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We prove the existence of canonical tubular neighbourhoods around complex submanifolds of K\"ahler manifolds that are adapted to both the holomorphic and symplectic structure. This is done by solving the complex Homogeneous Monge-Amp\`ere 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 rays in the space of (weak) K\"ahler 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.

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