Height of toric subschemes and Legendre-Fenchel duality
Toric varieties provide a wealth of examples for performing explicit calculations in algebraic geometry. Since they can be described by a combinatorial construction, many of their invariants can be computed by purely combinatorial considerations. For example, the degree of a toric variety with respect to an equivariant ample line bundle can be computed as the volume of a certain associated polytope. Recently, \textit{P. Philippon} and \textit{M. Sombra} [J. Inst. Math. Jussieu 7(2), 327--373 (2008; Zbl 1147.11033)] extended this yoga to the arithmetic setting in order to give a very concrete description of the canonical height of a toric variety. In the first part of the article under review, the authors give a formula for the height of a projective toric variety \(Y_{Q, \alpha}\) over a number field with respect to an equivariant semi-positive adelic line bundle \(\overline{L}_{\Psi}\). (They restrict to the case in which the metrics at all finite places are algebraic and positive.) Their formula is a sum of local integrals over the polytope \(\Delta_Q\) associated to the algebraic line bundle \(L_{\Psi}\), and the integrand is the Legendre-Fenchel dual of the equivariant local Green's function for \(\overline{L}_{\Psi}\). In the second part of the article, the authors explain the calculation of heights for several special classes of examples using the Fubini--Study metric. They give beautiful formulas for the height of a rational normal curve of degree-\(n\) in \(\mathbb P^n\) and for the height of a toric projective space bundle. This short article is remarkably clearly written, with proofs relegated to a later article. Few explicitly computable examples exist in the literature of arithmetic intersection theory, and the introduction of these elegant ones arising from toric geometry is exceedingly welcome.
scientific article; zbMATH DE number 5563757