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Arithmetic geometry of toric varieties metrics, measures and heights José Ignacio Burgos Gil, Patrice Philippon, Martín Sombra

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Burgos Gil, José I, 1962- [VerfasserIn]

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Materialtyp: Text

TextSprache: Englisch Reihen: ; 2 | ; 360 | Astérisque ; 360Gesamtaufnahme: Verlag: Paris Soc. Math. de France 2014Beschreibung: VI, 222 Seiten DiagrammeInhaltstyp:

Text

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ohne Hilfsmittel zu benutzen

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Band

ISBN:

9782856297834

Weitere Titel:

Géométrie arithmétique des variétés toriques métriques, mesures et hauteurs

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Andere physische Formen: Erscheint auch als: Arithmetic geometry of toric varietiesLOC-Klassifikation:

QA564

Andere Klassifikation:

17,1
SI 832
*14M25
14G40
52A41
31.51

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Zusammenfassung: 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-Ampère 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 formula 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 the height of some toric bundles"--Page 4 of coverAndere Ausgaben: Erscheint auch als (Online-Ausgabe): / Burgos Gil, José I., 1962 - : Arithmetic geometry of toric varieties

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Literaturverz. S. [207] - 212

Metrized line bundles and their associated heightsThe Legendre-Fenchel duality -- Toric varieties -- Metrics and measures on toric varieties -- Height of toric varieties -- Metrics from polytopes -- Variations on Fubini-Study metrics.

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-Ampère 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 formula 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 the height of some toric bundles"--Page 4 of cover

Burgos Gil, José I., 1962 - : Arithmetic geometry of toric varieties