Holomorphic tangent bundle

  (Redirected from Holomorphic tangent space)

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 of the complex manifold .

DefinitionEdit

Given a complex manifold   of complex dimension  , its tangent bundle as a smooth vector bundle is a real rank   vector bundle   on  . The integrable almost complex structure   corresponding to the complex structure on the manifold   is an endomorphism   with the property that  . After complexifying the real tangent bundle to  , the endomorphism   may be extended complex-linearly to an endomorphism   defined by   for vectors   in  .

Since  ,   has eigenvalues   on the complexified tangent bundle, and   therefore splits as a direct sum

 

where   is the  -eigenbundle, and   the  -eigenbundle. The holomorphic tangent bundle of   is the vector bundle  , and the anti-holomorphic tangent bundle is the vector bundle  .

The vector bundles   and   are naturally complex vector subbundles of the complex vector bundle  , and their duals may be taken. 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!) isomorphism  .

The holomorphic tangent bundle   is isomorphic as a real vector bundle of rank   to the regular tangent bundle  . The isomorphism is given by the composition   of inclusion into the complexified tangent bundle, and then projection onto the  -eigenbundle.

The canonical bundle is defined by  .

Alternative local descriptionEdit

In a local holomorphic chart   of  , one has distinguished real coordinates   defined by   for each  . These give distinguished complex-valued one-forms   on  . 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   are preserved, and so by covering   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. Similarly the complex-valued one-forms   and   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  . Explicitly, if we have two charts   with two sets of coordinates  , then

 

Since the coordinate functions are holomorphic, so are any derivatives of them, and so the transition functions of the holomorphic tangent bundle are also holomorphic. 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 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

 

Holomorphic vector fields and differential formsEdit

Since the holomorphic tangent and cotangent bundles have the structure of holomorphic vector bundles, there are distinguished holomorphic sections. A holomorphic vector field is a holomorphic section of  . A holomorphic one-form is a holomorphic section of  . By taking exterior powers of  , one can define holomorphic  -forms for integers  . The Cauchy-Riemann operator of   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  -forms that are annihilated by  . For more details see complex differential forms.

See alsoEdit

ReferencesEdit

  • Huybrechts, Daniel (2005). Complex Geometry: An Introduction. Springer. ISBN 3-540-21290-6.
  • Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523