Kobayashi-Hitchin correspondence for polarized fibrations

A Hermitian metric on a holomorphic vector bundle is said to be Hermite-Einstein if its mean curvature is proportional to the identity operator. The Kobayashi-Hitchin correspondence (or the Donaldson-Uhlenbeck-Yau theorem) asserts that a holomorphic vector bundle admits a Hermite-Einstein metric if and only if it satisfies the algebraic condition of slope polystability.

In this talk, I will describe a recent extension of the Kobayashi-Hitchin correspondence to general fibrations beyond holomorphic vector bundles. Specifically, for a polarized family of complex projective manifolds, we examine the so-called Wess-Zumino-Witten (WZW) equation, which specializes to the Hermite-Einstein equation, when the polarized fibration is associated with a projectivization of a holomorphic vector bundle. We establish that the existence of approximate solutions to this equation is equivalent to the asymptotic semistability of the direct image sheaves associated with high tensor powers of the polarizing line bundle. We also discuss a relation between this result and the conjecture of Demailly concerning the optimality of Holomorphic Morse Inequalities.