Analytic subvarieties with many rational points

TL;DR

This paper generalizes classical criteria in transcendence theory to show that certain holomorphic maps with many rational points have finite intersection with algebraic points, introducing LG-germs and conformally parabolic varieties.

Contribution

It introduces LG-germs and conformally parabolic varieties, extending value distribution theory and finiteness results for rational points on complex varieties.

Findings

Finite rational points on certain holomorphic maps are proven under new LG-germ conditions.

Conformally parabolic Kähler varieties are characterized and shown to support value distribution theory.

The results imply non-density of rational points in specific geometric contexts.

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 $K\subset \Bbb C$ be a number field and $X$ a quasi--projective variety defined over $K$. Let $\gamma\colon M\to X$ 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 Kh\"aler 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 Kh\"aler manifold is conformally parabolic; in particular every quasi projective variety is. Suppose that $A$ is conformally parabolic variety of dimension $m$ over $\Bbb C$ with Kh\"aler form $\omega$ and $\gamma\colon A\to X$ 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 $\int_AT\wedge\omega^{m-1}$ 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.