Homogeneous Monge–Ampère Equations and Canonical Tubular Neighbourhoods in Kähler Geometry
Accepted version
Peer-reviewed
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Repository DOI
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Authors
Ross, Julius
Nyström, David Witt
Abstract
We prove the existence of canonical tubular neighbourhoods around complex submanifolds of Kähler manifolds that are adapted to both the holomorphic and symplectic structure. This is done by solving the complex Homogeneous Monge–Ampère equation on the deformation to the normal cone of the submanifold. We use this to establish local regularity for global weak solutions, giving local smoothness to the (weak) geodesic ray in the space of (weak) Kähler potentials associated to a given complex submanifold. We also use it to get an optimal regularity result for naturally defined plurisubharmonic envelopes and for the boundaries of their associated equilibrium sets.
Description
Keywords
math.CV, math.CV, math.AG, math.DG, 32Q15
Journal Title
International Mathematics Research Notices
Conference Name
Journal ISSN
1687-0247
1687-0247
1687-0247
Volume Title
2017
Publisher
Oxford University Press
Publisher DOI
Sponsorship
Engineering and Physical Sciences Research Council (EP/J002062/1)
European Commission (329070)
European Commission (329070)
This work was supported by EPSRC Career Acceleration Fellowship [EP/J002062/1 to J.R.]; People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme [FP7/2007-2013] under REA grant agreement no 329070, and previously by Chalmers University and the University of Gothenburg to D.W.N.