Moduli spaces of Hecke modifications for rational and elliptic curves

TL;DR

This paper introduces moduli spaces of Hecke modifications for elliptic and rational curves, aiming to connect them to symplectic Khovanov homology of links in lens spaces and spheres.

Contribution

It defines complex manifolds as moduli spaces of Hecke modifications of rank 2 parabolic bundles over elliptic and rational curves, with explicit computations and embeddings.

Findings

Explicit description of Hecke modifications over elliptic curves.

Construction of embeddings into moduli spaces of stable parabolic bundles.

Identification of rational curve case with known link invariants.

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 $S^3$.