Holomorphic vector bundle

In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : EX is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle.

By Serre's GAGA, the category of holomorphic vector bundles on a smooth complex projective variety X (viewed as a complex manifold) is equivalent to the category of algebraic vector bundles (i.e., locally free sheaves of finite rank) on X.

Definition through trivializationEdit

Specifically, one requires that the trivialization maps

 

are biholomorphic maps. This is equivalent to requiring that the transition functions

 

are holomorphic maps. The holomorphic structure on the tangent bundle of a complex manifold is guaranteed by the remark that the derivative (in the appropriate sense) of a vector-valued holomorphic function is itself holomorphic.

The sheaf of holomorphic sectionsEdit

Let E be a holomorphic vector bundle. A local section s : UE|U is said to be holomorphic if, in a neighborhood of each point of U, it is holomorphic in some (equivalently any) trivialization.

This condition is local, meaning that holomorphic sections form a sheaf on X. This sheaf is sometimes denoted  , or abusively by E. Such a sheaf is always locally free of the same rank as the rank of the vector bundle. If E is the trivial line bundle   then this sheaf coincides with the structure sheaf   of the complex manifold X.

Basic ExamplesEdit

There are line bundles   over   whose global sections correspond to homogeneous polynomials of degree   (for   a positive integer). In particular,   corresponds to the trivial line bundle. If we take the covering   then we can find charts   defined by

 

We can construct transition functions   defined by

 

Now, if we consider the trivial bundle   we can form induced transition functions  . If we use the coordinate   on the fiber, then we can form transition functions

 

for any integer  . Each of these are associated with a line bundle  . Since vector bundles necessarily pull back, any holomorphic submanifold   has an associated line bundle  , sometimes denoted  .

Dolbeault operatorsEdit

Suppose E is a holomorphic vector bundle. Then there is a distinguished operator   defined as follows. In a local trivialisation   of E, with local frame  , any section may be written   for some smooth functions  . Define an operator locally by

 

where   is the regular Cauchy-Riemann operator of the base manifold. This operator is well-defined on all of E because on an overlap of two trivialisations   with holomorphic transition function  , if   where   is a local frame for E on  , then  , and so

 

because the transition functions are holomorphic. This leads to the following definition: A Dolbeault operator on a smooth complex vector bundle   is an  -linear operator

 

such that

  • (Cauchy-Riemann condition)  ,
  • (Leibniz rule) For any section   and function   on  , one has
 .

By an application of the Newlander-Nirenberg theorem, one obtains a converse to the construction of the Dolbeault operator of a holomorphic bundle:[1]

Theorem: Given a Dolbeault operator   on a smooth complex vector bundle  , there is a unique holomorphic structure on   such that   is the associated Dolbeault operator as constructed above.

With respect to the holomorphic structure induced by a Dolbeault operator  , a smooth section   is holomorphic if and only if  . This is similar morally to the definition of a smooth or complex manifold as a ringed space. Namely, it is enough to specify which functions on a topological manifold are smooth or complex, in order to imbue it with a smooth or complex structure.

Dolbeault operator has local inverse in terms of homotopy operator.[2]

The sheaves of forms with values in a holomorphic vector bundleEdit

If   denotes the sheaf of C differential forms of type (p, q), then the sheaf of type (p, q) forms with values in E can be defined as the tensor product

 

These sheaves are fine, meaning that they admit partitions of unity. A fundamental distinction between smooth and holomorphic vector bundles is that in the latter, there is a canonical differential operator, given by the Dolbeault operator defined above:

 

Cohomology of holomorphic vector bundlesEdit

If E is a holomorphic vector bundle, the cohomology of E is defined to be the sheaf cohomology of  . In particular, we have

 

the space of global holomorphic sections of E. We also have that   parametrizes the group of extensions of the trivial line bundle of X by E, that is, exact sequences of holomorphic vector bundles 0 → EFX × C → 0. For the group structure, see also Baer sum as well as sheaf extension.

By Dolbeault's theorem, this sheaf cohomology can alternatively be described as the cohomology of the chain complex defined by the sheaves of forms with values in the holomorphic bundle  . Namely we have

 

The Picard groupEdit

In the context of complex differential geometry, the Picard group Pic(X) of the complex manifold X is the group of isomorphism classes of holomorphic line bundles with group law given by tensor product and inversion given by dualization. It can be equivalently defined as the first cohomology group   of the sheaf of non-vanishing holomorphic functions.

Hermitian metrics on a holomorphic vector bundleEdit

Let E be a holomorphic vector bundle on a complex manifold M and suppose there is a hermitian metric on E; that is, fibers Ex are equipped with inner products <·,·> that vary smoothly. Then there exists a unique connection ∇ on E that is compatible with both complex structure and metric structure, called the Chern connection; that is, ∇ is a connection such that

(1) For any smooth sections s of E,   where π0,1 takes the (0, 1)-component of an E-valued 1-form.
(2) For any smooth sections s, t of E and a vector field X on M,
 
where we wrote   for the contraction of   by X. (This is equivalent to saying that the parallel transport by ∇ preserves the metric <·,·>.)

Indeed, if u = (e1, …, en) is a holomorphic frame, then let   and define ωu by the equation  , which we write more simply as:

 

If u' = ug is another frame with a holomorphic change of basis g, then

 

and so ω is indeed a connection form, giving rise to ∇ by ∇s = ds + ω · s. Now, since  ,

 

That is, ∇ is compatible with metric structure. Finally, since ω is a (1, 0)-form, the (0, 1)-component of   is  .

Let   be the curvature form of ∇. Since   squares to zero by the definition of a Dolbeault operator, Ω has no (0, 2)-component and since Ω is easily shown to be skew-hermitian,[3] it also has no (2, 0)-component. Consequently, Ω is a (1, 1)-form given by

 

The curvature Ω appears prominently in the vanishing theorems for higher cohomology of holomorphic vector bundles; e.g., Kodaira's vanishing theorem and Nakano's vanishing theorem.

NotesEdit

  1. ^ Kobayashi, S. (2014). Differential geometry of complex vector bundles (Vol. 793). Princeton University Press.
  2. ^ Kycia, Radosław Antoni. "The Poincare Lemma, Antiexact Forms, and Fermionic Quantum Harmonic Oscillator". Results in Mathematics. 75 (3): 122. doi:10.1007/s00025-020-01247-8. ISSN 1422-6383.
  3. ^ For example, the existence of a Hermitian metric on E means the structure group of the frame bundle can be reduced to the unitary group and Ω has values in the Lie algebra of this unitary group, which consists of skew-hermitian metrices.

ReferencesEdit

See alsoEdit

External linksEdit