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.