 1, (30)

computing ∇, | . | using the cone metric ι^{∗}(g′).

Theorem 6.5. Suppose L is an AC SL m-fold in ℂ^{m} with cone C. Let Σ, ι, ζ, U_{C}, Φ_{C}, K, T, φ, χ, χ^{1}, χ^{2} be as in Theorem 6.4. Then making T, K larger if necessary, there exists an open tubular neighbourhood U_{L} ⊂ T^{∗}L of the zero section L in T^{∗}L, such that under dφ : T^{∗}(Σ × (T, ∞)) → T^{∗}L we have

(dφ)^{∗}(U_{L}) = {(σ, r, τ, u) ∈ T^{∗}(Σ × (T, ∞)) : |(τ, u)| < ζr}, (31)

and there exists an embedding Φ_{L} : U_{L} → ℂ^{m} with Φ_{L}|_{L} = id : L → L and Φ∗_{L}(ω′) = ω̂, where ω̂ is the canonical symplectic structure on T^{∗}L, such that

Φ_{L} ∘ dφ(σ, r, τ, u) ≡ Φ_{C}(σ, r, τ + χ^{1}(σ, r), u + χ^{2}(σ, r)) (32)

for all (σ, r, τ, u) ∈ T^{∗}(Σ × (T, ∞)) with |(τ, u)| < ζr, computing | . | using ι^{∗}(g′).

Combining [17, Prop. 7.6] and [17, Th.s 7.7 & 7.11] gives an analogue of Theorem 4.4, on the regularity of L near infinity in ℂ^{m}. As in [17, Th. 7.11], the theorem can be strengthened when 0 ⩽ λ < min(D_{Σ} ∩ (0, ∞)).

Theorem 6.6. In Theorem 6.4 we have [χ] = Y (L) in H^{1}(Σ × (T, ∞), ℝ) ≅ H^{1}(Σ, ℝ), where Y (L) is as in Definition 6.2. Let γ be the unique 1-form on Σ with dγ = d^{∗}γ = 0 and [γ] = Y (L) ∈ H^{1}(Σ, ℝ), which exists by Hodge theory. Then χ = π^{∗}(γ) + dE, where π : Σ × (T, ∞) → Σ is the projection and E ∈ C^{∞}(Σ × (T, ∞)).

If either λ = λ′, or λ, λ′ lie in the same connected component of ℝ \ D_{Σ}, then L is an AC SL m-fold with rate λ′ and

|∇^{k}(φ − ι)| = O(r^{λ′−1−k}),  |∇^{k}χ| = O(r^{λ′−1−k}), (33)
|∇^{k+1}E| = O(r^{λ′−1−k}) for all k ⩾ 0, and

|E| = \begin{cases}O(r^{λ′}), & λ′ ≠ 0, \\ O(| log r|), & λ′ = 0. \end{cases}

Here ∇, | . | are computed using the cone metric ι^{∗}(g′) on Σ × (T, ∞).

6.3 Moduli spaces of AC SL m-folds

The deformation theory of asymptotically conical SL m-folds in ℂ^{m} has been studied independently by Pacini [28] and Marshall [23]. Pacini’s results are earlier, but Marshall’s are more complete.

Definition 6.7. Suppose L is an asymptotically conical SL m-fold in ℂ^{m} with cone C and rate λ < 2, as in Definition 6.1. Define the moduli space Mλ_{L} of deformations of L with rate λ to be the set of AC SL m-folds L̂ in ℂ^{m} with cone C and rate λ, such that L̂ is diffeomorphic to L and isotopic to L as an asymptotically conical submanifold of ℂ^{m}. One can define a natural topology on Mλ_{L}, in a similar way to the conical singularities case of [18, Def. 5.6].

Note that if L is an AC SL m-fold with rate λ, then it is also an AC SL m-fold with rate λ′ for any λ′ ∈ [λ, 2). Thus we have defined a 1-parameter family of moduli spaces Mλ′_{L} for L, and not just one. Since we did not impose any condition on λ in Definition 6.1 analogous to (14) in the conical singularities case, it turns out that Mλ_{L} depends nontrivially on λ.

The following result can be deduced from Marshall [23, Th. 6.2.15] and [23, Table 5.1]. (See also Pacini [28, Th. 2 & Th. 3].) It implies conjectures by the author in [9, Conj. 2.12] and [16, §10.2].

Theorem 6.8. Let L be an asymptotically conical SL m-fold in ℂ^{m} with cone C and rate λ < 2, and let Mλ_{L} be as in Definition 6.7. Set Σ = C ∩ S^{2m−1}, and let D_{Σ}, N_{Σ} be as in §3.1 and b^{k}(L), b_{cs}^{k}(L) as in §3.4. Then:

(a) If λ ∈ (0, 2) \ D_{Σ} then Mλ_{L} is a manifold with

dim Mλ_{L} = b^{1}(L) − b^{0}(L) + N_{Σ}(λ). (34)

Note that if 0 < λ < min(D_{Σ} ∩ (0, ∞)) then N_{Σ}(λ) = b^{0}(Σ).

(b) If λ ∈ (2−m, 0) then Mλ_{L} is a manifold of dimension b_{cs}^{1}(L) = b^{m−1}(L).

This is the analogue of Theorems 2.10 and 5.2 for AC SL m-folds. If λ ∈ (2−m, 2)\D_{Σ} then the deformation theory for L with rate λ is unobstructed and Mλ_{L} is a smooth manifold with a given dimension. This is similar to the case of nonsingular compact SL m-folds in Theorem 2.10, but different to the conical singularities case in Theorem 5.2.

6.4 Examples

Examples of AC SL m-folds L are constructed by Harvey and Lawson [5, §III.3], the author [10, 11, 12, 14], and others. Nearly all the known examples (up to translations) have minimum rate λ either 0 or 2 − m, which are topologically significant values by Proposition 6.3. For instance, all examples in [11] have λ = 0, and [10, Th. 6.4] constructs AC SL m-folds with λ = 2 − m in ℂ^{m} from any SL cone C in ℂ^{m}. The only explicit, nontrivial examples known to the author with λ ≠ 0, 2−m are in [12, Th. 11.6], and have λ = \frac{3}{2}.

We shall give three families of examples of AC SL m-folds L in ℂ^{m} explicitly. The first family is adapted from Harvey and Lawson [5, §III.3.A].

Example 6.9. Let Cm_{HL} be the SL cone in ℂ^{m} of Example 3.5. We shall define a family of AC SL m-folds in ℂ^{m} with cone Cm_{HL}. Let a_{1}, …, a_{m} ⩾ 0 with exactly two of the a_{j} zero and the rest positive. Write a = (a_{1}, …, a_{m}), and define

(35) La_{HL} = {(z_{1}, …, z_{m}) ∈ ℂ^{m} : i^{m+1}z_{1} ⋯ z_{m} ∈ [0, ∞),

|z_{1}|^{2} − a_{1} = ⋯ = |z_{m}|^{2} − a_{m}}.

Then La_{HL} is an AC SL m-fold in ℂ^{m} diffeomorphic to T^{m−2} × ℝ^{2}, with cone Cm_{HL} and rate 0. It is invariant under the U(1)^{m−1} group (10). It is surprising that equations of the form (35) should define a nonsingular submanifold of ℂ^{m} without boundary, but in fact they do.

Now suppose for simplicity that a_{1}, …, a_{m−2} > 0 and a_{m−1} = a_{m} = 0. As Σm_{HL} ≅ T^{m−1} we have H^{1}(Σm_{HL}, ℝ) ≅ ℝ^{m−1}, and calculation shows that Y (La_{HL}) = (πa_{1}, …, πa_{m−2}, 0) ∈ ℝ^{m−1} in the natural coordinates. Since La_{HL} ≅ T^{m−2}×ℝ^{2} we have H^{1}(La_{HL}, ℝ) = ℝ^{m−2}, and Y (La_{HL}) lies in the image ℝ^{m−2} ⊂ ℝ^{m−1} of H^{1}(La_{HL}, ℝ) in H^{1}(Σm_{HL}, ℝ), as in Definition 6.2. As b^{0}(Σm_{HL}) = 1, Proposition 6.3 shows that Z(La_{HL}) = 0.

Take C = Cm_{HL}, Σ = Σm_{HL} and L = La_{HL} in Theorem 6.8, and let 0 < λ < min(D_{Σ} ∩ (0, ∞)). Then b^{1}(L) = m − 2, b^{0}(L) = 1 and N_{Σ}(λ) = b^{0}(Σ) = 1, so part (a) of Theorem 6.8 shows that dim Mλ_{L} = m − 2. This is consistent with the fact that L depends on m − 2 real parameters a_{1}, …, a_{m−2} > 0.

The family of all La_{HL} has \frac{1}{2}m(m−1) connected components, indexed by which two of a_{1}, …, a_{m} are zero. Using the theory of §7, these can give many topologically distinct ways to desingularize SL m-folds with conical singularities with these cones.

Our second family, from [10, Ex. 9.4], was chosen for its simplicity.

Example 6.10. Let m, a_{1}, …, a_{m}, k and L_{0}^{a_{1},...,a_{m}} be as in Example 3.6. For 0 ≠ c ∈ ℝ define

(36) L_{c}^{a_{1},...,a_{m}} = {(ie^{ia_{1}θ}x_{1}, e^{ia_{2}θ}x_{2}, …, e^{ia_{m}θ}x_{m}) : θ ∈ [0, 2π),

x_{1}, …, x_{m} ∈ ℝ, a_{1}x_{1}^{2} + ⋯ + a_{m}x_{m}^{2} = c}.

Then L_{c}^{a_{1},...,a_{m}} is an AC SL m-fold in ℂ^{m} with rate 0 and cone L_{0}^{a_{1},...,a_{m}}. It is diffeomorphic as an immersed SL m-fold to (S^{k−1} ×ℝ^{m−k} ×S^{1})/ℤ_{2} if c > 0, and to (ℝ^{k} × S^{m−k−1} × S^{1})/ℤ_{2} if c < 0.

Our third family was first found by Lawlor [21], made more explicit by Harvey [4, p. 139–140], and discussed from a different point of view by the author in [11, §5.4(b)]. Our treatment is based on that of Harvey.

Example 6.11. Let m > 2 and a_{1}, …, a_{m} > 0, and define polynomials p, P by

p(x) = (1 + a_{1}x^{2}) ⋯ (1 + a_{m}x^{2}) − 1 and P (x) = \frac{p(x)}{x^{2}}.

Define real numbers ϕ_{1}, …, ϕ_{m} and A by

ϕ_{k} = a_{k} ∫_{−∞}^{∞} \frac{dx}{(1 + a_{k}x^{2})\sqrt{P (x)}} and A = ω_{m}(a_{1} ⋯ a_{m})^{−1/2}, (37)

where ω_{m} is the volume of the unit sphere in ℝ^{m}. Clearly ϕ_{k}, A > 0. But writing ϕ_{1} + ⋯ + ϕ_{m} as one integral gives

ϕ_{1} + ⋯ + ϕ_{m} = ∫_{0}^{∞} \frac{p′(x)dx}{(p(x) + 1)\sqrt{p(x)}} = 2 ∫_{0}^{∞} \frac{dw}{w^{2} + 1} = π,

making the substitution w = \sqrt{p(x)}. So ϕ_{k} ∈ (0, π) and ϕ_{1} +⋯+ϕ_{m} = π. This yields a 1-1 correspondence between m-tuples (a_{1}, …, a_{m}) with a_{k} > 0, and (m+1)-tuples (ϕ_{1}, …, ϕ_{m}, A) with ϕ_{k} ∈ (0, π), ϕ_{1} + ⋯ + ϕ_{m} = π and A > 0.

For k = 1, …, m and y ∈ ℝ, define a function z_{k} : ℝ → ℂ by

z_{k}(y) = e^{iψ_{k}(y)}\sqrt{a_{k}^{−1} + y^{2}},  where
ψ_{k}(y) = a_{k} ∫_{−∞}^{y} \frac{dx}{(1 + a_{k}x^{2})\sqrt{P (x)}} .

Now write ϕ = (ϕ_{1}, …, ϕ_{n}), and define a submanifold L^{ϕ,A} in ℂ^{m} by

L^{ϕ,A} = {(z_{1}(y)x_{1}, …, z_{m}(y)x_{m}) : y ∈ ℝ, x_{k} ∈ ℝ, x_{1}^{2} + ⋯ + x_{m}^{2} = 1}. (38)

Then L^{ϕ,A} is closed, embedded, and diffeomorphic to S^{m−1} × ℝ, and Harvey [4, Th. 7.78] shows that L^{ϕ,A} is special Lagrangian. One can also show that L^{ϕ,A} is asymptotically conical, with rate 2 − m and cone the union Π^{0} ∪ Π^{ϕ} of two special Lagrangian m-planes Π^{0}, Π^{ϕ} in ℂ^{m} given by

Π^{0} ={(x_{1}, …, x_{m}) : x_{j} ∈ ℝ}, Π^{ϕ}={(e^{iϕ_{1}}x_{1}, …, e^{iϕ_{m}}x_{m}) : x_{j} ∈ ℝ}. (39)

As λ = 2 − m < 0 we have Y (L^{ϕ,A}) = 0 by Proposition 6.3. Now L^{ϕ,A} ≅ S^{m−1} × ℝ so that H^{m−1}(L^{ϕ,A}, ℝ) ≅ ℝ, and Σ = (Π^{0} ∪ Π^{ϕ}) ∩ S^{2m−1} is the disjoint union of two unit (m−1)-spheres S^{m−1}, so H^{m−1}(Σ, ℝ) ≅ ℝ^{2}. The image of H^{m−1}(L^{ϕ,A}, ℝ) in H^{m−1}(Σ, ℝ) is {(x, −x) : x ∈ ℝ} in the natural coordinates. Calculation shows that Z(L^{ϕ,A}) = (A, −A) ∈ H^{m−1}(Σ, ℝ), which is why we defined A this way in (37).

Apply Theorem 6.8 with L = L^{ϕ,A} and λ ∈ (2 − m, 0). As L ≅ S^{m−1} × ℝ we have b_{cs}^{1}(L) = 1, so part (b) of Theorem 6.8 shows that dim Mλ_{L} = 1. This is consistent with the fact that when ϕ is fixed, L^{ϕ,A} depends on one real parameter A > 0. Here ϕ is fixed in Mλ_{L} as the cone C = Π^{0} ∪ Π^{ϕ} of L depends on ϕ, and all L̂ ∈ Mλ_{L} have the same cone C, by definition.

7. Desingularizing singular SL m-folds

We now discuss the work of [19, 20] on desingularizing compact SL m-folds with conical singularities. Here is the basic idea. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold, and X a compact SL mfold in M with conical singularities x_{1}, …, x_{n} and cones C_{1}, …, C_{n}. Suppose L_{1}, …, L_{n} are AC SL m-folds in ℂ^{m} with the same cones C_{1}, …, C_{n} as X.

If t > 0 then tL_{i} = {t x : x ∈ L_{i}} is also an AC SL m-fold with cone C_{i}. We construct a 1-parameter family of compact, nonsingular Lagrangian m-folds N^{t} in (M, ω) for t ∈ (0, δ) by gluing tL_{i} into X at x_{i}, using a partition of unity.

When t is small, N^{t} is close to special Lagrangian (its phase is nearly constant), but also close to singular (it has large curvature and small injectivity radius). We prove using analysis that for small t ∈ (0, δ) we can deform N^{t} to a special Lagrangian m-fold Ñ^{t} in M, using a small Hamiltonian deformation. The proof involves a delicate balancing act, showing that the advantage of being close to special Lagrangian outweighs the disadvantage of being nearly singular.

Doing this in full generality is rather complex. There are two kinds of obstructions to the existence of Ñ^{t}. Firstly, if Y (L_{i}) ≠ 0 then N^{t} may not exist as a Lagrangian m-fold. Secondly, if X′ is not connected then we may not be able to deform N^{t} to a special Lagrangian m-fold Ñ^{t} because of problems with small eigenvalues of the Laplacian Δ on N^{t}. In each case, Ñ^{t} exists for small t if the Y (L_{i}) or Z(L_{i}) satisfy an equation.

