Faltings Heights, Igusa Local Zeta Functions, and the Stability Conjectures in Kahler Geometry I

TL;DR

This paper explores the relationship between Faltings heights, Igusa local zeta functions, and stability conjectures in Kähler geometry, establishing criteria for the existence of constant scalar curvature Kähler metrics based on stability and height discrepancies.

Contribution

It introduces a new criterion linking height discrepancies and asymptotic stability to the existence of csck metrics in polarized manifolds.

Findings

Height discrepancy of O(d^2) implies existence of csck metric.

Asymptotic stability is equivalent to the existence of csck metrics under the given conditions.

Finite automorphism group assumption is crucial for the results.

Abstract

Let (X,L) be a polarized manifold. Assume that the automorphism group is finite. If the height discrepancy of (X,L) is O(d^2) then (X,L) admits a csck metric in the first chern class of L if and only if (X,L) is asymptotically stable.