Regular multi-types and the Bloom conjecture II

Abstract

Let $M\subset\mathbb{C}^n$ $(n\geq2)$ be a smooth pseudoconvex real hypersurface. We prove that, for every $1\leq s\leq n-1$, the regular $s$-contact type, the $s$-commutator type, and the $s$-Levi type coincide at every point of $M$, thereby resolving Bloom's conjecture in any dimension. The solution in dimension three was obtained in 2021 by the second and the third authors. Moreover, for any smooth rank $s$ subbundle $B\subset T^{1,0}M$, we establish the equality of the commutator type and the Levi type associated with $B$, giving an affirmative answer to the generalized D'Angelo conjecture.