Cao–Deng–Matsumura question on numerically flat bundles and holomorphic connections

Let $X$ be a compact complex threefold satisfying the ordinary $\partial\bar\partial$ -lemma, and let $E$ be a numerically flat holomorphic vector bundle of rank $2$ on $X$ . Must $E$ admit a holomorphic connection? Equivalently, does numerical flatness imply the existence of a holomorphic connection in this setting?

References

Primary source

arXiv

Additional references

A numerically flat rank-two bundle without a holomorphic connection on a ∂∂̄-threefold — arXiv

complex-geometry holomorphic-vector-bundles

Posted 4 weeks ago · edited 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A September 2026 preprint claims a counterexample showing that numerical flatness does not guarantee a holomorphic connection in the stated threefold setting.

Cao, Deng, and Matsumura asked whether numerical flatness implies the existence of a holomorphic connection. The question is claimed to have a negative answer for a compact complex threefold satisfying the ordinary $\partial\bar\partial$ -lemma.

September 2026 counterexample

A September 2026 arXiv preprint reports a numerically flat rank- $2$ bundle without a holomorphic connection on a $\partial\bar\partial$ -threefold. If correct, this disproves the proposed implication in the stated rank- $2$ threefold case; the report remains unverified.

Current status (as of September 2026): The implication is claimed false for the stated rank- $2$ compact complex threefold case, while independent verification and any broader conclusion remain open.

Sources

No solutions have been posted yet.