Let $g$ be a Riemannian metric on an almost complex manifold $(M,J)$. Suppose $g$ is Hermitian in the sense that
$$g(JX,JY) = g(X,Y)$$
Let $\Omega$ be the associated fundamental (Kahler) form
$$\Omega(X,Y) = g(JX,Y)$$
We may extend $g$ and $\Omega$ complex linearly to be defined on $TM^{\mathbb{C}}$.
I've seen it written that the following defines a Hermitian structure on $M$
$$h(X,Y) = g(X,Y) - i\Omega(X,Y)$$
My question is why?! I'm assuming this is defined on $TM^\mathbb{C}$: am I correct? How does it relate to the "standard" Hermitian structure on the holomorphic bundle $TM^+$ given by
$$h^+(X,Y) = g(X,\overline{Y})\textrm{?}$$
Attempted argument
We want to show $\overline{h(X,Y)}=h(Y,X)$. It's easy to prove $\overline{g(X,Y)}=g(\overline{X},\overline{Y})$. We write
$$\overline{h(X,Y)} = g(\overline{X},\overline{Y}) + ig(\overline{JX},\overline{Y})$$
We'll try cases. If $X$ and $Y$ are both in $TM^+$ or $TM^-$ one can easily show
$$g(X,Y) = g(\overline{X},\overline{Y})=0$$
Note : therefore this is very different from the "standard" Hermitian structure.
By symmetry we only need try $X\in TM^+, Y\in TM^-$. Then
$$\overline{h(X,Y)} = 2g(\overline{X},\overline{Y})$$
$$h(Y,X) = 2g(X,Y)$$
But these aren't equal necessarily! What on earth am I doing wrong here? It should be dead simple, but I'm just going round in circles. If someone could point out the silly mistake I'm making I'd be very grateful!
Assume $(M,J)$ is an almost complex manifold of real dimension $2n$: for example a complex manifold of complex dimension $n$.
Given a $J$-invariant riemannian structure $g$ on $(M,J)$, you derive from it:
a) A $J$-invariant hermitian structure $h$ defined by $h(X,Y)=g(X,Y)-ig(JX,Y)$ on $T(M)$.
Here T(M) is seen, by using $J$, as a complex vector bundle of complex rank $n$.
Hermitianity means that you have, among other conditions, $h(X,JY)=-ih(X,Y)$ .
b) An alternating $\mathbb C$-bilinear $2$-form $\omega (X,Y)\stackrel {def}{=}g(JX,Y)$ on $T(M)$
Here $\mathbb C$-bilinear means that you have, among other conditions, $\omega(X,JY)=i\omega(X,Y)$
These structures are related by the obvious equation $h(X,Y)=g(X,Y)-i\omega (X,Y)$ .
Everything happens on $T(M)$ provided with its complex $J$-structure and there is no need to complexify the vector bundle $T(M)$.