We call a formal morphism between completions of complex spaces convergent if it comes from a holomorphic mapping between the complex spaces. We assume always that the source space is compact. Then a formal morphism is either convergent everywhere or nowhere, under very general conditions.
"Convergence of formal morphisms of completions of complex spaces." J. Math. Soc. Japan 51 (3) 731 - 755, July, 1999. https://doi.org/10.2969/jmsj/05130731