The arithmetic volume of hypersurfaces in toric varieties and Mahler measures

TL;DR

This paper calculates the canonical arithmetic volume of hypersurfaces in smooth projective toric varieties and establishes a generalized Hodge index theorem, advancing understanding in algebraic geometry and number theory.

Contribution

It introduces a method to determine the arithmetic volume of hypersurfaces in toric varieties and proves a generalized Hodge index theorem for these hypersurfaces.

Findings

Computed the canonical arithmetic volume for hypersurfaces in toric varieties.

Proved a generalized Hodge index theorem in this context.

Abstract

In this paper we determine the canonical arithmetic volume of hypersurfaces in smooth projective toric varieties. As a consequence, we prove a generalized Hodge index theorem on hypersurfaces in smooth projective toric varieties.