Stability Of The Parabolic Poincar\'e Bundle

TL;DR

This paper proves that the Poincaré parabolic bundle over a Riemann surface and its moduli space remains parabolic stable under a natural polarization, enhancing understanding of stability in parabolic bundle theory.

Contribution

It establishes the parabolic stability of the Poincaré bundle on the product of a Riemann surface and its moduli space, a novel result in the study of parabolic bundles.

Findings

Poincaré parabolic bundle is parabolic stable.

Stability holds with respect to a natural polarization.

Results contribute to the theory of parabolic bundle stability.

Abstract

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