We begin by defining calibrations and calibrated submanifolds, following Harvey and Lawson [9].

Now let $\varphi$ be a closed $k$-form on $M$. We say that $\varphi$ is a calibration on $M$ if for every oriented $k$-plane $V$ on $M$ we have $\varphi|_{V}\leqslant\mathop{\rm vol}_{V}$. Here $\varphi|_{V}=\alpha\cdot\mathop{\rm vol}_{V}$ for some $\alpha\in\mathbin{\mathbb{R}}$, and $\varphi|_{V}\leqslant\mathop{\rm vol}_{V}$ if $\alpha\leqslant 1$. Let $N$ be an oriented submanifold of $M$ with dimension $k$. Then each tangent space $T_{x}N$ for $x\in N$ is an oriented tangent $k$-plane. We say that $N$ is a calibrated submanifold if $\varphi|_{T_{x}N}=\mathop{\rm vol}_{T_{x}N}$ for all $x\in N$.

It is easy to show that calibrated submanifolds are automatically minimal submanifolds [9, Th. II.4.2]. Here is the definition of special Lagrangian submanifolds in $\mathbin{\mathbb{C}}^{m}$, taken from [9, §III].

As in [10, 11] there is a more general definition of special Lagrangian $m$-fold involving a phase ${\rm e}^{i\theta}$, but we will not use it here. Harvey and Lawson [9, Cor. III.1.11] give the following alternative characterization of special Lagrangian submanifolds.

An $m$-dimensional submanifold $L$ in $\mathbin{\mathbb{C}}^{m}$ is called Lagrangian if $\omega|_{L}\equiv 0$. Thus special Lagrangian submanifolds are Lagrangian submanifolds satisfying the extra condition that $\mathop{\rm Im}\Omega|_{L}\equiv 0$, which is how they get their name.

Next we give a result characterizing SL 3-planes $\mathbin{\mathbb{R}}^{3}$ in $\mathbin{\mathbb{C}}^{3}$, which will be useful in §7. Define an anti-bilinear cross product $\times:\mathbin{\mathbb{C}}^{3}\times\mathbin{\mathbb{C}}^{3}\rightarrow\mathbin{\mathbb{C}}^{3}$ by

It is equivariant under the $\mathop{\rm SU}(3)$-action on $\mathbin{\mathbb{C}}^{3}$. Using this notation, we prove

Also we have $\omega({\bf r},{\bf s})=\omega({\bf r},{\bf r}\times{\bf s})=\omega({\bf s},{\bf r}\times{\bf s})=0$ by (4), so that $\langle{\bf r},{\bf s},{\bf r}\times{\bf s}\rangle_{\scriptscriptstyle\mathbb{R}}$ is a Lagrangian 3-plane. Then (6) shows that $\langle{\bf r},{\bf s},{\bf r}\times{\bf s}\rangle_{\scriptscriptstyle\mathbb{R}}$ is a special Lagrangian 3-plane, by Proposition 2.7. It is easy to see that this is the only SL 3-plane in $\mathbin{\mathbb{C}}^{3}$ containing $\langle{\bf r},{\bf s}\rangle_{\scriptscriptstyle\mathbb{R}}$. $\square$

Now special Lagrangian submanifolds in $\mathbin{\mathbb{C}}^{m}$ are the local model for special Lagrangian submanifolds in Calabi–Yau manifolds. We adopt the following definition of Calabi–Yau manifolds, which is not the usual one, but will be useful for our purposes.

Generally we will refer to a Calabi–Yau $m$-fold as $X$, taking $J,\omega,\Omega$ as given. Note that if $(X,J,\omega,\Omega)$ is a CY $m$-fold and $\theta\in[0,2\pi)$ then $(X,J,\omega,{\rm e}^{i\theta}\Omega)$ is also a CY $m$-fold. It can be shown that $g$ is Ricci-flat with holonomy group ${\textstyle\mathop{\rm Hol}}(g)$ contained in $\mathop{\rm SU}(m)$, and that $\nabla\omega=\nabla\Omega=0$, where $\nabla$ is the Levi-Civita connection of $g$. Furthermore, as $\Omega$ is a nonvanishing section of the canonical bundle $K_{X}=\Lambda^{m,0}X$ of $X$, we see that $K_{X}$ is trivial, so that the first Chern class $c_{1}(X)$ is zero.

