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Birational geometry of complete intersections. (English) Zbl 0907.14023
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.

##### MSC:
 14M10 Complete intersections 14E05 Rational and birational maps 14F17 Vanishing theorems in algebraic geometry
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##### References:
 [1] F. A. Bogomolov, Holomorphic symmetrical tensors on projective surfaces (Russian).Usp. Mat. Nauk. 33 (1978), 171–172. · Zbl 0417.15012 [2] R. Bott, Homogenous vector bundles.Ann. of Math. 66 (1957), 203–248. · Zbl 0094.35701 · doi:10.2307/1969996 [3] P. Brückmann, Some birational invariants of algebraic varieties. Proceedings of the Conference on Algebraic Geometry,Teubner-Texte Math. 92 (1985), 65–73. [4] —, The Euler-Poincaré-characteristic of the sheaf ofT-symmetrical tensor forms on complete intersections,to appear. [5] P. Brückmann andH.-G. Rackwitz,T-symmetrical tensor forms on complete intersections.Math. Ann. 288 (1990), 627–635. · Zbl 0724.14032 · doi:10.1007/BF01444555 [6] Ph. Griffiths andJ. Harris,Principles of Algebraic Geometry. John Wiley & Sons, New York, Chichester, Brisbane, Toronto, (1978). · Zbl 0408.14001 [7] F. Hirzebruch,Topological Methods in Algebraic Geometry. 3.Aufl., Grundl.131, Springer-Verlag, Heidelberg, (1966) · Zbl 0138.42001 [8] L. Manivel, Birational invariants of algebraic varieties.J. reine angew. Math. 458 (1995), 63–91. · Zbl 0811.14008 · doi:10.1515/crll.1995.458.63 [9] F. Sakai,Symmetric powers of the cotangent bundle and classification of algebraic varieties. Lect. Notes in Math.732, Springer-Verlag, Berlin-Heidelberg-New York, (1979). · Zbl 0415.14020 [10] M. Schneider, Symmetric differential forms as embedding obstructions and vanishing theorems.J. of Algebraic Geometry 1 (1992), 175–181. · Zbl 0790.14009
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