# Regular multi-types and the Bloom conjecture

## TL;DR

This paper proves the Bloom conjecture for complex dimension three and verifies it for higher dimensions when s=n-2, advancing understanding of pseudoconvexity in complex analysis.

## Contribution

It establishes the equality of vector field type and regular contact type, solving a long-standing open problem in complex geometry for specific cases.

## Key findings

- Proves equality of vector field and contact types in complex dimension three.
- Verifies Bloom conjecture for s=n-2 in higher dimensions.
- Provides the first positive result in pseudoconvexity-sensitive cases since 1981.

## Abstract

We prove equality of the vector field (iterated commutator) type and the regular contact type, which together with the Bloom theorem on equality of the Levi-form type and the regular contact type provides a complete solution of a long standing open problem of Bloom in the case of complex dimension three. For general dimensions, we verify the Bloom conjecture when $s=n-2$, which provides the first positive result in the pseudoconvexity sensitive case for a real hypersurface in ${\mathbb{C}}^n$ after his important work in 1981.