is also surjective: given a closed $(1,1)$-current $T$, one can find a locally finite open covering $U_j$ of $X$ such that $T={\partial\overline{\partial}}f_j$ is ${\partial\overline{\partial}}$-exact on $U_j$. If $\rho_j$ is a partition of unity associated to $U_j$ and $f:=\sum\rho_jf_j$, then $T-{\partial\overline{\partial}}f$ is smooth. Indeed, on $U_i$, it is just ${\partial\overline{\partial}}(\sum_j\rho_j(f_i-f_j)$, and each $f_i-f_j$ is smooth since it is even pluri-harmonic. As a consequence, a class $\alpha\in H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$ can be seen as an affine space of closed $(1,1)$-currents. We denote by $\{T\}\in H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$ the class of the current $T$. Remark that $i{\partial\overline{\partial}}$ is a real operator (on forms or currents), so that if $T$ is a real closed $(1,1)$-current, its class $\{T\}$ lies in ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ and consists in all the closed currents $T+i{\partial\overline{\partial}}\varphi$ where $\varphi$ is a real current of degree $0$.\
When $X$ is furthermore compact, it can be shown that $H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$ is finite dimensional. The operator ${\partial\overline{\partial}}$ from smooth functions to smooth closed (1,1)-forms is thus an operator between Fréchet spaces with finite codimensional range; it therefore has closed range, and the quotient map $\theta\mapsto\{\theta\}$ from smooth closed $(1,1)$-forms to $H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$ endowed with its unique finite-dimensional complex vector space Hausdorff topology is thus continuous and open.

General facts about currents
