  We recover results by Ullmo-Yafaev and Peterzil-Starchenko on the closure of
the image of an algebraic variety in a compact complex torus. Our approach uses
directed closed currents and allows us to extend the result for dimension 1
flows to the setting of commutative complex Lie groups which are not
necessarily compact. A version of the classical Ax-Lindemann-Weierstrass
theorem for commutative complex Lie groups is also given.
