The holomorphic tangent bundle to a complex manifold is given by its complexified tangent vectors
which are of type 
. In a coordinate chart 
,
the bundle is spanned by the local bundle
sections 
.
The antiholomorphic tangent bundle is locally spanned by 
, of type 
, where 
denotes the complex conjugate.
A holomorphic splitting 
specifies the subbundles as well as their
abstract isomorphism types. A summand is integrable
when its local holomorphic sections are closed under the Lie
bracket, so it defines a holomorphic foliation
(Höring 2007, §2.B).
