Moduli spaces of Hecke modifications for rational and elliptic curves

Boozer, David

doi:10.2140/agt.2021.21.543

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Moduli spaces of Hecke modifications for rational and elliptic curves

David Boozer

Algebraic & Geometric Topology 21 (2021) 543–600

DOI: 10.2140/agt.2021.21.543

Abstract

We propose definitions of complex manifolds $𝒫_{M} (X, m, n)$ that could potentially be used to construct the symplectic Khovanov homology of $n$ –stranded links in lens spaces. The manifolds $𝒫_{M} (X, m, n)$ 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 $𝒫_{M} (X, m, n)$ 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 $𝒫_{M} (X, 1, n)$ for $n = 0, 1, 2$ . For comparison, we present analogous results for the case of rational curves, for which a corresponding complex manifold $𝒫_{M} ({ℂ ℙ}^{1}, 3, n)$ is isomorphic for $n$ even to a space $𝒴 (S^{2}, n)$ defined by Seidel and Smith that can be used to compute the symplectic Khovanov homology of $n$ –stranded links in $S^{3}$ .

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

Volume 21, issue 2 (2021)