Metric properties of parabolic ample bundles

TL;DR

This paper develops a Hermitian metric framework for parabolic bundles, connecting differential geometry with algebraic geometry, and provides evidence supporting a Griffiths conjecture in this context.

Contribution

It introduces admissible Hermitian metrics on parabolic bundles, develops associated Chern-Weil theory, and proves the Griffiths conjecture for certain parabolic bundles on Riemann surfaces.

Findings

Metrics coincide with algebraic parabolic Chern classes

Griffiths conjecture holds on Riemann surfaces for specific bundles

Existence of metrics with positive Schur forms on stable bundles

Abstract

We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern-Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Griffiths conjecture in the parabolic setting and prove some results that provide evidence in its favour for certain kinds of parabolic bundles. For these kinds of parabolic structures, we prove that the conjecture holds on Riemann surfaces. We also prove that a Berndtsson-type result holds, and that there are metrics on stable bundles over surfaces whose Schur forms are positive.