Height of toric varieties

A complete toric variety $X$ of dimension $n$ is determined by a lattice $N$ and a complete integral fan $\Sigma$ in $N_R$. This variety has a model over the integers and is equipped with the action of a torus $T$. An equivariant ample line bundle $L$ on $X$ determines an integral polytope $P$ in the dual space $N_R^\vee$. Plenty of algebro-geometric properties of the pair $(X,L)$ can easily be read off from the polytope $P$. The exponential map determines a parametrization of the open orbit $X_0$ by $N_C$. Assume that $L$ is equipped with a positive Hermitian metric that is equivariant under the action of the compact torus. Then, minus the logarithm of the norm of a section of $L$, determines a strictly convex function $f$ on $N_R$. The stability set of this function turns out to be the polytope $P$ and the Legendre dual $g = f^\vee$ is a strictly convex function on $P$. This function $g$ is the symplectic potential in the Guillemin-Abreu theory. We prove that the height of $X$ with respect to the metrized line bundle $L$ is given by $(n + 1)!$ times the integral of $?g$ with respect to the normalized Haar measure of $N_R^\vee$. This is the arithmetic analogue of the expression of the degree of a toric variety as $n!$ times the volume of the polytope. We expect that many other Arakelov geometric properties of $X$ can be read from the function g. This is a report on joint work with J.I. Burgos (Barcelone) and P. Philippon (Paris).