The core of the Levi distribution [Le cœur de la distribution de Levi]

We introduce a new geometrical invariant of CR manifolds of hypersurface type, which we call the "Levi core" of the manifold. When the manifold is the boundary of a smooth bounded pseudoconvex domain, we show how the Levi core is related to two other important global invariants in several complex variables: the Diederich-Fornæss index and the D'Angelo class (namely the set of D'Angelo forms of the boundary). We also show that the Levi core is trivial whenever the domain is of finite-type in the sense of D'Angelo, or the set of weakly pseudoconvex points is contained in a totally real submanifold, while it is nontrivial if the boundary contains a local maximum set. As corollaries to the theory developed here, we prove that for any smooth bounded pseudoconvex domain with trivial Levi core the Diederich-Fornæss index is 1 and the ∂-Neumann problem is exactly regular (via a result of Kohn and its generalization by Harrington). Our work builds on and expands recent results of Liu and Adachi-Yum.