Pullback and direct image of parabolic Higgs bundles and parabolic connections with symplectic and orthogonal structures
David Alfaya, Indranil Biswas, Francois-Xavier Machu
TL;DR
This work establishes a robust, functorial framework for parabolic symplectic and orthogonal structures under two natural operations on curves: pullback by nonconstant maps and direct image by nonconstant maps. It shows that pullbacks and pushforwards preserve the corresponding symplectic/orthogonal structures and compatibility with parabolic Higgs fields and connections, while also preserving (semi)stability and polystability. A central achievement is proving that these constructions are compatible with the Nonabelian Hodge Correspondence for noncompact curves, yielding a coherent correspondence between polystable parabolic Higgs bundles and flat parabolic symplectic/orthogonal connections that respects pullback and pushforward. The results provide a rigorous, functorial toolkit for studying parabolic structures in the presence of additional symmetries and show the NAHC framework remains intact under these geometric operations.
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.