We also extend the results to desingularization in families of almost Calabi-Yau m-folds (M, J^{s}, ω^{s}, Ω^{s}). The cohomology classes [ω^{s}], [Im Ω^{s}] contribute to the obstruction equations in Y (L_{i}) and Z(L_{i}) for the existence of Ñ^{t}. Thus, a singular SL m-fold X which has no desingularizations Ñ^{t} in (M, J, ω, Ω) can still admit desingularizations Ñ^{s,t} in (M, J^{s}, ω^{s}, Ω^{s}) for small s ≠ 0.

We begin in §7.1 by explaining desingularization in the simplest case, in one almost Calabi-Yau m-fold (M, J, ω, Ω) when Y (L_{i}) = 0 and X′ is connected. Section 7.2 extends this to X′ not connected, and §7.3 to Y (L_{i}) ≠ 0, introducing the two kinds of obstructions. Section 7.4 discusses desingularization in families (M, J^{s}, ω^{s}, Ω^{s}).

7.1 Desingularization in the simplest case

Our simplest desingularization result is [19, Th. 6.13].

Theorem 7.1. Suppose (M, J, ω, Ω) is an almost Calabi-Yau m-fold and X a compact SL m-fold in M with conical singularities at x_{1}, …, x_{n} and cones C_{1}, …, C_{n}. Let L_{1}, …, L_{n} be asymptotically conical SL mfolds in ℂ^{m} with cones C_{1}, …, C_{n} and rates λ_{1}, …, λ_{n}. Suppose λ_{i} < 0 for i = 1, …, n, and X′ = X \ {x_{1}, …, x_{n}} is connected.

Then there exists ϵ > 0 and a smooth family {Ñ^{t} : t ∈ (0, ϵ]} of compact, nonsingular SL m-folds in (M, J, ω, Ω), such that Ñ^{t} is constructed by gluing tL_{i} into X at x_{i} for i = 1, …, n. In the sense of currents, Ñ^{t} →X as t→0.

Here is a sketch of the proof, divided into seven steps.

Step 1. Apply Theorem 4.1 to C_{i} for i = 1, …, n, and Theorem 4.3 to X, and Theorem 6.5 to L_{i} for i = 1, …, n. This gives Lagrangian neighbourhoods U_{C_{i}}, Φ_{C_{i}} for C_{i}, and U_{X′}, Φ_{X′} for X′, and U_{L_{i}}, Φ_{L_{i}} for L_{i}. Moreover U_{X′}, Φ_{X′} and U_{C_{i}}, Φ_{C_{i}} are related via Υ_{i} and an exact 1-form η_{i} on Σ_{i} × (0, R′) from Theorem 4.2, and U_{L_{i}}, Φ_{L_{i}} and U_{C_{i}}, Φ_{C_{i}} are related via an exact 1-form χ_{i} on Σ_{i} × (T, ∞) from Theorem 6.4.

Step 2. Let t > 0 be small. We define a nonsingular Lagrangian m-fold N^{t} in (M, ω), roughly as follows. Choose τ ∈ (0, 1) satisfying certain conditions. At distance at least 2t^{τ} from x_{1}, …, x_{n} we define N^{t} to be X′. At distance up to t^{τ} from x_{i} we define N^{t} to be Υ_{i}(tL_{i} ∩ B_{R}). Between distances t^{τ} and 2t^{τ} from x_{i} we define N^{t} to be a Lagrangian annulus Σ_{i} × (t^{τ} , 2t^{τ}) interpolating between X′ and Υ_{i}(tL_{i} ∩ B_{R}), using the Lagrangian neighbourhoods U_{C_{i}}, Φ_{C_{i}}, U_{X′}, Φ_{X′} and U_{L_{i}}, Φ_{L_{i}}. This is equivalent to choosing a closed 1form ξ_{i}^{t}(σ, r) on Σ_{i} × [t^{τ} , 2t^{τ}] interpolating between t^{2}χ_{i}(σ, t^{−1}r) at r = t^{τ} and η_{i}(σ, r) at r = 2t^{τ}.

Step 3. Let e^{iθ^{t}} be the phase function of N^{t}, so that N^{t} is special Lagrangian if sin θ^{t} ≡ 0. We bound various norms of ψ^{m} sin θ^{t} in terms of powers of t. These bounds imply that N^{t} is close to special Lagrangian when t is small. We also estimate other geometrical quantities, like the curvature and injectivity radius of N^{t}, in terms of powers of t.

Step 4. We glue together the Lagrangian neighbourhoods U_{C_{i}}, Φ_{C_{i}}, U_{X′}, Φ_{X′} and U_{L_{i}}, Φ_{L_{i}} to define a Lagrangian neighbourhood U_{N^{t}}, Φ_{N^{t}} for N^{t}.

Step 5. Let f ∈ C^{∞}(N^{t}). Then df is a 1-form on N^{t}, and the graph Γ(df) is a submanifold of T^{∗}N^{t}. If f is small in C^{1} then Γ(df) ⊂ U_{N^{t}} ⊂ T^{∗}N^{t}, and then Ñ^{t} = Φ_{N^{t}}(Γ(df)) is a nonsingular Lagrangian m-fold in (M, ω). Every small Hamiltonian deformation of N^{t} can be written in this way. We show that Ñ^{t} is special Lagrangian if and only if

(40) d^{∗}(ψ^{m} cos θ^{t}df)(x) = ψ^{m} sin θ^{t} + Q^{t}(x, df(x), ∇^{2}f(x))

for all x ∈ N^{t}, as in (25), (27), where Q^{t} is smooth and Q^{t}(x, y, z) = O(t^{−2}|y|^{2} + |z|^{2}) for small y, z.

Step 6. Working in the Sobolev space L_{3}^{2m}(N^{t}), we show that the operator

P^{t} : {u∈L_{3}^{2m}(N^{t}):∫_{N^{t}} u dV^{t} =0}→{v ∈L_{1}^{2m}(N^{t}):∫_{N^{t}} v dV^{t} =0} given by P^{t}(u) = d^{∗}(ψ^{m} cos θ^{t}du)

(41)

has an inverse (P^{t})^{−1} which is (in a rather weak sense) bounded independently of t. The restriction to u with ∫_{N^{t}} u dV^{t} = 0 is necessary as P^{t}(1) = 0, so P^{t} is not invertible on spaces including 1.

Step 7. We inductively construct a sequence (f_{k})_{k=0}^{∞} in L_{3}^{2m}(N^{t}) with f_{0} = 0, ∫_{N^{t}} f_{k} dV^{t} = 0 and f_{k} = (P^{t})^{−1}(ψ^{m} sin θ^{t} +Q^{t}(x, df_{k−1}, ∇^{2}df_{k−1})), so

d^{∗}(ψ^{m} cos θ^{t}df_{k}) ≡ ψ^{m} sin θ^{t} + Q^{t}(x, df_{k−1}(x), ∇^{2}f_{k−1}(x)). (42)

Using the bounds on ψ^{m} sin θ^{t} from Step 3 and on (P^{t})^{−1} from Step 6 we show that (f_{k})_{k=0}^{∞} exists and converges in L_{3}^{2m}(N^{t}) for small t. The limit f satisfies (40), and is smooth by elliptic regularity. Then Ñ^{t} = Φ_{N^{t}}(Γ(df)) is the SL m-fold we seek.

The condition λ_{i} < 0 in Theorem 6.3 is there for two reasons. Firstly, it forces Y (L_{i}) = 0 by Proposition 6.3, and therefore χ_{i} is an exact 1form on Σ_{i} × (T, ∞), since [χ_{i}] = Y (L_{i}) ∈ H^{1}(Σ_{i}, ℝ) by Theorem 6.6. This exactness makes it possible to define the closed 1-form ξ_{i}^{t} in Step 2.

Secondly, we need λ_{i} < 0 so that the contributions to ψ^{m} sin θ^{t} from tapering χ_{i} off to zero on the annulus Σ_{i} × [t^{τ} , 2t^{τ}] are small enough for the method to work. If λ_{i} > 0 then ∥ψ^{m} sin θ^{t}∥_{L^{2}} is too large, and we cannot prove that the sequence (f_{k})_{k=0}^{∞} in Step 7 converges.

7.2 Desingularization when X′ is not connected

In [19, Th. 7.10] we extend Theorem 7.1 to X′ not connected.

Theorem 7.2. Suppose (M, J, ω, Ω) is an almost Calabi-Yau mfold and X a compact SL m-fold in M with conical singularities at x_{1}, …, x_{n} and cones C_{1}, …, C_{n}. Define ψ : M → (0, ∞) as in (4). Let L_{1}, …, L_{n} be asymptotically conical SL m-folds in ℂ^{m} with cones C_{1}, …, C_{n} and rates λ_{1}, …, λ_{n}. Suppose λ_{i} < 0 for i = 1, …, n. Write X′ = X \ {x_{1}, …, x_{n}} and Σ_{i} = C_{i} ∩ S^{2m−1}.

Set q = b^{0}(X′), and let X′_{1}, …, X′_{q} be the connected components of X′. For i = 1, …, n let l_{i} = b^{0}(Σ_{i}), and let Σ_{i}^{1}, …, Σ_{i}^{l_{i}} be the connected components of Σ_{i}. Define k(i, j) = 1, …, q by Υ_{i} ∘ φ_{i}(Σ_{i}^{j} × (0, R′)) ⊂ X′_{k(i,j)} for i = 1, …, n and j = 1, …, l_{i}. Suppose that

∑_{\begin{matrix}1⩽i⩽n, 1⩽j⩽l_{i}: \\ k(i,j)=k \end{matrix}} ψ(x_{i})^{m}Z(L_{i}) · [Σ_{i}^{j} ] = 0 for all k = 1, …, q. (43)

Suppose also that the compact m-manifold N obtained by gluing L_{i} into X′ at x_{i} for i = 1, …, n is connected. A sufficient condition for this to hold is that X and L_{i} for i = 1, …, n are connected.

Then there exists ϵ > 0 and a smooth family {Ñ^{t} : t ∈ (0, ϵ]} of compact, nonsingular SL m-folds in (M, J, ω, Ω) diffeomorphic to N, such that Ñ^{t} is constructed by gluing tL_{i} into X at x_{i} for i = 1, …, n. In the sense of currents in geometric measure theory, Ñ^{t} → X as t → 0.

The new issues when X′ is not connected occur in Steps 6 and 7 of §7.1. Suppose b^{0}(X′) = q > 1, so that X′ has q connected components X′_{1}, …, X′_{q}. Then the operator P^{t} of (41) turns out to have q − 1 small positive eigenvalues λ_{1}, …, λ_{q−1} of size O(t^{m−2}). The corresponding eigenfunctions w_{1}, …, w_{q−1} are approximately constant on the parts of N^{t} coming from each X′_{k}, and change rapidly on the ‘small necks’ in between.

As (P^{t})^{−1}w_{k} = λ_{k}^{−1}w_{k} and λ_{k} = O(t^{m−2}) we see that (P^{t})^{−1} is O(t^{2−m}) on ⟨w_{1}, …, w_{q−1}⟩, and so cannot be bounded independently of t. To repair the proof, roughly speaking we set W^{t} = ⟨1, w_{1}, …, w_{q−1}⟩, and let (W^{t})^{⊥} be the orthogonal subspace to W^{t} in L^{2}(N^{t}). Then P^{t} maps

P^{t} : L_{3}^{2m}(N^{t}) ∩ (W^{t})^{⊥} → L_{1}^{2m}(N^{t}) ∩ (W^{t})^{⊥} (44)

and has an inverse (P^{t})^{−1} bounded independently of t on these spaces, in a weak sense. (Actually, we do something more complicated than this, in which W^{t} is an approximation to ⟨1, w_{1}, …, w_{q−1}⟩ defined explicitly in terms of bounded harmonic functions on L_{1}, …, L_{n}.)

In Step 7, the sequence (f_{k})_{k=0}^{∞} is constructed as before. The bound on the inverse of (44) can be used to inductively bound the components of f_{k} in (W^{t})^{⊥}. But we still need to bound the components π_{W^{t}}(f_{k}) of f_{k} in W^{t}. Since f_{0} = 0 and Q(x, 0, 0) = 0, Equation (42) gives P^{t}f_{1} = ψ^{m} sin θ^{t}, so that f_{1} = (P^{t})^{−1}(ψ^{m} sin θ^{t}). It turns out that we need π_{W^{t}}(f_{1}) = o(t^{2}) for f_{k} to remain small as k → ∞.

As (P^{t})^{−1} = O(t^{2−m}) on ⟨w_{1}, …, w_{q−1}⟩, this holds if π_{W^{t}}(ψ^{m} sin θ^{t}) = o(t^{m}). Calculation shows that the dominant term in π_{W^{t}}(ψ^{m} sin θ^{t}) is O(t^{m}), and proportional to the left-hand side of (43). Therefore π_{W^{t}}(ψ^{m} sin θ^{t}) = o(t^{m}) if and only if (43) holds, and this is the condition for the sequence (f_{k})_{k=0}^{∞} to remain bounded and converge to a small solution of (40).

If X′ is connected, so that q = 1, then k(i, j) ≡ 1 and (43) becomes

∑_{i=1}^{n} ψ(x_{i})^{m}Z(L_{i}) · ∑_{j=1}^{l_{i}}[Σ_{i}^{j} ] = 0.

But ∑_{j=1}^{l_{i}}[Σ_{i}^{j} ] = [Σ_{i}], and Z(L_{i}) · [Σ_{i}] = 0 as Z(L_{i}) is the image of a class in H^{m−1}(L_{i}, ℝ), and Σ_{i} is the boundary of L_{i}. Therefore (43) holds automatically when X′ is connected, and Theorem 7.2 reduces to Theorem 7.1 in this case.

7.3 Desingularization when Y (L_{i}) ≠ 0

In [20, Th. 6.13] we extend Theorem 7.1 to λ_{i} ⩽ 0, allowing Y (L_{i}) ≠ 0.

Theorem 7.3. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold for 2 < m < 6, and X a compact SL m-fold in M with conical singularities at x_{1}, …, x_{n} and cones C_{1}, …, C_{n}. Let L_{1}, …, L_{n} be asymptotically conical SL m-folds in ℂ^{m} with cones C_{1}, …, C_{n} and rates λ_{1}, …, λ_{n}. Suppose that λ_{i} ⩽ 0 for i = 1, …, n, that X′ = X \ {x_{1}, …, x_{n}} is connected, and that there exists ϱ ∈ H^{1}(X′, ℝ) such that (Y (L_{1}), …, Y (L_{n})) is the image of ϱ under the map H^{1}(X′, ℝ) → ⨁_{i=1}^{n} H^{1}(Σ_{i}, ℝ) in (16), where Σ_{i} = C_{i} ∩ S^{2m−1}.

Then there exists ϵ > 0 and a smooth family {Ñ^{t} : t ∈ (0, ϵ]} of compact, nonsingular SL m-folds in (M, J, ω, Ω), with Ñ^{t} constructed by gluing tL_{i} into X at x_{i} for i = 1, …, n. In the sense of currents, Ñ^{t} →X as t→0.

There is also [20, Th. 6.12] an analogue of Theorem 7.2, combining the modifications of Theorems 7.2 and 7.3, but for brevity we will not give it.

The new issues when Y (L_{i}) ≠ 0 come mostly in Step 2 of §7.1. As [χ_{i}] = Y (L_{i}) ∈ H^{1}(Σ_{i}, ℝ) by Theorem 6.6, if Y (L_{i}) ≠ 0 then χ_{i} is no longer an exact form. Therefore, in Step 2 we cannot choose a closed 1-form ξ_{i}^{t} on Σ_{i} × [t^{τ} , 2t^{τ}] interpolating between t^{2}χ_{i} at r = t^{τ} and η_{i} at r = 2t^{τ}, since t^{2}χ_{i} and η_{i} have different cohomology classes.

