Holomorphic tangent bundle

In mathematics, and especially complex geometry, the holomorphic tangent bundle of a complex manifold ${\displaystyle M}$ is the holomorphic analogue of the tangent bundle of a smooth manifold. The fibre of the holomorphic tangent bundle over a point is the holomorphic tangent space, which is the tangent space of the underlying smooth manifold, given the structure of a complex vector space via the almost complex structure ${\displaystyle J}$ of the complex manifold ${\displaystyle M}$.