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Arithmetic geometry of toric varieties : metrics, measures and heights / Jose Ignacio Burgos Gil, Patrice Philippon and Martín Sombra.

By: Contributor(s): Material type: TextTextSeries: Asterisque ; 360.Publication details: Paris : Societe Mathematique de France, 2014.Description: vi, 222 p. : illustrations ; 24 cmISBN:
  • 9782856297834 (pbk.)
Subject(s): DDC classification:
  • 510=4 23 As853
Contents:
1. Metrized line bundles and their associated heights -- 2. The Legendre-Fenchel duality -- 3. Toric varieties -- 4. Metrics and measures on toric varieties -- 5. Height of toric varieties -- 6. Metrics from polytopes -- 7. Variations on Fubini-Study metrics-- Bibliography-- List of symbols-- Index.
Abstract: 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.
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Holdings
Item type Current library Call number Status Date due Barcode Item holds
Books ISI Library, Kolkata 510=4 As853 (Browse shelf(Opens below)) Available C26465
Total holds: 0

Includes bibliographical references (pages 207-212) and index.

1. Metrized line bundles and their associated heights --
2. The Legendre-Fenchel duality --
3. Toric varieties --
4. Metrics and measures on toric varieties --
5. Height of toric varieties --
6. Metrics from polytopes --
7. Variations on Fubini-Study metrics--
Bibliography--
List of symbols--
Index.

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.

Abstract also in French.

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