In Huybrecht's text, he defines a Riemannian metric $g$ on a complex manifold $M$ to be a hermitian structure/metric on $M$ if for any point $x\in M$, the scalar product $g_x\in T_xM$ is compatible with the induced almost complex structure $I_x$, i.e. $g_x(v,w)=g_x(I(v),I(w))$ for all $v,w\in T_xM$. The induced real $(1,1)$-form $\omega=g(I(~),(~))$ is then called the fundamental form.
He refers to this form $\omega$ as being 'real' and of type $(1,1)$. I am not sure what he means here - does the real part refer to $\omega$ as defined as a real form above, whereas of type $(1,1)$ means the complexification of $\omega$ is of bidegree $(1,1)$? If so, how is one to define this complexification? I don't really understand the need to include the word 'real' here, since by definition $\omega$ is a real form. I'm slightly confused as to when we'd want to consider $\omega$ as a real form, and when we would want to consider it as a form of type $(1,1)$ - could someone please clarify this?
Thanks in advance.
A form of type $(1,1)$ is a linear combination of "balanced" forms $dz^j\wedge d\bar z^k$. To say it's real means literally that it equals its conjugate. So if we have $$\omega = \sum g_{j\bar k} dz^j\wedge d\bar z^k,$$ then $$\bar\omega = \sum \bar g_{j\bar k} d\bar z^j\wedge dz^k = -\sum\bar g_{j\bar k} dz^k\wedge d\bar z^j = -\sum\bar g_{k\bar j} dz^j\wedge d\bar z^k$$ is equal to $\omega$ if and only if $g_{j\bar k} = -\bar g_{k\bar j}$ for all $j,k$. This is equivalent to saying that the matrix $\sqrt{-1}\big[g_{j\bar k}\big]$ is hermitian.
(Note the simplest example: In dimension $1$, the $(1,1)$-form $\sqrt{-1}dz\wedge d\bar z = \sqrt{-1} (-2\sqrt{-1}dx\wedge dy) = 2dx\wedge dy$ is real.)