Let $M$ be a ALE $n$-manifold. Then it is known a folklore result that
the $L^2$ Hodge cohomology is given by:


* $L^2\mathcal H^k=H^k(M,\partial M)$ if $k < n$,
* $L^2\mathcal H^{n/2}=Im(H^{n/2}(M,\partial M)\to H^{n/2}(M))$, and
* $L^2\mathcal H^k=H^k(M)$ if $k>n$.


A proof can be found in "HODGE COHOMOLOGY OF
GRAVITATIONAL INSTANTONS" by Hunsicker, Hausel & Mazzeo. Their proof is very involved and it seems to me that a simpler proof should exist. Is a there a good simple proof? Maybe even published in the literature somewhere.



$L^2$ Hodge cohomology of ALE manifolds
I believe you can find this in the papers of Lockhart and McOwen. Specifically, check


Lockhart, Robert B.; McOwen, Robert C. Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 3, 409--447. MR0837256
[0837256](http://www.ams.org/mathscinet-getitem?mr=0837256)


or


Lockhart, Robert. Fredholm, Hodge and Liouville theorems on noncompact manifolds. Trans. Amer. Math. Soc. 301 (1987), no. 1, 1--35. MR0879560
[0879560](http://www.ams.org/mathscinet-getitem?mr=0879560)


At least, that's where I learned these things. A more recent and very clear exposition can be found in the thesis of Stephen Marshall, which is available on Dominic Joyce's website, but his treatment is specific to asympotically cylindrical or asympototically conical Riemannian manifolds. The ALE spaces are similar and are amenable to essentially the same treatment, if I recall correctly.


