Consider the exponential sequence of sheaves on $X$:
$0 \rightarrow \underline{\mathbb{Z}} \rightarrow \mathcal{O}_X \stackrel{\operatorname{exp}}{\rightarrow} \mathcal{O}_X^{\times} \rightarrow 0$.
The connecting map in sheaf cohomology gives a map
$H^1(X,\mathcal{O}_X^{\times}) \rightarrow H^2(X,\mathbb{Z})$.
In my experience, it is this map which is usually called the Chern class map. However, there is a natural (universal coefficent) map $H^2(X,\mathbb{Z}) \rightarrow H^2(X,\mathbb{C})$ and then we can use the Hodge decomposition $H^2(X,\mathbb{C}) = H^{0,2} \oplus H^{1,1} \oplus H^{2,0}$, where $H^{p,q} = H^q(X,\Omega^p)$. Under this decomposition the image of the Chern class map lands in $H^{1,1}$. So this is probably what is intended.