Here is how to construct examples of Calabi–Yau manifolds. Using algebraic geometry one can find many examples of compact complex manifolds $(X,J)$ with $c_{1}(X)=0$, for instance as hypersurfaces in toric varieties. If $X$ is simply-connected, $K_{X}$ is trivial, and has a nonvanishing holomorphic section $\Omega$.

If $X$ admits Kähler metrics, then as $c_{1}(X)=0$ Yau’s solution of the Calabi Conjecture shows that there exists a unique Ricci-flat Kähler metric $g$ in each Kähler class, with Kähler form $\omega$. It then follows that $\nabla\omega=\nabla\Omega=0$, and therefore that $\omega^{m}$ is a constant multiple of $\Omega\wedge\bar{\Omega}$. We can rescale $\Omega$ by a constant factor to make (7) hold, and then $(X,J,\omega,\Omega)$ is a CY $m$-fold.

We move on to discuss special Lagrangian submanifolds of Calabi–Yau manifolds. Following Definition 2.1, we define:

Then Proposition 2.3 gives:

Now $\omega$ and $\mathop{\rm Im}\Omega$ are closed forms on $X$, and so if $N$ is an $m$-dimensional submanifold of $X$ then we can consider the de Rham cohomology classes $[\omega|_{N}]$ in $H^{2}(N,\mathbin{\mathbb{R}})$ and $[\mathop{\rm Im}\Omega|_{N}]$ in $H^{m}(N,\mathbin{\mathbb{R}})$. Clearly, by the proposition, these have to be zero for $N$ to be an SL $m$-fold. Furthermore, they are invariant under continuous deformations of $N$ as a submanifold in $X$, and so they have to be zero for any deformation of $N$ to be special Lagrangian. So we prove:

This gives us a cohomological obstruction to finding special Lagrangian submanifolds in $X$. The deformation theory of special Lagrangian submanifolds was studied by McLean [14, §3], who proved the following result.

The idea in the proof of this theorem is that an infinitesimal deformation of $N$ as a submanifold in $X$ corresponds to a section of the normal bundle $\nu$ of $N$ in $X$. But because $N$ is Lagrangian, contracting with $\omega$ gives an isomorphism between the vector bundles $\nu$ and $T^{*}N$ over $N$. So there is a 1-1 correspondence between infinitesimal deformations of $N$ in $X$ and 1-forms $\alpha$ on $N$.

McLean shows that $\alpha$ corresponds to an infinitesimal deformation of $N$ as an SL submanifold if and only if ${\rm d}\alpha={\rm d}^{*}\alpha=0$. But as $N$ is compact, by Hodge theory the vector space of 1-forms $\alpha$ with ${\rm d}\alpha={\rm d}^{*}\alpha=0$ is isomorphic to $H^{1}(N,\mathbin{\mathbb{R}})$, and so has dimension $b^{1}(N)$.

Our next result concerns the stability of compact special Lagrangian submanifolds $N$ under small deformations of the underlying Calabi–Yau $m$-fold $(X,J,\omega,\Omega)$. McLean does not discuss this question, but it can be answered by the same techniques used to prove Theorem 2.9.

Note that the condition $[\tilde{\omega}|_{N}]=0$ in $H^{2}(N,\mathbin{\mathbb{R}})$ is automatically satisfied if the image of $H_{2}(N,\mathbin{\mathbb{R}})$ in $H_{2}(X,\mathbin{\mathbb{R}})$ is zero, and the condition $[\mathop{\rm Im}\tilde{\Omega}|_{N}]=0$ can always be satisfied by choosing the phase of $\tilde{\Omega}$ correctly. So the conditions $[\tilde{\omega}|_{N}]=[\mathop{\rm Im}\tilde{\Omega}|_{N}]=0$ are often not very stringent.

Later in the paper we will discuss fibrations of Calabi–Yau 3-folds by special Lagrangian 3-folds. We will be particularly interested in the question of what kind of fibrations one should expect in a generic Calabi–Yau 3-fold. Now when one speaks of a geometric object as generic, usually one thinks of it as chosen at random out of an infinite-dimensional family of similar geometric objects.

