Holomorphic tangent space

In mathematics, specifically in the field of complex geometry, the holomorphic tangent space of a complex manifold M is the tangent space of the smooth manifold Σ underlying M viewed as complex vector space via the almost complex structure J of M. The holomorphic tangent bundle (TΣ, J) is isomorphic to a subbundle of the complexification of the real tangent bundle, namely the eigenbundle of J for the eigenvalue +i (compare linear complex structure). The holomorphic tangent bundle is often denoted by T^(1,0)M.