Flat connections for numerically flat bundles on non-Kähler manifolds

Determine whether every numerically flat vector bundle on a compact complex manifold admits a holomorphic flat connection compatible with a filtration whose graded quotients are Hermitian flat, without assuming that the manifold is Kähler or satisfies the \(\partial\bar\partial\)-lemma.

Background

Theorem \ref{DPSnonkahler} proves that a numerically flat locally free sheaf on any compact complex manifold admits a filtration by holomorphic subbundles with Hermitian flat graded quotients. This extends the corresponding filtration result beyond the Kähler setting, but it does not itself produce a holomorphic flat connection.

The paper formulates the stronger connection question and explains that the answer is affirmative in the Kähler case by Simpson’s theorem and the explicit construction cited as \cite{Den21}. The unresolved issue is whether the same compatibility with the filtration persists on compact complex manifolds that do not satisfy the ∂∂ˉ\partial\bar\partial-lemma.

References

If E is numerically flat, does E admit a holomorphic flat connection compatible with the filtration filtra1? Recall that, when X is Kähler, Question \ref{flatconqq} has an affirmative answer by ; the key tool is the \partial\bar\partial-lemma. It is unclear whether Question \ref{flatconqq} remains true for manifolds that do not satisfy the \partial\bar\partial-lemma.

filtra1:

{0}=E0⊂E1⊂⋯⊂Ek=E\{0\}=E_0\subset E_1\subset\cdots\subset E_k=E

— A flatness criterion for pseudo-effective sheaves on compact Kähler spaces  (2609.05154 - Cao et al., 4 Sep 2026) in Question 6.2, Section 6 (labelled \ref{flatconqq}), followed by the concluding paragraph of Section 6