[Paper Review] On the volume of graded linear series and Monge-Ampère mass

Tomoyuki Hisamoto|arXiv (Cornell University)|Jan 20, 2012

Geometry and complex manifolds18 references4 citations

TL;DR

This paper establishes an analytic formula for the volume of a graded linear series on a smooth projective variety by expressing it as the Monge–Ampère mass of an equilibrium metric associated with a given smooth Hermitian metric on the line bundle. The key result shows that the volume equals the integral of the non-pluripolar Monge–Ampère measure of this equilibrium metric, generalizing prior results in complex geometry and providing a local, analytic tool for studying positivity and volume estimates.

ABSTRACT

We give an analytic description for the volume of a graded linear series.

Motivation & Objective

• To provide an analytic characterization of the volume of a graded linear series using pluripotential theory.
• To extend known results on line bundle volumes to proper graded linear series by introducing an equilibrium metric construction.
• To establish a connection between asymptotic algebraic invariants (volume) and singular metrics via Monge–Ampère theory.
• To investigate the regularity and continuity properties of the equilibrium metric under geometric assumptions.

Proposed method

• Define the equilibrium metric $ P_W\varphi $ as the upper-semicontinuous envelope of suprema of scaled log-norms of sections in $ W_k $, constrained by $ L^2 $-norm bounds.
• Use the Bedford–Taylor theory to define the non-pluripolar Monge–Ampère product $ \mathrm{MA}(P_W\varphi) = (dd^c P_W\varphi)^n $, which yields a positive measure with finite total mass.
• Apply the Ohsawa–Takegoshi $ L^2 $-extension theorem to construct test sections with controlled $ L^2 $-norms and pointwise lower bounds.
• Use the pushforward of the metric under a resolution map to relate the geometry of $ X $ to a target variety $ Y $, enabling extension techniques.
• Establish uniform estimates on the Bergman kernel via $ L^2 $-methods to show convergence of the metric $ \varphi_k $ to $ P_W\varphi $ in the sense of currents.
• Prove the volume formula by comparing the growth of $ \dim W_k $ with the mass of $ \mathrm{MA}(P_W\varphi) $, leveraging the birationality assumption on the rational map.

Experimental results

Research questions

• RQ1 Can the volume of a graded linear series be expressed as the Monge–Ampère mass of a canonical singular metric associated to a smooth Hermitian metric on the line bundle?
• RQ2 Under what geometric conditions does the equilibrium metric $ P_W\varphi $ exhibit continuity or regularity?
• RQ3 How does the equilibrium metric relate to the Okounkov body and asymptotic invariants of the graded linear series?
• RQ4 Is the volume formula via Monge–Ampère measure valid beyond the complete linear series case?
• RQ5 Can the regularity of the equilibrium metric be established for general finitely generated graded linear series?

Key findings

• The volume of a graded linear series $ W $ equals the total mass of the non-pluripolar Monge–Ampère measure of its equilibrium metric: $ \mathrm{vol}(W) = \int_X \mathrm{MA}(P_W\varphi) $, under the birationality assumption.
• The equilibrium metric $ P_W\varphi $ is plurisubharmonic and its Monge–Ampère current is well-defined and of finite mass.
• The equilibrium metric is continuous on a non-empty Zariski open subset when $ W $ is finitely generated and $ \mathrm{Proj}\, W $ is normal.
• The proof relies on $ L^2 $-extension techniques and avoids heavy Bergman kernel asymptotics, simplifying the argument compared to prior works.
• The volume formula holds even for proper subalgebras of the full section ring, generalizing results from complete linear series.
• The construction of test sections via Ohsawa–Takegoshi ensures uniform control over $ L^2 $-norms and pointwise lower bounds, enabling the volume estimate.