#### Volume 21, issue 2 (2021)

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Moduli spaces of Hecke modifications for rational and elliptic curves

### David Boozer

Algebraic & Geometric Topology 21 (2021) 543–600
##### Abstract

We propose definitions of complex manifolds ${\mathsc{𝒫}}_{M}\left(X,m,n\right)$ that could potentially be used to construct the symplectic Khovanov homology of $n$–stranded links in lens spaces. The manifolds ${\mathsc{𝒫}}_{M}\left(X,m,n\right)$ are defined as moduli spaces of Hecke modifications of rank $2$ parabolic bundles over an elliptic curve $X\phantom{\rule{-0.17em}{0ex}}$. To characterize these spaces, we describe all possible Hecke modifications of all possible rank $2$ vector bundles over $X\phantom{\rule{-0.17em}{0ex}}$, and we use these results to define a canonical open embedding of ${\mathsc{𝒫}}_{M}\left(X,m,n\right)$ 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 ${\mathsc{𝒫}}_{M}\left(X,1,n\right)$ for $n=0,1,2$. For comparison, we present analogous results for the case of rational curves, for which a corresponding complex manifold ${\mathsc{𝒫}}_{M}\left({ℂℙ}^{1},3,n\right)$ is isomorphic for $n$ even to a space $\mathsc{𝒴}\left({S}^{2},n\right)$ defined by Seidel and Smith that can be used to compute the symplectic Khovanov homology of $n$–stranded links in ${S}^{3}$.

##### Keywords
Hecke modifications, rational curves, elliptic curves, vector bundles, parabolic bundles, Khovanov homology
##### Mathematical Subject Classification 2010
Primary: 14H52, 14H99