Quasi-asymptotically conical Calabi–Yau manifolds

Conlon, Ronan¹ ; Degeratu, Anda² ; Rochon, Frédéric³

¹ Department of Mathematics and Statistics, Florida International University, Miami, FL, United States
² Fachbereich Mathematik, Universität Stuttgart, Stuttgart, Germany
³ Département de Mathématiques, Université du Québec à Montréal, Montréal, QC, Canada

Geometry & topology, Tome 23 (2019) no. 1, pp. 29-100.

Résumé

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.

Détail

DOI : 10.2140/gt.2019.23.29

Classification : 53C55, 58J05
Keywords: Calabi–Yau metrics, quasi-asymptotically conical metrics, manifolds with corners

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