Complex Geometry Seminar 30 September

[Jacopo Stoppa]

On Monday, 30 September at 11:00 in room 136 (SISSA) Carlo Scarpa (Lyon 1) will give the seminar:
"Curvature equations and stabilities of holomorphic vector bundles"
Abstract: 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 stability notions also detect the existence of Hermitian metrics satisfying special curvature properties. To answer this question, Dervan, McCarthy, and Sektnan introduced a class of geometric PDEs for Hermitian metrics called Z-critical equations, and showed that in some regimes these equations detect some stability properties of the bundle. In this talk, based on joint work with Julien Keller (arXiv:2405.03312[math.DG]), I will show that any bundle over a projective surface that admits a Z-positive, Z-critical Hermitian metric must be Z-stable and Z-positive. Time permitting, we will discuss a possible converse to this result.