Bloom and D’Angelo conjectures on relationships among contact, commutator, and Levi form types

Characterize the exact relationships among the contact type a(E,p), commutator type t(E,p), and Levi form type c(E,p) for smooth complex subbundles E of the (1,0) tangent bundle H^{1,0}S of pseudoconvex real hypersurfaces S ⊂ C^n; in particular, resolve Bloom’s and D’Angelo’s conjectures concerning the equivalence or precise comparison of these type invariants.

Background

The paper introduces three finiteness/type invariants for subbundles E ⊂ H^{1,0}S: the contact type a(E,p) (via formal orbits and contact orders), the commutator type t(E,p) (via iterated Lie brackets), and the Levi form type c(E,p) (via orders of nonzero Levi form derivatives). These notions trace back to work by Kohn, Bloom, and D’Angelo and are central in the analysis of the ∂-Neumann problem and CR geometry.

While partial results and recent advances (e.g., Huang–Yin) compare some of these invariants in specific settings, the authors emphasize that the comprehensive, general relationships among these types remain unresolved and are encapsulated by longstanding conjectures of Bloom and D’Angelo. Their resolution would unify different geometric approaches to type and have broad implications.