Narrow Results

Let E_* be a parabolic vector bundle over a smooth complex projective curve. We prove that E_* admits an algebraic connection if and only if the parabolic degree of every parabolic vector bundle which is a direct summand of E_* is zero. In particular, all parabolic semistable vector bundles of parabolic degree zero admit an algebraic connection.

We construct projectivization of a parabolic vector bundle and a tautological line bundle over it. It is shown that a parabolic vector bundle is ample if and only if the tautological line bundle is ample. This allows us to generalize the notion of a k-ample bundle, introduced by Sommese, to the context of parabolic bundles. A parabolic vector bundle E_* is defined to be k-ample if the tautological line bundle is k-ample. We establish some properties of parabolic k-ample bundles.

Let $k$ be an algebraically closed field of characteristic zero. We prove that the Brauer group of the moduli stack of stable parabolic $\text{PGL}(r,k)$-bundles on a smooth projective curve, with full flag quasi-parabolic structures at an arbitrary parabolic divisor, coincides with the Brauer group of the smooth locus of the corresponding coarse moduli space of parabolic $\text{PGL}(r,k)$-bundles. We also compute the Brauer group of the smooth locus of this coarse moduli for more general quasi-parabolic types and weights satisfying certain conditions.

In this paper, we show that the representation variety of the fundamental group of a 2n-punctured S² with different conjugacy classes in SU(2) along punctures is a symplectic stratified variety from the group cohomology point of view. The representation variety can be identified with the moduli space of s-equivalence classes of stable parabolic bundles over the 2n-punctured S² with corresponding weights along punctures, and also can be identified with the moduli space of gauge equivalence classes of SU(2)-flat connections with prescribed holonomies along punctures. We obtain an invariant of links (knots) from intersection theory on such a moduli space (a generalization of the signature of the link). We also study a SL₂(C)-character variety of a knot K in S³ with fixed holonomy μ + μ^-1 along the meridian of π₁(S³\ K) (μ ∈ C^*). The fixed-trace condition rules out the possibility of reducible representations with non-abelian image and the ideal point of irreducible representations via a generic perturbation. For knots without closed incompressible surfaces in S³ \ K, we show that there is a well-defined SL₂(C)-knot invariant which is also related to them A-polynomial for special values μ ∈ U(1)\ {± 1}.

Given a compact Riemann surface $X$ and a moduli space $M_{\alpha }(\mathrm{\Lambda })$ of parabolic stable bundles on it of fixed determinant of complete parabolic flags, we prove that the Poincaré parabolic bundle on $X\times M_{\alpha }(\mathrm{\Lambda })$ is parabolic stable with respect to a natural polarization on $X\times M_{\alpha }(\mathrm{\Lambda })$.