Thus we cannot choose N^{t} to coincide with X away from x_{i}, and work locally near x_{i}, as we did in §7.1. Instead, we define N^{t} away from x_{i} to be Φ_{X′}(Γ(t^{2}α)), where α is a 1-form on X′ satisfying dα = d^{∗}(ψ^{m}α) = 0, and |α| = O(r^{−1}) near x_{i}. We show using analysis on manifolds with ends that there is a unique such 1-form α with [α] = ϱ for each ϱ ∈ H^{1}(X′, ℝ).

To glue Υ_{i}(tL_{i} ∩ B_{R}) and Φ_{X′}(Γ(t^{2}α)) together as Lagrangian mfolds the cohomology classes of t^{2}χ_{i} and t^{2}α must agree in H^{1}(Σ_{i}, ℝ). This holds if the image of ϱ under the map H^{1}(X′, ℝ) → ⨁_{i=1}^{n} H^{1}(Σ_{i}, ℝ) is (Y (L_{1}), …, Y (L_{n})), as in the theorem. Thus, the existence of ϱ with this property is a necessary condition for the existence of N^{t} as a Lagrangian m-fold in (M, ω).

After constructing N^{t} we need to estimate norms of ψ^{m} sin θ^{t} in Step 3. The condition d^{∗}(ψ^{m}α) = 0 means that linear terms in t^{2}α contribute 0 to ψ^{m} sin θ^{t}. However, quadratic terms in t^{2}α contribute O(t^{4}) to ψ^{m} sin θ^{t} on most of N^{t}, so all norms of ψ^{m} sin θ^{t} are at least O(t^{4}).

Now to show that (f_{k})_{k=0}^{∞} converges in Step 7 we need ∥ψ^{m} sin θ^{t}∥_{L^{2}} = o(t^{1+m/2}) for small t. As ∥ψ^{m} sin θ^{t}∥_{L^{2}} has O(t^{4}) contributions, this is possible only if m < 6. Therefore we have to restrict to complex dimension m < 6 when Y (L_{i}) ≠ 0 for this method of proof to work.

7.4 Desingularization in families (M, J^{s}, ω^{s}, Ω^{s})

Next we explain the work of [20, §7–§8] on desingularization in families of almost Calabi-Yau m-folds (M, J^{s}, ω^{s}, Ω^{s}). The analogue of Theorem 7.1 is [20, Th. 7.15], but for brevity we will not give it. Here [20, Th. 7.14] is the families analogue of Theorem 7.2.

Theorem 7.4. Suppose (M, J, ω, Ω) is an almost Calabi-Yau mfold and X a compact SL m-fold in M with conical singularities at x_{1}, …, x_{n} and cones C_{1}, …, C_{n}. Define ψ : M → (0, ∞) as in (4). Let L_{1}, …, L_{n} be asymptotically conical SL m-folds in ℂ^{m} with cones C_{1}, …, C_{n} and rates λ_{1}, …, λ_{n}. Suppose λ_{i} < 0 for i = 1, …, n. Write X′ = X \ {x_{1}, …, x_{n}} and Σ_{i} = C_{i} ∩ S^{2m−1}.

Set q = b^{0}(X′), and let X′_{1}, …, X′_{q} be the connected components of X′. For i = 1, …, n let l_{i} = b^{0}(Σ_{i}), and let Σ_{i}^{1}, …, Σ_{i}^{l_{i}} be the connected components of Σ_{i}. Define k(i, j) = 1, …, q by Υ_{i} ∘ φ_{i}(Σ_{i}^{j} × (0, R′)) ⊂ X′_{k(i,j)} for i = 1, …, n and j = 1, …, l_{i}. Suppose the compact mmanifold N obtained by gluing L_{i} into X′ at x_{i} for i = 1, …, n is connected. A sufficient condition for this to hold is that X and L_{i} for i = 1, …, n are connected.

Suppose {(M, J^{s}, ω^{s}, Ω^{s}) : s ∈ F} is a smooth family of deformations of (M, J, ω, Ω), with base space F ⊂ ℝ^{d}. Let ι_{∗} : H_{2}(X, ℝ) → H_{2}(M, ℝ) be the natural inclusion. Suppose that

(45) [ω^{s}] · ι_{∗}(γ) = 0 for all s ∈ F and γ ∈ H_{2}(X, ℝ).

Define G ⊆ F × (0, 1) to be the subset of (s, t) ∈ F × (0, 1) with

[Im Ω^{s}] · [\overline{X′_{k}}] = t^{m} ∑_{\begin{matrix}1⩽i⩽n, 1⩽j⩽l_{i}: \\ k(i,j)=k \end{matrix}} ψ(x_{i})^{m}Z(L_{i}) · [Σ_{i}^{j} ] for k = 1, …, q. (46)

Then there exist ϵ ∈ (0, 1) and κ > 1 and a smooth family

{Ñ^{s,t} : (s, t) ∈ G, t ∈ (0, ϵ], |s| ⩽ t^{κ+m/2}}, (47)

such that Ñ^{s,t} is a compact, nonsingular SL m-fold in (M, J^{s}, ω^{s}, Ω^{s}) diffeomorphic to N, which is constructed by gluing tL_{i} into X at x_{i} for i = 1, …, n. In the sense of currents in geometric measure theory, Ñ^{s,t} → X as s, t → 0.

To prove it we modify Steps 1–7 of §7.1 in the following ways. In Step 1 we generalize Theorem 4.3 to give smooth families of maps Υ_{i}^{s} : B_{R} → M and Φs_{X′} : U_{X′} → M for small s ∈ F with (Υ_{i}^{s})^{∗}(ω^{s}) = ω′, (Φs_{X′})^{∗}(ω^{s}) = ω̂ and Υ_{i}^{0} = Υ_{i}, Φ0_{X′} = Φ_{X′}. Using these, in Step 2 we define a smooth family of Lagrangian m-folds N^{s,t} in (M, ω^{s}) for small s ∈ F and t ∈ (0, δ). In the rest of the proof we make everything depend on s ∈ F, and deform N^{s,t} to an SL m-fold Ñ^{s,t} in (M, J^{s}, ω^{s}, Ω^{s}) for small s ∈ F and t ∈ (0, δ).

To allow X′ not connected, as in §7.2, we introduce a vector subspace W^{s,t} ⊂ C^{∞}(N^{s,t}), and we need π_{W^{s,t}}(ψ^{m} sin θ^{s,t}) = o(t^{m}). The dominant terms in π_{W^{s,t}}(ψ^{m} sin θ^{s,t}) are of two kinds: O(t^{m}) terms involving the Z(L_{i}), as in §7.2, and also terms in [Im Ω^{s}] · [\overline{X′_{k}}]. Equation (46) requires these two terms to cancel, so that π_{W^{s,t}}(ψ^{m} sin θ^{s,t}) = o(t^{m}), and the rest of the proof works.

Here [20, Th. 8.10] is the families analogue of Theorem 7.3.

Theorem 7.5. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold for 2 < m < 6, and X a compact SL m-fold in M with conical singularities at x_{1}, …, x_{n} and cones C_{1}, …, C_{n}. Let L_{1}, …, L_{n} be asymptotically conical SL m-folds in ℂ^{m} with cones C_{1}, …, C_{n} and rates λ_{1}, …, λ_{n}. Suppose λ_{i} ⩽ 0 for i = 1, …, n, and X′ = X \{x_{1}, …, x_{n}} is connected.

Suppose {(M, J^{s}, ω^{s}, Ω^{s}) : s ∈ F} is a smooth family of deformations of (M, J, ω, Ω), with base space F ⊂ ℝ^{d}, satisfying

(48) [Im Ω^{s}] · [X] = 0 for all s ∈ F, where [X] ∈ H_{m}(M, ℝ).

Define ϖ ∈ H_{cs}^{2}(X′, ℝ) to be the image of (Y (L_{1}), …, Y (L_{n})) under the map ⨁_{i=1}^{n} H^{1}(Σ_{i}, ℝ) → H_{cs}^{2}(X′, ℝ) in (16). Define G ⊆ F × (0, 1) to be

G = {(s, t) ∈ F × (0, 1) : [ω^{s}] · ι_{∗}(γ) = t^{2}ϖ · γ for all γ ∈ H_{2}(X, ℝ)}, (49)

where ι_{∗} : H_{2}(X, ℝ) → H_{2}(M, ℝ) is the natural inclusion.

Then there exist ϵ ∈ (0, 1), κ > 1 and ϑ ∈ (0, 2) and a smooth family

{Ñ^{s,t} : (s, t) ∈ G, t ∈ (0, ϵ], |s| ⩽ t^{ϑ}}, (50)

such that Ñ^{s,t} is a compact, nonsingular SL m-fold in (M, J^{s}, ω^{s}, Ω^{s}), which is constructed by gluing tL_{i} into X at x_{i} for i = 1, …, n. In the sense of currents in geometric measure theory, Ñ^{s,t} → X as s, t → 0.

In §7.3 we saw that when Y (L_{i}) ≠ 0 there is a topological obstruction to defining N^{t} as a Lagrangian m-fold in (M, ω), so that N^{t} exists only if the Y (L_{i}) satisfy an equation. In this case there is also an obstruction to defining N^{s,t} as a Lagrangian m-fold in (M, ω^{s}), but now the condition in (49) for N^{s,t} to exist involves both Y (L_{i}), which determine ϖ, and [ω^{s}].

Here is how to understand the relation between the conditions for N^{t} to exist in Theorem 7.3, and for N^{s,t} to exist in Theorem 7.5. As (16) is exact, (Y (L_{1}), …, Y (L_{n})) is the image of ϱ ∈ H^{1}(X′, ℝ) if and only if the image ϖ of (Y (L_{1}), …, Y (L_{n})) in H_{cs}^{2}(X′, ℝ) is zero.

Now ω^{0} = ω and [ω] · ι_{∗}(γ) = 0 for all γ ∈ H_{2}(X, ℝ), as X′ is Lagrangian in (M, ω). Thus when s = 0, Equation (49) reduces to t^{2}ϖ · γ = 0 for all γ. But H_{cs}^{2}(X′, ℝ) ≅ H_{2}(X, ℝ)^{∗} by (18). Thus when s = 0 Equation (49) is equivalent to ϖ = 0, which is equivalent to the existence of ϱ in Theorem 7.3.

Note that as the conditions (46) and (49) for the existence of Ñ^{s,t} involve both s and t, it can happen that an SL m-fold X with conical singularities admits no desingularizations Ñ^{t} in (M, J, ω, Ω), but does admit desingularizations Ñ^{s,t} in (M, J^{s}, ω^{s}, Ω^{s}) for small s ≠ 0. Thus we can overcome obstructions to the existence of desingularizations by varying the underlying almost Calabi-Yau m-fold (M, J, ω, Ω).

We also prove a theorem [20, Th. 8.9] combining Theorems 7.4 and 7.5, desingularizing in families when Y (L_{i}) ≠ 0 and X′ is not connected. However, for technical reasons it is not as strong as the author would like, in that we must assume both sides of (46) are zero rather than just that (46) holds.

8. Discussion: How moduli spaces fit together

We now consider the boundary ∂M_{N} of a moduli space M_{N} of SL m-folds.

Definition 8.1. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold, N a compact, nonsingular SL m-fold in M, and M_{N} the moduli space of deformations of N in M. Then M_{N} is a smooth manifold of dimension b^{1}(N), by Theorem 2.10. In general M_{N} will be a noncompact manifold, but we can construct a natural compactification \overline{M}_{N} as follows.

Regard M_{N} as a moduli space of special Lagrangian integral currents in the sense of geometric measure theory, as discussed in [17, §6]. An introduction to geometric measure theory can be found in Morgan [27]. Let \overline{M}_{N} be the closure of M_{N} in the space of integral currents. As elements of M_{N} have uniformly bounded volume, \overline{M}_{N} is compact by [27, 5.5].

Define the boundary ∂M_{N} to be \overline{M}_{N} \ M_{N}. Then elements of ∂M_{N} are singular special Lagrangian integral currents. Essentially, they are singular SL m-folds X in M which are limits of nonsingular N̂ ∈ M_{N} in an appropriate sense. By a result of Almgren [1], the singular set of each X ∈ ∂M_{N} has Hausdorff dimension at most m − 2.

In good cases, say if (M, J, ω, Ω) is suitably generic, it seems reasonable that ∂M_{N} should be divided into a number of strata, each of which is a moduli space of singular SL m-folds with singularities of a particular type, and is itself a manifold with singularities. In particular, some or all of these strata could be moduli spaces M_{X} of SL m-folds with isolated conical singularities, as in §5.

In this case, using §7 for each X̂ ∈ M_{X} we can try to construct desingularizations Ñ^{t} in M_{N} by gluing in AC SL m-folds L̂_{i} at the singular points x̂_{i} of X̂ for i = 1, …, n. In good cases, say when the cones C_{i} of X̂ are stable, every N̂ ∈ M_{N} close to M_{X} might be constructed uniquely from some X̂, L̂_{1}, …, L̂_{n}, and so we could identify an open neighbourhood of M_{X} in M_{N} with a submanifold of the prod-

uct M_{X} × M0_{L_{1}} × ⋯ × M0_{Ln}.

The goal of this section is to work towards such a description of M_{N} near a boundary stratum M_{X} which is a moduli space of SL m-folds with conical singularities. Our treatment will be informal or conjectural in places, and is far from giving a complete picture of ∂M_{N}.

8.1 Topological calculations and dimension counting

We shall consider the following situation.

Definition 8.2. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold for 2 < m < 6. Here the assumption m < 6 is only so that we can apply Theorem 7.3 and [20, Th. 6.12], and all of the topological calculations below actually hold when m > 2. Define ψ : M → (0, ∞) as in (4).

Let X be a compact SL m-fold in M with conical singularities x_{1}, …, x_{n} and cones C_{1}, …, C_{n}. Write Σ_{i} = C_{i} ∩ S^{2m−1} and X′ = X \ {x_{1}, …, x_{n}}. Let M_{X} be the moduli space of deformations of X in M, as in Definition 5.1. Write X̂ for a general element of M_{X}. Let I_{X′} be the image of H_{cs}^{1}(X′, ℝ) in H^{1}(X′, ℝ), as in Theorem 5.2.

Let L_{1}, …, L_{n} be asymptotically conical SL m-folds in ℂ^{m} with cones C_{1}, …, C_{n} and rate 0. Let M0_{L_{i}} be the moduli space of deformations of L_{i} with rate 0, as in Definition 6.7. Write L̂_{i} for a general element of M0_{L_{i}}.

Let q = b^{0}(X′) and X′_{1}, …, X′_{q} be the connected components of X′. For i = 1, …, n let l_{i} = b^{0}(Σ_{i}), and let Σ_{i}^{1}, …, Σ_{i}^{l_{i}} be the connected components of Σ_{i}. Define k(i, j) = 1, …, q by Υ_{i} ∘ φ_{i}(Σ_{i}^{j} × (0, R′)) ⊂ X′_{k(i,j)}, as usual.

Define Y_{i} ⊂ H^{1}(Σ_{i}, ℝ) and Z_{i} ⊂ H^{m−1}(Σ_{i}, ℝ) to be the images of the map H^{k}(L_{i}, ℝ) → H^{k}(Σ_{i}, ℝ) of (29) for k = 1, m − 1. Define

maps π_{Y_{i}} : M0_{L_{i}} → Y_{i} and π_{Z_{i}} : M0_{L_{i}} → Z_{i} by π_{Y_{i}}(L̂_{i}) = Y (L̂_{i}) and

π_{Z_{i}}(L̂_{i}) = Z(L̂_{i}). These are well-defined as Y (L_{i}), Z(L_{i}) are images of classes in H^{k}(L_{i}, ℝ) by Definition 6.2. Write general elements of Y_{i} as γ_{i}, and of Z_{i} as δ_{i}.

