Description
The parabolic Higgs bundle moduli spaces on the n-punctured sphere are hyperkahler manifolds with integrable system structures. Star-shaped Nakajima quiver varieties are hyperkahler manifolds which were used by Kronheimer and Nakajima to model ALE spaces. In higher dimensional cases, they were recently shown by Dimakis and Rochon to be quasi-asymptotically conical (QAC).
We generalize a construction by Rayan and Schaposnik to create a map T from a given star-shaped quiver variety X to a parabolic Hitchin moduli space M. We verify that T preserves stability and we show that it is a homeomorphism onto a Zariski open subspace of M. We then prove that T preserves the natural holomorphic symplectic forms on the two spaces, generalizing work by Biswas-Florentino-Godinho-Mandini from the rank 2, full flag, strongly parabolic case to the rank r, partial flag, weakly parabolic case. Finally, we discus some interesting corollaries.