# The Wirtinger theorem proof

Let $M$ be a complex hermitian manifold with symplectic form $\Omega$ and $N \subseteq M$ be its smooth $2n$-dimensional (real dimensions, $2n \geq 2$) compact submanifold. I want to show the inequality $$\text{vol}(N) \geq \frac{1}{n!} \int\limits_N \Omega^n.$$ If $N$ is a complex submanifold then choosing in the neighborhood of each point holomorphic coordinates $z_1$, $\ldots$, $z_n$, such that in this neighborhood the hermitian metric is equal to $\delta_{kl}$ we can obtain $$\Omega = \frac{i}{2} \sum_k dz_k \wedge d\bar z_k, \quad d\text{vol} = (i/2)^n dz_1 \wedge d\bar z_1\wedge \cdots \wedge dz_n \wedge d\bar z_n,$$ and we can see that $\Omega^n = n!d\text{vol}$. Hence $$\text{vol}(N) = \frac{1}{n!} \int\limits_N \Omega^n$$ for the complex submanifold $N$. But how to show that the inequality holds even for smooth submanifolds $N$?