According to a result of Fujita, complex projective space has maximal degree (volume) among all K-semistable Fano varieties of a given dimension. In this talk, which is based on a joint work with Rolf Andreasson, I will discuss a conjectural arithmetic analog of Fujita's result, saying that the height (arithmetic volume) of a K-semistable metrized arithmetic Fano variety of given dimension is maximal for 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, up to relative dimension six. The extension to higher dimensions is conditioned on a conjectural “gap hypothesis” for the degree. Connections to Kähler-Einstein metrics and Odaka's modular invariant will also be pointed out.

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