$C^{1,1}$ 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.

37 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