= Holomorphic tangent connection and mixed torsion criterion
{title2=$D^{0,1}=\bar\partial_{T'}\iff T_\nabla(T^{0,1},T^{1,0})=0$}
A real <connection on a vector bundle> on $TM$ with $\nabla J=0$ restricts after complexification to a connection $D$ on the <holomorphic tangent bundle>. Conversely $D$ and its conjugate determine a unique such real connection. In holomorphic coordinates, compatibility with the holomorphic structure means $D_{\partial_{\bar z^i}}\partial_{z^j}=0$. Zero mixed <torsion tensor> says that this derivative equals $\nabla_{\partial_{z^j}}\partial_{\bar z^i}$; the two belong to opposite eigentypes, so both vanish. Reality proves the reverse implication. Preserving $J$ alone does not imply this condition.
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