Uhlenbeck spaces via affine Lie algebras
arXiv:math/0301176
Abstract
Let be an almost simple simply connected group over $\BC$, and let $\Bun^a_G(\BP^2,\BP^1)$ be the moduli scheme of principal -bundles on the projective plave $\BP^2$, of second Chern class , trivialized along a line $\BP^1\subset \BP^2$. We define the Uhlenbeck compactification $\fU^a_G$ of $\Bun^a_G(\BP^2,\BP^1)$, which classifies, roughly, pairs $(\F_G,D)$, where is a 0-cycle on $\BA^2=\BP^2-\BP^1$ of degree , and $\F_G$ is a point of $\Bun^{a-b}_G(\BP^2,\BP^1)$, for varying . In addition, we calculate the stalks of the Intersection Cohomology sheaf of $\fU^a_G$. To do that we give a geometric realization of Kashiwara's crystals for affine Kac-Moody algebras.
Erratum added: the statements of Proposition 15.2, Theorems 7.10, 16.7, 16.8 corrected in Section 19