Sharp bounds on the height of K-semistable toric Fano varieties

Abstract

Inspired by Fujita's algebro-geometric result that complex projective space has maximal degree among all K-semistable Fano varieties, we conjecture that the height of a K-semistable metrized arithmetic Fano variety X of relative dimension n is maximal when X is the projective space over the integers, endowed with the Fubini-Study metric. Our main result establishes the conjecture for the canonical integral model of a toric Fano variety when n ≤ 6 (the extension to higher dimensions is conditioned on a conjectural "gap hypothesis" for the degree). Translated into toric Kähler geometry this result yields a sharp lower bound on a toric invariant introduced by Donaldson, defined as the minimum of the toric Mabuchi functional. We furthermore reformulate our conjecture as an optimal lower bound on Odaka's modular height. Along the way we show how to control the height of the canonical toric model X , with respect to the Kähler-Einstein metric, by the algebrogeometric degree of X in any dimension n. Logarithmic generalizations of the results are also given.