only if all of the $\theta_j=0$ which means that $P$ is complex.

Putting together Theorem \[cxcalib.thm\] and Theorem \[calibmin.thm\] yields the following.

$\!\!$Compact complex submanifolds of Kähler manifolds are homologically volume-minimizing.

We know that complex submanifolds are defined by holomorphic functions; i.e. solutions to the Cauchy–Riemann equations, which are a first-order PDE system, as one would expect for calibrated submanifolds.

$N=\{(z,\frac{1}{z})\in\C^2\,:\,z\in\C\setminus\{0\}\}$ is a complex curve in $\C^2$, and thus is calibrated.

An important non-trivial example of a Kähler manifold is $\C\P^n$, where the zero set of a system of polynomial equations defines a (possibly singular) complex submanifold.

Special Lagrangians
===================

Complex submanifolds are very familiar, but can we find any other interesting classes of calibrated submanifolds? The answer is that indeed we can, particularly when the manifold has special holonomy. We begin with the case of holonomy $\operatorname{SU}(n)$ – so-called *Calabi–Yau manifolds*. The model example for Calabi–Yau manifolds is $\C^n$ with complex structure $J$, Kähler form $\omega$ and holomorphic volume form $$\Upsilon=\d z_1\wedge\ldots\wedge\d z_n,$$ if $z_1,\ldots,z_n$ are complex coordinates on $\C^n$.

Let $M$ be a Calabi–Yau manifold with holomorphic volume form $\Upsilon$. Then $\operatorname{Re}(e^{-i\theta}\Upsilon)$ is a calibration for any $\theta\in\R$.

Since $\d\Upsilon=0$, the result follows immediately from the following.

\[Lagcalib.thm\] 