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Sectional genus and cohomology of projective varieties. (English) Zbl 0783.14010
Generalizing his previous results for surfaces [Arch. Math. 50, No. 1, 68-72 (1988; Zbl 0705.14019)], the author obtains upper bounds for the cohomology \(H^ i( \mathbb{P}^ d,{\mathcal F}(n)\) \((n \geq 0,i>0)\) of a sheaf on \(\mathbb{P}^ d\). The bounds are too complicated to be stated here. They depend on the cohomology of the restrictions of \({\mathcal F}\) to sufficiently general projective subspaces and the linear spans of the nonmaximal associated primes of \({\mathcal F}\). The bounds for left trivial sheaves, i.e. sheaves with vanishing cohomology for \(n<0\) and \(i<\dim \bigl( \sup({\mathcal F}) \bigr)\), are simpler and can be applied to smooth varieties in characteristic zero.

MSC:
14F99 (Co)homology theory in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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