Let $(X,h)$ be a Hermitian complex manifold, let $\omega$ be its associated real $(1,1)$-form, and let $\nabla^C$ be the Chern connection of $h$ on the holomorphic tangent bundle $T^{1,0}X$. Denote by $T^C$ the torsion tensor of $\nabla^C$. Then
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\begin{align*}
T^C = 0
\end{align*}
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if and only if
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\begin{align*}
d\omega = 0.
\end{align*}
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Equivalently, a Hermitian metric is Kähler if and only if its Chern torsion vanishes.