= Solution
For $(0,1)$ vector fields $V,W$, direct evaluation gives
$$
(\bar\partial^2f)(V,W)=df\bigl([V,W]^{1,0}\bigr)
$$
up to the harmless overall sign fixed by the exterior-derivative convention. If $J$ is integrable, $(0,1)$ fields are closed under bracket, so the right side vanishes. Conversely, if it vanishes for every smooth complex function $f$, differentials separate tangent vectors and therefore $[V,W]^{1,0}=0$. Involutivity of $T^{0,1}X$, equivalently vanishing of the Nijenhuis tensor, is the Newlander-Nirenberg criterion for an <integrable almost complex structure>. Thus $J$ is integrable exactly when $\bar\partial^2f=0$ for every $f$.
Back to article page
