Regularity of the singular set in the fully nonlinear obstacle problem

For the obstacle problem involving a convex fully nonlinear elliptic operator, we show that the singular set in the free boundary stratifies. The top stratum is locally covered by a $C^{1,\alpha }$-manifold, and the lower strata are covered by $C^{1,{\log }^{\epsilon }}$-manifolds. This essentially recovers the regularity result obtained by Figalli-Serra when the operator is the Laplacian.