About Wess–Zumino–Witten Equation and Harder–Narasimhan Potentials

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–Ampère 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.