Pullback and direct image of parabolic Higgs bundles and parabolic connections with symplectic and orthogonal structures

TL;DR

This paper proves that symplectic and orthogonal structures on parabolic bundles are preserved under pullback and direct image operations, including compatibility with Higgs fields and connections, and are consistent with the Nonabelian Hodge Correspondence.

Contribution

It establishes the preservation of symplectic and orthogonal structures on parabolic bundles under pullback and direct image, extending to Higgs fields, connections, and the Nonabelian Hodge Correspondence.

Findings

Pullback and direct image preserve symplectic and orthogonal structures.

Compatibility with Higgs fields and connections is maintained under these operations.

Results are consistent with the Nonabelian Hodge Correspondence.

Abstract

Given a symplectic (respectively, orthogonal) parabolic vector bundle over a compact Riemann surface, we prove that its pullback and direct image through a map between compact Riemann surfaces inherit a natural symplectic (respectively, orthogonal) structure. If the parabolic bundle is endowed with a parabolic Higgs field or a parabolic connection which are compatible with the symplectic (respectively, orthogonal) structure, then its pullback and direct image are also compatible with the resulting symplectic (respectively, orthogonal) structure. We also show that these constructions are preserved through the Nonabelian Hodge Correspondence.