Analytic subvarieties with many rational points |
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Authors: | C Gasbarri |
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Institution: | 1. Dipartimento di Matematica, dell’Università di Roma “Tor Vergata”, Viale della Ricerca Scientifica, 00133, Rome, Italy
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Abstract: | 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 \mathbb{C}}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 ì \mathbbC{K \subset \mathbb{C}} be a number field and X a quasi-projective variety defined over K. Let γ : 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 ? X(K)?g(M){p\in X(K)\cap\gamma(M)} the formal germ of M near P is an LG-germ, then we prove that X(K)?g(M){X(K)\cap\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
\mathbbC{\mathbb{C}} with K?hler form ω and γ : 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 ? X(K)?g(A){p\in X(K)\cap \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 òATùwm-1{\int_AT\wedge\omega^{m-1}} explicitly bounded and with Lelong number bigger or equal then one on each point in γ
−1(X(K)). In particular if A is affine γ
−1(X(K)) is not Zariski dense. |
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