Let the vector subspace Y in Y_{1} × ⋯ × Y_{n} be the intersection of Y_{1} × ⋯ × Y_{n} with the image of the map H^{1}(X′, ℝ) → ⨁_{i=1}^{n} H^{1}(Σ_{i}, ℝ) in (16). Let the vector subspace Z in Z_{1} × ⋯ × Z_{n} be the set of all (δ_{1}, …, δ_{n}) for which

∑_{\begin{matrix}1⩽i⩽n, 1⩽j⩽l_{i}: \\ k(i,j)=k \end{matrix}} ψ(x_{i})^{m}δ_{i} · [Σ_{i}^{j} ] = 0 for all k = 1, …, q. (51)

Suppose (Y (L_{1}), …, Y (L_{n})) ∈ Y. This is equivalent to the existence of ϱ in Theorem 7.3. Suppose (Z(L_{1}), …, Z(L_{n})) ∈ Z. This is equivalent to Equation (43) of Theorem 7.2. Let N be the compact m-manifold obtained by gluing L_{i} into X′ at x_{i} for i = 1, …, n, as in Theorem 7.2. Suppose N is connected.

Let Ñ^{t} for t ∈ (0, ϵ] be the desingularizations of X constructed in Theorem 7.2 when Y (L_{i}) = 0 (as then L_{i} is actually AC with rate λ_{i} < 0 by [17, Th. 7.11(b)]), and in Theorem 7.3 when X′ is connected, and in [20, Th. 6.12] in the general case. Then each Ñ^{t} is a compact SL m-fold in M diffeomorphic to N. Let M_{N} be the moduli space of deformations of Ñ^{t}, which is independent of t. Then M_{N} is a smooth manifold of dimension b^{1}(N), by Theorem 2.10.

The next four results compute the dimensions of various spaces.

Lemma 8.3. In the situation above we have dim I_{X′} = b_{cs}^{1}(X′) + q − ∑_{i=1}^{n} l_{i}.

Proof. From (16) we see that I_{X′} fits into an exact sequence

0 → H^{0}(X′, ℝ) → ⨁_{i=1}^{n} H^{0}(Σ_{i}, ℝ) → H_{cs}^{1}(X′, ℝ) → I_{X′} → 0.

The lemma follows by alternating sum of dimensions. q.e.d.

Proposition 8.4. In the situation above, M0_{L_{i}} is a smooth manifold with

dim Y_{i} = b^{1}(L_{i}) − b^{0}(L_{i}) + l_{i} − b_{cs}^{1}(L_{i}), dim Z_{i} = l_{i} − b^{0}(L_{i})

and dim M0_{L_{i}} = b^{1}(L_{i}) − b^{0}(L_{i}) + l_{i}. (52)

Also the projection π_{Y_{i}} × π_{Z_{i}} : M0_{L_{i}} → Y_{i} × Z_{i} is a smooth submersion.

Proof. From (29) we see that Y_{i} fits into an exact sequence

0 → H^{0}(L_{i}, ℝ) → H^{0}(Σ_{i}, ℝ) → H_{cs}^{1}(L_{i}, ℝ) → H^{1}(L_{i}, ℝ) → Y_{i} → 0.

Taking alternating sums of dimensions gives dim Y_{i} in (52). Similarly, Z_{i} fits into an exact sequence

0 → Z_{i} → H^{m−1}(Σ_{i}, ℝ) → H_{cs}^{m}(L_{i}, ℝ) → H^{m}(L_{i}, ℝ) → 0.

But Poincaré duality gives b^{m−1}(Σ_{i}) = b^{0}(Σ_{i}) = l_{i}, b_{cs}^{m}(L_{i}) = b^{0}(L_{i}) and b^{m}(L_{i}) = b_{cs}^{0}(L_{i}) = 0, so we deduce dim Z_{i} in (52).

Suppose 0 < λ_{i} < min (D_{Σ_{i}} ∩ (0, ∞)). Then [17, Th. 7.11(b)] shows that any AC SL m-fold L̂_{i} with cone C_{i} and rate λ_{i} is also asymptotically conical with rate 0. Hence Mλ_{i}_{L_{i}} = M0_{L_{i}}, in the notation of Definition 6.7. Part (a) of Theorem 6.8 then shows that M0_{L_{i}} is smooth with dimension given in (52).

We can deduce that π_{Y_{i}} ×π_{Z_{i}} is a smooth submersion from the proof of Theorem 6.8 in Marshall [23, §6]. Smoothness holds for fairly general reasons. To show that π_{Y_{i}} × π_{Z_{i}} is a submersion we need to verify that the natural projection T_{L_{i}}M0_{L_{i}} → Y_{i} × Z_{i} is surjective, and this follows from the determination of T_{L_{i}}M0_{L_{i}} in [23, §5.2]. q.e.d.

Proposition 8.5. dim Z = 1−q+∑_{i=1}^{n} dim Z_{i} = 1−q+∑_{i=1}^{n} l_{i} − ∑_{i=1}^{n} b^{0}(L_{i}).

Proof. Let δ_{i} ∈ Z_{i} for i = 1, …, n. Then δ_{i} · [Σ_{i}] = 0, since δ_{i} is the image of a class in H^{1}(L_{i}) and Σ_{i} is a boundary in L_{i}. As [Σ_{i}] = ∑_{j=1}^{l_{i}}[Σ_{i}^{j}], summing the left hand side of (51) over k = 1, …, q yields ∑_{i=1}^{n} ψ(x_{i})^{m}δ_{i} · [Σ_{i}], which is zero. Thus, for any (δ_{1}, …, δ_{n}) ∈ Z_{1} × ⋯ × Z_{n}, the sum of (51) over k = 1, …, q holds automatically. That is, the q Equations (51) on δ_{1}, …, δ_{n} are dependent, and represent at most q − 1 independent restrictions on δ_{1}, …, δ_{n}.

We claim that (51) is exactly q − 1 independent restrictions on δ_{1}, …, δ_{n}. Then dim Z = 1 − q + ∑_{i=1}^{n} dim Z_{i}, and the proposition follows from (52). To see this, note that X′ has q connected components X′_{1}, …, X′_{q}, which are joined into one connected N by gluing in L_{1}, …, L_{n}. Define a link to be a triple (X′_{j}, X′_{k}, L_{i}^{l}), where 1 ⩽ j < k ⩽ q and L_{i}^{l} is a connected component of some L_{i} which is glued into both X′_{j} and X′_{k} at x_{i}.

Then we can choose a minimal set of q−1 links which join X′_{1}, ..., X′_{q} into one component. It is not difficult to show that from (X′_{j}, X′_{k}, L_{i}^{l}) we can construct δ_{i} ∈ Z_{i} as the image of a class in H^{m−1}(L_{i}^{l}, ℝ), such that the q − 1 classes δ_{i} obtained from the minimal set of q − 1 links give linearly independent left-hand sides of (51), thought of as vectors in ℝ^{q}. Hence (51) is at least q −1 independent restrictions on δ_{1}, …, δ_{n}, and the proof is complete. q.e.d.

Proposition 8.6. b^{1}(N) = dim Y + 1 + b_{cs}^{1}(X′) + ∑_{i=1}^{n} b_{cs}^{1}(L_{i}) − ∑_{i=1}^{n} l_{i}.

Proof. Regard X′ as the interior of a compact manifold X′ with boundary ∐_{i=1}^{m} Σ_{i}, and L_{i} as the interior of a compact manifold L_{i} with boundary Σ_{i}. Then N is constructed by gluing X′ and L_{1}, …, L_{n} together along Σ_{1}, …, Σ_{n}.

Thus the disjoint union ∐_{i=1}^{n} Σ_{i} is a subset of N, with N \ ∐_{i=1}^{n} Σ_{i} diffeomorphic to the disjoint union of X′ and L_{1}, …, L_{n}. The pair (N; ∐_{i=1}^{n} Σ_{i}) gives an exact sequence in cohomology:

\begin{aligned}⋯ & → H^{k−1}(N, ℝ) → ⨁_{i=1}^{n} H^{k−1}(Σ_{i}, ℝ) → H_{cs}^{k}(X′, ℝ) ⊕ ⨁_{i=1}^{n} H_{cs}^{k}(L_{i}, ℝ) \\ & → H^{k}(N, ℝ) → ⋯, \end{aligned} (53)

since H^{k}(N; ∐_{i=1}^{n} Σ_{i}, ℝ) ≅ H_{cs}^{k}(X′, ℝ) ⊕ ⨁_{i=1}^{n} H_{cs}^{k}(L_{i}, ℝ) by excision.

Now from the definitions of Y_{i}, Y and exactness of (16) and (29) we find that the kernel of ⨁_{i=1}^{n} H^{1}(Σ_{i}, ℝ) → H_{cs}^{2}(X′, ℝ) ⊕ ⨁_{i=1}^{n} H_{cs}^{2}(L_{i}, ℝ) in (53) is Y. Thus as b_{cs}^{0}(X′) = b_{cs}^{0}(L_{i}) = 0 we have an exact sequence

0 → H^{0}(N, ℝ) → ⨁_{i=1}^{n} H^{0}(Σ_{i}, ℝ) → H_{cs}^{1}(X′, ℝ) ⊕ ⨁_{i=1}^{n} H_{cs}^{1}(L_{i}, ℝ)
→ H^{1}(N, ℝ) → Y → 0.

The result follows by alternating sums of dimensions, as b^{0}(N) = 1. q.e.d.

8.2 Describing the moduli space M_{N} near its boundary

We continue to use the notation of §8.1. From Lemma 8.3 and Propositions 8.4–8.6 we deduce the following theorem. Smoothness of F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} and the first line of (55) follow as π_{Y_{i}} × π_{Z_{i}} is a smooth submersion by Proposition 8.4, and the rest of (55) from the dimension formulae above.

Theorem 8.7. In the situation of Definition 8.2, define a family F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} of n-tuples of AC SL m-folds by

\begin{matrix}F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} = \{ & (L̂_{1}, …, L̂_{n}) ∈ M0_{L_{1}} × ⋯ × M0_{Ln} : \\ (Y (L̂_{1}), …, Y (L̂_{n})) ∈ Y, (Z(L̂_{1}), …, Z(L̂_{n})) ∈ Z\}. \end{matrix} (54)

Then F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} is a smooth manifold with

dim F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} = dim Y + dim Z + ∑_{i=1}^{n}(dim M0_{L_{i}} − dim Y_{i} − dim Z_{i})
= dim Y + 1 − q + ∑_{i=1}^{n} b_{cs}^{1}(L_{i})
= b^{1}(N) − dim I_{X′}.

(55)

The significance of the theorem is that F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} is the family of ntuples of AC SL m-folds L̂_{1}, …, L̂_{n} which can be used to desingularize X using the results of §7. Now M_{N} is smooth with dim M_{N} = b^{1}(N) by Theorem 2.10. If the cones C_{i} are stable then Corollary 5.3 shows that M_{X} is smooth with dim M_{X} = dim I_{X′}. So we see from Theorem 8.7 that:

Corollary 8.8. Suppose the SL cones C_{1}, …, C_{n} are stable, in the sense of Definition 3.4. Then the moduli spaces M_{X}, M_{N} and F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} are smooth manifolds with dim M_{X} + dim F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} = dim M_{N}.

We claim that in the stable singularities case, M_{N} is roughly speaking locally diffeomorphic to M_{X} × F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} near M_{X} ⊂ ∂M_{N}. That is, each N̂ in this region of M_{N} can be constructed from some unique

X̂ ∈ M_{X} and (L̂_{1}, …, L̂_{n}) ∈ F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} by gluing L̂_{i} into X̂ at x̂_{i}. This

is the reason for the formula dim M_{X} + dim F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} = dim M_{N}. To explain why, we make the following definition:

Definition 8.9. Suppose (M, J, ω, Ω) is Calabi-Yau, so that ψ ≡ 1. Choose X̂ ∈ M_{X} with singular points x̂_{1}, …, x̂_{n} and (L̂_{1}, …, L̂_{n}) ∈ F\begin{matrix}X \\ L_{1},...,Ln \end{matrix}. Then the definition of F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} implies that X̂, L̂_{i} satisfy the hypotheses of Theorem 7.2 if X(L̂_{i}) = 0, or Theorem 7.3 if q = 1, or [20, Th. 6.12] in the general case. Thus, these theorems give ϵ > 0 such that for t ∈ (0, ϵ] there exists a compact SL m-fold Ñ^{t} in M constructed by gluing L̂_{i} into X̂ at x̂_{i}.

Observe that (tL̂_{1}, …, tL̂_{n}) ∈ F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} for t > 0. Define a subset U

in M_{X} × F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} and Ψ : U → M_{N} by (X̂, (tL̂_{1}, …, tL̂_{n})) ∈ U if t ∈

(0, ϵ], where ϵ > 0 depends on X̂, L̂_{i} as above, and then Ψ(X̂, (tL̂_{1}, …, tL̂_{n})) = Ñ^{t}.

To make Ψ well-defined we have to ensure that Ñ^{t} is independent of choices made in its construction. Actually the choice of ϱ ∈ H^{1}(X′, ℝ) in Theorem 7.3 does affect Ñ^{t}. Now ϱ is unique up to addition of the kernel of H^{1}(X′, ℝ) → ⨁_{i=1}^{n} H^{1}(Σ_{i}, ℝ) in (16). Choose a vector subspace of H^{1}(X′, ℝ) transverse to this kernel, and restrict ϱ to lie in this subspace.

This gives a way to choose ϱ uniquely. Once this is done the N^{t} are independent of choices up to a small Hamiltonian isotopy, and Ñ^{t} is the unique SL m-fold in this Hamiltonian isotopy class close to N^{t}, so Ñ^{t} is independent of the remaining choices.

Here is why we assumed M is Calabi-Yau above. If M is only almost Calabi-Yau, then ψ need not be constant. But Z depends on ψ(x_{i}) by (51). Thus, if we vary X to X̂ ∈ M_{X} then we should define F X̂ ; L_{1},...,Ln using Ẑ defined with ψ(x̂_{i}) in (51) instead of Z. So the family F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} should vary with X̂ ∈ M_{X} rather than being constant, but only in a rather trivial way.

We claim that when the C_{i} are stable, the map Ψ is a local diffeomorphism from the interior U^{∘} of U to its image in M_{N}. One can justify this as follows. By [16, §9.4] we can define natural coordinates on M_{N}, local diffeomorphisms M_{N} → H^{1}(N, ℝ) defined uniquely up to translations in H^{1}(N, ℝ). In the same way [18, §6.5] defines local diffeomorphisms M_{X} → I_{X′} uniquely up to translations in I_{X′}, and a similar thing applies for M0_{L_{i}}, so that we can construct natural coordinate systems on F\begin{matrix}X \\ L_{1},...,Ln \end{matrix}.

Using the topological calculations of §8.1, one can show that for C_{i} stable the natural coordinate systems on M_{N} can be identified with products of the natural coordinate systems on M_{X} and F\begin{matrix}X \\ L_{1},...,Ln \end{matrix}, and Ψ is just the product map in these coordinates. Thus Ψ is a local diffeomorphism on U^{∘}.

When the C_{i} are not stable, things are more complicated. Then M_{X} may be singular. If X is transverse, which we expect for generic (M, J, ω, Ω) by Conjecture 5.12, then M_{X} is smooth of dimension dim I_{X′} − dim O_{X′} near X. In this case Theorem 8.7 implies that dim M_{X} + dim F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} = dim M_{N} − dim O_{X′}. We expect Ψ to be a smooth immersion wherever M_{X} is smooth, with image of codimension dim O_{X′} in M_{N}.

Thus, when the C_{i} are not stable and X is transverse, the desingularization results of §7 do not yield the whole of M_{N} locally, but only a subset of codimension dim O_{X′} = ∑_{i=1}^{n} s-ind(C_{i}). Where do these extra degrees of freedom in M_{N} come from? A rough answer is that the ∑_{i=1}^{n} s-ind(C_{i}) reduction in dim M_{X} reappears as an extra s-ind(C_{i}) degrees of freedom to deform each L_{i} as an AC SL m-fold, but with rate λ_{i} < 2 rather than rate 0.

Choose λ_{i} with max(D_{Σ_{i}} ∩ [0, 2)) < λ_{i} < 2. Then by part (a) of Theorem 6.8, the moduli space Mλ_{i}_{L_{i}} of deformations of L_{i} with rate λ_{i} is smooth with

(56)  dim Mλ_{i}_{L_{i}} = b^{1}(L_{i}) − b^{0}(L_{i}) + N_{Σ_{i}}(λ_{i})

= dim M0_{L_{i}} + N_{Σ_{i}}(2) − m_{Σ_{i}}(2) − b^{0}(Σ_{i}),

using the notation of Definition 3.3, and the fact that N_{Σ_{i}} is monotone increasing and upper semicontinuous, and increases by m_{Σ_{i}}(2) at 2.

