Moduli spaces of Hecke modifications for rational and elliptic curves

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Abstract

We propose definitions of complex manifolds PM(X, m, n) that could potentially be used to construct the symplectic Khovanov homology of n-stranded links in lens spaces. The manifolds PM(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 PM.(X, m, n) into Ms.(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 PM.(X, 1, n) for n D 0; 1; 2. For comparison, we present analogous results for the case of rational curves, for which a corresponding complex manifold PM.CP1; 3; n/ is isomorphic for n even to a space Y.(S2; n) defined by Seidel and Smith that can be used to compute the symplectic Khovanov homology of n-stranded links in S3.

Original languageEnglish (US)
Pages (from-to)543-600
Number of pages58
JournalAlgebraic and Geometric Topology
Volume21
Issue number2
DOIs
StatePublished - 2021

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

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