Rackwitz, H.-G. Birational geometry of complete intersections. (English) Zbl 0907.14023 Abh. Math. Semin. Univ. Hamb. 66, 263-271 (1996). Let \(Y\subset \mathbb{P}^n\) be a smooth complete intersection. The author studies the cohomology groups of symmetric differential forms \(S^r \Omega^1_Y\) on \(Y\). Some vanishing theorems and explicit formulas for \(\dim H^q(Y,S^r\Omega^1_Y)\) are proved. In particular, it is shown that the birational invariants \(\dim H^0 (Y,S^r \Omega^1_Y \otimes \omega^s_Y)\) are independent of classical invariants. Reviewer: Yuri G.Prokhorov (MR 97g:14016) Cited in 1 Document MSC: 14M10 Complete intersections 14E05 Rational and birational maps 14F17 Vanishing theorems in algebraic geometry Keywords:complete intersection; symmetric differential forms; vanishing theorems; birational invariants PDFBibTeX XMLCite \textit{H. G. Rackwitz}, Abh. Math. Semin. Univ. Hamb. 66, 263--271 (1996; Zbl 0907.14023) Full Text: DOI References: [1] Bogomolov, F. A., Holomorphic symmetrical tensors on projective surfaces (Russian), Usp. Mat. Nauk., 33, 171-172 (1978) · Zbl 0417.15012 [2] Bott, R., Homogenous vector bundles, Ann. of Math., 66, 203-248 (1957) · Zbl 0094.35701 · doi:10.2307/1969996 [3] Brückmann, P., Some birational invariants of algebraic varieties. Proceedings of the Conference on Algebraic Geometry, Teubner-Texte Math., 92, 65-73 (1985) · Zbl 0637.14010 [4] —, The Euler-Poincaré-characteristic of the sheaf ofT-symmetrical tensor forms on complete intersections,to appear. [5] Brückmann, P.; Rackwitz, H.-G., T-symmetrical tensor forms on complete intersections, Math. Ann., 288, 627-635 (1990) · Zbl 0724.14032 · doi:10.1007/BF01444555 [6] Griffiths, Ph.; Harris, J., Principles of Algebraic Geometry (1978), New York, Chichester, Brisbane, Toronto: John Wiley & Sons, New York, Chichester, Brisbane, Toronto · Zbl 0408.14001 [7] Hirzebruch, F., Topological Methods in Algebraic Geometry (1966), Heidelberg: Springer-Verlag, Heidelberg · Zbl 0138.42001 [8] Manivel, L., Birational invariants of algebraic varieties, J. reine angew. Math., 458, 63-91 (1995) · Zbl 0811.14008 [9] Sakai, F., Symmetric powers of the cotangent bundle and classification of algebraic varieties (1979), Berlin-Heidelberg-New York: Springer-Verlag, Berlin-Heidelberg-New York · Zbl 0415.14020 [10] Schneider, M., Symmetric differential forms as embedding obstructions and vanishing theorems, J. of Algebraic Geometry, 1, 175-181 (1992) · Zbl 0790.14009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.