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 hq(X,Lp⊗ E)=Oε(pr+ε) for every ε>0 and every Dolbeault degree, with the sharp bounds h0,hn=O(pr) 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 κ(L|X0)≤ r on each connected component X0 and gives the endpoint bounds.
Related
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BibTeX
@misc{qin2026dolbeault,
title = {{Dolbeault Cohomology Growth under Curvature Rank Bounds}},
author = {Xiangsen Qin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09295},
eprint = {2610.09295},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09295},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Dolbeault Cohomology Growth under Curvature Rank Bounds
AU - Xiangsen Qin
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.09295
UR - https://arxiv.org/abs/2610.09295
AB - 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⊗ E)=O_ε(p^r+ε) for every ε>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 κ(L|_X_0)≤ r on each connected component X_0 and gives the endpoint bounds.
ER -