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

TL;DR

This paper explores the connections between height bounds on K-semistable Fano varieties, Peyre's conjecture on rational points, and related inequalities, linking algebraic geometry, number theory, and differential geometry.

Contribution

It establishes analogs of height inequalities for real points and relates them to K"ahler-Einstein metrics, advancing understanding of height bounds and rational points.

Findings

Relation between height bounds and Peyre's conjecture

Height inequalities established for the real projective line

Connections to K"ahler-Einstein metrics

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.