On the symplectic forms of Groechenig's Higgs moduli over an elliptic curve

Abstract

Gorsky, Nekrasov, and Rubtsov introduced the moduli space of marked Higgs bundles over an elliptic curve $E$ and identified it with the Hilbert scheme of points on the cotangent bundle. Later, Groechenig constructed four analogous isomorphisms on marked rational curves, of affine Dynkin types ${\tilde{D}}_4$, ${\tilde{E}}_6$, ${\tilde{E}}_7$, and ${\tilde{E}}_8$, for the $\Gamma$-Hilbert schemes of $T^{\ast }E$ for cyclic groups $\Gamma$ with $|\Gamma |\in \{2,3,4,6\}$. We prove that the isomorphisms in all five cases are holomorphic symplectomorphisms.

20 pages, Comments welcome