Regular multi-types and the Bloom conjecture

We prove that the commutator type, the regular contact type and the Levi form type of order s=(n−2) are the same for a smooth pseudoconvex real hypersurface in Cⁿ with n≥3. In particular, this provides, in the case of complex dimension three, a complete solution of a long standing conjecture of Bloom formulated in his famous and important 1981 paper [12]. When n≥4, our theorem provides the first result along the lines of the Bloom conjecture in any dimensions in a case where the pseudoconvexity assumption of the hypersurface starts to be crucial.