By Norman Earl Steenrod
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Extra info for Cohomology Operations and Obstructions to Extending Continuous Functions
In Section 3, we prove the theorem of Segre. Essentially the same argument, with minor modifications to be made afterwards, proves the following stronger theorem of Manin (1966). 1 For an arbitrary normal projective variety the Picard number is defined as the rank of the N´eron–Severi group, the group of Cartier divisors up to numerical equivalence. For varieties with h 1 (X, O X ) = 0 the two definitions coincide. 2. Two smooth cubic surfaces defined over a perfect field, each of Picard number one, are birationally equivalent if and only if they are projectively equivalent.
For arbitrary X , the spaces (X, ⊗m X ) are usually hard to compute because the ⊗m X have quite high rank. Therefore it is important to have similar criteria which involve line bundles only. The natural candidate is the canonical bundle ω X = ∧n X of highest degree K¨ahler differential forms, which is always defined over the fixed ground field. For smooth X , the canonical bundle is represented by a divisor K X defined over the given ground field, and it is convenient to denote it by O X (K X ).
Prove that X is rational. 7 Numerical criteria for nonrationality Rationality and unirationality force strong numerical constraints on a variety. Let X = X/k be the sheaf of regular differential forms (K¨ahler differentials) on a variety X over k. 52. If a smooth projective variety X is rational, then it has no nontrivial global K¨ahler one-forms. In fact, the space of global sections ⊗m (X, ⊗m X ) of the sheaf X is zero for all m ≥ 1. The same holds for unirational X , provided the ground field has characteristic zero.