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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)
##### Keywords:
upper bounds for cohomology
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##### References:
 [1] Brodmann, M.: Bounds on the cohomological Hilbert Functions of a Projective Variety. J. Algebra109, 352–380 (1987) · Zbl 0578.14015 · doi:10.1016/0021-8693(87)90144-X [2] Brodmann, M.: Bounds on the Serre Cohomology of a Projective Variety. Ĉas. Peŝtovóní Mat.112, 238–244 (1987) · Zbl 0646.14016 [3] Brodmann, M.: A Bound for the first Cohomology of a Projective Surface. Arch. Math.50, 68–72 (1988) · Zbl 0705.14019 · doi:10.1007/BF01313497 [4] Brodmann, M.: A priori Bounds of Castelnuovo Type for Cohomological Hilbert Functions. Comment. Math. Helv.65, 478–518 (1990) · Zbl 0728.14014 · doi:10.1007/BF02566622 [5] Hartshorne, P.: Algebraic Geometry. (Grad. Texts Math., vol. 52) Berlin Heidelberg New York: Springer 1977 · Zbl 0367.14001 [6] Jouanolou, J.P.: Théorèmes de Bertini et Applications. (Prog. Math., vol. 42) Boston Basel Stuttgart: Birkhäuser 1983 · Zbl 0519.14002 [7] Kodaira, K.: On a Differential Geometric Method in the Theory of Analytic Stacks. Proc. Natl. Acad. Sci. USA39, 1268–1273 (1953) · Zbl 0053.11701 · doi:10.1073/pnas.39.12.1268 [8] Matsumura, H.: Commutative Algebra. New York: Benjamin 1970 · Zbl 0211.06501 [9] Mumford, D.: Pathologies III. Am. J. Math.89, 94–104 (1967) · Zbl 0146.42403 · doi:10.2307/2373099 [10] Ramanujam, C.: Remarks on the Kodaira Vanishing Theorem. J. Indian Math. Soc.36, 41–51 (1972) · Zbl 0276.32018 [11] Serre, J.P.: Faisceaux Algébriques Cohérents. Ann. Math.61, 192–278 (1955) · Zbl 0067.16201 · doi:10.2307/1969915 [12] Serre, J.P.: Géométrique Algébrique et Géométrie Analytique. Ann. Inst. Fourier6, 1–42 (1956)
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