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This paper deals with an inverse problem for a non-self-adjoint Schrödinger equation on a compact Riemannian manifold. Our goal is to stably determine a real vector field from the dynamical Dirichlet-to-Neumann map. We establish in dimension N ≥ 2, an Hölder type stability estimate for the inverse problem under study. The proof is mainly based on the reduction to an equivalent problem for an electro-magnetic Schrödinger equation and the use of a Carleman estimate designed for elliptic operators. [ABSTRACT FROM AUTHOR]