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.



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

> 
> 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](https://doi.org/10.1007/BF01444534). 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](https://doi.org/10.1007/s00220-009-0863-8).
Comm. Math. Phys. 290 (2009), no. 2, 719–736.


and on arbitrary ALF space:


Cherkis, Sergey A.
[Instantons on gravitons](https://doi.org/10.1007/s00220-011-1293-y).
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.


