Degenerations of Bundle Moduli

TL;DR

This paper constructs a degeneration of the moduli space of SL(n,C) bundles over a family of curves, connecting complex algebraic and symplectic perspectives, and describes the limit space in terms of simpler building blocks.

Contribution

It introduces a new framework for understanding degenerations of bundle moduli spaces over nodal curves, linking algebraic and symplectic approaches with explicit quotient descriptions.

Findings

Degeneration of bundle moduli spaces into quotient spaces associated with three-pointed spheres.

Compatibility of algebraic and symplectic degenerations.

Explicit description of the limit space as a quotient of a product space.

Abstract

Over a family $\mathbb X$ of genus $g$ complete curves, which gives the degeneration of a smooth curve into one with nodal singularities, we build a moduli space which is the moduli space of ${\rm SL}(n, \mathbb C)$ bundles over the generic smooth curve $X_t$ in the family, and is a moduli space of bundles equipped with extra structure at the nodes for the nodal curves in the family. This moduli space is a quotient by $(\mathbb C^*)^s$ of a moduli space on the desingularisation. Taking a "maximal" degeneration of the curve into a nodal curve built from the glueing of three-pointed spheres, we obtain a degeneration of the moduli space of bundles into a $(\mathbb C^*)^{(3g-3)(n-1)}$-quotient of a $(2g-2)$-th power of a space associated to the three-pointed sphere. Via the Narasimhan-Seshadri theorem, the moduli of bundles on the smooth curve is a space of representations of the fundamental group into ${\rm SU}(n)$ (the "symplectic picture"). We obtain the degenerations also in this symplectic context, in a way that is compatible with the holomorphic degeneration, so that our limit space is also a $(S^1)^{(3g-3)(n-1)}$ symplectic quotient of a $(2g-2)$-th power of a space associated to the three-pointed sphere.