Hyperkähler structure of framed instantons over $\smash{\overline{\mathbb{C}P}}^2$

Question

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.

Answer 1

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

The "decay constraint" condition is called "gravitational instanton". A gravitational instanton is a complete hyperkähler metric on a 4-manifold with $L^2$-integrable curvature. There is a lot of literature on gravitational instantons and on instanton bundles on gravitational instantons. The gravitational instantons are classified according to the asymptotic growth of a volume of a ball: when it grows as a 4-th degree (like on Euclidean space), it's ALE instanton, degree 3 is ALF, and the classification becomes very detailed and intricate for smaller degrees. ALF and ALE spaces can be constucted as hyperkähler quotients of flat hyperkähler spaces, for lower degrees it's probably false.

The space of instantons on ALE space is hyperkähler:

P.B. Kronheimer, H. Nakajima, Yang–Mills instantons and ALE gravitational instantons. Math. Ann. 288 (1990), 263–307.

The same is true for instantons on Taub-NUT space, which is ALF:

Cherkis, Sergey A. Moduli spaces of instantons on the Taub-NUT space. Comm. Math. Phys. 290 (2009), no. 2, 719–736.

and on arbitrary ALF space:

Cherkis, Sergey A. Instantons on gravitons. Comm. Math. Phys. 306 (2011), no. 2, 449–483.

There is no framing in these results, you need to consider instantons with $L^2$-integrable curvature.

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?

Mostly not, in fact, it can easily be odd-dimensional. One could imagine results like that (for instance, $CP^2$ is not hyperkähler, but instantons on $CP^2$ are hyperkähler, if the framing is chosen right), but this would probably mean that the complement to the curve where you have chosen framing is hyperkähler.