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Complex vector bundle on a complex manifold
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. 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[edit]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 sections[edit]Let E be a holomorphic vector bundle. A local section s : U → E|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 O ( E ) {\displaystyle {\mathcal {O}}(E)} , or abusively by E. Such a sheaf is always locally free and of the same rank as the rank of the vector bundle. If E is the trivial line bundle C _ , {\displaystyle {\underline {\mathbf {C} }},} then this sheaf coincides with the structure sheaf O X {\displaystyle {\mathcal {O}}_{X}} of the complex manifold X.
There are line bundles O ( k ) {\displaystyle {\mathcal {O}}(k)} over C P n {\displaystyle \mathbb {CP} ^{n}} whose global sections correspond to homogeneous polynomials of degree k {\displaystyle k} (for k {\displaystyle k} a positive integer). In particular, k = 0 {\displaystyle k=0} corresponds to the trivial line bundle. If we take the covering by the open sets U i = { [ x 0 : ⋯ : x n ] : x i ≠ 0 } {\displaystyle U_{i}=\{[x_{0}:\cdots :x_{n}]:x_{i}\neq 0\}} then we can find charts ϕ i : U i → C n {\displaystyle \phi _{i}:U_{i}\to \mathbb {C} ^{n}} defined by
ϕ i ( [ x 0 : ⋯ : x i : ⋯ : x n ] ) = ( x 0 x i , … , x i − 1 x i , x i + 1 x i , … , x n x i ) = C i n {\displaystyle \phi _{i}([x_{0}:\cdots :x_{i}:\cdots :x_{n}])=\left({\frac {x_{0}}{x_{i}}},\ldots ,{\frac {x_{i-1}}{x_{i}}},{\frac {x_{i+1}}{x_{i}}},\ldots ,{\frac {x_{n}}{x_{i}}}\right)=\mathbb {C} _{i}^{n}}
We can construct transition functions ϕ i j | U i ∩ U j : C i n ∩ ϕ i ( U i ∩ U j ) → C j n ∩ ϕ j ( U i ∩ U j ) {\displaystyle \phi _{ij}|_{U_{i}\cap U_{j}}:\mathbb {C} _{i}^{n}\cap \phi _{i}(U_{i}\cap U_{j})\to \mathbb {C} _{j}^{n}\cap \phi _{j}(U_{i}\cap U_{j})} defined by
ϕ i j = ϕ i ∘ ϕ j − 1 ( z 1 , … , z n ) = ( z 1 z i , … , z i − 1 z i , z i + 1 z i , … , z j z i , 1 z i , z j + 1 z i , … , z n z i ) {\displaystyle \phi _{ij}=\phi _{i}\circ \phi _{j}^{-1}(z_{1},\ldots ,z_{n})=\left({\frac {z_{1}}{z_{i}}},\ldots ,{\frac {z_{i-1}}{z_{i}}},{\frac {z_{i+1}}{z_{i}}},\ldots ,{\frac {z_{j}}{z_{i}}},{\frac {1}{z_{i}}},{\frac {z_{j+1}}{z_{i}}},\ldots ,{\frac {z_{n}}{z_{i}}}\right)}
Now, if we consider the trivial bundle L i = ϕ i ( U i ) × C {\displaystyle L_{i}=\phi _{i}(U_{i})\times \mathbb {C} } we can form induced transition functions ψ i , j {\displaystyle \psi _{i,j}} . If we use the coordinate z {\displaystyle z} on the fiber, then we can form transition functions
ψ i , j ( ( z 1 , … , z n ) , z ) = ( ϕ i , j ( z 1 , … , z n ) , z i k z j k ⋅ z ) {\displaystyle \psi _{i,j}((z_{1},\ldots ,z_{n}),z)=\left(\phi _{i,j}(z_{1},\ldots ,z_{n}),{\frac {z_{i}^{k}}{z_{j}^{k}}}\cdot z\right)}
for any integer k {\displaystyle k} . Each of these are associated with a line bundle O ( k ) {\displaystyle {\mathcal {O}}(k)} . Since vector bundles necessarily pull back, any holomorphic submanifold f : X → C P n {\displaystyle f:X\to \mathbb {CP} ^{n}} has an associated line bundle f ∗ ( O ( k ) ) {\displaystyle f^{*}({\mathcal {O}}(k))} , sometimes denoted O ( k ) | X {\displaystyle {\mathcal {O}}(k)|_{X}} .
