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 : E → X is holomorphic.
Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent 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.
This is equivalent to requiring that the transition functions are holomorphic maps.
This condition is local, meaning that holomorphic sections form a sheaf on X.
whose global sections correspond to homogeneous polynomials of degree
corresponds to the trivial line bundle.
we can form induced transition functions
on the fiber, then we can form transition functions
Since vector bundles necessarily pull back, any holomorphic submanifold
Suppose E is a holomorphic vector bundle.
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
This leads to the following definition: A Dolbeault operator on a smooth complex vector bundle
on a smooth complex vector bundle
is the associated Dolbeault operator as constructed above.With respect to the holomorphic structure induced by a Dolbeault operator
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.
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: 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 → E → F → X × 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 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.
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.
, 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
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.