Dolbeault Cohomology Growth under Curvature Rank Bounds

Abstract

Let $L$ be a smooth Hermitian holomorphic line bundle on a compact complex manifold $X$ of dimension $n$, and let $r$ be its maximal curvature rank. For every fixed holomorphic vector bundle $E$, we prove $h^q(X,L^p\otimes E)=O_{\epsilon }(p^{r+\epsilon })$ for every $\epsilon >0$ and every Dolbeault degree, with the sharp bounds $h^0,h^n=O(p^r)$ at the endpoints. The curvature may have variable rank and arbitrary signature, and no global foliation or Kähler assumption is imposed. The all-degree estimate follows from local rescaling and holomorphic restriction estimates that remain uniform across rank changes. An independent maximum-principle argument locates the maximal rank of a section map at a semipositive curvature point; it yields $\kappa (L{|}_{X_0})\le r$ on each connected component $X_0$ and gives the endpoint bounds.