Nous étudions les fibrés vectoriels relativement semi-stables sur des varietés non-kählériennes qui sont des fibrés principaux elliptiques. Les principaux outils techniques utilisés sont la transformée de Fourier-Mukai tordue et une construction de revêtement spectral. Pour un exemple important de ces fibrés principaux, nous calculons les invariants numériques des fibrés elliptiques sur une surface de Kodaira primaire.
We study relatively semi-stable vector bundles and their moduli on non-Kähler principal elliptic bundles over compact complex manifolds of arbitrary dimension. The main technical tools used are the twisted Fourier-Mukai transform and a spectral cover construction. For the important example of such principal bundles, the numerical invariants of a 3-dimensional non-Kähler elliptic principal bundle over a primary Kodaira surface are computed.
Classification : 14J60, 32L05, 14D22, 14F05, 32J17, 32Q25
Mots clés : Fibrés elliptiques principaux non-kählériens, varietés de dimension 3 de type Calabi-Yau, fibrés vectoriels holomorphes, espaces de modules
@article{AIF_2013__63_3_1033_0, author = {Br\^\i nz\u anescu, Vasile and Halanay, Andrei D. and Trautmann, G\"unther}, title = {Vector bundles on non-Kaehler elliptic principal bundles}, journal = {Annales de l'Institut Fourier}, pages = {1033--1054}, publisher = {Association des Annales de l'institut Fourier}, volume = {63}, number = {3}, year = {2013}, doi = {10.5802/aif.2783}, mrnumber = {3137479}, zbl = {1299.14037}, language = {en}, url = {www.numdam.org/item/AIF_2013__63_3_1033_0/} }
Brînzănescu, Vasile; Halanay, Andrei D.; Trautmann, Günther. Vector bundles on non-Kaehler elliptic principal bundles. Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 1033-1054. doi : 10.5802/aif.2783. http://www.numdam.org/item/AIF_2013__63_3_1033_0/
[1] The Derived Category of the Intersection of Four Quadrics, 2009 (arXiv:0904.1764)
[2] Spinor sheaves on singular quadrics, Proc. Amer. Math. Soc., Volume 139 (2011) no. 11, pp. 3867-3879 | Article | MR 2823033 | Zbl 1235.14038
[3] Vector bundles over an elliptic curve, Proc. London Math. Soc. (3), Volume 7 (1957), pp. 414-452 | Article | MR 131423 | Zbl 0084.17305
[4] Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension finie, Fonctions de plusieurs variables complexes, II (Sém. François Norguet, 1974–1975), Springer, Berlin, 1975, p. 1-158. Lecture Notes in Math., Vol. 482 | MR 399503 | Zbl 0331.32008
[5] Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Volume 4, Springer-Verlag, Berlin, 1984 | MR 749574 | Zbl 1036.14016
[6] Fourier-Mukai and Nahm transforms in geometry and mathematical physics, Progress in Mathematics, Volume 276, Birkhäuser Boston Inc., Boston, MA, 2009 | Article | MR 2511017 | Zbl 1186.14001
[7] Mirror symmetry on surfaces via Fourier-Mukai transform, Comm. Math. Phys., Volume 195 (1998) no. 1, pp. 79-93 | Article | MR 1637405 | Zbl 0930.14028
[8] Compactifications of heterotic theory on non-Kähler complex manifolds. I, J. High Energy Phys. (2003) no. 4, pp. 007, 60 pp. (electronic) | Article | MR 1989858
[9] Twisting derived equivalences, Trans. Amer. Math. Soc., Volume 361 (2009) no. 10, pp. 5469-5504 | Article | MR 2515820 | Zbl 1177.14037
[10] Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Volume 302, Springer-Verlag, Berlin, 2004 | MR 2062673 | Zbl 0779.14012
[11] Semiorthogonal decomposition for algebraic varieties, 1995 (eprinthttp://arxiv.org/abs/alg-geom/9506012)
[12] Fourier-Mukai transforms for elliptic surfaces, J. Reine Angew. Math., Volume 498 (1998), pp. 115-133 | Article | MR 1629929 | Zbl 0905.14020
[13] Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc., Volume 31 (1999) no. 1, pp. 25-34 | Article | MR 1651025 | Zbl 0937.18012
[14] Fourier-Mukai transforms for and elliptic fibrations, J. Algebraic Geom., Volume 11 (2002) no. 4, pp. 629-657 | Article | MR 1910263 | Zbl 1066.14047
[15] Stable bundles on non-Kähler elliptic surfaces, Comm. Math. Phys., Volume 254 (2005) no. 3, pp. 565-580 | Article | MR 2126483 | Zbl 1071.32009
[16] Twisted Fourier-Mukai transforms and bundles on non-Kähler elliptic surfaces, Math. Res. Lett., Volume 13 (2006) no. 4, pp. 501-514 | Article | MR 2250486 | Zbl 1133.14040
