Very simple question about the space of Kähler forms

Question

On Page 117 of Daniel Huybrechts' book, "Complex Geometry: An Introduction", Corollary 3.1.8 says

Corollary 3.1.8 The set of all Kähler forms on a compact complex manifold X is an open convex cone in the linear space $\{\omega \in \mathcal{A}^{1,1}(X) \cap \mathcal{A}^{2}(X) \, | \, d\omega = 0\}$.

Question: Why does one have to consider the intersection of $\mathcal{A}^{1,1}(X)$ and $\mathcal{A}^{2}(X)$? Sometimes, in the physics setting where these manifolds are used, I've seen $\mathcal{A}^2(X)$ suppressed.

My question stems from the fact that in the Hodge decomposition, $\mathcal{A}^{1,1}(X)$ would be seen to be a part of $\mathcal{A}^{2}(X)$ anyway --am I missing something?

EDIT: Is the intersection written to emphasize that the Kähler form is real?

Answer 1

The comment in your edit is correct. While $\mathcal{A}^{1,1}(X)$ denotes complex $(1, 1)$-forms of type $(1, 1)$, $\mathcal{A}^2(X)$ denotes real two forms.