Let $E$ be a holomorphic vector bundle on a complex manifold $X$, and let $D: E \to E \otimes \Omega_X$ be a holomorphic connection on $E$ i.e. $$ D(fs) = s\otimes \partial(f)+ fD(s), $$ for any local holomorphic function $f$ and local section $s$. Locally such conections look like $$ D = \partial + A, $$ where $A$ is a holomorphic section of $\operatorname{End}(E)\otimes \Omega_X$.

If $D$ is a holomorphic connection on $E$ then $D+\bar{\partial}$ is an "ordinary" connection on $E$.

How can one show that if $E$ admits a holomorphic connection then it admits a flat connection? In other words there is $B \in\operatorname{End}(E)\otimes \Omega_X$ s.t. $$ (\bar{\partial} +D+B)^2=0. $$

  • $\begingroup$ Why do you think this is true? For example, the Chern classes of a holomorphic vector bundle generally do not vanish. (By the Chern-Weil formula, the cohomology class is independent of connection.) $\endgroup$ – Ted Shifrin Feb 3 '15 at 14:08
  • $\begingroup$ Chern classes of a holomorphic vector bundle with holomorphic connection all vanish. At least if we assume that $X$ is Kahler. $\endgroup$ – Alex Feb 3 '15 at 15:11
  • $\begingroup$ Yeah, sorry, I wasn't paying enough attention. Do you know this paper? $\endgroup$ – Ted Shifrin Feb 3 '15 at 17:40
  • $\begingroup$ No, I did not know about that paper. But in that paper more subtle question is solved: existence of a compatible flat connection. I want much less, any flat connection on the same holomorphic bundle will be enough. $\endgroup$ – Alex Feb 3 '15 at 19:14

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