Type decomposition of the complexified tangent bundle ($T_{\mathbb{C}}X=T^{1,0}X\oplus T^{0,1}X$)

An almost complex structure $J$ gives the $+i$ and $-i$ eigenbundles of its complex-linear extension. Their projections are $(I-iJ)/2$ and $(I+iJ)/2$. For an integrable structure the first is the holomorphic tangent bundle.