Intermediate-degree cohomology bound without curvature regularity

Determine whether, for every compact complex manifold X of dimension n≥3, every smooth Hermitian holomorphic line bundle (L,h) whose curvature has maximal rank 1≤r<n, and every fixed holomorphic vector bundle E, the intermediate Dolbeault cohomology dimensions satisfy h^q(X,L^p⊗E)=O(p^r) for all 1≤q≤n−1, without assuming positivity, constant curvature rank, or a regular foliation in the curvature kernel.

Background

The paper proves the bound h^q(X,L^p⊗E)=O_ε(p^{r+ε}) for every ε>0 and obtains the sharper O(p^r) estimate in degrees q=0 and q=n. It also establishes the O(p^r) bound in every degree on compact complex surfaces. The unresolved issue concerns intermediate degrees in dimensions at least three.

Bouche’s earlier result gives the desired O(p^r) estimate under the additional hypothesis that the curvature kernel contains the tangent bundle of a regular holomorphic foliation. The question asks whether that geometric hypothesis, as well as positivity and constant-rank assumptions, can be removed entirely. The paper’s fixed-depth rescaling argument does not resolve this endpoint exponent because its constants are not uniform as the number of rescaling steps tends to infinity.