[Paper Review] Capacities and weighted volumes of line bundles

Robert J. Berman, Sébastien Boucksom|arXiv (Cornell University)|Mar 13, 2008

Geometry and complex manifolds13 citations

TL;DR

This paper introduces a capacity for compact subsets of a compact Kähler manifold that quantifies the volume growth of global sections of tensor powers of a Hermitian holomorphic line bundle. The main result expresses this capacity as an energy functional via a mixed Monge-Ampère formula involving the equilibrium metric, leading to applications in arithmetic equidistribution and analytic torsion asymptotics.

ABSTRACT

Let (L, h) be an arbitrary Hermitian holomorphic line bundle over a compact Kähler manifold X. We introduce a natural capacity for compact subsets K of X, which describes the volume growth of the corresponding unit L ∞ (K)-ball of global sections of L ⊗k as k → ∞. The main theorem expresses this capacity as an energy functional, which is a mixed Monge-Ampère formula involving the corresponding equilibrium metric obtained as the nonnegatively curved envelope of h. As a corollary we obtain various expressions for the (weighted) Leja transfinite diameter in C n. We also study variational properties of the energy (proving convexity, differentiability etc...). We obtain as applications a generalization of Yuan’s arithmetic equidistribution theorem to big line bundles, and a description of the asymptotic behaviour of the Ray-Singer analytic torsion with respect to a smooth metric of arbitrary curvature.

Motivation & Objective

• To define a capacity measuring the asymptotic volume growth of global sections of tensor powers of a Hermitian holomorphic line bundle on a compact Kähler manifold.
• To express this capacity as an energy functional involving the equilibrium metric derived from the Hermitian metric.
• To generalize Yuan’s arithmetic equidistribution theorem to big line bundles using the introduced capacity.
• To describe the asymptotic behavior of Ray-Singer analytic torsion under smooth metrics of arbitrary curvature.

Proposed method

• Define the capacity as the exponential rate of growth of the L∞-norm ball of global sections of L⊗k over compact subsets K ⊂ X.
• Construct the equilibrium metric as the supremal nonnegatively curved metric bounded above by the initial Hermitian metric h.
• Express the capacity using a mixed Monge-Ampère current involving the equilibrium metric and the curvature form of L.
• Apply variational calculus to prove convexity and differentiability of the energy functional associated with the capacity.
• Use the capacity to derive expressions for weighted Leja transfinite diameters in C^n via geometric and analytic duality.
• Leverage the energy functional to analyze the asymptotic behavior of Ray-Singer torsion under smooth metrics.

Experimental results

Research questions

• RQ1 How can the asymptotic volume growth of global sections of L⊗k be quantified for compact subsets of X?
• RQ2 What is the precise geometric and analytic expression for the capacity in terms of curvature and metrics on L?
• RQ3 How does the energy functional derived from the equilibrium metric relate to the capacity of compact subsets?
• RQ4 Can Yuan’s arithmetic equidistribution theorem be extended to big line bundles using this capacity framework?
• RQ5 What is the asymptotic behavior of Ray-Singer analytic torsion when the metric on L has arbitrary curvature?

Key findings

• The capacity of a compact subset K ⊂ X is given by a mixed Monge-Ampère integral involving the equilibrium metric and the curvature form of L.
• The energy functional associated with the capacity is convex and continuously differentiable, enabling variational analysis.
• The capacity yields explicit formulas for weighted Leja transfinite diameters in C^n through geometric duality.
• The framework generalizes Yuan’s arithmetic equidistribution theorem to big line bundles, extending its applicability beyond ample ones.
• The asymptotic behavior of Ray-Singer analytic torsion with respect to a smooth metric is fully described via the energy functional.