Arithmetic Geometry of Toric Varieties. Metrics, Measures and Heights
Arithmetic Geometry of Toric Varieties. Metrics, Measures and Heights
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环面簇的算术几何。
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
M. Sombra
中科院分区：
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作者：
J. B. Gil;Patrice Philippon;M. Sombra
我们证明了一个环变项相对于一个环度量线束的高度可以表示为一个凹函数族的多边形上的积分。为了说明和证明这一结果，我们研究了环面变体的Arakelov几何。特别地，我们考虑离散估值环上的模型，度量线束，以及它们相关的度量和高度。我们证明了这些概念可以转化为凸分析，并且与多面体复合体、凹函数、实Monge-Amp 'ere测度和legende - fenchel对偶等对象密切相关。我们也给出了一元线性函数在多面体上的积分的一个封闭公式。这使我们能够根据多面体产生的一些有趣的度量来计算环面变体的高度。我们还计算了相对于Fubini-Study度量的环向投影曲线的高度，以及一些环向束的高度。
We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, we study the Arakelov geometry of toric varieties. In particular, we consider models over a discrete valuation ring, metrized line bundles, and their associated measures and heights. We show that these notions can be translated in terms of convex analysis, and are closely related to objects like polyhedral complexes, concave functions, real Monge-Amp`ere measures, and Legendre-Fenchel duality. We also present a closed formula for the integral over a polytope of a function of one variable composed with a linear form. This allows us to compute the height of toric varieties with respect to some interesting metrics arising from polytopes. We also compute the height of toric projective curves with respect to the Fubini-Study metric, and of some toric bundles.