However, the family of Calabi–Yau structures $(X,J,\omega,\Omega)$ on a compact 6-manifold $X$, up to diffeomorphism, is only finite-dimensional, of dimension $h^{1,1}(X)+2h^{2,1}(X)+1$. Thus the family of Calabi–Yau structures on a compact 6-manifold is too small for the device of picking a generic Calabi–Yau 3-fold $X$ to be really useful in a proof.

To get round this we introduce the following generalizations of Calabi–Yau and special Lagrangian geometry.

The difference between this and Definition 2.2 is that we do not require $\omega$ and $\Omega$ to satisfy equation (7). For some purposes it may be useful to restrict to real analytic Kähler forms $\omega$, but we will not worry about this in this paper. Here is the appropriate definition of SL $m$-folds in ACY $m$-folds.

By Proposition 2.7, if $(X,J,\omega,\Omega)$ is Calabi–Yau rather than almost Calabi–Yau, then $N$ is special Lagrangian in the sense of Definition 2.2. Thus, this is a genuine extension of the idea of special Lagrangian submanifold.

The idea of extending special Lagrangian geometry to almost Calabi–Yau manifolds is not new. It appears in the work of Goldstein, who introduced the term ‘almost Calabi–Yau’, and Bryant [1, §1], who uses the term ‘special Kähler’ instead of ‘almost Calabi–Yau’.

Many of the good properties of special Lagrangian submanifolds in Calabi–Yau manifolds also apply in almost Calabi–Yau manifolds. For instance:

This is because the proofs of these results only really depend on the conditions $\omega|_{N}\equiv\mathop{\rm Im}\Omega|_{N}\equiv 0$, and the pointwise connection (7) between $\omega$ and $\Omega$ is not important. Let $(X,J,\omega,\Omega)$ be an ACY $m$-fold, with metric $g$. In general, SL $m$-folds in $X$ are neither calibrated nor minimal with respect to $g$.

However, let $f:X\rightarrow(0,\infty)$ be the unique smooth function such that $f^{2m}\omega^{m}/m!=(-1)^{m(m-1)/2}(i/2)^{m}\Omega\wedge\bar{\Omega}$, and define $\tilde{g}$ to be the conformally equivalent metric $f^{2}g$ on $X$. Then it is easy to show that $\mathop{\rm Re}\Omega$ is a calibration on the Riemannian manifold $(X,\tilde{g})$, and that SL $m$-folds $N$ in $(X,J,\omega,\Omega)$ are calibrated with respect to it, so that they are minimal with respect to $\tilde{g}$.

We can also give a volume bound for compact SL $m$-folds in $X$ using these ideas. If $N$ is an SL $m$-fold in $X$ then $\mathop{\rm Re}\Omega|_{T_{x}N}=f(x)^{m}\mathop{\rm vol}_{T_{x}N}$ for each $x\in N$, where $\mathop{\rm vol}_{T_{x}N}$ is computed using $g$. Integrating this over $N$ yields

This is a bound on the volume of $N$ using $g$, depending only on $(X,J,\omega,\Omega)$ and the homology class of $N$.

Another important point about SL $m$-folds in ACY $m$-folds is that locally, in a small neighbourhood of any point, they look like SL $m$-folds in CY $m$-folds. Therefore we expect the singularities of SL $m$-folds in ACY $m$-folds to behave in the same way as singularities of SL $m$-folds in CY $m$-folds.

Almost Calabi–Yau $m$-folds are useful in problems involving special Lagrangian fibrations because it is very easy to write down explicit examples of ACY $m$-folds, and so to study fibrations on them, whereas Calabi–Yau structures on the same $m$-folds are usually given only by an existence theorem, with no explicit formula for the metric or Kähler form.

The reason why almost Calabi–Yau manifolds are useful in discussing problems involving generic Calabi–Yau manifolds is the following. Although the family of Calabi–Yau structures on a compact manifold $X$ is finite-dimensional, the family of almost Calabi–Yau structures on the same manifold is infinite-dimensional. So arguments involving choosing a generic almost Calabi–Yau structure on $X$ will be far more powerful.

Here is an example of the kind of thing the author has in mind, which we will return to in §3.1. Let $(X,J,\omega,\Omega)$ be a CY 3-fold, and $N$ a compact, nonsingular, immersed SL 3-fold in $X$. If $N$ is generically placed in $X$ as an immersed submanifold then it will intersect itself in only finitely many points, but if $N$ is very nongeneric it could intersect itself in a 1-dimensional set such as a circle.

