A few splitting criteria for vector bundles. (English) Zbl 1232.14010
Summary: We prove a few splitting criteria for vector bundles on a quadric hypersurface and Grassmannians. We give also some cohomological splitting conditions for rank 2 bundles on multiprojective spaces. The tools are monads and a Beilinson’s type spectral sequence generalized by L. Costa and R. Miró-Roig [Mich. Math. J. 55, No. 2, 417–436 (2007; Zbl 1135.14034)].
MSC:
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
Citations:
Zbl 1135.14034References:
| [1] | Barth W., Hulek K. (1978) Monads and moduli of vector bundles. Manuscripta Math. 25: 323–447 · Zbl 0395.14007 · doi:10.1007/BF01168047 |
| [2] | Beilinson A.A. (1979) Coherent sheaves on $${\(\backslash\)mathbb{P}\^n}$$ and problems of linear algebra. Funkt. Anal. Appl. 12: 214–216 · Zbl 0424.14003 · doi:10.1007/BF01681436 |
| [3] | Costa L., Miró-Roig R.M. (2005) Cohomological characterization of vector bundles on multiprojective spaces. J. Algebra 294: 73–96 · Zbl 1093.14060 · doi:10.1016/j.jalgebra.2005.08.035 |
| [4] | Costa, L., Miró-Roig, R.M.: Monads and regularity of vector bundles on projective varieties. Preprint (2007) · Zbl 1135.14034 |
| [5] | Horrocks G. (1964) Vector bundles on the punctured spectrum of a ring. Proc. Lond. Math. Soc. 3(14): 689–713 · Zbl 0126.16801 · doi:10.1112/plms/s3-14.4.689 |
| [6] | Kapranov M.M. (1985) On the derived category of coerent sheaves on Grassmann manifolds. Math. USSR Izvestiya 24: 183–192 · Zbl 0564.14023 · doi:10.1070/IM1985v024n01ABEH001221 |
| [7] | Kapranov M.M. (1988) On the derived category of coerent sheaves on some homogeneous spaces. Invent. Math. 92: 479–508 · Zbl 0651.18008 · doi:10.1007/BF01393744 |
| [8] | Knörrer H. (1987) Cohen-Macaulay modules of hypersurface singularities I. Invent. Math. 88: 153–164 · Zbl 0617.14033 · doi:10.1007/BF01405095 |
| [9] | Mohan Kumar N., Peterson C., Rao A.P. (2003) Monads on projective spaces. Manuscripta Math. 112: 183–189 · Zbl 1030.14008 · doi:10.1007/s00229-003-0389-x |
| [10] | Malaspina, F.: Monads and vector bundles on quadrics. Adv. Geom. (to appear). Available at math.AG/0612512 |
| [11] | Ottaviani G. (1987) Criteres de scindage pour les fibres vectoriel sur les grassmanniennes et les quadriques. C. R. Acad. Sci. Paris 305: 257–260 · Zbl 0629.14012 |
| [12] | Ottaviani G. (1988) Spinor bundles on quadrics. Trans. Am. Math. Soc. 307: 301–316 · Zbl 0657.14006 · doi:10.1090/S0002-9947-1988-0936818-5 |
| [13] | Ottaviani G. (1989) Some extension of Horrocks criterion to vector bundles on Grassmannians and quadrics. Ann. Math. Pura Appl. (IV) 155: 317–341 · Zbl 0718.14010 · doi:10.1007/BF01765948 |
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