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

Tristan C. Collins (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Adam Jacob (Department of Mathematics, University of California at Davis)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

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