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Potential theory and Lefschetz theorems for arithmetic surfaces. (English) Zbl 0931.14014
From the abstract: “We prove an arithmetic analogue of the so-called Lefschetz theorem which asserts that, if $D$ is an effective divisor in a projective normal surface $X$ which is nef and big, then the inclusion map from the support $\left|D\right|$ of $D$ in $X$ induces a surjection from the (algebraic) fundamental group of $|D|$ onto the one of $X$. In the arithmetic setting, $X$ is a normal arithmetic surface, quasi-projective over Spec$(\Bbb{Z})$, $D$ is an effective divisor in $X$, proper over Spec$(\Bbb{Z})$, and furthermore one is given an open neighborhood $\Omega$ of $|D|(\Bbb{C})$ on the Riemann surface $X(\Bbb{C})$ such that the inclusion map $|D|(\Bbb{C})\hookrightarrow\Omega$ is a homotopy equivalence. Then we may consider the equilibrium potential $g_{D,\Omega}$ of the divisor $D(\Bbb{C})$ in $\Omega$ and the Arakelov divisor $(D,g_{D,\Omega})$, and we show that if the latter is nef and big in the sense of Arakelov geometry, then the fundamental group of $|D|$ still surjects onto the one of $X$. This result extends an earlier theorem of Ihara, and is proved by using a generalization of Arakelov intersection theory on arithmetic surfaces, based on the use of Green functions, up to logarithmic singularities, belonging to the Sobolev space $L_{2}^{1}$”.

MSC:
14G40Arithmetic varieties and schemes; Arakelov theory; heights
31C45Nonlinear potential theory, etc.
14G25Global ground fields
14J25Special surfaces
WorldCat.org
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