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By Manin Yu.I.

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Low Dimensional Topology

During this quantity, that is devoted to H. Seifert, are papers in keeping with talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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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 .

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