## Bounds on the cohomological Hilbert functions of a projective variety.(English)Zbl 0578.14015

We give bounds on the cohomological Hilbert functions $$h^ i_{{\mathcal F}}(n)=\dim_ kH^ i(X,{\mathcal F}(n))$$, where $${\mathcal F}$$ is a coherent sheaf over a projective variety over an algebraically closed field k. These bounds are given in terms of the functions $$h^ i_{{\mathcal F}| H}$$, where H runs through a linear system of hyperplane sections of X which is in general position.
Applying these bounds to vector bundles over projective spaces, we get estimates on their cohomological Hilbert functions which depend only from the first two Chern classes and the ”span” of the generic splitting type. Similar bounds have been given by G. Elencwajg and O. Forster [Math. Ann. 246, 251-270 (1980; Zbl 0432.14011)]. - Applying our general method to a very ample divisor $${\mathcal L}$$ over a complete irreducible variety X, we obtain a bound $$n_ 0$$ such that $$h^ 1(X,{\mathcal L}^ n)$$ takes the same value for all $$n\leq n_ 0$$. Thereby $$n_ 0$$ depends only on $$h^ 1(X,{\mathcal O}_ X)$$ and on local data of X, but not on the characteristic of the ground field k.

### MSC:

 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14C05 Parametrization (Chow and Hilbert schemes)

Zbl 0432.14011
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### References:

 [1] Barth, W, Some properties of stable rank-2 vector bundles on Pn, Math. ann., 226, 323-347, (1977) [2] {\scW. Barth and G. Elencwajg}, Concernant la cohomologie des fibrés algébriquement stables sur $$P$$_{n}($$C$$), in “Variétés Analytiques Compactes, Nice 1977.” Lecture Notes in Math. No. 683, Springer-Verlag, Berlin. · Zbl 0381.55005 [3] Barth, W; Van de Ven, A, A decomposability criterion for algebraic 2-bundles on projective spaces, Invent. math., 25, 91-106, (1974) · Zbl 0295.14006 [4] Brodmann, M, Finiteness of ideal transforms, J. algebra, 63, 162-185, (1980) · Zbl 0396.13018 [5] Brodmann, M, Kohomologische eigenschaften von aufblasungen an lokal vollständigen durchschnitten, Habilitationsschrift, (1980), Münster [6] Brodmann, M, A lifting result for local cohomology of graded modules, (), 221-229 · Zbl 0493.14013 [7] {\scM. Brodmann}, Local cohomology and the connectedness in algebraic varieties, preprint. · Zbl 0578.14019 [8] {\scM. Brodmann}, Bounds on the Serre cohomology of projective varieties, preprint. · Zbl 0646.14016 [9] Elencwajg, G; Forster, O, Bounding cohomology groups of vector bundles on $$P$$_{n}, Math. ann., 246, 251-270, (1980) · Zbl 0432.14011 [10] Grauert, H; Riemenschneider, O, Verschwindungssätze für analytische kohomologiegruppen auf komplexen Räumen, Invent. math., 11, 263-292, (1970) · Zbl 0202.07602 [11] Grothendieck, A, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. math., 79, 121-138, (1957) · Zbl 0079.17001 [12] Grothendieck, A, La théorie des classes de Chern, Bull. soc. math. France, 86, 137-154, (1958) · Zbl 0091.33201 [13] Grothendieck, A, Ega iv, Publ. math. IHES, No. 24, (1968) [14] Grothendieck, A, Sga ii, (1968), North-Holland Amsterdam [15] Grothendieck, A, Local cohomology, () · Zbl 0185.49202 [16] Hartshorne, R, Algebraic geometry, (1977), Springer Heidelberg · Zbl 0367.14001 [17] Hirzebruch, F, Topological methods in algebraic geometry, (1966), Springer Heidelberg · Zbl 0138.42001 [18] Hochster, M; Eagon, J.A, Cohen-Macaulay rings, invariant theory and the generic perfection of determinantal loci, Amer. J. math., 93, 1020-1058, (1971) · Zbl 0244.13012 [19] Horrocks, G, Vector bundles on the punctured spectrum of local ring, (), 689-713 · Zbl 0126.16801 [20] Jouanolou, J.P, Théorèmes de Bertini et applications, (1983), Birkhäuser Boston · Zbl 0519.14002 [21] Kodaira, K, On a differential-geometric method in the theory of analytic stacks, (), 1268-1273 · Zbl 0053.11701 [22] Matsumura, H, Commutative algebra, (1970), Benjamin New York · Zbl 0211.06501 [23] Mumford, D, Lectures on curves on an algebraic surface, () · Zbl 0187.42701 [24] Mumford, D, Pathologies III, Amer. J. math., 89, 94-104, (1967) · Zbl 0146.42403 [25] Okonek, C; Schneider, M; Spindler, H, Vector bundles on complex projective space, () [26] Ramanujam, C, Remarks on the Kodaira vanishing theorem, J. Indian math. soc. (N. S.), 36, 41-51, (1972) · Zbl 0276.32018 [27] Raynaud, M, Contre-example au “vanishing theorem” en caracteristique p⪢0, () · Zbl 0441.14006 [28] Serre, J.P, Faisceaux algébriques cohérents, Ann. math., 61, 197-278, (1955) · Zbl 0067.16201 [29] Serre, J.P, Géometrie algébrique et géometrie analytique, Ann. inst. Fourier, 6, 1-42, (1956) · Zbl 0075.30401 [30] Spindler, H, Der satz von grauert-Mülich für beliebige semistabile holomorphe vektorbündel über dem n-dimensionalen komplex-projektiven raum, Math. ann., 243, 131-141, (1979) · Zbl 0435.32018
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