A properness result for degenerate Quadratic and Symplectic Bundles on a smooth projective curve

Abstract

Let $(V,q)$ be a vector bundle on a smooth projective curve $X$ together with a quadratic form $q: \mathrm{Sym}^2(V) \ra \mathcal{O}_X$ (respectively symplectic form $q: Λ^2V \ra \mathcal{O}_X$). Fixing the degeneracy locus of the quadratic form induced on $V/\ker (q)$, we construct a coarse moduli of such objects. Further, we prove semi-stable reduction theorem for equivalence classes of such objects. In particular, the case when degeneracies of $q$ are higher than one is that of principal interest. We also provide a proof of properness of polystable orthogonal bundles without appealing to Bruhat-Tits theory in any characteristic.

44 pages

References in corpus (2)

• The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations
• Moduli of parahoric $\mathcal{G}$--torsors on a compact Riemann surface