Let $X = V/\Lambda$ be a complex torus, where $V$ is a complex vector space and $\Lambda \subset V$ is a full-rank lattice. We can identify the cohomology group $H^{2}(X, \mathbb{Z})$ with alternating forms $E: \Lambda \times \Lambda \to \mathbb{Z}$. So every such form we may identify as the first Chern class of some $C^{\infty}$ complex line bundle on $X$.
Now, it seems to be well-known that $E$ is the first Chern class of a holomorphic line bundle if and only if $E_{\mathbb{R}} (iu, iv) = E_{\mathbb{R}}(u,v)$ for all $u,v \in V$. Here, $E_{\mathbb{R}}: V \times V \to \mathbb{R}$ is the $\mathbb{R}$-linear extension of $E$.
Unless I'm seriously mistaken, one should then be able to show that $E_{\mathbb{R}} (iu, iv) = E_{\mathbb{R}}(u,v)$ holds if and only if the image of $E_{\mathbb{R}}$ under the Hodge decomposition lies in $H^{1,1}(X)$. Is this true? If so, is it straightforward to prove? I've tried showing it directly, and failed.