Moduli spaces of Hecke modifications for rational and elliptic curves

Abstract

We propose definitions of complex manifolds ${\mathcal{𝒫}}_M\left(X,m,n\right)$ that could potentially be used to construct the symplectic Khovanov homology of $n$–stranded links in lens spaces. The manifolds ${\mathcal{𝒫}}_M\left(X,m,n\right)$ 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 ${\mathcal{𝒫}}_M\left(X,m,n\right)$ into $M^s\left(X,m+n\right)$, the moduli space of stable rank $2$ parabolic bundles over $X$ with trivial determinant bundle and $m+n$ marked points. We explicitly compute ${\mathcal{𝒫}}_M\left(X,1,n\right)$ for $n=0,1,2$. For comparison, we present analogous results for the case of rational curves, for which a corresponding complex manifold ${\mathcal{𝒫}}_M\left({ℂ\mathbb{P}}^1,3,n\right)$ is isomorphic for $n$ even to a space $\mathcal{𝒴}\left(S^2,n\right)$ defined by Seidel and Smith that can be used to compute the symplectic Khovanov homology of $n$–stranded links in $S^3$.