Let $X$ be compact Kahler and $E \to X$ a holomorphic vector bundle. Then $E$ has an Atiyah class, $At(E)$, valued in the sheaf cohomology $H^1(\Omega\_X \otimes \operatorname{End} E)$. Suppose the topological Chern classes of $E$ vanish rationally. Evaluating $At(E)$ on an invariant polynomial gives a class in $\bigoplus\_k H^k(\Omega\_X^k)$, which must vanish in non-zero degrees since, by the Hodge decomposition, they correspond to polynomials in the Chern classes of $E$.


My question is:



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> Is it the case that the full Atiyah class itself, $At(E)$, is necessarily zero in $H^1(\Omega\_X \otimes \operatorname{End} E)$?
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Atiyah classes of holomorphic vector bundles with trivial Chern classes
No. A counter-example : the vector bundle $\mathcal{O}\_{\mathbb{P}^1}(p)\oplus \mathcal{O}\_{\mathbb{P}^1}(-p)$ on $\mathbb{P}^1$ has zero Chern class, but does not admit a holomorphic connection if $p>0$ (by a theorem of Weil, a vector bundle on a curve admits a holomorphic connection iff its indecomposable summands have degree zero), hence its Atiyah class is nonzero. If you want a higher-dimensional example just pull back this one by a morphism to $\mathbb{P}^1$.


