Hermitian-Einstein connections on polystable orthogonal and symplectic parabolic Higgs bundles (Q414516)

Let \(X\) be an irreducible smooth complex projective curve, and let \(S \subset X\) be a fixed finite subset. The notion of parabolic vector bundle on \(X\) with \(S\) as the parabolic divisor was introduced by Seshadri, while parabolic bundles equipped with Higgs fields were introduced by Simpson under the name of filtered regular Higgs bundles.NEWLINENEWLINEAn orthogonal or symplectic parabolic bundle is a parabolic vector bundle equipped with a symmetric or alternating form, respectively, with values in a parabolic line bundle; this form is required to be non-degenerate in a suitable sense. In the case of rational parabolic weights, this coincides with the notion of parabolic principal \(G\)-bundle, where \(G\) is the orthogonal or symplectic group, respectively.NEWLINENEWLINEThe main objective of the paper under review is to define Higgs fields on orthogonal and symplectic parabolic bundles and to generalize the Hitchin-Kobayashi correspondence to this context. The main result of the paper is stated as:NEWLINENEWLINETheorem 1. Let \((E_{*}, \varphi, \theta)\) be an orthogonal or symplectic parabolic Higgs bundle. If \((E_{*}, \varphi, \theta)\) is polystable, then it admits a Hermitian-Einstein connection. Conversely, if \((E_{*}, \varphi, \theta)\) admits a Hermitian-Einstein connection lying in the space \(\mathcal{A}\), then it is polystable.