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\(T\)-symmetrical tensor forms on complete intersections. (English) Zbl 0724.14032
Let \(\Omega^ 1\) be the sheaf of regular 1-forms over a smooth projective variety V and let \({\mathcal T}^ r=\otimes^{r}_{1}\Omega^ 1 \) be its r-th tensor product. The authors consider a splitting of \({\mathcal T}^ r\) into the direct sum \({\mathcal T}^ r=\oplus {\mathcal T}^ T \) of subsheaves \({\mathcal T}^ T\) where each \({\mathcal T}^ T\) is assigned to a standard Young tableau with r cells (these tableaux are connected with irreducible representations of the symmetric group \(S_ r).\)
The main results of the paper are some vanishing theorems for the cohomology groups of a smooth complete intersection Y with coefficients in the twisted sheaves \({\mathcal T}^ T(p)\). - Besides, there are some Lefschetz type theorems concerning the restrictions of tensor forms on the section by hypersurfaces.

MSC:
14M10 Complete intersections
14F17 Vanishing theorems in algebraic geometry
14E05 Rational and birational maps
14M17 Homogeneous spaces and generalizations
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References:
[1] Bogomolov, F.A.: Holomorphic symmetrical tensors on projective surfaces (Russian). Usp. Mat. Nauk.33, 171-172 (1978) · Zbl 0417.15012
[2] Brückmann, P.: Differential tensor forms on algebraic varieties (Russian). Izv. Akad. Nauk. SSSR, Ser. Mat.35, 1008-1036 (1971) · Zbl 0221.14012
[3] Brückmann, P.: Zur Kohomologie von projektiven Hyperflächen. Beitr. Algebra Geom.2, 87-101 (1974) · Zbl 0288.14008
[4] Brückmann, P.: Zur Kohomologie von vollständigen Durchschnitten mit Koeffizienten in der Garbe der Keime der Differentialformen I, II. Math. Nachr.71, 203-210 (1976);77, 307-318 (1977) · Zbl 0318.14015 · doi:10.1002/mana.19760710116
[5] Brückmann, P.: Some birational invariants of algebraic varieties. Proceedings of the Conference on Algebraic Geometry, Teubner-Texte Math.92, 65-73 (1985)
[6] Ebeling, W.: An example of two homeomorphic, non diffeomorphic complete intersection surfaces. Invent. Math.99, 651-654 (1990) · Zbl 0707.14045 · doi:10.1007/BF01234435
[7] Sakai, F.: Symmetric powers of the cotangent bundle and classification of algebraic varieties. (Lect. Notes in Math. vol. 732), Berlin Heidelberg New York: Springer 1979 · Zbl 0415.14020
[8] Sakai, F.: Differential tensor forms in Algebraic Geometry. Preprint Dep. of Math., Fac. of Sci., Saitama Univ., Urawa, Japan (1979)
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