Let $(V,J)$ be an almost complex vector space. $V_\mathbb{C}$ is the complexification of $V$.
I have the following questions:
$1.$ A $k$-forms on $V_\mathbb{C}$ is a $k$–$\mathbb{C} $ linear and alternate function into $\mathbb{C}$.
Is this the correct definition?
$2.$ Let the extension of $J$ to $V_\mathbb{C}$ be again denoted by $J$. Now look the at eigenspaces $V^{1,0}=E_i{(V)}$ and $V^{0,1}= E_{-i}(V)$. Now these two are complex vector spaces. Now we can talk about their exterior algebras.
Now a form of bidegree $(p,q)$ is an element of $\Lambda^p V^{1,0}\otimes \Lambda^q V^{0,1}$. Now in the book "Complex Geometry An Introduction" the writer Huybrechts writes that this is naturally subspace of $\Lambda^{p+q}V_\mathbb{C}$. In the proof he says that $v_{I_1}\otimes w_{I_2}$ is a basis for $\Lambda^p V^{1,0}\otimes \Lambda^q V^{0,1}$ and then he probably means that these elements are also elements of $\Lambda^{p+q}V_\mathbb{C}$. To establish the last claim, we need to show that $v_{I_1}\otimes w_{I_2}$ is alternating. I am unable to show that.
Note that the second question depends on the definition in $1.$
Best Answer
Warning: I personally found (and to some degree still find) this stuff confusing. Hopefully I didn't screw something up. Don't despair and don't hesitate to ask additional questions.
In 1. you have the correct definition. Alternatively, you could say a $k$-form on $V_{\mathbb C}$ is an element $\omega \in \Lambda^k V_{\mathbb C}^*$ (where $(-)^*$ denotes the dual vector space) or a linear map $\omega: \Lambda^k V_{\mathbb C} \to \mathbb C$. Also note that there is a natural isomorphism $\Lambda^k (V_{\mathbb C}^*) \cong (\Lambda^k V_{\mathbb C})^*$, which is why I'm omitting brackets.
Regarding 2., I think you have to dualize somewhere. The decomoposition $$V_{\mathbb C} = V^{1,0} \oplus V^{0,1}$$ induces a dual decomposition $$\label{decomp}V^*_{\mathbb C} = (V^{1,0})^* \oplus (V^{0,1})^*.\tag{1}$$ Note that naturally, the inclusions $V^{s,t} \to V_{\mathbb C}$ induce quotients $V_{\mathbb C}^* \to (V^{s,t})^*$, and we split those quotients by prescribing $\alpha \mapsto 0$ for all $\alpha \in V^{t,s}$. Applying exterior powers to \eqref{decomp} we obtain $$\label{decomp-ext}\Lambda^k V_{\mathbb C}^* = \bigoplus_{p+q=k} \Lambda^p (V^{1,0})^* \otimes \Lambda^q (V^{0,1})^*.\tag{2}$$
Now a form of type $(p,q)$ is an element $\omega \in \Lambda^p(V^{1,0})^* \otimes \Lambda^q(V^{0,1})^*$, and using that decomposition we can consider $\omega$ as an element of $\Lambda^{p+q} V_{\mathbb C}^*$.
Now that is all pretty abstract. Boiled down it means that a form $\omega$ of type $(p,q)$ is an alternating $p+q$-form on $V_{\mathbb C}$, which can only be non-zero if you plug in $p$ vectors from $V^{1,0}$ and $q$ vectors from $V^{0,1}$. In terms of bases $v_1, \dotsc, v_n \in V^{1,0}$, $w_1, \dotsc, w_n \in V^{0,1}$ it means that only terms of the form $\omega(v_{I_1} \wedge w_{I_2})$ for $|I_1| = p, |I_2| = q$ are non-zero.