C1,1 regularity of geodesics of singular Kähler metrics

We show the optimal (Formula presented.) 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 (Formula presented.) 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.