Symplectic geometry of Higgs moduli and the Hilbert scheme of points over an elliptic curve

TL;DR

This paper demonstrates a symplectomorphism between the moduli space of specific parabolic Higgs bundles on an elliptic curve and the Hilbert scheme of points on its cotangent bundle, revealing deep geometric connections.

Contribution

It establishes a symplectic isomorphism linking Higgs bundle moduli spaces and Hilbert schemes over elliptic curves, highlighting new geometric structures.

Findings

The moduli space of certain parabolic Higgs bundles is symplectomorphic to the Hilbert scheme of points.

The natural symplectic structures on these spaces are preserved under the isomorphism.

This connection provides new insights into the geometry of Higgs bundles and Hilbert schemes.

Abstract

We show that the isomorphism between the moduli space of certain parabolic Higgs bundles over an elliptic curve and the Hilbert scheme of n points of the cotangent bundle of the elliptic curve is a symplectomorphism with respect to their natural symplectic structures.