Suppose C_{i} is rigid, as in Definition 3.4. Then (9) and (56) give

(57) dim Mλ_{i}_{L_{i}} = dim M0_{L_{i}} + s-ind(C_{i}) + 2m.

Thus, deforming L_{i} with rate λ_{i} rather than 0 gives an extra s-ind(C_{i})+ 2m degrees of freedom. Here the 2m comes from translations in ℂ^{m}, since the AC SL m-folds of rate λ ⩾ 1 are closed under translations, and the s-ind(C_{i}) from new, nontrivial deformations of L_{i}.

So when X is transverse and the C_{i} are rigid, we can understand the difference in dimension ∑_{i=1}^{n} s-ind(C_{i}) between M_{X} × F\begin{matrix}X \\ L_{1},...,Ln \end{matrix} and M_{N} as coming from an extra s-ind(C_{i}) nontrivial deformations of L_{i} an an AC SL m-fold with rate λ_{i} rather than 0. If the C_{i} are not rigid, we should take into account also extra infinitesimal deformations of C_{i} as an SL cone.

8.3 The index of singularities of SL m-folds

We can now make more rigorous some speculations by the author in [16, §10.3]. Suppose, as above, that M_{N} is a moduli space of compact, nonsingular SL m-folds in (M, J, ω, Ω), and that M_{X} is a moduli space of singular SL m-folds in ∂M_{N} with singularities of a particular type, and X ∈ M_{X}.

Define the index of the singularities of X to be ind(X) = dim M_{N} − dim M_{X}, provided M_{X} is smooth near X so dim M_{X} is well-defined. Note that ind(X) depends not just on X and its singularities, but also on N through dim M_{N} = b^{1}(N). Thus there could be topologically distinct desingularizations N_{1}, N_{2}, … yielding different values of ind(X).

We can also work in families F of almost Calabi-Yau m-folds. Defining M\begin{matrix}F \\ N \end{matrix} as in Definition 2.15 and M\begin{matrix}F \\ X \end{matrix} as in Definition 5.6, the index of X in F is ind^{F}(X) = dim M\begin{matrix}F \\ N \end{matrix} − dim M\begin{matrix}F \\ X \end{matrix}. Note that ind(X) ⩽ dim M_{N} = b^{1}(N), as dim M_{X} ⩾ 0, and ind^{F}(X) ⩽ dim M\begin{matrix}F \\ N \end{matrix}.

Combining Corollary 5.5 and Theorem 8.7, we can compute ind(X) when X is transverse with conical singularities. In the families case, if X is transverse in F then a similar proof shows that ind^{F}(X) is given by the same formula (58).

Theorem 8.10. Let X be a compact, transverse SL m-fold in (M, J, ω, Ω) with conical singularities at x_{1}, …, x_{n} and cones C_{1}, …, C_{n}. Construct desingularizations N of X by gluing AC SL m-folds L_{i} in at x_{i} for i = 1, …, n, as in §7. Let q, Y be as in Definition 8.2. Then

(58) ind(X) = dim Y + 1 − q + ∑_{i=1}^{n} b_{cs}^{1}(L_{i}) + ∑_{i=1}^{n} s-ind(C_{i}).

When n = 1 this proves [9, Conj. 2.13], in the transverse case.

Suppose C_{i} is not rigid, for instance if Σ_{i} is not connected. Then C_{i} may lie in a smooth, connected moduli space C_{i} of SL cones in ℂ^{m}, upon which SU(m) does not act transitively. In this case, as in [18, §8.3], it is better to define M_{X} to be the moduli space of X̂ with cones Ĉ_{i} ∈ C_{i}, rather than with fixed cones C_{i}. Under suitable transversality assumptions, this increases the dimension of M_{X} by the codimension of the SU(m) orbit of C_{i} in C_{i}, so this codimension should be subtracted from the r.h.s. of (58).

Here is why the index is an important idea. As ind(X) is the codimension of M_{X} in M_{N}, the largest pieces of ∂M_{N} are the M_{X} with smallest index. So we argue that singularities with small index are the most generic, and the most interesting. In good cases, we expect \overline{M}_{N} to be a compact manifold with singular boundary. Thus the largest pieces of ∂M_{N} should have codimension 1 in M_{N}, and hence index 1. If ∂M_{N} has no index 1 strata, then \overline{M}_{N} is roughly a compact, singular manifold without boundary.

As tools, ind(X) or ind^{F}(X) probably work best together with a genericity assumption on (M, J, ω, Ω) or F. Suppose that ω is generic in its Kähler class. Then as in Conjecture 5.12, we expect SL m-folds X in (M, J, ω, Ω) with conical singularities to be transverse, so we can compute ind(X) using (58).

Since ind(X) ⩽ b^{1}(N), this places strong restrictions on the kinds of singularities that can occur in ∂M_{N}. In the families case, when F is generic in a suitable sense we expect SL m-folds X in (M, J^{s}, ω^{s}, Ω^{s}) with conical singularities to be transverse in F. Then we can compute ind^{F}(X) using (58). Since ind^{F}(X) ⩽ dim M\begin{matrix}F \\ N \end{matrix}, this again places strong restrictions on the kinds of singularities that can occur in ∂M\begin{matrix}F \\ N \end{matrix}.

For some problems we only need to know about singularities with index up to a certain value. For example, in [9] the author proposed to define an invariant of almost Calabi-Yau 3-folds by a weighted count of SL homology 3-spheres in a given homology class. To understand how this invariant transforms as we deform (M, J, ω, Ω), we restrict to a generic 1-dimensional family F, and then we meet only singularities with index 1.

The index may also be useful in the SYZ Conjecture [29]. This explains mirror symmetry of (almost) Calabi-Yau 3-folds M, M̌ in terms of fibrations by SL 3-tori N, Ň. The corresponding moduli spaces M_{N}, MŇ have dimension 3, so that ∂M_{N}, ∂MŇ can only contain singularities of index 1, 2 or 3.

9. Applications to connected sums

We shall now apply the results of §7 to the case where the SL m-fold X with conical singularities is actually a nonsingular, immersed SL mfold, the singular points x_{i} are self-intersection points of X satisfying an angle criterion, and the AC SL m-folds L_{i} are chosen from the L^{ϕ,A} of Example 6.11 due to Lawlor [21]. The desingularizations N are multiple connected sums of X with itself.

For the connected sum X_{1}#X_{2} of two SL 3-folds at one point in a Calabi-Yau 3-fold, our results were conjectured by the author in [9, §5–§6]. Butscher [3] proves existence of SL connected sums X_{1}#X_{2} at one point by gluing in Lawlor necks L^{ϕ,A}, where X_{1}, X_{2} are compact SL m-folds in ℂ^{m} with boundary.

Closer to our results, Lee [8] considers a compact, connected, immersed SL m-fold X in a Calabi-Yau m-fold M, whose self-intersection points x_{i} satisfy an angle criterion. She glues in L^{ϕ,A} at x_{i} to get a family of compact, embedded SL m-folds in M. Her result is re-proved in Theorem 9.5 below.

9.1 Transverse intersections of SL planes Π^{+}, Π^{−} in ℂ^{m}

Let Π^{+}, Π^{−} be two SL m-planes ℝ^{m} in ℂ^{m}. We call Π^{±} transverse if they intersect transversely, that is, if Π^{+} ∩ Π^{−} = {0}. Note that this has nothing to do with the use of ‘transverse’ in §5. We now classify transverse pairs of SL m-planes up to the action of SU(m). Related results may be found in [3, §1] and [8, §2], but we believe our approach is clearer.

Proposition 9.1. Let Π^{+}, Π^{−} be transverse SL m-planes ℝ^{m} in ℂ^{m}. Then there exists B ∈ SU(m) and ϕ_{1}, …, ϕ_{m} ∈ (0, π) such that B(Π^{+}) = Π^{0} and B(Π^{−}) = Π^{ϕ}, where ϕ = (ϕ_{1}, …, ϕ_{m}) and

Π^{0} ={(x_{1}, …, x_{m}) : x_{j} ∈ ℝ}, Π^{ϕ}={(e^{iϕ_{1}}x_{1}, …, e^{iϕ_{m}}x_{m}) : x_{j} ∈ ℝ}, (59)

as in (39). Moreover ϕ_{1}, …, ϕ_{m} are unique up to order, so that we can make them unique by assuming that ϕ_{1} ⩽ ϕ_{2} ⩽ ⋯ ⩽ ϕ_{m}, and ϕ_{1} + ⋯ + ϕ_{m} = kπ for some k = 1, …, m − 1.

To prove this, first find B′ ∈ SU(m) with B′(Π^{+}) = Π^{0}, and then find B′′ ∈ SO(m) which ‘diagonalizes’ B′(Π^{−}) to get Π^{ϕ}, and set B = B′′B′. The process is like diagonalizing a real quadatic form on ℝ^{m} with an SO(m) matrix. We use the proposition to divide transverse pairs Π^{±} into types.

Definition 9.2. Let Π^{+}, Π^{−} be transverse SL m-planes ℝ^{m} in ℂ^{m}. Then Proposition 9.1 gives B ∈ SU(m) and unique 0 < ϕ_{1} ⩽ ϕ_{2} ⩽ ⋯ ⩽ ϕ_{m} < π with ϕ_{1} + ⋯ + ϕ_{m} = kπ for some k = 1, …, m − 1, such that B(Π^{+}) = Π^{0} and B(Π^{−}) = Π^{ϕ}. We say that the intersection point 0 of Π^{+}, Π^{−} is of type k.

Note that this depends on the order of Π^{+}, Π^{−}. Exchanging Π^{±} replaces ϕ_{j} by π − ϕ_{m+1−j} for j = 1, …, m, and therefore Π^{+}, Π^{−} intersect with type k if and only if Π^{−}, Π^{+} intersect with type m − k.

When k = 1 in Proposition 9.1, Example 6.11 gives AC SL m-folds L^{ϕ,A} for A > 0 with cone Π^{0} ∪ Π^{ϕ}. Thus L^{±,A} = B^{−1}L^{ϕ,A} is an AC SL m-fold with cone Π^{+} ∪ Π^{−}. When k = m − 1 we can exchange Π^{±} to get k = 1, and do the same thing. So combining Example 6.11 and Proposition 9.1 gives:

Proposition 9.3. Let Π^{+}, Π^{−} be transverse SL m-planes ℝ^{m} in ℂ^{m}, for m > 2. Regard C = Π_{+} ∪ Π_{−} as an SL cone in ℂ^{m} with isolated singularity at 0. Then Σ = C ∩ S^{2m−1} is a disjoint union Σ^{+} ∪ Σ^{−}, where Σ^{±} are the unit spheres S^{m−1} in Π^{±}. Then:

(a) Suppose Π^{+}, Π^{−} intersect with type 1. Then there is a 1-parameter family of AC SL m-folds L^{±,A} in ℂ^{m} asymptotic to C with rate 2−m for A > 0, with Z(L^{±,A})·[Σ^{+}] = A and Z(L^{±,A})·[Σ^{−}] = −A.

(b) Suppose Π^{+}, Π^{−} intersect with type m − 1. Then there is a 1parameter family of AC SL m-folds L^{±,A} in ℂ^{m} asymptotic to C with rate 2−m for A > 0, with Z(L^{±,A})·[Σ^{+}] = −A and Z(L^{±,A})· [Σ^{−}] = A.

The L^{±,A} are images of the L^{ϕ,A} of Example 6.11 under SU(m) rotations.

We will call the L^{±,A} Lawlor necks. They are diffeomorphic to S^{m−1} × ℝ.

9.2 Desingularizing immersed SL m-folds

Here is some notation for self-intersection points of immersed SL mfolds.

Definition 9.4. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold for m > 2, and define ψ : M → (0, ∞) as in (4). Let X be a compact, nonsingular immersed SL m-fold in M. That is, X is a compact mmanifold (not necessarily connected) and ι : X → M an immersion, with special Lagrangian image.

Call x ∈ M a self-intersection point of X if ι^{∗}(x) is at least two points in X. Call such an x transverse if ι^{∗}(x) is exactly two points x^{+}, x^{−} in X, and ι_{∗}(T_{x^{+}}X) ∩ ι_{∗}(T_{x^{−}}X) = {0} in T_{x}M.

Let x be a transverse self-intersection point of X, and x^{±} as above. Choose an isomorphism υ : ℂ^{m} → T_{x}M with υ^{∗}(ω) = ω′ and υ^{∗}(Ω) = ψ(x)^{m}Ω′, where ω′, Ω′ are as in (2). Define Π^{±} = υ^{∗}(ι_{∗}(T_{x^{±}}X)). Then Π^{+}, Π^{−} are transverse SL planes ℝ^{m} in ℂ^{m}.

Define the type k = 1, …, m − 1 of x to be the type of Π^{+}, Π^{−}, in the sense of Definition 9.2. This is independent of the choice of υ, but it does depend on the order of x^{+}, x^{−}, and exchanging x^{±} replaces k by m − k.

We now apply Theorem 7.1 to desingularize connected, immersed SL m-folds X in M. Our result is equivalent to Lee’s main result [8, Theorems 3 & 4], save that Lee also allows m = 2, and considers only the Calabi-Yau case.

Theorem 9.5. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold for m > 2, and X a compact, connected, immersed SL m-fold in M. Suppose x_{1}, …, x_{n} are transverse self-intersection points of X with type 1 or m − 1.

Then there exists ϵ > 0 and a smooth family {Ñ^{t} : t ∈ (0, ϵ]} of compact, immersed SL m-folds in (M, J, ω, Ω), such that Ñ^{t} is constructed by gluing a Lawlor neck L^{±,t^{m}A_{i}} into X at x_{i} for i = 1, …, n, and so is a multiple connected sum of X with itself. In the sense of currents, Ñ^{t} → X as t → 0. If x_{1}, …, x_{n} are the only self-intersection points of X then Ñ^{t} is embedded.

Proof. For each i = 1, …, n let ι^{∗}(x_{i}) be x_{i}^{+}, x_{i}^{−}, and define υ_{i} : ℂ^{m} → T_{x_{i}}M and Π_{i}^{±} as in Definition 9.4. Then Π_{i}^{+}, Π_{i}^{−} are transverse SL planes with type 1 or m − 1, by assumption. Therefore Proposition 9.3 gives a 1-parameter family L_{i}^{±,A} for A > 0 of AC SL m-folds in ℂ^{m} asymptotic to Π_{i}^{+} ∪ Π_{i}^{−}.

Choose some A_{1}, …, A_{n} > 0, for instance A_{i} ≡ 1, and let L_{i} = L_{i}^{±,A_{i}} for i = 1, …, n. Apply Theorem 7.1 to X with conical singular points x_{i}, cones C_{i} = Π_{i}^{+} ∪ Π_{i}^{−} and AC SL m-folds L_{i} for i = 1, …, n. As X is connected X′ = X \ {x_{1}^{±}, …, x_{n}^{±}} is connected, and L_{i} has rate λ_{i} = 2 − m < 0, so the hypotheses hold.

Thus Theorem 7.1 gives ϵ > 0 and the family Ñ^{t}, and most of the theorem follows. If X has other self-intersection points than x_{1}, …, x_{n} then the Ñ^{t} are immersed. But if x_{1}, ..., x_{n} are the only self-intersection points then X′ and the L_{i} are embedded, so the Ñ^{t} are embedded for small t. q.e.d.

When m = 3 the only possible types are 1 and m − 1, giving:

Corollary 9.6. Let X be a compact, connected, immersed SL 3-fold with transverse self-intersection points in an almost Calabi-Yau 3-fold M. Then X is a limit of embedded SL 3-folds.

Next we apply Theorem 7.2 to desingularize non-connected X. If a self-intersection point x_{i} has type m − 1 we can swap x_{i}^{+}, x_{i}^{−} to get type 1, so for simplicity we suppose the x_{i} all have type 1.

