[Paper Review] Differentiability of relative volumes over an arbitrary non-Archimedean field

Sébastien Boucksom, Walter Gubler+1 more|arXiv (Cornell University)|Apr 8, 2020

Geometry and complex manifolds36 references11 citations

TL;DR

This paper establishes the differentiability of relative volumes for continuous psh metrics on an ample line bundle over a geometrically reduced projective scheme over an arbitrary non-Archimedean field, extending prior results from discretely valued fields. The key contribution is a variational formula linking the derivative of relative volume to the non-Archimedean Monge–Ampère measure, enabling solutions to Monge–Ampère equations and generalizing equidistribution results for Fekete points via determinant of cohomology and Deligne pairings.

ABSTRACT

Given an ample line bundle $L$ on a geometrically reduced projective scheme defined over an arbitrary non-Archimedean field, we establish a differentiability property for the relative volume of two continuous metrics on the Berkovich analytification of $L$, extending previously known results in the discretely valued case. As applications, we provide fundamental solutions to certain non-Archimedean Monge--Amp\\`ere equations, and generalize an equidistribution result for Fekete points. Our main technical input comes from determinant of cohomology and Deligne pairings.

Motivation & Objective

• To extend the differentiability of relative volumes from discretely valued to arbitrary non-Archimedean fields.
• To provide a variational approach to solving non-Archimedean Monge–Ampère equations.
• To generalize equidistribution results for Fekete points beyond the discretely valued case.
• To establish a foundational differentiability property for relative volumes in the context of Berkovich geometry over arbitrary non-Archimedean fields.
• To handle non-Noetherian valuation rings using Deligne pairings and determinant of cohomology.

Proposed method

• Uses additive notation for line bundles and metrics, defining relative volume via logarithmic ratios of determinants of cohomology norms.
• Applies the determinant of cohomology and Deligne pairings to handle non-Noetherian valuation rings in the non-discretely valued setting.
• Employs a variational approach inspired by Abbes–Bouche and Yuan, extending techniques from Arakelov geometry.
• Relies on the continuity of envelopes and the orthogonality of psh metrics to control convergence and differentiability.
• Establishes the differentiability of the energy functional via uniform convergence of envelopes and the use of psh metric regularization.
• Uses the non-Archimedean Monge–Ampère measure $(dd^c heta)^n$ to characterize the derivative of relative volume.

Experimental results

Research questions

• RQ1 Does the relative volume functional remain differentiable over arbitrary non-Archimedean fields, not just discretely valued ones?
• RQ2 Can the variational method for solving Monge–Ampère equations be extended beyond the discretely valued case?
• RQ3 How does the differentiability of relative volumes relate to the non-Archimedean Monge–Ampère measure?
• RQ4 What is the role of Deligne pairings and determinant of cohomology in handling non-Noetherian valuation rings?
• RQ5 Can equidistribution results for Fekete points be generalized to non-discretely valued non-Archimedean fields?

Key findings

• The relative volume functional is differentiable at any continuous psh metric, with derivative given by the integral of the test function against the non-Archimedean Monge–Ampère measure: $\frac{d}{dt}\big|_{t=0} \operatorname{vol}(L,\phi+tf,\phi) = \int_{X^{\mathrm{an}}} f \, (dd^c\phi)^n$.
• Fundamental solutions to non-Archimedean Monge–Ampère equations are constructed via the variational approach, with the equilibrium metric $\phi_x = \operatorname{P}(\{x\},\phi)$ satisfying $V^{-1}(dd^c\phi_x)^n = \delta_x$ for $L$-regular points.
• The differentiability result holds over any complete non-Archimedean field, including trivially and densely valued fields, extending previous results restricted to discrete or residue characteristic zero settings.
• The energy functional $\operatorname{E} \circ \operatorname{P}$ is differentiable at every continuous psh metric, with the derivative expressed via the Monge–Ampère measure.
• The proof relies on the continuity of envelopes and uniform convergence of envelopes, ensuring the validity of the variational argument in the general non-discretely valued case.
• The result generalizes equidistribution of Fekete points beyond the discretely valued case, under the assumption of continuity of envelopes.