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Asymptotics of Fubini-Study Currents for Sequences of Line Bundles

Melody Wolff

Abstract

We study the Fubini-Study currents and equilibrium metrics of continuous Hermitian metrics on sequences of holomorphic line bundles over a fixed compact Kähler 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.

Asymptotics of Fubini-Study Currents for Sequences of Line Bundles

Abstract

We study the Fubini-Study currents and equilibrium metrics of continuous Hermitian metrics on sequences of holomorphic line bundles over a fixed compact Kähler 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.

Paper Structure

This paper contains 9 sections, 15 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasi-plurisubharmonic functions
  4. Global weights and the equilibrium metric
  5. Fubini-Study currents and Bergman kernels
  6. Main Results
  7. Proofs of Theorems \ref{['T1.1']}, \ref{['T1.2']}, and \ref{['T1.3']}
  8. Proofs of Proposition \ref{['Prop']} and Theorems \ref{['T1.4']}
  9. Applications

Key Result

Theorem 1.1

Let $(X,\omega)$ and $(L_p,h_p)$ be as in $(A)$ and $(B)$. If every $x\in X$ has a neighborhood $U$ with local frames $e_p$ of $L_p$, such that the families of local weights $\{\phi_p/A_p\}$ and $\{\rho_p/A_p\}$ are uniformly bounded in $L^1(U)$, and there exists $M>0$ such that then

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Corollary 4.1
  • ...and 7 more