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Every vector bundle over a variety X gives a projective bundle by taking the projective spaces of the fibers, but not all projective bundles arise in this way: there is an obstruction in the cohomology group H 2 (X,O*). To see why, recall that a projective bundle comes equipped with transition functions on double intersections of a suitable ...
In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre.The basic version applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations, for example to singular varieties.
The Z-cohomology of RP 2a+1 is the same together with an extra copy of Z in degree 2a+1. [10] The cohomology ring of complex projective space CP n is Z[x]/(x n+1), with x in degree 2. [9] Here x is the class of a hyperplane CP n−1 in CP n. More generally, x j is the class of a linear subspace CP n−j in CP n.
In the special case where is projective over , this is proved by reducing to the case of line bundles on projective space, discussed above. In the general case of a proper scheme over a field, Grothendieck proved the finiteness of cohomology by reducing to the projective case, using Chow's lemma.
The Hodge decomposition writes the complex cohomology of a complex projective variety as a sum ... is the ring of stable equivalence classes of complex vector bundles ...
In particular, a projective bundle is defined to be zero in the Brauer group if and only if it is the projectivization of some vector bundle. The cohomological Brauer group of a quasi-compact scheme X is defined to be the torsion subgroup of the étale cohomology group H 2 (X, G m).
Dolbeault cohomology of vector bundles [ edit ] If E is a holomorphic vector bundle on a complex manifold X , then one can define likewise a fine resolution of the sheaf O ( E ) {\displaystyle {\mathcal {O}}(E)} of holomorphic sections of E , using the Dolbeault operator of E .
The cohomology groups in the finite case can be derived from the long exact sequence for bundles and the above fact that SU(n) is a / bundle over PU(n). The cohomology in the infinite case was argued above from the isomorphism with the cohomology of the infinite complex projective space.