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Spinor bundles on quadrics. (English) Zbl 0657.14006
By means of the spinor varieties (parametrising the families of k-planes in a smooth quadric \(Q_{2k}\) or \(Q_{2k+1}\) in \({\mathbb{C}}P^{n+1})\), we can embed the quadric in a Grassmannian. A spinor bundle on Q is what the author calls the pull back of the universal bundle of the Grassmannian under this embedding. This gives a stable vectorbundle S of degree \(2^{k-1}\) \((n=2k\) or 2k-1). - Actually, in the case of n even, there are two spinor bundles corresponding to the two different families of linear spaces of maximal dimension in the quadric. These generalise the universal bundle and the dual of the quotient bundle on \(Q_ 4\simeq Gr(1,3).\)
Work concerning these two bundles on Q by R. Hernandez and I. Sols [J. Reine Angew. Math. 360, 124-135 (1985)] is extended to \(Q_ 5\) and \(Q_ 6\) using the new spinor bundles. In particular, every rank 3 stable vectorbundle on \(Q_ 5\), whose Chern classes are all 2, arises as the quotient E of the dual of the \(Q_ 5\)-spinor bundle S in a sequence \(0\to {\mathcal O}\to S^*\to E\to 0.\)
Finally \(CP^ 7\setminus Q_ 6\) is the fine moduli space for all such stable bundles on \(Q_ 5\) (as well as for stable bundles on \(Q_ 6\) with \(c_ 1=2=c_ 2\), \(c_ 3=(2,0)\) or (0,2)).
Reviewer: P.Bryant

MSC:
14D20 Algebraic moduli problems, moduli of vector bundles
14M15 Grassmannians, Schubert varieties, flag manifolds
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14D22 Fine and coarse moduli spaces
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[1] M. Atiyah, On the Krull-Schmidt theorem with application to sheaves, Bull. Soc. Math. France 84 (1956), 307 – 317. · Zbl 0072.18101
[2] Edoardo Ballico, Uniform vector bundles on quadrics, Ann. Univ. Ferrara Sez. VII (N.S.) 27 (1981), 135 – 146 (1982) (English, with Italian summary). · Zbl 0495.14008
[3] I. N. Bernšteĭn, I. M. Gel\(^{\prime}\)fand, and S. I. Gel\(^{\prime}\)fand, Algebraic vector bundles on \?\(^{n}\) and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66 – 67 (Russian). A. A. Beĭlinson, Coherent sheaves on \?\(^{n}\) and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68 – 69 (Russian).
[4] Klaus Fritzsche, Linear-uniforme Bündel auf Quadriken, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 2, 313 – 339 (German). · Zbl 0527.14017
[5] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. · Zbl 0408.14001
[6] Rafael Hernández and Ignacio Sols, On a family of rank 3 bundles on \?\?(1,3), J. Reine Angew. Math. 360 (1985), 124 – 135.
[7] R. Lazarsfeld and A. Van de Ven, Topics in the geometry of projective space, DMV Seminar, vol. 4, Birkhäuser Verlag, Basel, 1984. Recent work of F. L. Zak; With an addendum by Zak. · Zbl 0564.14007
[8] Masaki Maruyama, Moduli of stable sheaves. I, J. Math. Kyoto Univ. 17 (1977), no. 1, 91 – 126. · Zbl 0374.14002
[9] Masaki Maruyama, Boundedness of semistable sheaves of small ranks, Nagoya Math. J. 78 (1980), 65 – 94. · Zbl 0456.14011
[10] Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. · Zbl 0438.32016
[11] Giorgio Ottaviani, A class of \?-bundles on \?\?(\?,\?), J. Reine Angew. Math. 379 (1987), 182 – 208. · Zbl 0611.14015
[12] -, Some extensions of Horrocks criterion to vector bundles on Grassmannians and quadrics, preprint. · Zbl 0718.14010
[13] Manfred Steinsiek, Transformation groups on homogeneous-rational manifolds, Math. Ann. 260 (1982), no. 4, 423 – 435. , https://doi.org/10.1007/BF01457022 Manfred Steinsiek, Über homogen-rationale Mannigfaltigkeiten, Schriftenreihe des Mathematischen Instituts der Universität Münster, Ser. 2 [Series of the Mathematical Institute of the University of Münster, Ser. 2], vol. 23, Universität Münster, Mathematisches Institut, Münster, 1982 (German). · Zbl 0491.32025
[14] Jacques Tits, Sur la trialité et certains groupes qui s’en déduisent, Inst. Hautes Études Sci. Publ. Math. 2 (1959), 13 – 60 (French). · Zbl 0088.37204
[15] Hiroshi Umemura, On a theorem of Ramanan, Nagoya Math. J. 69 (1978), 131 – 138. · Zbl 0345.14017
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