We prove approximation results, on projective algebraic and Stein manifolds, of positive closed currents of bidegree (1,1) by rational divisors with a control on their support and Lelong numbers. We define a notion of (strong-)rational convexity on complex manifolds and show that a totally real compact submanifold \(S\) of a projective algebraic (resp. a Stein) manifold \(X\) is rationally convex iff there is a Hodge form \(\omega\) on \(X\) s.t. \(\omega|_S\equiv 0\), generalizing a theorem of Duval and Sibony.