I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19):

Wirtinger's Inequality.

Let $L$ be a complex linear space and let $M$ be a real even-dimensional subspace. Let $H$ be a positive definite Hermitian form on $L$ . Then $H = S + iA$ where $S$ is symmetric and $A$ is alternating . Let $\{ m_1,...,m_{2k} \} $ be a basis of $M$ which is orthonormal with respect to $S$ . Then $$ | A ^k ( m_1 , . . . , m_{2 > k} ) | \le k !$$ with equality holding precisely when $M$ is a complex $k$-dimensional subspace of $L$ . (Here $A^k$ is the $k$-th exterior power of $A$ .)

By a suitable shuffling of the basis it can be arranged that $A ^k ( > m_1 , . . . , m_{2 k} ) \ge 0$

Let $z_1, . . . , z_n$ be coordinates on complex $n$ - space $\mathbb{C}^n$ and set

$$\omega = \frac{i}{2} \sum_{j=1}^n dz_j \wedge d \overline{z}_j$$

This is the fundamental $2$-form of the standard Kahler structure on $\mathbb{C}^n$ and at each point $p \in \mathbb{C}^n$ $\omega_p$ is the alternating part of the positive definite Hermetian form:

$$\sum_{j=1}^n dz_{j(p)} \cdot d \overline{z}_{j(p)}$$

on the tangent space to $\mathbb{C}^n$ at $p$.

Therefore , if $\mathcal{M}$ is any smoothn $2k$-dimensional manifold immersed in $\mathbb{C}^ n$ , Wirtinger's Inequality implies immediately that:

$$\int_{\mathcal{M}} \frac{1}{k!} \omega ^k \le \int_{\mathcal{M}} 1 \ d > \mathcal{M} = \text{Volume} _{2k}(\mathcal{M})$$

with equality is and only if $ \mathcal{M}$ is a complex $k$-dimensional manifold.

Also each $\frac{1}{k!} \omega ^k $ is an exact $2k$ form.

My questions are:

Could you explain to me how we use the fact that $\omega$ is the fundamental $2$-form of the standard Kahler structure on $\mathbb{C}^n$ to apply the Wirtinger inequality here?

Is this somehow connected to this Second fundamental form ?

Could you explain to me what the fundamental $2$-form of the standard Kahler structure is? Or recommend a good source in which I could read about it?

In Werner Ballmann's Lectures on Kahler Manifolds the author defines the *associated Kahler form* (which I presume could be the same as the fundamental $2$-form) in this way:

Let $M$ be a complex manifold with complex structure $J$ and compatible Riemannian metric $g = < \cdot, \cdot > $ (so $<JX, JY>= < X, Y > $). The alternating $2$-form $\omega(X, Y ) := g(JX, Y )$ is called the associated Kahler form. We say that $g$ is a Kahler metric and if $\omega$ is closed, we say that $(M, g)$ is a Kahler manifold.

On page 48, the author states that we can

view $TM$ together with $J$ as a complex vector bundle over $M$, and let $h$ be a Hermitian metric on $TM$. Then $g = Re h$ is a compatible metric on $M$ and $Imh$ is the associated Kahler form: $$g(JX, Y ) = \Re h(JX, Y ) = \Re h(iX, Y ) = \Re(−ih(X, Y )) = \Im h(X, Y )$$ If $g$ is a compatible Riemannian metric on $M$ and $\omega$ is the associated Kahler form, then $h = g + i\omega$ is a Hermitian metric on $TM$.

Also, a Riemannian metric is a general notion. Looking at what I've pasted into the frame above, do we need to consider Riemannian metrics compatible with the complex structure in general or not necessarily?

I would be very grateful for all your insight.