Complex geometry seminar

A fundamental result in Complex Geometry is the Kobayashi-Hitchin correspondence, stating that a holomorphic vector bundle on a Kähler manifold is poly-stable (as defined by Mumford, Takemoto) if and only if it admits a Hermitian metric solving the Hermite-Einstein equation. It has now become clear that there exist many possible different stability notions for vector bundles, that are of great interest in Algebraic Geometry and String Theory. It is natural to wonder if these stabilities are also tied to the existence of Hermitian metrics with special curvature properties. In this talk, based on joint work with Julien Keller (arXiv:2405.03312[math.DG]), we will consider a class of “polynomial” equations for the curvature of rank 2 holomorphic vector bundles on compact projective surfaces, and a corresponding class of polynomial stability conditions for these bundles. We will then explain how each of these stability conditions is related to the existence of a Hermitian metric satisfying the corresponding equation. This refines and partially confirms a conjectural correspondence between Bridgeland stability conditions and PDEs on holomorphic vector bundles, formulated by Dervan, McCarthy, and Sektnan.