A boundedness theorem for principal bundles on curves

TL;DR

The paper addresses the boundedness of principal $G$-bundles on a smooth projective curve when the associated $V$-bundle admits a section into the GIT stable locus $V^{\mathrm s}(\theta)$. The main result shows that, for a fixed degree $d=\deg_{\mathscr C}(\mathscr P\times_{G} \mathbb C_{\theta})$, the relevant family of bundles is bounded, enabling boundedness results for $\varepsilon$-stable quasimaps and $\Omega$-stable LG-quasimaps in a nonabelian setting. The approach reduces boundedness to controlling canonical reductions to parabolic subgroups and to constructing a finite set $\Omega$ of possible degrees for associated Levi factors, leveraging convexity arguments and the Hilbert–Mumford criterion. This provides the nonabelian boundedness piece missing in the broader theory of LG-quasimaps and stable quasimaps, with implications for related moduli problems in mixed-field-type settings.

Abstract

Let $G$ be a reductive group acting on an affine scheme $V$. We study the set of principal $G$-bundles on a smooth projective curve $\mathcal C$ such that the associated $V$-bundle admits a section sending the generic point of $\mathcal C$ into the GIT stable locus $V^{\mathrm{s}}(θ)$. We show that after fixing the degree of the line bundle induced by the character $θ$, the set of such principal $G$-bundles is bounded. The statement of our theorem is made slightly more general so that we deduce from it the boundedness for $ε$-stable quasimaps and $Ω$-stable LG-quasimap.

Theorems & Definitions (37)

• Theorem 1.1: Simplified version
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3: ramanathan1975stable
• Proposition 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof