Pullback and Direct Image of Parabolic Connections and Parabolic Higgs Bundles.

The article focuses on explicit algebraic constructions of pullbacks and direct images of parabolic bundles, parabolic Higgs bundles, and parabolic connections via nonconstant holomorphic maps between compact connected Riemann surfaces. It establishes that these operations preserve semistability and polystability of the parabolic objects and proves their compatibility with the nonabelian Hodge correspondence, which relates parabolic Higgs bundles and connections to representations of the fundamental group. The work provides detailed constructions of these functorial operations, including the behavior of parabolic weights and residues under pullback and push-forward, and demonstrates that the induced harmonic bundles correspond appropriately under these maps. Tables summarizing the transformation of parabolic jumps and eigenvalues under pullbacks and direct images are also presented.