Moduli spaces of Hecke modifications for rational and elliptic curves

Abstract

We propose definitions of complex manifolds ${\mathcal{P}}_M(X,m,n)$ that could potentially be used to construct the symplectic Khovanov homology of $n$-stranded links in lens spaces. The manifolds ${\mathcal{P}}_M(X,m,n)$ are defined as moduli spaces of Hecke modifications of rank 2 parabolic bundles over an elliptic curve $X$. To characterize these spaces, we describe all possible Hecke modifications of all possible rank 2 vector bundles over $X$, and we use these results to define a canonical open embedding of ${\mathcal{P}}_M(X,m,n)$ into $M^s(X,m+n)$, the moduli space of stable rank 2 parabolic bundles over $X$ with trivial determinant bundle and $m+n$ marked points. We explicitly compute ${\mathcal{P}}_M(X,1,n)$ for $n=0,1,2$. For comparison, we present analogous results for the case of rational curves, for which a corresponding complex manifold ${\mathcal{P}}_M({\mathbb{CP}}^1,3,n)$ is isomorphic for $n$ even to a space $\mathcal{Y}(S^2,n)$ defined by Seidel and Smith that can be used to compute the symplectic Khovanov homology of $n$-stranded links in $S^3$.

29 pages, 1 figure; extensively revised; new methods are used to describe Hecke modifications in order to obtain more concise proofs

References in corpus (1)

• Stability of projective Poincare and Picard bundles

Cited by in corpus (1)

• Moduli spaces of parabolic bundles over ${\mathbb{P}}^1$ with five marked points