Wikipedia / Holomorphic vector bundle
Wikipedia / Holomorphic vector bundle
In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map 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 trivialization
Specifically, one requires that the trivialization maps
are biholomorphic maps. This is equivalent to requiring that the transition functionss
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 sections
Let be a holomorphic vector bundle. A local section is said to be holomorphic if, in a neighborhood of each point of , it is holomorphic in some (equivalently any) trivialization.
This condition is local, meaning that holomorphic sections form a sheaf on , or abusively by . Such a sheaf is always locally free and of the same rank as the rank of the vector bundle. If then this sheaf coincides with the structure sheaf of the complex manifold .
Basic examples
corresponds to the trivial line bundle defined by defined by on the fiber, then we can form transition functions.
Dolbeault operators
Suppose of . Define an operator locally by
is the regular Cauchy–Riemann operator of the base manifold. This operator is well-defined on all of is a local frame for , and so
because the transition functions are holomorphic. This leads to the following definition: A Dolbeault operator-linear operator
such that
(Cauchy–Riemann condition),
(Leibniz rule), 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:
Theorem: is the associated Dolbeault operator as constructed above.. 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.
The sheaves of forms with values in a holomorphic vector bundle
denotes the sheaf of differential forms of type , then the sheaf of type forms with values in 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 bundles
If is a holomorphic vector bundle, the cohomology of is defined to be the sheaf cohomology. In particular, we have
the space of global holomorphic sections of parametrizes the group of extensions of the trivial line bundle of by , that is, exact sequences of holomorphic vector bundles . 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. Namely we have
The Picard group
In the context of complex differential geometry, the Picard group of the complex manifold of the sheaf of non-vanishing holomorphic functions.
Hermitian metrics on a holomorphic vector bundle
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,
for the contraction by X. (This is equivalent to saying that the parallel transport by ∇ preserves the metric <·,·>.)
Indeed, if u = (e1, …, en and define ωu, 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,
.
be the curvature form 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, it also has no (2, 0)-component. Consequently, Ω is a (1, 1)-form given by
The curvature Ω appears prominently in the vanishing theoremss for higher cohomology of holomorphic vector bundles; e.g., Kodaira's vanishing theorem and Nakano's vanishing theorem.
See also
Notes
References
External links