On the smoothness of moduli spaces for quiver bundles

TL;DR

The paper addresses the smoothness of moduli spaces of finite quiver bundles on smooth complex projective curves. It develops a hypercohomology framework for Ext groups via a two-term complex $F^\bullet(E_\bullet,E'\_\bullet)$ and uses deformation theory to identify the tangent space with $\mathbb{H}^1([\cdot,\phi_\bullet])$ and obstructions with $\mathbb{H}^2([\cdot,\phi_\bullet])$. The main result shows that $\mathcal{M}_\alpha^s(t)$ is smooth under the stability-parameter bound $\alpha_{at}-\alpha_{ah} \ge 2g-2$, with local dimension given by $1-\chi(E_\bullet,E_\bullet)$. The work also provides vanishing criteria for obstructions, explicit dimension formulas, and a construction of a projective (Poincaré-type) bundle over the moduli space, enhancing the understanding of the geometry of quiver-bundle moduli.

Abstract

In this article, we study the smoothness of the moduli space of finite quiver vector bundles over the smooth complex projective curves.

Theorems & Definitions (21)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Proposition 3.1
• proof
• Proposition 3.3
• proof
• Corollary 3.4
• Proposition 3.5
• Proposition 3.6