Theorem 9.7. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold for m > 2, and X a compact, immersed SL m-fold in M. Define ψ : M → (0, ∞) as in (4). Suppose x_{1}, …, x_{n} ∈ M are transverse self-intersection points of X with type 1, and let x_{i}^{±} ∈ X be as in Definition 9.4. Set q = b^{0}(X), and let X_{1}, …, X_{q} be the connected components of X. Suppose A_{1}, …, A_{n} > 0 satisfy

∑_{\begin{matrix}i=1,...,n: \\ x_{i}^{+}∈X_{k} \end{matrix}} ψ(x_{i})^{m}A_{i} = ∑_{\begin{matrix}i=1,...,n: \\ x_{i}^{−}∈X_{k} \end{matrix}} ψ(x_{i})^{m}A_{i} for all k = 1, …, q. (60)

Let N be the oriented multiple connected sum of X with itself at the pairs of points x_{i}^{+}, x_{i}^{−} for i = 1, …, n. Suppose N is connected.

Then there exists ϵ > 0 and a smooth family {Ñ^{t} : t ∈ (0, ϵ]} of compact, immersed SL m-folds in (M, J, ω, Ω) diffeomorphic to N, such that Ñ^{t} is constructed by gluing a Lawlor neck L^{±,t^{m}A_{i}} into X at x_{i} for i = 1, …, n. In the sense of currents, Ñ^{t} → X as t → 0. If x_{1}, …, x_{n} are the only self-intersection points of X then Ñ^{t} is embedded.

Proof. Use the notation of the proof of Theorem 9.5. The assumption that N is connected above is one of the hypotheses of Theorem 7.2. Part (a) of Proposition 9.3 gives Z(L_{i}) · [Σ_{i}^{±}] = ±A_{i}, and using this we find that (43) is equivalent to (60). The other hypotheses of Theorem 7.2 are established in the proof of Theorem 9.5. Thus Theorem 7.2 applies, and the rest of the proof follows Theorem 9.5. q.e.d.

To decide whether desingularizations Ñ^{t} of X exist, we need to know when (60) admits solutions A_{i} > 0. Since q − 1 of Equations (60) are independent as in the proof of Proposition 8.5, there can only exist nonzero solutions A_{i} if n ⩾ q.

Here is a graphical method for deciding. Draw q vertices, numbered 1, …, q. For i = 1, …, n draw a directed edge from vertex j to vertex k, where x_{i}^{+} ∈ X_{j} and x_{i}^{−} ∈ X_{k}. Then (60) admits solutions A_{i} > 0 if and only if whenever we divide the q vertices into two nonempty disjoint subsets B and C, there is at least one directed edge going from B to C, and at least one going from C to B.

As in §8 we can compare the dimensions of moduli spaces M_{X}, M_{N} in Theorem 9.7. Since m > 2 it is easy to show that each connected sum either reduces b^{0} by 1 and fixes b^{1}, or fixes b^{0} and increases b^{1} by 1. As b^{0}(X) = q, b^{0}(N) = 1, and dim M_{X} = b^{1}(X) by Theorem 2.10 in the immersed case, we see that

dim M_{N} = b^{1}(N) = n + 1 − q + b^{1}(X) = n + 1 − q + dim M_{X}. (61)

If n = q then dim M_{N} = 1 + dim M_{X}, and \overline{M}_{N} is a manifold with boundary M_{X} near X, and the singularities of X have index ind(X) = 1 in the sense of §8.3. Note that (58) does not give the right answer for ind(X) in this case, as C_{i} = Π_{i}^{+} ∪ Π_{i}^{−} is not rigid, and to get the right definition of M_{X} we have to allow singular cones Ĉ_{i} not just some fixed Π_{i}^{+} ∪ Π_{i}^{−}, but any union Π̂_{i}^{+} ∪ Π̂_{i}^{−} of transverse SL m-planes Π̂_{i}^{±} in ℂ^{m} with type 1.

9.3 Desingularizing immersed SL m-folds in families

Next we extend Theorem 9.7 to families of almost Calabi-Yau m-folds. Following the proof of Theorem 9.7 and using Theorem 7.4 we prove:

Theorem 9.8. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold for m > 2, and X a compact, immersed SL m-fold in M with immersion ι. Define ψ : M → (0, ∞) as in (4). Suppose x_{1}, …, x_{n} ∈ M are transverse self-intersection points of X with type 1, and let x_{i}^{±} ∈ X be as in Definition 9.4. Set q = b^{0}(X), and let X_{1}, …, X_{q} be the connected components of X. Let N be the oriented multiple connected sum of X with itself at the pairs of points x_{i}^{+}, x_{i}^{−} for i = 1, …, n. Suppose N is connected.

Suppose {(M, J^{s}, ω^{s}, Ω^{s}) : s ∈ F} is a smooth family of deformations of (M, J, ω, Ω) with ι^{∗}([ω^{s}]) = 0 in H^{2}(X, ℝ) for all s ∈ F. Let A_{1}, …, A_{n} > 0. Define G ⊆ F × (0, 1) to be the subset of (s, t) ∈ F × (0, 1) with

[Im Ω^{s}] · [X_{k}] = t^{m} ∑_{\begin{matrix}i=1,...,n: \\ x_{i}^{+}∈X_{k} \end{matrix}} ψ(x_{i})^{m}A_{i} − t^{m} ∑_{\begin{matrix}i=1,...,n: \\ x_{i}^{−}∈X_{k} \end{matrix}} ψ(x_{i})^{m}A_{i} (62)

for all k = 1, …, q. Then there exist ϵ ∈ (0, 1), κ > 1 and a smooth family

{Ñ^{s,t} : (s, t) ∈ G, t ∈ (0, ϵ], |s| ⩽ t^{κ+m/2}}, (63)

such that Ñ^{s,t} is a compact, nonsingular SL m-fold in (M, J^{s}, ω^{s}, Ω^{s}) diffeomorphic to N, constructed by gluing a Lawlor neck L^{±,t^{m}A_{i}} into X at x_{i} for i = 1, …, n. In the sense of currents, Ñ^{s,t} → X as s, t → 0. If x_{1}, …, x_{n} are the only self-intersection points of X then Ñ^{s,t} is embedded.

Thus, the main condition for the existence of desingularizations Ñ^{s,t} of X in (M, J^{s}, ω^{s}, Ω^{s}) is that there should exist solutions A_{1}, …, A_{n} > 0 to (62). Note that the sum of (62) over k = 1, …, q gives [Im Ω^{s}]·[X] = 0, which is clearly a necessary condition for Ñ^{s,t} to exist with [Ñ^{s,t}] = [X].

In [9] the author proposed to define an invariant I_{3} : H_{3}(M, ℤ) → ℚ of almost Calabi-Yau 3-folds (M, J, ω, Ω) by counting SL homology 3spheres in a given homology class with a topological weight. Theorem 9.8 will be important for this programme, because it determines the transformation rules I_{3} satisfies as we deform (M, J, ω, Ω) so that J passes through certain real hypersurfaces in the complex structure moduli space.

To explain this we need the idea of SL m-folds with phase e^{iθ}, as in [9, 10].

Definition 9.9. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold, and N an oriented real m-dimensional submanifold of M. Fix θ ∈ ℝ. We call N a special Lagrangian submanifold, or SL m-fold for short, with phase e^{iθ} if

(64) ω|_{N} ≡ 0 and (sin θ Re Ω − cos θ Im Ω)|_{N} ≡ 0,

and cos θ Re Ω + sin θ Im Ω is a positive m-form on the oriented mfold N.

If N is compact it easily follows that [Ω] · [N] = Re^{iθ}, where [Ω] ∈ H^{m}(M, ℝ) and [N] ∈ H_{m}(M, ℤ), and R = ∫_{N} cos θ Re Ω + sin θ Im Ω > 0. Thus the homology class [N] determines the phase e^{iθ} of N.

The definition of SL m-fold used in the rest of the paper, Definition 2.9, is of SL m-fold with phase 1. If N has phase e^{iθ} in (M, J, ω, Ω) then it has phase 1 in (M, J, ω, e^{−iθ}Ω), so if we are dealing with SL m-folds in only one homology class then we can rescale Ω to make the phase 1. But when we consider several SL m-folds N_{1}, N_{2}, … we cannot always take them to have phase 1.

Using this notation, we rewrite Theorem 9.8 when n = 1 and q = 2, so that we take the connected sum X_{1}#X_{2} of SL m-folds X_{1}, X_{2} at one point x.

Theorem 9.10. Let (M, J, ω, Ω) be an almost Calabi-Yau m-fold for m > 2, and X_{1}, X_{2} be compact, connected SL m-folds in M with the same phase e^{iθ}, which intersect transversely at x ∈ M with type 1. Suppose {(M, J^{s}, ω^{s}, Ω^{s}) : s∈F} is a smooth family of deformations of (M, J, ω, Ω) with ι^{∗}([ω^{s}]) = 0 in H^{2}(X_{k}, ℝ) for all k = 1, 2 and s ∈ F. Write

[Ω^{s}] · [X_{k}] = R_{k}^{s}e^{iθ_{k}^{s}} for k = 1, 2 and [Ω^{s}] · ([X_{1}] + [X_{2}]) = R^{s}e^{iθ^{s}}, (65)

where R_{k}^{s}, R^{s} > 0 and θ_{k}^{s}, θ^{s} ∈ ℝ depend continuously on s with θ_{k}^{0} = θ^{0} = θ. Make F smaller if necessary so that R_{k}^{s}, θ_{k}^{s}, R^{s}, θ^{s} are welldefined. Define

G = {(s, t) ∈ F × (0, 1) : R_{1}^{s} sin(θ_{1}^{s} − θ^{s}) = t^{m}ψ(x)^{m}}. (66)

Then there exist ϵ ∈ (0, 1), κ > 1 and a smooth family

{Ñ^{s,t} : (s, t) ∈ G, t ∈ (0, ϵ], |s| ⩽ t^{κ+m/2}}, (67)

such that Ñ^{s,t} is a compact SL m-fold with phase e^{iθ^{s}} in (M, J^{s}, ω^{s}, Ω^{s}) diffeomorphic to X_{1}#X_{2}, constructed by gluing a Lawlor neck L^{±,t^{m}} into X_{1} ∪ X_{2} at x. In the sense of currents, Ñ^{s,t} → X as s, t → 0. If X_{1}, X_{2} are embedded and x is their only intersection point then Ñ^{s,t} is embedded.

This follows from Theorem 9.8 with X = X_{1} ∪ X_{2}, n = 1, x_{1} = x and A_{1} = 1, replacing (M, J, ω, Ω), (M, J^{s}, ω^{s}, Ω^{s}) by (M, J, ω, e^{−iθ}Ω), (M, J^{s}, ω^{s}, e^{−iθ^{s}}Ω^{s}) so that X_{1}, X_{2} and Ñ^{s,t} have phase 1. Equation (62) becomes

[e^{−iθ^{s}} Im Ω^{s}] · [X_{1}] = t^{m}ψ(x)^{m} and [e^{−iθ^{s}} Im Ω^{s}] · [X_{2}] = −t^{m}ψ(x)^{m}.

By (65) these are equivalent to

(68) R_{1}^{s} sin(θ_{1}^{s} − θ^{s}) = t^{m}ψ(x)^{m} and R_{2}^{s} sin(θ_{2}^{s} − θ^{s}) = −t^{m}ψ(x)^{m}.

But R_{1}^{s} sin(θ_{1}^{s} − θ^{s}) = −R_{2}^{s} sin(θ_{2}^{s} − θ^{s}) as R_{1}^{s}e^{iθ_{1}^{s}} + R_{2}^{s}e^{iθ_{2}^{s}} = R^{s}e^{iθ^{s}} by (65). Thus both equations of (68) are equivalent, so we use only the first in (66).

We can interpret Theorem 9.10 like this. From (65) we see that θ^{s} always lies between θ_{1}^{s} and θ_{2}^{s} for small s ∈ F. Thus making F smaller if necessary we can divide F into three regions:

\begin{matrix}F^{+} = \{s ∈ F : θ_{1}^{s} > θ^{s} > θ_{2}^{s}\}, F^{−} = \{s ∈ F : θ_{1}^{s} < θ^{s} < θ_{2}^{s}\}, \\ and F^{0} = \{s ∈ F : θ_{1}^{s} = θ^{s} = θ_{2}^{s}\}. \end{matrix} (69)

If [X_{1}], [X_{2}] are linearly dependent in H_{m}(M, ℝ) then θ_{1}^{s} ≡ θ_{2}^{s} ≡ θ^{s}, giving G = ∅ in (66), and the theorem is trivial.

So suppose [X_{1}], [X_{2}] are linearly independent. Then for F suitably generic F^{0} will be a smooth real hypersurface in F, which divides F \F^{0} into two open regions F^{±}. Call F^{+} the positive side and F^{−} the negative side of F^{0}. Now θ_{1}^{s} − θ^{s} is small close to F^{0} in F, and so sin(θ_{1}^{s} − θ^{s}) has the same sign as θ_{1}^{s} − θ^{s}. Therefore R_{1}^{s} sin(θ_{1}^{s} − θ^{s}) = t^{m}ψ(x)^{m} in (66) admits a unique solution t > 0 for small s if and only if θ_{1}^{s} > θ^{s}, that is, if and only if s ∈ F^{+}.

We thus have the following picture, described when m = 3 in Conjecture 6.5 of [9], which we have now proved. By Theorem 2.16 we can extend X_{1}, X_{2} to families of SL m-folds X_{k}^{s} with phase e^{iθ_{k}^{s}} in (M, J^{s}, ω^{s}, Ω^{s}) for k = 1, 2 and small s ∈ F, such that X_{1}^{s}, X_{2}^{s} intersect transversely with type 1 at x^{s} ∈ M close to x. On the hypersurface F^{0} in F the phases of X_{1}^{s}, X_{2}^{s} are equal.

On the positive side F^{+} of F^{0} there exist SL m-fold connected sums X_{1}^{s}#X_{2}^{s} with phase e^{iθ^{s}}. On the negative side F^{−} there are no such X_{1}^{s}#X_{2}^{s}. Thus, as we cross hypersurfaces F^{0} in F where the phases of two SL m-folds X_{1}, X_{2} become equal, we create or destroy new SL m-folds X_{1}#X_{2} by connected sum at points x where X_{1}, X_{2} intersect transversely with type 1 or m − 1.

The conjectured invariant I_{3} : H_{3}(M, ℤ) → ℚ of [9] should change in a predictable fashion as we cross hypersurfaces F^{0}, owing to the creation and destruction of SL homology 3-spheres. Theorem 9.8 also gives criteria for the existence of multiple connected sums X_{1}#X_{2}# ⋯ #X_{q} of SL m-folds. Using this I can derive a complete set of transformation rules for I_{3}, and also extend the programme to all m ⩾ 3. I am writing papers about this.

10. Stable T^{2}-cone singularities in SL 3-folds

We now study SL 3-folds with conical singularities modelled on the stable T^{2}-cone C = C3_{HL} of Example 3.5, given by

C = {(z_{1}, z_{2}, z_{3}) ∈ ℂ^{m} : z_{1}z_{2}z_{3} ∈ [0, ∞), |z_{1}| = |z_{2}| = |z_{3}|}. (70)

Example 6.9 gives three families of AC SL 3-folds La_{HL} with rate 0 and cone C, which we will write as L_{j}^{a} for j = 1, 2, 3 and a > 0, given by

L_{1}^{a} = {(z_{1}, z_{2}, z_{3}) ∈ ℂ^{3} : z_{1}z_{2}z_{3} ∈ [0, ∞), |z_{1}|^{2} − a = |z_{2}|^{2} = |z_{3}|^{2}},
L_{2}^{a} = {(z_{1}, z_{2}, z_{3}) ∈ ℂ^{3} : z_{1}z_{2}z_{3} ∈ [0, ∞), |z_{1}|^{2} = |z_{2}|^{2} − a = |z_{3}|^{2}},
L_{3}^{a} = {(z_{1}, z_{2}, z_{3}) ∈ ℂ^{3} : z_{1}z_{2}z_{3} ∈ [0, ∞), |z_{1}|^{2} = |z_{2}|^{2} = |z_{3}|^{2} − a}.

Then L_{j}^{a} is diffeomorphic to S^{1} × ℝ^{2}.

Identify Σ = C ∩ S^{5} with T^{2} = U(1)^{2} by the map

(e^{iθ_{1}}, e^{iθ_{2}}) ⟼ (\frac{1}{\sqrt{3}} e^{iθ_{1}}, \frac{1}{\sqrt{3}} e^{iθ_{2}}, \frac{1}{\sqrt{3}} e^{−iθ_{1}−iθ_{2}}). (71)

This identifies H^{1}(Σ, ℝ) ≅ H^{1}(T^{2}, ℝ) = ℝ^{2}. Under this identification, as in Example 6.9 we have Z(L_{j}^{a}) = 0 for all j, a and

Y (L_{1}^{a}) = (πa, 0), Y (L_{2}^{a}) = (0, πa) and Y (L_{3}^{a}) = (−πa, −πa). (72)

For all j, a define a holomorphic disc D_{j}^{a} with ∂D_{j}^{a} ⊂ L_{j}^{a} and area πa by

D_{j}^{a} = {(z_{1}, z_{2}, z_{3}) ∈ ℂ^{3} : |z_{j}|^{2} ⩽ a, z_{k} = 0 for j ≠ k}. (73)

The cone C is interesting as it has three topologically distinct families of AC SL 3-folds asymptotic to it, giving three different ways to desingularize singularities of SL 3-folds with cone C. It is also significant as it is the only nontrivial example of a stable SL cone in ℂ^{m} known to the author. Mark Haskins [7, Th. A] has proved that C is the only stable T^{2}-cone in ℂ^{3} up to SU(3) isomorphisms. But SL m-folds with stable conical singularities have particularly good properties, as in Corollaries 5.3 and 8.8.

The author discussed singular SL 3-folds with cone C in [9, §3–§4] and [13]. We can now prove some of the conjectures in these papers, and give nontrivial applications of Theorems 7.3 and 7.5. For simplicity we consider only SL 3-folds with one or two singular points, but the results of §7 apply to arbitrarily many singularities x_{1}, …, x_{n}.

10.1 SL 3-folds with one T^{2}-cone singularity

We shall need a lemma on 3-manifolds with boundary.

Lemma 10.1. Let N be a compact, oriented 3-manifold with boundary Σ. Then the natural map H^{1}(N, ℝ) → H^{1}(Σ, ℝ) has image of dimension \frac{1}{2}b^{1}(Σ).

Proof. Let N^{∘} = N \ Σ. The pair (N, Σ) has an exact sequence in cohomology

⋯ → H^{1}(N, ℝ) = H^{1}(N^{∘}, ℝ) → H^{1}(Σ, ℝ) → H_{cs}^{2}(N^{∘}, ℝ) → · · · . (74)

But H^{1}(N, ℝ) ≅ H_{cs}^{2}(N^{∘}, ℝ)^{∗} and H^{1}(Σ, ℝ) ≅ H^{1}(Σ, ℝ)^{∗} by Poincaré duality for N and Σ. These isomorphisms identify the map H^{1}(N, ℝ) → H^{1}(Σ, ℝ) with the dual of the map H^{1}(Σ, ℝ) → H_{cs}^{2}(N^{∘}, ℝ) in (74). Hence the image of H^{1}(N, ℝ) → H^{1}(Σ, ℝ) has the same dimension as the cokernel of H^{1}(Σ, ℝ) → H_{cs}^{2}(N^{∘}, ℝ), and the lemma follows by exactness in (74). q.e.d.

Consider the following situation, as in [9, §4].

Definition 10.2. Let (M, J, ω, Ω) be an almost Calabi-Yau 3-fold, and X a compact, connected SL 3-fold with exactly one conical singularity at x, with cone C in (70). Then X′ = X \ {x} is connected. Let Σ = C ∩ S^{5}, and identify H^{1}(Σ, ℝ) ≅ ℝ^{2} as above. Since X′ is the interior of a compact, oriented 3-manifold X′ with boundary Σ ≅ T^{2}, Lemma 10.1 shows that the map H^{1}(X′, ℝ) → H^{1}(Σ, ℝ) of (16) has image ℝ.

Similarly, the map H^{1}(X′, ℚ) → H^{1}(Σ, ℚ) has image ℚ ⊂ ℚ^{2}. Thus there exist coprime integers k_{1}, k_{2}, such that (k_{2}, −k_{1}) ∈ ℤ^{2} ⊂ ℚ^{2} ⊂ ℝ^{2} generates the images of H^{1}(X′, ℝ) → H^{1}(Σ, ℝ) and H^{1}(X′, ℚ) → H^{1}(Σ, ℚ). Define k_{3} = −k_{1} − k_{2}. By exactness in (16), the map ℝ^{2} = H^{1}(Σ, ℝ) → H_{cs}^{2}(X′, ℝ) has kernel ⟨(k_{2}, −k_{1})⟩. Therefore this map is given by

(75) (y_{1}, y_{2}) ↦ (k_{1}y_{1} + k_{2}y_{2})χ for some nonzero χ ∈ H_{cs}^{2}(X′, ℝ).

Then k_{1}, k_{2}, k_{3} and χ are unique up to an overall change of sign.

The integers k_{1}, k_{2}, k_{3} were introduced in [9, Def. 4.3]. We now carry out the topological calculations of §8 for desingularizing X by gluing in L_{j}^{a} at x.

Proposition 10.3. In the situation of Definition 10.2, fix j = 1, 2 or 3 and let N_{j} be the compact 3-manifold obtained by gluing L_{j}^{a} into X′ at x. Use the notation of §8, with n = 1 and L_{1} = L_{j}^{a}. Then Z_{1} = Z = {0} and

dim I_{X′} = b_{cs}^{1}(X′), Y_{1} = \begin{cases}⟨(1, 0)⟩, & j = 1, \\ ⟨(0, 1)⟩, & j = 2, \\ ⟨(1, 1)⟩, & j = 3, \end{cases} Y = \begin{cases}Y_{1}, & k_{j} = 0, \\ \{0\}, & k_{j} ≠ 0, \end{cases} (76)

F\begin{matrix}X \\ L_{1} \end{matrix} = \begin{cases}\{L_{j}^{a} : a > 0\}, & k_{j} = 0, \\ ∅, & k_{j} ≠ 0, \end{cases} b^{1}(N_{j}) = \begin{cases}b_{cs}^{1}(X′) + 1, & k_{j} = 0, \\ b_{cs}^{1}(X′), & k_{j} ≠ 0. \end{cases} (77)

Proof. In §8.1 as L_{1} ≅ S^{1} × ℝ^{2} and Σ_{1} ≅ T^{2} we have l_{1} = 1, b^{0}(L_{1}) = 1 and b_{cs}^{1}(L_{1}) = 0, and q = 1 as X′ is connected. Thus (52) gives dim Z_{1} = 0, and Z ⊆ Z_{1}, so Z_{1} = Z = {0}, and Lemma 8.3 gives dim I_{X′} = b_{cs}^{1}(X′).

Now Y_{1} ⊂ H^{1}(Σ, ℝ) = ℝ^{2} is the image of H^{1}(L_{j}^{a}, ℝ) → H^{1}(Σ, ℝ) by Definition 8.2, and calculation shows it is as in (76). But the image of H^{1}(X′, ℝ) → H^{1}(Σ, ℝ) is ⟨(k_{2}, −k_{1})⟩ by Definition 10.2, and Y is the intersection of this image with Y_{1}. As k_{3} = −k_{1} − k_{2}, we see that Y is as given in (76).

Equation (52) gives dim M0_{L_{1}} = 1, so M0_{L_{1}} = {L_{j}^{a} : a > 0}. But by (54), as Z_{1} = \{0\} we see that F\begin{matrix}X \\ L_{1} \end{matrix} is the subset of L̂_{1} ∈ M0_{L_{1}} with Y (L̂_{1}) ∈ Y, so (76) gives the first equation of (77). The second follows from Proposition 8.6, since dim Y is 1 if k_{j} = 0 and 0 if k_{j} ≠ 0 by (76).

q.e.d.

Now from §8.2, F\begin{matrix}X \\ L_{1} \end{matrix} is the family of AC SL 3-folds L_{1} which satisfy the hypotheses of the desingularization results of §7. Thus if k_{j} = 0, applying Theorem 7.3 with L_{1} = L_{j}^{1} gives:

Theorem 10.4. Suppose (M, J, ω, Ω) is an almost Calabi-Yau 3fold, and X a compact, connected SL 3-fold with exactly one conical singularity at x, with cone C in (70). Let k_{1}, k_{2}, k_{3} be as in Definition 10.2, and suppose k_{j} = 0 for j = 1, 2 or 3. Then there exists ϵ > 0 and a smooth family {Ñ_{j}^{t} : t ∈ (0, ϵ]} of compact SL 3-folds in (M, J, ω, Ω) constructed by gluing L\begin{matrix}t^{2} \\ j \end{matrix} into X at x. In the sense of currents, Ñ_{j}^{t} → X as t → 0.

If k_{j} ≠ 0 then for topological reasons there exist no Lagrangian 3folds N_{j}^{t} constructed by gluing tL_{j}^{1} into X at x, and hence no SL 3-folds Ñ_{j}^{t}. As the k_{j} are not all zero and k_{1} + k_{2} + k_{3} = 0 there can be at most one j with k_{j} = 0.

In the situation of Theorem 10.4, by Corollary 5.3 and (76) the moduli space M_{X} of deformations of X is a smooth manifold of dimension b_{cs}^{1}(X′), and by Theorem 2.10 and (77) the moduli space M_{N_{j}} of deformations of Ñ_{j}^{t} is a smooth manifold of dimension b_{cs}^{1}(X′) + 1.

Hence the singularities of X have index one in the sense of §8.3, and \overline{M}_{N_{j}} is near X a nonsingular manifold with boundary M_{X}. So in this case we have a very good understanding of the boundary ∂M_{N_{j}} of M_{N_{j}}, as discussed in §8.

Following [9, §3.3] we can explain why Ñ_{j}^{t} becomes singular as t → 0, using the D_{j}^{a} of (73). As Ñ_{j}^{t} is made by gluing L\begin{matrix}t^{2} \\ j \end{matrix} into X at x, and there is a holomorphic disc D\begin{matrix}t^{2} \\ j \end{matrix} with area πt^{2} and ∂D\begin{matrix}t^{2} \\ j \end{matrix} ⊂ L_{j}^{t^{2}}, we expect there to exist a holomorphic disc D̃^{t} with area πt^{2} and ∂D̃^{t} ⊂ Ñ_{j}^{t}, for small t.

As t → 0 the area of D̃^{t} goes to zero, and D̃^{t} collapses to a point. Its boundary S^{1} in Ñ_{j}^{t} also collapses to a point, giving the singular SL 3-fold X. The author expects that singularities with cone C are the generic singularity of SL 3-folds occurring when areas of holomorphic discs become zero.

10.2 SL 3-folds with one T^{2}-cone singularity in families

Next we apply Theorem 7.5 to desingularize SL 3-folds with one singularity x with cone C in families of almost Calabi-Yau 3-folds (M, J^{s}, ω^{s}, Ω^{s}).

Theorem 10.5. Suppose (M, J, ω, Ω) is an almost Calabi-Yau 3fold, and X a compact, connected SL 3-fold with exactly one conical singularity at x, with cone C in (70). Let k_{1}, k_{2}, k_{3}, χ be as in Definition 10.2. Suppose {(M, J^{s}, ω^{s}, Ω^{s}) : s ∈ F} is a smooth family of deformations of (M, J, ω, Ω) with

(78) [Im Ω^{s}] · [X] = 0 for all s ∈ F, where [X] ∈ H_{3}(M, ℝ).

Let ι_{∗} : H_{2}(X, ℝ) → H_{2}(M, ℝ) be the inclusion, fix j = 1, 2 or 3, and define

\begin{aligned}G_{j} = \{(s, t) ∈ F × (0, 1) : [ω^{s}] · ι_{∗}(γ) = π & t^{2}k_{j}(χ · γ) \\ & for all γ ∈ H_{2}(X, ℝ)\}. \end{aligned} (79)

Then there exist ϵ ∈ (0, 1), κ > 1 and ϑ ∈ (0, 2) and a smooth family

{Ñ_{j}^{s,t} : (s, t) ∈ G_{j}, t ∈ (0, ϵ], |s| ⩽ t^{ϑ}}, (80)

such that Ñ_{j}^{s,t} is a compact SL 3-fold in (M, J^{s}, ω^{s}, Ω^{s}) constructed by gluing L\begin{matrix}t^{2} \\ j \end{matrix} into X at x. In the sense of currents, Ñ_{j}^{s,t} → X as s, t → 0.

Proof. We apply Theorem 7.5, with L_{1} = L_{j}^{1}. As X is connected, X′ is connected. Equation (78) gives (48). The values of Y (L_{j}^{a}) in H^{1}(Σ, ℝ) ≅ ℝ^{2} are given in (72), and the map H^{1}(Σ, ℝ) → H_{cs}^{2}(X′, ℝ) is given in (75). Combining these two shows that the image of Y (L_{j}^{a}) in H_{cs}^{2}(X′, ℝ) is πak_{j} χ. Thus putting a = 1 as L_{1} = L_{j}^{1}, we have ϖ = πk_{j} χ in Theorem 7.5, and so G_{j} in (79) agrees with G in (49). The result then follows from Theorem 7.5. q.e.d.

We now specialize to the case that b_{cs}^{1}(X′) = 0. Then b^{2}(X′) = 0 by (17), so (16) gives an exact sequence

0 → H^{1}(X′, ℝ) → H^{1}(Σ_{i}, ℝ) ≅ ℝ^{2} → H_{cs}^{2}(X′, ℝ) → 0.

As b^{1}(X′) = b_{cs}^{2}(X′) by (17), this gives b^{1}(X′) = b_{cs}^{2}(X′) = 1.

Now H_{2}(X, ℝ) ≅ H_{cs}^{2}(X′, ℝ)^{∗} ≅ ℝ by (18), and χ ≠ 0 by (75). Thus there exists a unique γ_{0} ∈ H_{2}(X, ℝ) with χ · γ_{0} = 1. The condition [ω^{s}] · ι_{∗}(γ) = πt^{2}k_{j}(χ · γ) for all γ ∈ H_{2}(X, ℝ) in (79) then becomes the single real equation [ω^{s}] · ι_{∗}(γ_{0}) = πt^{2}k_{j}.

As in (69), let us divide the family F into three regions:

\begin{aligned}F^{+} & = \{s ∈ F : [ω^{s}] · ι_{∗}(γ_{0}) > 0\}, \\ F^{−} & = \{s ∈ F : [ω^{s}] · ι_{∗}(γ_{0}) < 0\}, \\ and F^{0} & = \{s ∈ F : [ω^{s}] · ι_{∗}(γ_{0}) = 0\}. \end{aligned} (81)

If ι_{∗}(γ_{0}) ≠ 0 and F is sufficiently generic, then F^{0} will be a smooth real hypersurface in F, which divides F \ F^{0} into two open regions F^{±}. Call F^{+} the positive side and F^{−} the negative side of F^{0}.

We investigate the existence of deformations X^{s} and desingularizations Ñ_{j}^{s,t} of X in (M, J^{s}, ω^{s}, Ω^{s}), for small s in each of these regions.

(a) From §5, a necessary condition for the existence of any SL 3-fold X^{s} with a conical singularity isotopic to X in (M, J^{s}, ω^{s}, Ω^{s}) is that ι_{∗}(γ) · [ω^{s}] = 0 for all γ ∈ H_{2}(X, ℝ). Thus, such X^{s} can exist only if s ∈ F^{0}.

As dim I_{X′} = b_{cs}^{1}(X′) = 0, Corollary 5.8 then shows that for small s ∈ F^{0}, there is a unique deformation X^{s} of X close to X in (M, J^{s}, ω^{s}, Ω^{s}).

