When underlying $4$ manifold is compact and hyperkahler, the philosophy of infinite dimensional moment map tells us that its instantons moduli space is also hyperkahler. I'm curious about the following two questions.

(1) When our hyperkahler $4$ fold is no longer compact, is its instantons moduli space(maybe with some constriant about decaying condition) still hyperkahler?

(2) For some noncompact kahler surface that are not hyperkahler, for example blowup of $\mathbb{C}^2$ at origin, does its framed instantons moduli space still admit hyperkahler structure? I know this space can be constructed by finite dimensional symplectic quotient, but the configuration space does not admit a natural hyperkahler structure in genral.

Thank you for your answer.

When underlying $4$ manifold is compact and hyperkähler, the philosophy of infinite dimensional moment map tells us that its instantons moduli space is also hyperkähler. I'm curious about the following two questions.

1. When our hyperkähler $4$ fold is no longer compact, is its instantons moduli space(maybe with some constriant about decaying condition) still hyperkähler?

2. For some noncompact kähler surface that are not hyperkähler, for example blowup of $\mathbb{C}^2$ at origin, does its framed instantons moduli space still admit a hyperkähler structure? I know this space can be constructed by a finite dimensional symplectic quotient, but the configuration space does not admit a natural hyperkähler structure in general.