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# Communications in Analysis and Geometry

## Volume 27 (2019)

### Number 3

### Poisson metrics on flat vector bundles over non-compact curves

Pages: 529 – 597

DOI: https://dx.doi.org/10.4310/CAG.2019.v27.n3.a2

#### Authors

#### Abstract

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