arXiv 0811.31952008-11-19math.AGmath.NT

具有多个有理点的解析子簇

Analytic subvarieties with many rational points

Carlo Gasbarri

备注：: 44 pages. Submitted

摘要：

我们给出了超越理论中经典的Bézieri-Schneider-Lang准则的一个推广。本文给出了$LG$ -胚的一个局部概念，它类似于$E$ -函数的概念和Gevrey条件，并且推广（和代替）了上述定理中关于导数的条件。设$K\subset \Bbb C$是数域，$X$是定义在$K$上的拟投射簇.设$γ\colon M\to X$是从抛物Riemann曲面到$X$的有限级全纯映射，使得它的象的Zapriki闭包严格大于1.设对每个$p\in X(K)\capγ(M)$，$M$在$P$附近的形式芽是$LG$ -芽，则证明了$X(K)\capγ(M)$是有限集.然后我们定义共形抛物Khäler簇的概念;这推广了抛物黎曼曲面的概念。我们表明，在这些品种，我们可以定义一个值分布理论。紧凯勒流形上的因子的补是共形抛物的;特别是每个拟投射簇都是。设$A$是$\Bbb C$上维数为$m$的共形抛物簇，具有Khäler形式$ω$，$γ\colon A\to X$是有限阶全纯映射，使得象的Zerkki闭包严格大于$m$ .假设对于每一个$p\in X(K)\cap γ(A)$，$A$的映像都是一个$LG$ --胚芽。然后证明了在二次$(1,1)$的$A$上存在当前$T$，使得$\int_AT\wedgeω^{m-1}$显式有界，且Lelong数在$γ^{-1}(X(K))$中的每一点上大于或等于1.特别地，如果$A$是仿射的，则$γ^{-1}(X(K))$不是Zurski稠密的。

英文摘要：

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 $γ\colon M\to X$ 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 $p\in X(K)\capγ(M)$ the formal germ of $M$ near $P$ is an $LG$-- germ, then we prove that $X(K)\capγ(M)$ is a finite set. Then we define the notion of conformally parabolic Khäler 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 Khäler 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 Khäler form $ω$ and $γ\colon A\to X$ is an holomorphic map of finite order such that the Zariski closure of the image is strictly bigger then $m$. Suppose that for every $p\in X(K)\cap γ(A)$, 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 $\int_AT\wedgeω^{m-1}$ 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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