Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges to a compact reduced subspace $C$ in the sense of the analytic topology on complex points of the Hilbert scheme (resp. Douady space).

**Question.** Is it true that $\{C_i\}$ converges to $C$ in the sense of currents, namely for any smooth differential form $\omega$ on $X$ one has
$$\lim_{i\to\infty}\int_{C_i}\omega=\int_C\omega?$$
A reference would be helpful.

**Added:** In the case when $C$ is not reduced, the right hand side in the last equality should probably be multiplied by an appropriate multiplicity.