$\newcommand{\conjugate}{\overline} \newcommand{\tensor}{\otimes} \newcommand{\IC}{\mathbb{C}} \newcommand{\IR}{\mathbb{R}} \newcommand{\blank}{{-}} \newcommand{\dual}{\vee} \newcommand{\i}{\sqrt{-1}}$ I am trying to read the notes http://sebastien.boucksom.perso.math.cnrs.fr/notes/L2.pdf . My question is about linear algebra, more specifically about Section 5.1. I am not used to complex geometry and my question is basically, whether I am interpreting everything correctly (because I actually thought I had understood the subject of linear algbera). The parts in quote stem more or less verbatim from Boucksom's notes.
Let $V$ be a finite dimensional complex vector space and denote by $$\Lambda = \bigoplus_{p,q\in \mathbb{N}} \Lambda^{p,q}V^\dual$$ the corresponding bigraded exterior algebra.
First question: Is it correct that $\Lambda^{p,q}V^\dual$ can be defined by $$\Lambda^{p,q}V^\dual = (\bigwedge^p V \tensor_\IC \bigwedge^q\conjugate{V})^\dual?$$ Here by $\conjugate{V}$ I denote the same real vector space as $V$, but with conjugate complex scalar multiplication. This would mean that it consists of maps $V^{p+q} \to \IC$ which are alternating and linear in the first $p$ variables and alternating and antilinear in the latter $q$ variables.
Now suppose that $V$ is equipped with a Hermitian sclar product, and denote by $\omega \in \Lambda^{1,1}$ the corresponding positive $(1,1)$-form. Given an orthonormal basis $(\xi_i)$ of $\Lambda^{1,0} \cong V^\dual$, $(\xi_I \wedge \conjugate{\xi}_J)_{|I|=p,|J|=q}$ is an orthonormal basis of $\Lambda^{p,q}$, and $$\omega = \i · \sum_j \xi_j \wedge \conjugate{\xi}_j.$$
Second question: Am I completely wrong in thinking that (1) the scalar product $\langle \blank, \blank \rangle$ is an element of $\Lambda^{1,1}V^\dual$ and that (2) we simply have $\omega = \i · \langle \blank, \blank \rangle$? This cannot be intended, can it?
In other sources (e.g. Hodge Theory and Complex Analytic Geometry by Claire Voisin, Section 3.1), I find things that seem to be related, but I cannot quite match them together. For example, in Lemma 3.3, she associates to a Hermitian form (in particular to a Hermitian scalar product) $\langle \blank, \blank\rangle$ a “form” (I don't quite know what kind of form this is supposed to be) $\omega$ defined by $$\omega(v,w) = -\mathrm{Im}(\langle v, w \rangle)$$ and calls this a positive $(1,1)$-form if $\langle \blank, \blank\rangle$ was indeed a scalar product.
Third question: Do Boucksom and Voisin mean the same thing when they write $\omega$?
Fourth question: Do I translate the notation of Voisin correctly into the notation of Boucksom if I think that she claims $\omega \in \Lambda^{1,1}V^\dual$? But surely this cannot be correct if my interpretation of $\Lambda^{1,1}V^\dual$ from question 1 is correct? Obviously, $\omega$ takes only real values, but if it is given by an element of $\Lambda^{1,1}V^\dual$, i.e. a complex linear map $V \tensor_\IC \conjugate{V} \to \IC$, then it must vanish identically. Also, one can easily check by hand that e.g. $\omega$ is not complex-linear in the first variable.
All that I can read off from Voisin's $\omega$ is that it should be an element of $\bigwedge^2|V|_\IR^\dual$, where by $|V|_\IR$ I mean restriction of scalars to the real numbers.