$C^{1,1}$ regularity of geodesics of singular Kähler metrics

Abstract

We show the optimal $C^{1,1}$ regularity of geodesics in nef and big cohomology class on Kähler manifolds away from the non-Kähler locus, assuming sufficiently regular initial data. As a special case, we prove the $C^{1,1}$ regularity of geodesics of Kähler metrics on compact Kähler varieties away from the singular locus. Our main novelty is an improved boundary estimate for the complex Monge-Ampère equation that does not require strict positivity of the reference form near the boundary. We also discuss the case of some special geodesic rays.

arXiv comment37 pages; Corrected an incorrect formula in the proof of the main result; Results are unchanged; Exposition is improved and more background material is added; Final version, to appear in J. Lond. Math. Soc

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• The Monge-Ampère operator and geodesics in the space of Kähler potentials
• On the singularity type of full mass currents in big cohomology classes
• On the $C^{1,1}$ regularity of geodesics in the space of Kähler metrics
• Geodesics in the space of Kähler metrics
• The Monge-Ampère equation for non-integrable almost complex structures
• $C^{1,1}$ regularity for degenerate complex Monge-Ampère equations and geodesic rays
• Geometry and Topology of the space of Kähler metrics on singular varieties
• Regularity of envelopes in Kähler classes
• Envelopes with Prescribed Singularities