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We establish an upper bound on the growth of Dolbeault cohomology groups $h^q(X,L^p\otimes E)$ by $O_\varepsilon(p^{r+\varepsilon})$, where $r$ is the maximal rank of the curvature of a Hermitian holomorphic line bundle $L$ on a compact complex manifold $X$, and $E$ is any fixed holomorphic vector bundle. This result holds without imposing positivity, constant-rank, or K\"ahler conditions, and implies that the Kodaira--Iitaka dimension is bounded by $r$ on any connected component of $X$. The proof leverages rank identities in finite Taylor jets under rescalings and bounded Hermitian kernels, alongside exact Dolbeault restriction identities for finite-rank approximations.
Dolbeault cohomology growth is polynomially bounded by the curvature rank, even without positivity or K\"ahler assumptions.
Let $L$ be a smooth Hermitian holomorphic line bundle on a compact complex manifold $X$, and let $r$ be the maximal rank of its curvature. We prove that $h^q(X,L^p\otimes E)=O_\varepsilon(p^{r+\varepsilon})$ for every fixed holomorphic vector bundle $E$, every $\varepsilon>0$, and every degree $q$. No positivity, constant-rank, or K\"ahler assumption is imposed. Consequently, the Kodaira--Iitaka dimension satisfies $\kappa(L|_{X_0})\le r$ on every connected component $X_0$ of $X$. The proof retains rank identities in finite Taylor jets under successive rescalings and factors bounded Hermitian kernels without a positive lower bound on the absolute values of the nonzero curvature eigenvalues. Exact Dolbeault restriction identities yield finite-rank approximations with additive rank bounds, which control cohomology.