Asymptotics of Fubini-Study Currents for Sequences of Line Bundles

TL;DR

This paper investigates the asymptotic behavior of Fubini-Study currents associated with sequences of line bundles on compact Kähler manifolds, establishing convergence results related to equilibrium metrics and curvature.

Contribution

It provides new conditions under which scaled Fubini-Study currents converge weakly to the curvature of equilibrium metrics on line bundles.

Findings

Scaled Fubini-Study currents converge to equilibrium metric curvature

Difference between currents and curvature tends to zero in the sense of currents

Sufficient conditions for weak convergence of scaled currents

Abstract

We study the Fubini-Study currents and equilibrium metrics of continuous Hermitian metrics on sequences of holomorphic line bundles over a fixed compact K\"ahler manifold. We show that the difference between the Fubini-Study currents and the curvature of the equilibrium metric, when appropriately scaled, converges to 0 in the sense of currents. As a consequence, we obtain sufficient conditions for the scaled Fubini-Study currents to converge weakly.