Analytic subvarieties with many rational points

We give a generalization of the classical Bombieri-Schneider-Lang criterion in transcendence theory. We give a local notion of LG-germ, which is similar to the notion of E-function and Gevrey condition, and which generalize (and replace) the condition on derivatives in the theorem quoted above. Let K subset of C be a number field and X a quasi-projective variety defined over K. Let gamma : M -> X be an holomorphic map of finite order from a parabolic Riemann surface to X such that the Zariski closure of the image of it is strictly bigger then one. Suppose that for every p is an element of X(K) boolean AND gamma (M) the formal germ of M near P is an LG-germ, then we prove that X(K) boolean AND gamma (M) is a finite set. Then we define the notion of conformally parabolic K hler varieties; this generalize the notion of parabolic Riemann surface. We show that on these varieties we can define a value distribution theory. The complementary of a divisor on a compact K hler manifold is conformally parabolic; in particular every quasi projective variety is. Suppose that A is conformally parabolic variety of dimension m over C with Kahler form omega and gamma : A -> X is an holomorphic map of finite order such that the Zariski closure of the image is strictly bigger then m. Suppose that for every p is an element of X(K) boolean AND gamma (A), the image of A is an LG-germ. then we prove that there exists a current T on A of bidegree (1, 1) such that integral(A) A boolean AND omega(m-1) explicitly bounded and with Lelong number bigger or equal then one on each point in gamma(-1)(X(K)). In particular if A is affine gamma(-1)(X(K)) is not Zariski dense.