# Pullback and direct image of parabolic connections and parabolic Higgs bundles

## TL;DR

This paper develops explicit algebraic methods for transforming parabolic bundles, Higgs bundles, and connections via surface maps, ensuring stability properties are maintained and compatibility with nonabelian Hodge theory.

## Contribution

It introduces explicit algebraic constructions for pullback and direct image of parabolic structures, preserving stability and aligning with nonabelian Hodge correspondence.

## Key findings

- Constructions preserve semistability and polystability.
- Compatibility with nonabelian Hodge correspondence established.
- Explicit algebraic formulas provided for transformations.

## Abstract

We provide an explicit algebraic construction for the pullback and direct image of parabolic bundles, parabolic Higgs bundles and parabolic connections through maps between Riemann surfaces. We show that these constructions preserve semistability and polystability, and we prove that they are compatible with the nonabelian Hodge correspondence.