By Saemann C.
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Extra info for Aspects of twistor geometry and supersymmetric field theories within superstring theory (hep-th 0603098)
2 Vector bundles and sheaves 39 It is easily checked that the fab constructed in this way are holomorphic. Furthermore, they define holomorphic vector bundles which are topologically trivial, but not holomorˇ 1 (M, H) → H ˇ 1 (M, S). phically trivial. Thus, they belong to the kernel of a map ρ : H Conversely, given a transition function fab of a topologically trivial vector bundle on the intersection Ua ∩ Ub , we have ¯ ab = ∂(ψ ¯ −1 ψb ) = ψ −a (ψa ∂ψ ¯ −1 − ψb ∂ψ ¯ −1 )ψb = ψ −1 (Aa − Ab )ψb .
K ) over U defining a basis for each fibre over U , we can represent a connection by a collection of one-forms ωij ∈ Γ(U, Λ1 U ): ∇σi = ωij ⊗ σj . The components of the corresponding curvature F∇ σi = Fij ⊗ σj are easily calculated to be Fij = dωij + ωik ∧ ωkj . Roughly speaking, the curvature measures the difference between the parallel transport along a loop and the identity. ¯ one immediately sees from the Identifying ∇0,1 with the holomorphic structure ∂, theorem §7 that the (0, 2)-part of the curvature of a holomorphic vector bundle has to vanish.
29 Theorem. (Grothendieck) Any holomorphic bundle E over P 1 can be decomposed into a direct sum of holomorphic line bundles. This decomposition is unique up to permutations of holomorphically equivalent line bundles. The Chern numbers of the line bundles are holomorphic invariants of E, but only their sum is also a topological invariant. 3 Dolbeault and Cech cohomology There are two convenient descriptions of holomorphic vector bundles: the Dolbeault and ˇ the Cech description. Since the Penrose-Ward transform (see chapter VII) heavily relies on both of them, we recollect here the main aspects of these descriptions and comment on their equivalence.