PreprintCC BY 4.0arXiv
A numerically flat rank-two bundle without a holomorphic connection on a ∂∂̅-threefold
Abstract
We construct a numerically flat holomorphic vector bundle of rank two on a compact complex threefold satisfying the ordinary ∂∂̅-lemma and prove that it admits no holomorphic connection. This gives, in particular, a negative answer to a question posed by Cao--Deng--Matsumura. In contrast, for compact simply connected complex manifolds X, we prove that the answer is affirmative if H1(X,𝒪X)=0, and hence if the Frölicher spectral sequence of X degenerates at E1.
Related
- Positivity of the canonical bundle and Hermitian metrics with quasi-negative holomorphic sectional curvature
- Wronskian curvature positivity is equivalent to the existence of a holomorphic projective connection
- Gauss-Manin Connection in Disguise: Holomorphic Anomaly Equations and Higher Genus Real Gromov-Witten Invariants
- Holomorphic functions on complete Hermitian manifolds with flat Chern connection, II
- Holomorphic 1-forms without zeros on Kähler manifolds
BibTeX
@misc{hu2026numerically,
title = {{A numerically flat rank-two bundle without a holomorphic connection on a $\partial\bar\partial$-threefold}},
author = {Tianzhi Hu and Runze Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08757},
eprint = {2609.08757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08757},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A numerically flat rank-two bundle without a holomorphic connection on a ∂∂̅-threefold
AU - Tianzhi Hu
AU - Runze Zhang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.08757
UR - https://arxiv.org/abs/2609.08757
AB - We construct a numerically flat holomorphic vector bundle of rank two on a compact complex threefold satisfying the ordinary ∂∂̅-lemma and prove that it admits no holomorphic connection. This gives, in particular, a negative answer to a question posed by Cao--Deng--Matsumura. In contrast, for compact simply connected complex manifolds X, we prove that the answer is affirmative if H^1(X,𝒪_X)=0, and hence if the Frölicher spectral sequence of X degenerates at E_1.
ER -