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

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 Kähler-Einstein metrics.