Spinor sheaves on singular quadrics

Addington, Nicolas

doi:10.1090/S0002-9939-2011-10819-0

Spinor sheaves on singular quadrics
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by Nicolas Addington PDF

Proc. Amer. Math. Soc. 139 (2011), 3867-3879 Request permission

Abstract:

We define, using matrix factorizations of the equation of $Q$, reflexive sheaves on a singular quadric $Q$ that generalize the spinor bundles on smooth quadrics. We study the first properties of these spinor sheaves, give a Horrocks-type criterion, and show that they are semi-stable, and indeed stable in some cases.

References

N. Addington, The derived category of the intersection of four quadrics, preprint, arXiv:0904.1764.
E. Ballico, Splitting criteria for vector bundles on singular quadrics, Int. J. Contemp. Math. Sci. 2 (2007), no. 29-32, 1549–1551. MR 2378570, DOI 10.12988/ijcms.2007.07162
A. A. Beĭlinson, Coherent sheaves on $\textbf {P}^{n}$ and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68–69 (Russian). MR 509388
José Bertin, Clifford algebras and matrix factorizations, Adv. Appl. Clifford Algebr. 18 (2008), no. 3-4, 417–430. MR 2490565, DOI 10.1007/s00006-008-0079-6
Ragnar-Olaf Buchweitz, David Eisenbud, and Jürgen Herzog, Cohen-Macaulay modules on quadrics, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 58–116. MR 915169, DOI 10.1007/BFb0078838
Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1450870, DOI 10.1007/978-3-663-11624-0
M. M. Kapranov, Derived category of coherent bundles on a quadric, Funktsional. Anal. i Prilozhen. 20 (1986), no. 2, 67 (Russian). MR 847146
Anton Kapustin and Yi Li, D-branes in Landau-Ginzburg models and algebraic geometry, J. High Energy Phys. 12 (2003), 005, 44. MR 2041170, DOI 10.1088/1126-6708/2003/12/005
Alexander Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008), no. 5, 1340–1369. MR 2419925, DOI 10.1016/j.aim.2008.03.007
Adrian Langer, $D$-affinity and Frobenius morphism on quadrics, Int. Math. Res. Not. IMRN 1 (2008), Art. ID rnm 145, 26. MR 2417792, DOI 10.1093/imrn/rnm145
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
Giorgio Ottaviani, Spinor bundles on quadrics, Trans. Amer. Math. Soc. 307 (1988), no. 1, 301–316. MR 936818, DOI 10.1090/S0002-9947-1988-0936818-5
Giorgio Ottaviani, Some extensions of Horrocks criterion to vector bundles on Grassmannians and quadrics, Ann. Mat. Pura Appl. (4) 155 (1989), 317–341. MR 1042842, DOI 10.1007/BF01765948