(b) If s ∈ F^{0} then [ω^{s}] · ι_{∗}(γ_{0}) = πt^{2}k_{j} has solutions t > 0 if and only if k_{j} = 0, and then any t > 0 is a solution. For small s ∈ F^{0} we may apply Theorem 10.4 to the unique X^{s} above to get a 1parameter family of SL 3-folds Ñ_{j}^{s,t} in (M, J^{s}, ω^{s}, Ω^{s}) for small t > 0, with b^{1}(Ñ_{j}^{s,t}) = b^{1}(N_{j}) = 1.

(c) If s ∈ F^{+} then [ω^{s}] · ι_{∗}(γ_{0}) = πt^{2}k_{j} has a unique solution t > 0 if and only if k_{j} > 0. Theorem 10.5 then shows that a desingularization Ñ_{j}^{s,t} exists provided t ⩽ ϵ and |s| ⩽ t^{ϑ}, which is unique as b^{1}(Ñ_{j}^{s,t}) = b^{1}(N_{j}) = 0. By applying Theorem 10.5 not just to X but to X^{s} for small s ∈ F^{0}, one can show that such Ñ_{j}^{s,t} actually exist for all small s ∈ F^{+}.

(d) If s ∈ F^{−} then [ω^{s}] · ι_{∗}(γ_{0}) = πt^{2}k_{j} has a unique solution t > 0 if and only if k_{j} < 0. As for s ∈ F^{+}, we find that a unique desingularization Ñ_{j}^{s,t} then exists for small s ∈ F^{−}.

As k_{1}, k_{2}, k_{3} are not all zero with k_{1} + k_{2} + k_{3} = 0, there is at least one k_{j} < 0, and at least one k_{j} > 0. Suppose k_{1} < 0 and k_{2}, k_{3} > 0, for example. Imagine s ∈ F moving along a curve near 0 starting in F^{−}, crossing F^{0} and ending in F^{+}. Then initially there is one SL homology 3-sphere Ñ_{1}^{s,t} in (M, J^{s}, ω^{s}, Ω^{s}). As s crosses F^{0} this SL 3-fold collapses to a singular SL 3-fold X^{s}, with a conical singularity with cone C.

As s moves into F^{+} it is desingularized in two topologically distinct ways to give two SL homology 3-spheres Ñ_{2}^{s,t}, Ñ_{3}^{s,t}. We have found a process by which one SL homology 3-sphere can turn into two SL homology 3-spheres as we deform the underlying almost Calabi-Yau 3fold (M, J^{s}, ω^{s}, Ω^{s}). This was described in [9, §4.2], and we have now proved [9, Conj. 4.4].

In [9, Prop. 4.5] we compute H_{1}(X, ℤ), H_{1}(N_{j}, ℤ) and show:

Proposition 10.6. In the situation of Definition 10.2, let N_{j} be the compact, nonsingular 3-manifold obtained by gluing L_{j}^{a} into X′ at x, for j = 1, 2, 3. Suppose b_{cs}^{1}(X′) = 0. Then H_{1}(X, ℤ) is finite. If k_{j} ≠ 0 then H_{1}(N_{j}, ℤ) is also finite with |H_{1}(N_{j}, ℤ)| = |k_{j}| · |H_{1}(X, ℤ)|.

In the situation above with k_{1} < 0 and k_{2}, k_{3} > 0, as k_{1} +k_{2} +k_{3} = 0 we have |k_{1}| = |k_{2}| + |k_{3}|, and therefore |H_{1}(N_{1}, ℤ)| = |H_{1}(N_{2}, ℤ)| + |H_{1}(N_{3}, ℤ)| by Proposition 10.6. Now |H_{1}(N_{j}, ℤ)| is the number of flat U(1)-connections on N_{j}. Thus when Ñ_{1}^{s,t} turns into Ñ_{2}^{s,t} and Ñ_{3}^{s,t}, the number of SL homology 3-spheres with flat U(1)-connections does not change.

This is physically significant since 3-branes in string theory correspond in a classical limit to SL 3-folds with flat U(1)-connections, as in [29] for instance. The proposal of [9] is to count SL homology 3-spheres with flat U(1)-connections in a given homology class. We have shown that this number is conserved under a nontrivial kind of transition in the family of SL homology 3-spheres.

10.3 SL 3-folds with two T^{2}-cone singularities

Next we study SL 3-folds with two singularities with cone C.

Definition 10.7. Let (M, J, ω, Ω) be an almost Calabi-Yau 3-fold, and X a compact, connected SL 3-fold with exactly two conical singularities at x_{1}, x_{2}, both with cone C in (70). Then X′ = X \ {x_{1}, x_{2}} is connected. Write Σ_{1}, Σ_{2} for the two copies of Σ = C ∩ S^{5} at x_{1}, x_{2}, and identify H^{1}(Σ_{i}, ℝ) ≅ ℝ^{2} as above. Write elements of H^{1}(Σ_{1}, ℝ) ⊕ H^{1}(Σ_{2}, ℝ) = ℝ^{2} ⊕ ℝ^{2} as ((u, v), (y, z)).

Since X′ is the interior of a compact, oriented 3-manifold X′ with boundary Σ_{1} ∪ Σ_{2}, the map H^{1}(X′, ℝ) → H^{1}(Σ_{1}, ℝ) ⊕ H^{1}(Σ_{2}, ℝ) of (16) has image ℝ^{2} by Lemma 10.1. Choose a basis ((u_{1}, v_{1}), (y_{1}, z_{1})), ((u_{2}, v_{2}), (y_{2}, z_{2})) for this image. As it is also a basis over ℝ for the image of H^{1}(X′, ℚ) → H^{1}(Σ_{1}, ℚ) ⊕ H^{1}(Σ_{2}, ℚ) we can take u_{1}, …, z_{2} ∈ ℚ.

Let α_{1}, α_{2} be closed 1-forms on X′ such that the images of [α_{1}], [α_{2}] in H^{1}(Σ_{1}, ℝ) ⊕ H^{1}(Σ_{2}, ℝ) are this basis. Then α_{1} ∧ α_{2} is a closed 2-form on X′, an oriented 3-manifold with boundary Σ_{1} ∪ Σ_{2}, so by Stokes’ Theorem we have ∫_{Σ_{1}∪Σ_{2}} α_{1} ∧α_{2} = 0. This gives the consistency condition

u_{1}v_{2} − u_{2}v_{1} + y_{1}z_{2} − y_{2}z_{1} = 0. (82)

Applying Theorem 7.3 gives a necessary and sufficient criterion for when we can desingularize X by gluing in AC SL 3-folds L\begin{matrix}a_{1} \\ j_{1} \end{matrix}, L\begin{matrix}a_{2} \\ j_{2} \end{matrix} at x_{1}, x_{2}.

Theorem 10.8. Suppose (M, J, ω, Ω) is an almost Calabi-Yau 3fold, and X a compact, connected SL 3-fold with exactly two conical singularities at x_{1}, x_{2}, both with cone C in (70). Let u_{1}, …, z_{2} be as in Definition 10.7. Choose j_{1}, j_{2} = 1, 2, 3 and a_{1}, a_{2} > 0, and set L_{i} = L\begin{matrix}a_{i} \\ j_{i} \end{matrix} for i = 1, 2. Then there exists ϵ > 0 and a smooth family {Ñ^{t} : t ∈ (0, ϵ]} of compact SL 3-folds in (M, J, ω, Ω) constructed by gluing tL_{i} into into X at x_{i} if and only if

(Y (L_{1}), Y (L_{2})) ∈ ⟨((u_{1}, v_{1}), (y_{1}, z_{1})), ((u_{2}, v_{2}), (y_{2}, z_{2}))⟩ ⊂ ℝ^{2} ⊕ ℝ^{2}, (83)

where Y (L_{i}) are given in (72). In the sense of currents, Ñ^{t} → X as t → 0.

Here ϱ exists in Theorem 7.3 if and only if (Y (L_{1}), Y (L_{2})) lies in the image of H^{1}(X′, ℝ) → H^{1}(Σ_{1}, ℝ) ⊕ H^{1}(Σ_{2}, ℝ), that is, if and only if (83) holds. We are interested in how many topologically distinct ways of desingularizing X there are, and in the index of the singularities of X.

Let us use the notation of §8, so that I_{X′} is the image of H_{cs}^{1}(X′, ℝ) in H^{1}(X′, ℝ), and M_{X} the moduli space of deformations of X in (M, J, ω, Ω) as in §5, and N the compact 3-manifold obtained by gluing L_{i} = L\begin{matrix}a_{i} \\ j_{i} \end{matrix} into X at x_{i} for i = 1, 2, and M_{N} the moduli space of deformations of Ñ^{t} in (M, J, ω, Ω).

Corollary 5.3 and Theorem 2.10 show that M_{X}, M_{N} are smooth with

(84) dim M_{X} = dim I_{X′} and dim M_{N} = b^{1}(N).

As q = l_{1} = l_{2} = 1, Lemma 8.3 gives dim I_{X′} = b_{cs}^{1}(X′) − 1. Equation (52) shows that Z_{i} = Z = {0} and dim Y_{i} = 1, so that Y_{i} = ⟨Y (L_{i})⟩. Therefore

Y = ⟨(Y (L_{1}), 0), (0, Y (L_{2}))⟩ ∩ ⟨((u_{1}, v_{1}), (y_{1}, z_{1})), ((u_{2}, v_{2}), (y_{2}, z_{2}))⟩. (85)

Proposition 8.6 then gives

(86) b^{1}(N) = dim Y + b_{cs}^{1}(X′) − 1 = dim Y + dim I_{X′},

so that dim M_{N} = dim M_{X} +dim Y, and ind(X) = dim Y in the sense of §8.3.

For generic u_{1}, …, z_{2} there will be no choices of j_{1}, j_{2}, a_{1}, a_{2} for which (83) holds, and so no way to desingularize X in (M, J, ω, Ω). Here are four examples where other things happen.

Example 10.9. Suppose u_{1}, …, z_{2} are given by

((u_{1}, v_{1}), (y_{1}, z_{1}))=((1, 0), (0, 0)), ((u_{2}, v_{2}), (y_{2}, z_{2}))=((0, 0), (1, 0)), (87)

so that (82) holds. Then we can desingularize X using L_{1} = L\begin{matrix}a_{1} \\ 1 \end{matrix} and L_{2} = L\begin{matrix}a_{2} \\ 1 \end{matrix} for any a_{1}, a_{2} > 0, with ind(X) = dim Y = 2. We must have j_{1} = j_{2} = 1, so there is only one topological possibility.

Example 10.10. Let ((u_{1}, v_{1}), (y_{1}, z_{1})) = ((1, 0), (r, 0)) for r > 0 in ℚ, and ((u_{2}, v_{2}), (y_{2}, z_{2})) be generic with v_{2} + rz_{2} = 0. The only way to desingularize X is with L_{1} = L\begin{matrix}a \\ 1 \end{matrix} and L_{2} = L\begin{matrix}ra \\ 1 \end{matrix} for a > 0, with ind(X) = dim Y = 1.

Example 10.11. Let r > 0 be in ℚ with r ≠ 1, and let

((u_{1}, v_{1}), (y_{1}, z_{1}))=((1, 0), (0, r)), ((u_{2}, v_{2}), (y_{2}, z_{2}))=((0, r), (1, 0)). (88)

Then there are exactly two topologically distinct ways to desingularize X:

(a) j_{1} = 1, j_{2} = 2, a_{1} = a > 0, a_{2} = ra > 0, L_{1} = L\begin{matrix}a \\ 1 \end{matrix} and L_{2} = L\begin{matrix}ra \\ 2 \end{matrix},

(b) j_{1} = 2, j_{2} = 1, a_{1} = ra > 0, a_{2} = a > 0, L_{1} = L\begin{matrix}ra \\ 2 \end{matrix} and L_{2} = L\begin{matrix}a \\ 1 \end{matrix}.

Both have ind(X) = dim Y = 1.

Example 10.12. Suppose u_{1}, …, z_{2} are given by

((u_{1}, v_{1}), (y_{1}, z_{1}))=((1, 0), (0, 1)), ((u_{2}, v_{2}), (y_{2}, z_{2}))=((0, 1), (1, 0)), (89)

which is the case r = 1 in Example 10.11. Then there are exactly three topologically distinct ways to desingularize X, each with ind(X)= dim Y =1:

(a) j_{1} = 1, j_{2} = 2, a_{1} = a_{2} = a > 0, L_{1} = L\begin{matrix}a \\ 1 \end{matrix} and L_{2} = L\begin{matrix}a \\ 2 \end{matrix},

(b) j_{1} = 2, j_{2} = 1, a_{1} = a_{2} = a > 0, L_{1} = L\begin{matrix}a \\ 2 \end{matrix} and L_{2} = L\begin{matrix}a \\ 1 \end{matrix},

(c) j_{1} = j_{2} = 3, a_{1} = a_{2} = a > 0, and L_{1} = L_{2} = L\begin{matrix}a \\ 3 \end{matrix}.

We can explain this using the holomorphic discs D_{j}^{a} of (73). For each desingularization Ñ^{t} when t is small, we expect there to exist unique holomorphic discs D̃_{1}^{t}, D̃_{2}^{t} in (M, J), where D̃_{i}^{t} is near x_{i}, has area πa_{i} and boundary ∂D̃_{i}^{t} ⊂ Ñ^{t} for i = 1, 2.

In Example 10.9 the homology classes [∂D̃_{1}^{t}], [∂D̃_{2}^{t}] ∈ H_{1}(Ñ^{t}, ℝ) are linearly independent. Therefore by deforming Ñ^{t} as an SL 3-fold we can vary the areas of D̃_{1}^{t}, D̃_{2}^{t} independently. These two areas give two real parameters for the desingularization of X, which is why ind(X) = 2.

However, in Examples 10.10–10.12 the homology classes [∂D̃_{1}^{t}], [∂D̃_{2}^{t}] are proportional in H_{1}(Ñ^{t}, ℝ). This forces area(D̃_{2}^{t}) = c · area(D̃_{1}^{t}) to hold under deformations of Ñ^{t}, where c = r in Example 10.10 and part (a) of Example 10.11, c = r^{−1} in part (b) of Example 10.11, and c = 1 in Example 10.12. In particular, the areas of D̃_{1}^{t}, D̃_{2}^{t} can only become zero simultaneously.

Therefore the two singularities x_{1}, x_{2} in X are not independent, but coupled together. For an SL m-fold X with conical singularities x_{1}, …, x_{n} one might naïvely expect each x_{i} to be desingularized separately, and ind(X) to be the sum of contributions from each x_{i}. But in Examples 10.10–10.12 we see that x_{1}, x_{2} can only be desingularized together, and ind(X) = 1 is not a sum of separate contributions from x_{1}, x_{2}.

In Examples 10.10–10.12 the moduli spaces M_{X}, M_{N} are smooth with dim M_{N} = dim M_{X} + 1. Therefore \overline{M}_{N} is near X a nonsingular manifold with boundary M_{X}. So we have a good understanding of the boundary ∂M_{N} of M_{N}, as in §8. It is also interesting that in Examples 10.11 and 10.12 we have two or three different moduli spaces M_{N} with common boundary M_{X}.

Finally we discuss the ideas of [13] on SL fibrations of (almost) Calabi-Yau 3-folds, as required by the SYZ Conjecture [29]. Let (M, J, ω, Ω) be an almost Calabi-Yau 3-fold and f : M → B an SL fibration. That is, B is a compact 3-manifold, f is continuous and piecewise smooth, and for some Δ ⊂ B with B \ Δ open and dense the fibres X_{b} = f^{−1}(b) are SL 3-tori for b ∈ B \ Δ, and singular SL 3-folds for b ∈ Δ.

The idea of [13, §7–§8] is that for ω generic in its Kähler class, Δ should be of codimension 1 in B, and for b ∈ Δ generic X_{b} should have 2 (or 2n) singularities with cone C, as in Definition 10.7. Then Δ is locally a hypersurface which divides B \ Δ into two pieces. These are two different moduli spaces of SL 3-tori with common boundary Δ, as in Examples 10.11 and 10.12. Our results provide a partial proof of speculations in [13, §8.2(a)].

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Lincoln College, Oxford