[17] Néron-Severi group for torus quasi bundles over curves, Moduli of vector bundles (Sanda, 1994; Kyoto, 1994) (Lecture Notes in Pure and Appl. Math.) Volume 179, Dekker, New York, 1996, pp. 11-32 | MR 1397977 | Zbl 0883.14015
[18] On a relative Fourier-Mukai transform on genus one fibrations, Manuscripta Math., Volume 120 (2006) no. 3, pp. 283-306 | Article | MR 2243564 | Zbl 1105.18011
[19] Derived categories of twisted sheaves on elliptic threefolds, J. Reine Angew. Math., Volume 544 (2002), pp. 161-179 | Article | MR 1887894 | Zbl 0995.14012
[20] Non-birational twisted derived equivalences in abelian GLSMs, Comm. Math. Phys., Volume 294 (2010) no. 3, pp. 605-645 | Article | MR 2585982 | Zbl 1231.14035
[21] Derived categories of twisted sheaves on Calabi-Yau manifolds, ProQuest LLC, Ann Arbor, MI, 2000 http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9967459 (Thesis (Ph.D.)–Cornell University) | MR 2700538
[22] Non-Kähler string backgrounds and their five torsion classes, Nuclear Phys. B, Volume 652 (2003) no. 1-3, pp. 5-34 | Article | MR 1959324 | Zbl 1010.83063
[23] Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Inst. Hautes Études Sci. Publ. Math. (1968) no. 35, pp. 259-278 | Numdam | MR 244265 | Zbl 0159.22501
[24] Spectral covers, Current topics in complex algebraic geometry (Berkeley, CA, 1992/93) (Math. Sci. Res. Inst. Publ.) Volume 28, Cambridge Univ. Press, Cambridge, 1995, pp. 65-86 | MR 1397059 | Zbl 0877.14026
[25] Principal bundles on elliptic fibrations, Asian J. Math., Volume 1 (1997) no. 2, pp. 214-223 | MR 1491982 | Zbl 0927.14006
[26] Torus fibrations, gerbes, and duality, Mem. Amer. Math. Soc., Volume 193 (2008) no. 901, pp. vi+90 (With an appendix by Dmitry Arinkin) | MR 2399730 | Zbl 1140.14001
[27] Flatness and privilege, Enseignement Math. (2), Volume 14 (1968), pp. 47-74 | MR 236420 | Zbl 0183.35102
[28] Commutative algebra, Graduate Texts in Mathematics, Volume 150, Springer-Verlag, New York, 1995 (With a view toward algebraic geometry) | Article | MR 1322960 | Zbl 0819.13001
[29] Rank two vector bundles over regular elliptic surfaces, Invent. Math., Volume 96 (1989) no. 2, pp. 283-332 | Article | MR 989699 | Zbl 0671.14006
[30] Vector bundles over elliptic fibrations, J. Algebraic Geom., Volume 8 (1999) no. 2, pp. 279-401 | MR 1675162 | Zbl 0937.14004
[31] Geometric model for complex non-Kähler manifolds with structure, Comm. Math. Phys., Volume 251 (2004) no. 1, pp. 65-78 | Article | MR 2096734 | Zbl 1085.32009
[32] Algebraic geometry, Springer-Verlag, New York, 1977 (Graduate Texts in Mathematics, No. 52) | MR 463157 | Zbl 0531.14001
[33] Topological methods in algebraic geometry, Third enlarged edition. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. Die Grundlehren der Mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966 | MR 1335917 | Zbl 0376.14001
[34] Remarks on torus principal bundles, J. Math. Kyoto Univ., Volume 33 (1993) no. 1, pp. 227-259 | MR 1203897 | Zbl 0788.32023
[35] Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2006 | Article | MR 2244106 | Zbl 1095.14002
[36] The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997 | MR 1450870 | Zbl 0872.14002
[37] Vertex algebras, mirror symmetry, and D-branes: the case of complex tori, Comm. Math. Phys., Volume 233 (2003) no. 1, pp. 79-136 | Article | MR 1957733 | Zbl 1051.17017
[38] Derived categories of quadric fibrations and intersections of quadrics, Adv. Math., Volume 218 (2008) no. 5, pp. 1340-1369 | Article | MR 2419925 | Zbl 1168.14012
[39] Duality between and with its application to Picard sheaves, Nagoya Math. J., Volume 81 (1981), pp. 153-175 http://projecteuclid.org/getRecord?id=euclid.nmj/1118786312 | MR 607081 | Zbl 0417.14036
[40] Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay, 1970 | MR 282985 | Zbl 0223.14022
[41] Derived categories of coherent sheaves and equivalences between them, Uspekhi Mat. Nauk, Volume 58 (2003) no. 3(351), pp. 89-172 | Article | MR 1998775 | Zbl 1118.14021
[42] Théorème de Douady au-dessus de , Ann. Scuola Norm. Sup. Pisa (3), Volume 23 (1969), pp. 451-459 | Numdam | MR 257402 | Zbl 0186.14003
[43] Stable sheaves on elliptic fibrations, J. Geom. Phys., Volume 43 (2002) no. 2-3, pp. 163-183 | Article | MR 1919209 | Zbl 1068.14051
[44] Semistable bundles over an elliptic curve, Adv. Math., Volume 98 (1993) no. 1, pp. 1-26 | Article | MR 1212625 | Zbl 0786.14021