In mathematics, and especially complex geometry, the holomorphic tangent bundle of a complex manifold
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
The integrable almost complex structure
corresponding to the complex structure on the manifold
After complexifying the real tangent bundle to
may be extended complex-linearly to an endomorphism
are naturally complex vector subbundles of the complex vector bundle
The holomorphic cotangent bundle is the dual of the holomorphic tangent bundle, and is written
Similarly the anti-holomorphic cotangent bundle is the dual of the anti-holomorphic tangent bundle, and is written
The holomorphic and anti-holomorphic (co)tangent bundles are interchanged by conjugation, which gives a real-linear (but not complex linear!)
is isomorphic as a real vector bundle of rank
of inclusion into the complexified tangent bundle, and then projection onto the
These give distinguished complex-valued one-forms
Dual to these complex-valued one-forms are the complex-valued vector fields (that is, sections of the complexified tangent bundle), Taken together, these vector fields form a frame for
, the restriction of the complexified tangent bundle to
As such, these vector fields also split the complexified tangent bundle into two subbundles Under a holomorphic change of coordinates, these two subbundles of
by holomorphic charts one obtains a splitting of the complexified tangent bundle.
This is precisely the splitting into the holomorphic and anti-holomorphic tangent bundles previously described.
provide the splitting of the complexified cotangent bundle into the holomorphic and anti-holomorphic cotangent bundles.
From this perspective, the name holomorphic tangent bundle becomes transparent.
Namely, the transition functions for the holomorphic tangent bundle, with local frames generated by the
, are given by the Jacobian matrix of the transition functions of
Thus the holomorphic tangent bundle is a genuine holomorphic vector bundle.
Similarly the holomorphic cotangent bundle is a genuine holomorphic vector bundle, with transition functions given by the inverse transpose of the Jacobian matrix.
Notice that the anti-holomorphic tangent and cotangent bundles do not have holomorphic transition functions, but anti-holomorphic ones.
In terms of the local frames described, the almost-complex structure
acts by or in real coordinates by Since the holomorphic tangent and cotangent bundles have the structure of holomorphic vector bundles, there are distinguished holomorphic sections.
may be extended from functions to complex-valued differential forms, and the holomorphic sections of the holomorphic cotangent bundle agree with the complex-valued differential
For more details see complex differential forms.