Abstract
Let X be a smooth, projective, d-dimensional subvariety of ℙn(ℂ). Barth's theorem says that H q(X, Ωp X )=0 when p≠q and q+p≤2d−n (if p=0 we must have q>0). It is very interesting to look for analogous vanishing theorems for H q(X, Ωp X (m)), m ∈ ℤ (see [S-S], [F], [S]). In this paper we prove some vanishing theorems for H q(X, Ωp X (1)), for H q(X, Ωp X (m)) when m≤−1, and, if dim(X)=n−2, for H q(X, Ω2 X (m)) and H q(X, S kΩ1 X (m)). We use standard techniques and some of our previous results.
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References
Alzati, A. and Ottaviani, G. ‘Small codimension subvarieties of ℙn, Boll. Un. Mat. Ital. (7), 2-A (1988), 81–89.
Alzati, A. and Ottaviani, G. ‘A vanishing theorem for the ideal sheaf of codimension two subvarieties of ℙn’, Rend. Istit. Mat. Univ. Trieste.
Barth, R. ‘Transplanting cohomology classes in complex projective space’, Amer. J. Math. 92 (1970), 951–957.
Bruckmann, P. ‘Some birational invariants of algebraic varieties’ Proc. Internat. Cong. Algebraic Geometry, Berlin, 1985; Teubner-Texte 92 (1986), 65–73.
Chang, M. C. and Ran, Z. ‘A vanishing theorem for symmetric differentials on projective varieties of small codimension’ (Preprint).
Ein, L. ‘Vanishing theorems for varieties of low codimension’ in Algebraic Geometry Sundance 1986. Lecture Notes in Math. 1311, Springer, Berlin, Heidelberg, New York, 1988.
G. Faltings, ‘Verschwindungssatze und Untermannigfaltigkeiten kleiner Kodimension des Komplex-projektiven Raumes’, J. reine angew. Math. 326 (1980), 136–151.
J. Le Potier, ‘Annulation de la cohomologie à valeurs dans un fibré vectoriel holomorphe positif de rang quelconque’, Math. Ann. 218 (1975), 35–53.
Ran, Z. ‘Systèmes linéaires complets de sections hypersurfaces sur les variétés projectives de codimension 2’, C.R. Acad. Sci. Paris Série I, 298, 6 (1984), 111–112.
Schneider, M. ‘Symmetric differential forms as embedding obstructions and vanishing theorems’ (preprint).
Shiffman, B. and Sommese, A. J. ‘Vanishing theorems on complex manifolds’ Progr. Math. 56 (1975).
Zak, F. L. ‘Projections of algebraic varieties’, Mat. Sb. 116, No. 4 (1981), 593–602; English translation: Math. USSR-Sb 44, No. 4 (1983), 535–544.
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An erratum to this article is available at http://dx.doi.org/10.1007/s10711-009-9382-1.
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Alzati, A. Barth type vanishing theorems. Geom Dedicata 44, 159–168 (1992). https://doi.org/10.1007/BF00182947
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DOI: https://doi.org/10.1007/BF00182947