Analytic torsion, vortices and positive Ricci curvature

TL;DR

This paper characterizes maximizers of a non-local functional on positively curved metrics, applying it to prove conjectures in analytic torsion, vortices, and Kahler geometry, and providing new proofs of classical results.

Contribution

It introduces a new approach to analyze positive Ricci curvature metrics, generalizing existing functionals, and applies this to solve longstanding conjectures and derive new geometric inequalities.

Findings

Proved conjectures on regularized determinants of Laplacians on the sphere.

Established inequalities related to Kahler-Einstein metrics.

Provided new proofs of lower bounds on Mabuchi's K-energy and uniqueness of Kahler-Einstein metrics.

Abstract

We characterize the global maximizers of a certain non-local functional defined on the space of all positively curved metrics on an ample line bundle L over a Kahler manifold X. This functional is an adjoint version, introduced by Berndtsson, of Donaldson's L-functional and generalizes the Ding-Tian functional whose critical points are Kahler-Einstein metrics of positive Ricci curvature. Applications to (1) analytic torsions on Fano manifolds (2) Chern-Simons-Higgs vortices on tori and (3) Kahler geometry are given. In particular, proofs of conjectures of (1) Gillet-Soul\'e and Fang (concerning the regularized determinant of Dolbeault Laplacians on the two-sphere) (2) Tarantello and (3) Aubin (concerning Moser-Trudinger type inequalities) in these three settings are obtained. New proofs of some results in Kahler geometry are also obtained, including a lower bound on Mabuchi's K-energy and the uniqueness result for Kahler-Einstein metrics on Fano manifolds of Bando-Mabuchi. This paper is a substantially extended version of the preprint arXiv:0905.4263 which it supersedes.