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Height of cycles in toric varieties

Abstract : We investigate in this work the relation between suitable Arakelov heights of a cycle in a toric variety and the arithmetic features of its defining Laurent polynomials. To this purpose, we associate to a Laurent polynomial certain concave functions which we call Ronkin functions and upper functions. We give upper bounds for the height of a complete intersection in terms of the associated upper functions. For a hypersurfaces, we prove a formula relating its height to the Ronkin function of the associated Laurent polynomial. We conjecture an analogous equality for a suitable average height in higher codimensions and indicate a strategy for the proof of a particular case. In all the treatment, we deal with convex geometrical objects such as polytopes, real Monge-Ampère measures and Legendre-Fenchel duality of concave functions. We suggest an algebraic framework for such a study and deepen the understanding of mixed integrals.
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Roberto Gualdi. Height of cycles in toric varieties. General Mathematics [math.GM]. Université de Bordeaux; Universitat internacional de Catalunya, 2018. English. ⟨NNT : 2018BORD0139⟩. ⟨tel-01931089⟩

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