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

2023/05/12 by Berman, Robert J. · 1 citation
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2305.07272

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 Kähler-Einstein metrics.

Cited by

• On the volume of K-semistable Fano manifolds

Related

• Freeness alone is insufficient for Manin-Peyre
• Sharp bounds on the height of K-semistable Fano varieties II, the log case
• Height of rational points on random Fano hypersurfaces
• Manin's conjecture for M-points
• Manin's conjecture for integral points on toric varieties