By Manin Yu.I.
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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 . 44) Hence on Ua ∩ Ub , we have Aa = Ab and we have defined a global (0, 1)-form A0,1 := ¯ −1 . ψ ∂ψ The bijection between the moduli spaces of both descriptions is easily found. 45) ¯ −1 ∈ A and δ1 is where i denotes the embedding of H in S, δ0 is the map S ∋ ψ → ψ ∂ψ 0,1 0,1 0,1 0,1 ¯ the map A ∋ A → ∂A + A ∧ A .
16 Calculating Chern classes. A simple method for calculating Chern classes is available if one can diagonalize F by an element g ∈ GL(k, ) such that g−1 Fg = diag(x1 , . . , xn ) =: D. One then easily derives that det(½ + D) = det(diag(1 + x1 , . . , 1 + xn )) = 1 + tr D + 21 (( tr D)2 − tr D 2 ) + . . + det D . 25) §17 Theorem. Consider two complex vector bundles E → M and F → M with total Chern classes c(E) and c(F ). Then the total Chern class of a Whitney sum bundle4 (E ⊕ F ) → M is given by c(E ⊕ F ) = c(E) ∧ c(F ).
Z5 ) = c2 (z0 , . . 60) where c1 and c2 are homogeneous cubic polynomials. §11 Calabi-Yau manifolds from vector bundles over P 1 . A very prominent class of local Calabi-Yau manifolds can be obtained from the vector bundles O(a) ⊕ O(b) → P 1 , where the Calabi-Yau condition of vanishing first Chern class amounts to a+b = −2. To describe these bundles, we will always choose the standard inhomogeneous coor1 dinates λ± on the patches U± covering the base P 1 , together with the coordinates z± 2 in the fibres over the patches U .