2021/09/27 by Biswas, Indranil, Hurtubise, Jacques
#14D20 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2109.13358
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 Xt 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 (S1)(3g-3)(n-1) symplectic quotient of a (2g-2)-th power of a space associated to the three-pointed sphere.