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.