Iterated destabilizing modifications for vector bundles with connection

TL;DR

This paper introduces an iterative process for vector bundles with connections on curves to produce a Hodge-like filtration, revealing a stratification of the moduli space with Lagrangian fibrations, connecting stability and geometric structures.

Contribution

It develops a novel iterative modification method for non-semistable bundles with connections, leading to a stratification of the moduli space and insights into its geometric structure.

Findings

Constructs a Hodge-like filtration satisfying Griffiths transversality.

Establishes a stratification of the moduli space with minimal stratum of opers.

Identifies fibrations with Lagrangian fibers within the moduli space.

Abstract

Given a vector bundle with integrable connection $(V,\nabla)$ on a curve, if $V$ is not itself semistable as a vector bundle then we can iterate a construction involving modification by the destabilizing subobject to obtain a Hodge-like filtration $F^p$ which satisfies Griffiths transversality. The associated graded Higgs bundle is the limit of $(V,t\nabla)$ under the de Rham to Dolbeault degeneration. We get a stratification of the moduli space of connections, with as minimal stratum the space of opers. The strata have fibrations whose fibers are Lagrangian subspaces of the moduli space.