Quasi-asymptotically conical Calabi–Yau manifolds

Abstract

We construct new examples of quasi-asymptotically conical ($QAC$) Calabi–Yau manifolds that are not quasi-asymptotically locally Euclidean ($QALE$). We do so by first providing a natural compactification of $QAC$–spaces by manifolds with fibered corners and by giving a definition of $QAC$–metrics in terms of an associated Lie algebra of smooth vector fields on this compactification. Thanks to this compactification and the Fredholm theory for elliptic operators on $QAC$–spaces developed by the second author and Mazzeo, we can in many instances obtain Kähler $QAC$–metrics having Ricci potential decaying sufficiently fast at infinity. This allows us to obtain $QAC$ Calabi–Yau metrics in the Kähler classes of these metrics by solving a corresponding complex Monge–Ampère equation.