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We develop new techniques to compute the weighted L2-cohomology of quasi- fibered boundary metrics (QFB-metrics). Combined with the decay of L2-harmonic forms obtained in a companion paper, this allows us to compute the reduced L2- cohomology for various classes of QFB-metrics. Our results applies in particular to the Nakajima metric on the Hilbert scheme of n points on C2, for which we can show that the Vafa-Witten conjecture holds. Using the compactification of the monopole moduli space announced by Fritzsch, the first author and Singer, we can also give a proof of the Sen conjecture for the monopole moduli space of magnetic charge 3. 

Manifolds with fibered corners arise as resolutions of stratified spaces, as ‘many-body’ compactifications of vector spaces, and as compactifications of certain moduli spaces includ- ing those of non-abelian Yang–Mills–Higgs monopoles, among other settings. However, Cartesian products of manifolds with fibered corners do not generally have fibered corners themselves and thus fail to reflect the appropriate structure of products of the underlying spaces in the above settings. Here, we determine a resolution of the Cartesian product of fibered corners manifolds by blow-up which we call the ‘ordered product,’ which leads to a well-behaved category of fibered corners manifolds in which the ordered product satisfies the appropriate universal property. In contrast to the usual category of manifolds with cor- ners, this category of fibered corners not only has all finite products, but all finite transverse fiber products as well, and we show in addition that the ordered product is a natural product for wedge (aka incomplete edge) metrics and quasi-fibered boundary metrics, a class which includes QAC and QALE metrics.