Quasi-asymptotically conical Calabi–Yau manifolds

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.

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Keywords

Calabi–Yau metrics, quasi-asymptotically conical metrics, manifolds with corners

Mathematical Subject Classification 2010

Primary: 53C55, 58J05

References

Publication

Received: 21 March 2017

Revised: 12 February 2018

Accepted: 14 June 2018

Published: 5 March 2019

Proposed: Simon Donaldson

Seconded: Tobias H Colding, Gang Tian

Authors

Ronan J Conlon
Department of Mathematics and Statistics Florida International University Miami, FL United States

Anda Degeratu
Fachbereich Mathematik Universität Stuttgart Stuttgart Germany

Frédéric Rochon
Département de Mathématiques Université du Québec à Montréal Montréal, QC Canada

Ronan J Conlon


Frédéric Rochon


Lars Sektnan
Département de Mathématiques Université du Québec à Montréal Montréal, QC Canada

Volume 23, issue 1 (2019)

Ronan J Conlon, Anda Degeratu and Frédéric Rochon

Appendix: Ronan J Conlon, Frédéric Rochon and Lars Sektnan

Abstract