Canonical metrics on families of vector bundles
Shing Tak Lam
TL;DR
The paper introduces the family Hermite--Einstein equation as a canonical metric notion for families of holomorphic vector bundles over a Kähler base, incorporating a first-order vertical deformation via a moment-map term.It develops a comprehensive analytic framework: deformation theory (Kuranishi), the geometry of vertically FE metrics, and an expansion/perturbation strategy to produce FE metrics on the total space in adiabatic (large k) limits, assuming a FE family metric exists and a mild automorphism condition.A parabolic FE flow is proven to exist for all time with unique solutions, and a Dirichlet boundary problem for the flow is solved, both leveraging moment-map geometry and maximum principle arguments.Together, these results provide a pathway toward a Hitchin--Kobayashi-type stability theory for families and connect the FE framework to broader moduli and gauge-theoretic contexts.
Abstract
We introduce a geometric partial differential equation for families of holomorphic vector bundles, generalising the theory of Hermite--Einstein metrics. We consider families of holomorphic vector bundles which each admit Hermite--Einstein metrics, together with a first order deformation. On such families, we define the family Hermite--Einstein equation for Hermitian metrics, which we view as a notion of a canonical metric in this setting. We prove two main results concerning family Hermite--Einstein metrics. Firstly, we construct Hermite--Einstein metrics in adiabatic classes on product manifolds, assuming the existence of a family Hermite--Einstein metric. Secondly, we prove that the associated parabolic flow admits a unique smooth solution for all time, and use this to show that the Dirichlet problem always admits a unique solution.