ag.algebraic geometry - Transition Functions and Complex Structure

For a real manifold $M$ the transition functions of the tangent bundle $T(M)$ come from the Jacobian of the change-of-coordinate maps.

When $M$ is complex, it has a complex tangent bundle $T_{mathbb{C}}M$, which can be identified with the holomorphic vector bundle $T^{(1,0)} subset TM otimes mathbb{C}$. The transition functions on $T_{mathbb{C}}M$ are given by the (complex) Jacobian of the change-of-coordinate maps, so the same is true for $T^{(1,0)}$.

Since $T^{(0,1)}$ is the complex conjugate bundle, its transition functions are the complex conjugate of the same Jacobian.

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