. Moreover, geometric properties of X are reflected by the ones of . For example, the latter is a complex manifold if and only if X is smooth; it is compact if and only if X is proper over . Relation to complex Kähler manifolds Complex projective space is a Kähler manifold. This implies that, for any projective algebraic variety X, is a compact Kähler manifold. The converse is not in general true, but the Kodaira embedding theorem gives a criterion for a Kähler manifold to be projective. In low dimensions, there are the following results: (Riemann) A compact Riemann surface (i.e., compact complex manifold of dimension one) is a projective variety. By the Torelli theorem, it is uniquely determined by its Jacobian. (Chow-Kodaira) A compact complex manifold of dimension two with two algebraically independent meromorphic functions is a projective variety. GAGA and Chow's theorem Chow's theorem provides a striking way to go the other way, from analytic to algebraic geometry. It states that every analytic subvariety of a complex projective space is algebraic. The theorem may be interpreted to saying that a holomorphic function satisfying certain growth condition is necessarily algebraic: "projective" provides this growth condition. One can deduce from the theorem the following: Meromorphic functions on the complex projective space are rational. If an algebraic map between algebraic varieties is an analytic isomorphism, then it is an (algebraic) isomorphism. (This part is a basic fact in complex analysis.) In particular, Chow's theorem implies that a holomorphic map between projective varieties is algebraic. (consider the graph of such a map.) Every holomorphic vector bundle on a projective variety is induced by a unique algebraic vector bundle. Every holomorphic line bundle on a projective variety is a line bundle of a divisor. Chow's theorem can be shown via Serre's GAGA principle. Its main theorem states: Let X be a projective scheme over . Then the functor associating the coherent sheaves on X to the coherent sheaves on the corresponding complex analytic space Xan is an equivalence of categories. Furthermore, the natural maps are isomorphisms for all i and all coherent sheaves on X. Complex tori vs. complex abelian varieties The complex manifold associated to an abelian variety A over is a compact complex Lie group. These can be shown to be of the form and are also referred to as complex tori. Here, g is the dimension of the torus