Now if we choose a generic Calabi–Yau structure on $X$, we cannot guarantee that $N$ will not intersect itself in a circle, because we cannot completely eliminate the possibility that by a miracle, all the Calabi–Yau structures on $X$ happen to have this property. (Indeed, this is to be expected when $X$ is a product $K3\times T^{2}$). However, one can show that in a generic almost Calabi–Yau 3-fold, any compact, nonsingular, immersed SL 3-fold $N$ intersects itself in only finitely many points.

One of the key claims of this paper is that generic special Lagrangian fibrations are not smooth. So we should begin by defining what we mean by ‘generic’ here.

Here is the reasoning behind this definition. We intend to call a property of a special Lagrangian fibration generic if it holds for fibrations of all nearby almost Calabi–Yau 3-folds $(X,\tilde{J},\tilde{\omega},\tilde{\Omega})$. Now if $N$ is a smooth fibre of $f$, then Corollary 2.8 and Theorem 2.10 show that the only obstructions to finding an SL 3-fold in $(X,\tilde{J},\tilde{\omega},\tilde{\Omega})$ near $N$ are that $[\tilde{\omega}|_{N}]\equiv[\mathop{\rm Im}\tilde{\Omega}|_{N}]\equiv 0$.

To make sure this holds, we restrict our attention to ACY 3-folds $(X,\tilde{J},\tilde{\omega},\tilde{\Omega})$ with $[\tilde{\omega}]=[\omega]$ in $H^{2}(X,\mathbin{\mathbb{R}})$ and $[\mathop{\rm Im}\tilde{\Omega}]=[\mathop{\rm Im}\Omega]$ in $H^{3}(X,\mathbin{\mathbb{R}})$. But one can show that if $[\mathop{\rm Im}\tilde{\Omega}]=[\mathop{\rm Im}\Omega]$ and $(X,J,\Omega)$, $(X,\tilde{J},\tilde{\Omega})$ are close, then they are isomorphic. So we may as well fix $\tilde{J}=J$ and $\tilde{\Omega}=\Omega$, and just vary the Kähler form $\tilde{\omega}$ within the Kähler class of $\omega$.

It could be asked why Definition 3.1 is a good definition, if we are only interested in Calabi–Yau and not in almost Calabi–Yau 3-folds. My answer is that anything that is true of special Lagrangian fibrations of generic Calabi–Yau 3-folds with holonomy $\mathop{\rm SU}(3)$ really ought to be true of nearby generic almost Calabi–Yau 3-folds as well, as we know of no relevant geometric properties of such CY 3-folds that do not also hold for ACY 3-folds.

We now give some reasons why generic special Lagrangian fibrations $f:X\rightarrow B$ cannot be smooth. Gross [6, §1] gives the following rough argument why $f$ should be smooth. Let $X_{b}=f^{-1}(b)$ be a singular fibre for $b\in B$. Then $X_{b}$ is nonsingular at a general point $x\in X_{b}$. Using the exponential map at $x$ on the normal vector space $\nu_{x}$ to $X_{b}$ at $x$ gives a natural, smooth local section for $f:X\rightarrow B$. Projecting this down to $B$ using $f$, we define the structure of a smooth manifold on $B$ near $b$. Hopefully $f$ will be smooth with respect to this.

The problem with this argument is as follows. For two different nonsingular points $x,y$ in $X_{b}$, the maps $f\circ\exp_{x}:\nu_{x}\rightarrow B$ and $f\circ\exp_{y}:\nu_{y}\rightarrow B$ do define smooth structures on $B$ near $b$. However, in general these will be different smooth structures. There will be no one smooth structure near $b$ such that $f$ is smooth at every point of $X_{b}$, even at every nonsingular point.

