Derived categories of projective bundles
HTML articles powered by AMS MathViewer
- by L. Costa and R. M. Miró-Roig PDF
- Proc. Amer. Math. Soc. 133 (2005), 2533-2537 Request permission
Abstract:
The goal of this short note is to prove that any projective bundle $\mathbb {P}(\mathcal {E}) \rightarrow X$ has a tilting bundle whose summands are line bundles whenever the same holds for $X$.References
- I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Algebraic vector bundles on $\textbf {P}^{n}$ and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66–67 (Russian). MR 509387
- A. I. Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 25–44 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 23–42. MR 992977, DOI 10.1070/IM1990v034n01ABEH000583
- Alexei Bondal and Dmitri Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math. 125 (2001), no. 3, 327–344. MR 1818984, DOI 10.1023/A:1002470302976
- Tom Bridgeland and Antony Maciocia, Complex surfaces with equivalent derived categories, Math. Z. 236 (2001), no. 4, 677–697. MR 1827500, DOI 10.1007/PL00004847
- Tom Bridgeland, Fourier-Mukai transforms for elliptic surfaces, J. Reine Angew. Math. 498 (1998), 115–133. MR 1629929, DOI 10.1515/crll.1998.046
- L. Costa, R.M. Miró-Roig, Tilting bundles on toric varieties, Preprint, Univ. Barcelona (2003).
- A. L. Gorodentsev and A. N. Rudakov, Exceptional vector bundles on projective spaces, Duke Math. J. 54 (1987), no. 1, 115–130. MR 885779, DOI 10.1215/S0012-7094-87-05409-3
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- A. King, Tilting bundles on some rational surfaces, Preprint at http://www.maths.bath.ac.uk/masadk/papers/.
- Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120–139. MR 1403918
- Antony Maciocia, Generalized Fourier-Mukai transforms, J. Reine Angew. Math. 480 (1996), 197–211. MR 1420564, DOI 10.1515/crll.1996.480.197
- Shigeru Mukai, Fourier functor and its application to the moduli of bundles on an abelian variety, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 515–550. MR 946249, DOI 10.2969/aspm/01010515
- D. O. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 852–862 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 133–141. MR 1208153, DOI 10.1070/IM1993v041n01ABEH002182
- Helices and vector bundles, London Mathematical Society Lecture Note Series, vol. 148, Cambridge University Press, Cambridge, 1990. Seminaire Rudakov; Translated from the Russian by A. D. King, P. Kobak and A. Maciocia. MR 1074776
Additional Information
- L. Costa
- Affiliation: Facultat de Matemàtiques, Departament d’Algebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
- Email: costa@ub.edu
- R. M. Miró-Roig
- Affiliation: Facultat de Matemàtiques, Departament d’Algebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
- MR Author ID: 125375
- ORCID: 0000-0003-1375-6547
- Email: miro@ub.edu
- Received by editor(s): July 15, 2003
- Received by editor(s) in revised form: May 17, 2004
- Published electronically: April 8, 2005
- Additional Notes: The first author was partially supported by MTM2004-00666
The second author was partially supported by MTM2004-00666 - Communicated by: Michael Stillman
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 2533-2537
- MSC (2000): Primary 14F05; Secondary 14M25
- DOI: https://doi.org/10.1090/S0002-9939-05-07846-9
- MathSciNet review: 2146195