In mathematics, specifically in the field of complex geometry

Complex geometry

In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis....

, the holomorphic tangent space of a complex manifold

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

 M is the tangent space

Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....

 of the smooth manifold Σ underlying M viewed as complex vector space via the almost complex structure

Almost complex manifold

In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost...

 J of M. The holomorphic tangent space (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

Linear complex structure

In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector space.Complex structures have...

). The holomorphic tangent space is often denoted by T^(1,0)M.