Next, we discuss the codimension of the set of singular fibres in the base $B$, and the dimension of the singular set in a generic singular fibre $X_{b}$. The assumption that $f:X\rightarrow B$ is smooth has strong consequences for these. In particular, Gross [6, p. 10] proves:

His proof uses the fact that the fibres are both Lagrangian and minimal. The Lagrangian assumption is used to prove [6, Prop. 2.2] that if $x\in X_{b}$ and $\mathop{\rm rank}{\rm d}_{x}f:T_{x}X\rightarrow T_{b}B$ is $k$, then $X_{b}$ contains a $k$-dimensional submanifold through $x$ on which $\mathop{\rm rank}{\rm d}f$ is $k$. But by a result of Almgren, the singularities of a minimal submanifold are of Hausdorff codimension at least two. Combining these two shows that $\mathop{\rm rank}{\rm d}_{x}f$ cannot be $m-1$, so that if $x$ is a singular point of $X_{b}$ then $\mathop{\rm rank}{\rm d}_{x}f\leqslant m-2$.

Using these ideas, one can show that if $f:X\rightarrow B$ is a smooth special Lagrangian fibration of an (almost) Calabi–Yau 3-fold, and $\Delta$ the set of $b\in B$ with $X_{b}=f^{-1}(b)$ singular, then under good circumstances we expect the following properties:

• (i)

  $\Delta$ is a union $\Delta_{0}\cup\Delta_{1}$, where $\Delta_{0}$ is a finite set of points, and $\Delta_{1}$ a finite set of open intervals. Essentially, $\Delta$ is a graph in $B$.

• (ii)

  For each $b\in\Delta_{1}$, the singular set of $X_{b}$ is a finite number of circles ${\mathcal{S}}^{1}$, and the singularities are locally modelled on $L\times\mathbin{\mathbb{R}}$ in $\mathbin{\mathbb{C}}^{2}\times\mathbin{\mathbb{C}}$, where $L$ is a special Lagrangian 2-fold in $\mathbin{\mathbb{C}}^{2}$ with an isolated singularity at 0.

That is, singular fibres occur in codimension two in the base, and the generic singular fibre has a one-dimensional singular set.

Ruan [18, §4] argues that as in two dimensions special Lagrangian fibrations have singular fibres of codimension two in the base, it is reasonable to expect this in three dimensions as well. The problem with this argument is that two dimensions is a special case: SL 2-folds in a Calabi–Yau 2-fold $X$ are equivalent to complex curves with respect to an alternative complex structure on $X$. So singularities occur in complex codimension one, which is real codimension two. But for $m\geqslant 3$ there is no such complex interpretation of SL $m$-folds.

Now in the fibrations we shall define later in the paper, singular fibres occur in codimension one in the base, and all the singular fibres have zero-dimensional singular sets. We claim that is what one should expect of generic special Lagrangian fibrations in three dimensions.

Here is a heuristic argument why fibrations satisfying (i) and (ii) above cannot be generic. Suppose $X$ is a generic almost Calabi–Yau 3-fold, $f:X\rightarrow B$ a special Lagrangian fibration satisfying (i) and (ii), and let $b\in\Delta_{1}$. Then the singular set of $X_{b}$ is a finite number of circles ${\mathcal{S}}^{1}$, with singularities locally modelled on $L\times\mathbin{\mathbb{R}}$ in $\mathbin{\mathbb{C}}^{2}\times\mathbin{\mathbb{C}}$, where $L$ is an SL 2-fold in $\mathbin{\mathbb{C}}^{2}$ with an isolated singularity at 0.

As $X$ is generic, it is reasonable to expect that $L$ should be the most generic kind of singular SL 2-fold. But the most generic singularity of SL 2-folds is the normal crossing, where $L$ is the union of two distinct SL 2-planes $\mathbin{\mathbb{R}}^{2}$ in $\mathbin{\mathbb{C}}^{2}$ intersecting at 0. Assume the singularities of $X_{b}$ are of this kind.

Then $X_{b}$ is in fact nonsingular as an immersed 3-submanifold. So we can regard $X_{b}$ as a compact, nonsingular, immersed SL 3-fold in $X$. It intersects itself in a collection of circles, but a generic immersed 3-submanifold in $X$ should intersect itself in finitely many points. Thus, as an immersed 3-submanifold $X_{b}$ is not generic, and indeed highly non-generic, as submanifolds of this kind are of infinite codimension in the family of all immersed 3-submanifolds.

