About Wess-Zumino-Witten equation and Harder-Narasimhan potentials

TL;DR

This paper investigates algebraic obstructions to solutions of the Wess-Zumino-Witten equation in complex geometry, introduces a generalized Monge-Ampère equation, and demonstrates approximate solutions related to Harder-Narasimhan filtrations, with applications to a conjecture by Demailly.

Contribution

It identifies algebraic obstructions for approximate solutions, generalizes the Wess-Zumino-Witten equation with a Monge-Ampère type equation, and proves an asymptotic converse to the Andreotti-Grauert theorem.

Findings

Identified algebraic obstructions to approximate solutions of the Wess-Zumino-Witten equation.

Established the existence of approximate solutions to a generalized Monge-Ampère equation.

Proved an asymptotic converse to the Andreotti-Grauert theorem in a fibered setting.

Abstract

For a polarized family of complex projective manifolds, we identify the algebraic obstructions that govern the existence of approximate solutions to the Wess-Zumino-Witten equation. When this is specialized to the fibration associated with a projectivization of a vector bundle, we recover a version of Kobayashi-Hitchin correspondence. More broadly, we demonstrate that a certain auxiliary Monge-Amp\`ere type equation, generalizing the Wess-Zumino-Witten equation by taking into account the weighted Bergman kernel associated with the Harder-Narasimhan filtrations of direct image sheaves, admits approximate solutions over any polarized family. These approximate solutions are shown to be the closest counterparts to true solutions of the Wess-Zumino-Witten equation whenever the latter do not exist, as they minimize the associated Yang-Mills functional. As an application, in a fibered setting, we prove an asymptotic converse to the Andreotti-Grauert theorem conjectured by Demailly.