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.