A numerically flat rank-two bundle without a holomorphic connection on a $\partial \bar{\partial }$-threefold

Abstract

We construct a numerically flat holomorphic vector bundle of rank two on a compact complex threefold satisfying the ordinary $\partial \bar{\partial }$-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 $H^1(X,{\mathcal{O}}_X)=0$, and hence if the Frölicher spectral sequence of $X$ degenerates at $E_1$.

arXiv commentv2: 9 pages. Added an affirmative answer to the question under additional assumptions and expanded the references. Comments are welcome!