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Analytic subvarieties with many rational points. (English) Zbl 1244.11072

Summary: 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 \({K \subset \mathbb{C}}\) be a number field and \(X\) a quasi-projective variety defined over \(K\). Let \(\gamma : M \rightarrow X\) 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 \({p\in X(K)\cap\gamma(M)}\) the formal germ of \(M\) near \(P\) is an \(LG\)-germ, then we prove that \({X(K)\cap\gamma(M)}\) is a finite set. Then we define the notion of conformally parabolic Kähler 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 Kähler manifold is conformally parabolic; in particular every quasi projective variety is. Suppose that \(A\) is conformally parabolic variety of dimension \(m\) over \({\mathbb{C}}\) with Kähler form \(\omega \) and \(\gamma : A \rightarrow X\) is an holomorphic map of finite order such that the Zariski closure of the image is strictly bigger then \(m\). Suppose that for every \({p\in X(K)\cap \gamma (A)}\), 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 \({\int_A T\wedge\omega^{m-1}}\) 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.

MSC:

11J97 Number-theoretic analogues of methods in Nevanlinna theory (work of Vojta et al.)
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
32B10 Germs of analytic sets, local parametrization
32H30 Value distribution theory in higher dimensions
32Q15 Kähler manifolds
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References:

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