TY - JOUR

T1 - Moduli spaces of Hecke modifications for rational and elliptic curves

AU - Boozer, David

N1 - Funding Information:
The author would like to express his gratitude towards Ciprian Manolescu for suggesting the problem considered in this paper and for providing invaluable guidance, Joel Kamnitzer for helpful discussions, Burt Totaro for helpful advice and for offering extensive comments on an earlier version of this paper that led to significant changes in the current version, and Chris Woodward for sharing his personal notes on the Woodward embedding described in Section 4.6.2. The author was partially supported by NSF grant number DMS-1708320.
Publisher Copyright:
© 2021, Mathematical Science Publishers. All rights reserved.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

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U2 - 10.2140/agt.2021.21.543

DO - 10.2140/agt.2021.21.543

M3 - Article

AN - SCOPUS:85108950702

VL - 21

SP - 543

EP - 600

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

SN - 1472-2747

IS - 2

ER -