On K-stability, height bounds and the Manin-Peyre conjecture

Author:

Robert J. Berman

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Differential Geometry (math.DG), Number Theory (math.NT)

journal:

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date:

2023-05-11 16:00:00

Abstract

This note discusses some intriguing connections between height bounds on complex K-semistable Fano varieties X and Peyre's conjectural formula for the density of rational points on X. Relations to an upper bound for the smallest rational point, proposed by Elsenhans-Jahnel, are also explored. These relations suggest an analog of the height inequalities, adapted to the real points, which is established for the real projective line and related to K\"ahler-Einstein metrics.