Communications in Analysis and Geometry
Volume 27 (2019)
Poisson metrics on flat vector bundles over non-compact curves
Pages: 529 – 597
Let $(E, \nabla, \Pi) \to (M,g)$ be a flat vector bundle with a parabolic structure over a punctured Riemann surface. We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as the dimension reduction of the Hermitian–Yang–Mills equation for holomorphic vector bundles on $K3$ surfaces in the large complex structure limit. We define a notion of slope stability, and show that if the flat connection $\nabla$ has regular singularities, and the Riemannian metric $g$ has finite volume then $E$ admits a Poisson metric with asymptotics determined by the parabolic structure if and only if $(E, \nabla, \Pi)$ is slope polystable.
Received 3 October 2015
Accepted 5 April 2017
Published 3 September 2019