Holomorphic tangent bundle ($T^{1,0}X$)

The holomorphic tangent bundle of a complex manifold $X$ is locally spanned in holomorphic coordinates by $\partial /\partial z^1,\dots ,\partial /\partial z^n$. Under a holomorphic coordinate change $w=w(z)$, the chain rule giveswhose coefficient matrix is holomorphic; these frames therefore define a holomorphic vector bundle.

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.

The dual of the holomorphic tangent bundle is a holomorphic vector bundle with local frames $d{z}^1,\dots ,d{z}^n$. Coordinate changes have holomorphic invertible Jacobian matrices, so these frames have holomorphic transition functions. Its holomorphic sections are exactly the holomorphic one-forms.