Complex manifolds, vector bundles and Hodge theory by Foth

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CPn We also recall that a section of L⊗m over an open set W ⊂ CPn is a holomorphic function over q −1 (W ) which is homogeneous of degree m. , zn ) is a homogeneous polynomial in Cn+1 of degree m + ˇ k(p + 1). We have then a natural subspaces of Cech cochains Cˇ p (k) ⊂ p Cˇ consisting of those having poles of order at most k in CPn . 3. COHOMOLOGY OF PROJECTIVE SPACE 43 Our strategy is to compute the cohomology of subcomplexes Cˇ • (k) and then the cohomology of Cˇ • (U, L⊗m ) will be the direct limit of the cohomologies of subcomplexes.

1 (Riemann-Roch-Hirzebruch) χ(X, F ) = ch(F ) ∪ T d(X), [X] , where the pairing , is the standard duality pairing between homology and cohomology. Remark. In the case X is a singular algebraic variety it is still possible to define the Todd class but now it will be an element of homology of X rather then cohomology and the above theorem will have ch(F ), T d(X) in the right hand side. The version of the RiemannRoch-Hirzebruch theorem for singular algebraic varieties is due to Baum, Fulton and MacPherson.

Contrary to the smooth case, such a sequence as we saw before is not always split, and if we take the corresponding spaces of global sections then we get a shortened version of the previous exact sequence α 0 → Γ(X, S) → Γ(X, E) → Γ(X, Q), where the map α is not necessarily surjective. In these cases people say that the global section functor is left exact, which is a weaker condition 31 32 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES than exact. We have to require some additional properties from X to guarantee that the above sequence may be continued to the right by zero that is the surjectivity of α.

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