Dolbeault operators[edit]Suppose E is a holomorphic vector bundle. Then there is a distinguished operator ∂ ¯ E {\displaystyle {\bar {\partial }}_{E}} defined as follows. In a local trivialisation U α {\displaystyle U_{\alpha }} of E, with local frame e 1 , … , e n {\displaystyle e_{1},\dots ,e_{n}} , any section may be written s = ∑ i s i e i {\displaystyle s=\sum _{i}s^{i}e_{i}} for some smooth functions s i : U α → C {\displaystyle s^{i}:U_{\alpha }\to \mathbb {C} } . Define an operator locally by
where ∂ ¯ {\displaystyle {\bar {\partial }}} 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 U α , U β {\displaystyle U_{\alpha },U_{\beta }} with holomorphic transition function g α β {\displaystyle g_{\alpha \beta }} , if s = s i e i = s ~ j f j {\displaystyle s=s^{i}e_{i}={\tilde {s}}^{j}f_{j}} where f j {\displaystyle f_{j}} is a local frame for E on U β {\displaystyle U_{\beta }} , then s i = ∑ j ( g α β ) j i s ~ j {\displaystyle s^{i}=\sum _{j}(g_{\alpha \beta })_{j}^{i}{\tilde {s}}^{j}} , and so
because the transition functions are holomorphic. This leads to the following definition: A Dolbeault operator on a smooth complex vector bundle E → M {\displaystyle E\to M} is a C {\displaystyle \mathbb {C} } -linear operator
such that
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 ∂ ¯ E {\displaystyle {\bar {\partial }}_{E}} on a smooth complex vector bundle E {\displaystyle E} , there is a unique holomorphic structure on E {\displaystyle E} such that ∂ ¯ E {\displaystyle {\bar {\partial }}_{E}} is the associated Dolbeault operator as constructed above.
With respect to the holomorphic structure induced by a Dolbeault operator ∂ ¯ E {\displaystyle {\bar {\partial }}_{E}} , a smooth section s ∈ Γ ( E ) {\displaystyle s\in \Gamma (E)} is holomorphic if and only if ∂ ¯ E ( s ) = 0 {\displaystyle {\bar {\partial }}_{E}(s)=0} . 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 bundle[edit]If E X p , q {\displaystyle {\mathcal {E}}_{X}^{p,q}} 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 O ( E ) {\displaystyle {\mathcal {O}}(E)} . In particular, we have
the space of global holomorphic sections of E. We also have that H 1 ( X , O ( E ) ) {\displaystyle H^{1}(X,{\mathcal {O}}(E))} 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 E {\displaystyle E} . 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 H 1 ( X , O X ∗ ) {\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})} of the sheaf of non-vanishing holomorphic functions.
Hermitian metrics on a holomorphic vector bundle[edit]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
Indeed, if u = (e1, …, en) is a holomorphic frame, then let h i j = ⟨ e i , e j ⟩ {\displaystyle h_{ij}=\langle e_{i},e_{j}\rangle } and define ωu by the equation ∑ h i k ( ω u ) j k = ∂ h i j {\displaystyle \sum h_{ik}\,{(\omega _{u})}_{j}^{k}=\partial h_{ij}} , 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 ω ¯ T = ∂ ¯ h ⋅ h − 1 {\displaystyle {\overline {\omega }}^{T}={\overline {\partial }}h\cdot h^{-1}} ,
That is, ∇ is compatible with metric structure. Finally, since ω is a (1, 0)-form, the (0, 1)-component of ∇ s {\displaystyle \nabla s} is ∂ ¯ E s {\displaystyle {\bar {\partial }}_{E}s} .
Let Ω = d ω + ω ∧ ω {\displaystyle \Omega =d\omega +\omega \wedge \omega } be the curvature form of ∇. Since π 0 , 1 ∇ = ∂ ¯ E {\displaystyle \pi _{0,1}\nabla ={\bar {\partial }}_{E}} 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.
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