It follows from Proposition 5.6, that ${\rm trop}_{U}({\rm supp}(\alpha))={\rm supp}(\alpha_{U})$ (see [CD12], Corollaire 3.2.3). Note however that not every differential form $\alpha$ on the tropical chart $(V,\varphi_{U})$ is given by a single $\alpha_{U}\in A^{p,q}({\rm trop}_{U}(V))$ as in Proposition 5.6.