The weighted Hermite--Einstein equation

TL;DR

This paper introduces a weighted Hermite--Einstein equation that generalizes known metrics, establishes stability conditions, and applies these concepts to derive curvature bounds and new equations in complex geometry.

Contribution

The paper develops a new weighted Hermite--Einstein equation, links it to stability notions, and extends curvature bounds, unifying various geometric structures under a common framework.

Findings

Weighted Hermite--Einstein equation characterizes vector bundle stability.

Solutions are unique up to scaling.

Extended Tian's Ricci curvature bound to modified Ricci curvature.

Abstract

We introduce a new weighted version of the Hermite--Einstein equation, along with notions of weighted slope (semi/poly)stability, and prove that a vector bundle admits a weighted Hermite--Einstein metric if and only if it is weighted slope polystable. The new equation encompasses several well-known examples of canonical Hermitian metrics on vector bundles, including the usual Hermite--Einstein metrics, K\"ahler--Ricci solitons, and transversally Hermite--Einstein metrics on certain Sasaki manifolds. We prove that the equation arises naturally as a moment map, that solutions to the equation are unique up to scaling, and demonstrate a weighted Kobayashi--L\"ubke inequality satisfied by vector bundles admitting a weighted Hermite--Einstein metric. As an application of our techniques, we extend a bound of Tian on the Ricci curvature to a bound on a modified Ricci curvature, related to the existence of K\"ahler--Ricci solitons. Along the way, we introduce a new weighted vortex equation, as well as a weighted analogue of Gieseker stability. A key technical point is the application of a new extension of Inoue's equivariant intersection numbers to arbitrary weight functions on the moment polytope of a K\"ahler manifold with Hamiltonian torus action.