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A spectral sequence and nef vector bundles of the first Chern class two on hyperquadrics. (English) Zbl 1408.14133
Summary: We deduce from Bondal’s theorem a spectral sequence, which yields a description, such as a resolution, of a coherent sheaf in terms of a full strong exceptional sequence. We then apply the sequence to the case of a vector bundle given with some cohomological data on a projective space; we obtain a resolution of the vector bundle in terms of exceptional line bundles, resolution which is different from that obtained by the Beilinson spectral sequence. Finally we list all known nef vector bundles of the first Chern class two on a hyperquadric of dimension greater than three.

MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14N30 Adjunction problems 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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References:
 [1] Assem, I., Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras, vol. 1, London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006). Techniques of representation theory · Zbl 0651.18008 [2] Beilinson, AA, Coherent sheaves on $${P}^n$$ and problems in linear algebra, Funktsional. Anal. i Prilozhen., 12, 214-216, (1978) · Zbl 0424.14003 [3] Bondal, AI, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Math., 53, 25-44, (1989) [4] Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and reconstructions. Izv. Akad. Nauk SSSR Ser. Math. 53(6), 1183-1205, 1337 (1989) · Zbl 0703.14011 [5] Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford (2006) · Zbl 1095.14002 [6] Kapranov, MM, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math., 92, 479-508, (1988) · Zbl 0651.18008 [7] Ottaviani, G, Spinor bundles on quadrics, Trans. Am. Math. Soc., 307, 301-316, (1988) · Zbl 0657.14006 [8] Peternell, T., Szurek, M., Wiśniewski, J.A.: Numerically effective vector bundles with small chern classes. In: Hulek, T.S.M., Peternell, K., Schreyer, F.O. (eds.) Complex Algebraic Varieties, Proceedings, Bayreuth, 1990, no. 1507 in Lecture Notes in Math., pp. 145-156. Springer, Berlin (1992)
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