# Compactness of the moduli space of bundles with fixed determinant

The moduli space of semistable holomorphic vector bundles of fixed rank and fixed determinant line bundle on a compact Riemann surface is known to be compact itself. (In particular, when the rank is $1$, this space is just a single point, in the Picard group.)

When the rank is $2$ or more, is there a direct way to see that the moduli space is compact?

By "direct", I mean I'd like to be able to see this without appealing to flat connections, representations of the fundamental group, or Yang-Mills theory.

• Isn't the compactness equivalent to the notion of 'boundedness', meaning that set theoretic familes of semistable bundles with fixed Hilbert polynomial and bounded coefficients (I'm being vague here) can be put into families? I would think that @AsalBeagDubh would know. – Alex Youcis Mar 5 '15 at 9:00
• Also, @Brenin might know :) – Alex Youcis Mar 5 '15 at 10:00