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Let $E\to X$ be a holomorphic vector bundle over a Kähler manifold. The vanishing of the Atiyah class, $At(E)=0$, is equivalent to the existence of a holomorphic connection on $E$.

Moreover, it is well known that $At(E)=0$ implies that all the rational Chern classes of $E$, of degree at least one, vanish. Therefore, there is no topological obstruction to the existence of a flat connection on $E$.

There is a paper by I. Biswas analyzing the related question of the existence of a flat connection compatible with the holomorphic structure of $E$.

However I have been unable to find an answer to the more general question regarding the existence of a flat connection on $E$, not necessarily compatible with the holomorphic structure of $E$.

Notice however, that in the end this is equivalent to asking wether there exists a flat connection on $E$ compatible with an eventual change in the holomorphic structure of $E$.

So please, can anybody provide a counterexample or a reference proving that all holomorphic vector bundles $E\to X$ with $At(E)=0$ are flat.

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  • $\begingroup$ Could you add a reference to Biswas' paper and maybe a definition of At(E)? $\endgroup$
    – Misha
    Nov 5 '15 at 18:32
  • $\begingroup$ @Misha: There is already a link to the paper by Biswas in my question. The word "paper" has the link to it and appears in a slightly blueish color to indicate that. The Atiyah class of E is the Dolbeault cohomology class $At(E)\in H^1(X,\Omega_X^1\otimes\operatorname{End} E)$ defined by the $(1,1)$ part of the curvature of any connection compatible with the holomorphic structure of $E$. $\endgroup$
    – user6319
    Nov 5 '15 at 19:42
  • $\begingroup$ see the related question mathoverflow.net/questions/161894/… in particular, read Pavel Safronov's comments. $\endgroup$
    – Malkoun
    Nov 25 '15 at 20:58

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