By John Bonnycastle

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Aq } of sections of G defined ˇ on nonempty intersections Ua0 ∩ . . ∩ Uaq . Furthermore, we define the sets of Cech 0and 1-cocycles by Z 0 (U, G) := { ψ ∈ C 0 (U, G) | ψa = ψb on Ua ∩ Ub = ∅} = Γ(U, G) , Z 1 (U, S) := { χ ∈ C 1 (U, G) | χab = χ−1 ba on Ua ∩ Ub = ∅, χab χbc χca = ½ on Ua ∩ Ub ∩ Uc = ∅} . 37) ˇ This definition implies that the Cech 0-cocycles are independent of the covering: it is 0 0 ˇ ˇ 0 (M, G) := Z (U, G) = Z (M, G), and we define the zeroth Cech cohomology set by H 0 Z (M, G).

29) ch(F) = tr exp 2π and the j-th Chern character as a part of the corresponding Taylor expansion 1 tr chj (F) = j! iF 2π j . 30) Note that ch(F) is a polynomial of finite order on a finite-dimensional manifold. g. ch1 (F) = c1 (F) 4 and ch2 (F) = 2 1 2 (c1 (F) − 2c2 (F)) . 31) A Whitney sum of two vector bundles over a manifold M yields the vector bundle whose fibres are the direct sums of the fibres of the original two bundles. 2 Vector bundles and sheaves 35 The zeroth Chern character ch0 (F) is simply the dimension of the vector bundle associated to the curvature two-form F.

Then the total Chern class of a Whitney sum bundle4 (E ⊕ F ) → M is given by c(E ⊕ F ) = c(E) ∧ c(F ). In particular, the first Chern classes add: c1 (E ⊕ F ) = c1 (E) + c1 (F ). §18 Whitney product formula. 26) we have a splitting B = A ⊕ C and together with the above theorem, we obtain the formula c(A) ∧ c(C) = c(B). 323). §19 Further rules for calculations. Given two vector bundles E and F over a complex manifold M , we have the following formulæ: c1 (E ⊗ F ) = rk(F )c1 (E) + rk(E)c1 (F ) , c1 E ∨ = −c1 (E) .