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Homogeneous Monge–Ampère Equations and Canonical Tubular Neighbourhoods in Kähler Geometry

Accepted version
Peer-reviewed

Type

Article

Change log

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

Volume Title

2017

Publisher

Oxford University Press
Sponsorship
Engineering and Physical Sciences Research Council (EP/J002062/1)
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.