Now the local deformation theory of compact immersed SL 3-folds is well understood (see for example Theorems 2.9 and 2.10). It is easy to show that in a generic almost Calabi–Yau 3-fold, compact immersed SL 3-folds should be of at most finite codimension in the family of all immersed 3-submanifolds. Therefore $X_{b}$ is an SL 3-fold of a kind that should not occur in a generic almost Calabi–Yau 3-fold, which is a contradiction. The author believes that this argument could be upgraded to a rigorous proof without difficulty.

We now describe some aspects of the work of Wei-Dong Ruan in [18, 19, 20, 21]. This is based on the following idea. Suppose we are given a family of Calabi–Yau hypersurfaces $X$ in some projective toric variety. As in [18, 19, 21] we take this to be a pencil of quintics $\bigl\{X_{c}:c\in\mathbin{\mathbb{C}}\cup\{\infty\}\bigr\}$ in $\mathbb{CP}^{4}$, where

and $p,q$ are homogeneous, linearly independent quintic polynomials.

Choose a Kähler metric $g$ on $\mathbb{CP}^{4}$, with Kähler form $\omega$. Let $s$ be the meromorphic function $p(z_{0},\ldots,z_{4})/q(z_{0},\ldots,z_{4})$ on $\mathbb{CP}^{4}\setminus X_{\infty}$, and let $f=\mathop{\rm Re}(s)$. Define a vector field $v$ on $\mathbb{CP}^{4}$ by $v^{a}=|{\rm d}f|^{-2}g^{ab}({\rm d}f)_{b}$, using the index notation for tensors. Note that $v$ becomes infinite on $X_{\infty}$, as $f$ is infinite there, and also on the set of points where ${\rm d}f=0$. Ruan shows that flowing along the vector field $v$ for time $t$ takes $X_{c}$ to $X_{c+t}$ for each $c\in\mathbin{\mathbb{C}}$, at least where $v$ is finite. Furthermore, the flow takes Lagrangian submanifolds of $X_{c}$ to Lagrangian submanifolds of $X_{c+t}$.

Ruan’s method is to set $p(z_{0},\ldots,z_{4})=z_{0}z_{1}z_{2}z_{3}z_{4}$, so that $X_{0}$ is the union of five copies of $\mathbb{CP}^{3}$ in $\mathbb{CP}^{4}$, a very degenerate, singular quintic. He defines an explicit Lagrangian fibration of $X_{0}$, with respect to the Fubini–Study metric on $\mathbb{CP}^{4}$. Then he uses the flow from $X_{0}$ to $X_{t}$ to translate this fibration to a Lagrangian fibration of the general, nonsingular quintic $X_{t}$ for $t\in\mathbin{\mathbb{R}}\setminus\{0\}$. One has to consider carefully what happens when $v$ is infinite, and around the singularities of $X_{0}$. But it turns out that these do not spoil things, and we end up with a genuine Lagrangian fibration of $X_{t}$.

Part of the motivation for Ruan’s construction is that $X_{0}$ is considered to be the ‘large complex structure limit’ of Calabi–Yau quintics. Thus, the construction starts with an explicit fibration of the singular ‘large complex structure limit’ 3-fold, and deforms it to a fibration of nonsingular 3-folds close to this limit. This is quite a natural thing to do from the String Theory point of view, and others such as Zharkov and Goldstein have tried similar ideas.

Now we are interested in the nature of the set of singular fibres in Ruan’s fibrations, and in their singularities. Ruan proves [19, Th. 2.2]:

The properties of Ruan’s fibrations given above are very similar to the fibrations we shall propose later in the paper. In particular, versions of parts (i) and (iii)–(vi) will hold for our fibrations. For part (ii), the set of singular points in the fibrations we shall discuss need not be a holomorphic curve, but it will be a real 2-manifold in $X$ that is close to being holomorphic. Only in part (vii) do we seriously diverge from Ruan, as our fibrations will not contain fibres in which $T^{2}$ collapses to a point.

Ruan himself, however, appears to regard these properties of his fibrations as a problem (see for instance [21, Conj. 1.1], where he conjectures that special Lagrangian fibrations are always smooth, the ‘Precise SYZ mirror conjecture’ in [20, §9], and many other places), and spends much effort in showing how to deform his fibrations to smooth Lagrangian fibrations. One moral of this paper may be that Ruan’s construction gives something quite close to the right answer, and it might even be possible to modify it to yield genuine special Lagrangian fibrations of (almost) Calabi–Yau manifolds.