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 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 = 0,1,2. For comparison, we present analogous results for the case of rational curves, for which a corresponding complex manifold 𝒫M(1,3,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.

Keywords
Hecke modifications, rational curves, elliptic curves, vector bundles, parabolic bundles, Khovanov homology
Mathematical Subject Classification 2010
Primary: 14H52, 14H99
References
Publication
Received: 6 July 2018
Revised: 28 April 2020
Accepted: 1 June 2020
Published: 25 April 2021
Authors
David Boozer
Department of Mathematics
Princeton University
Princeton, NJ
United States