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2012 | 10 | 4 | 1422-1441
Tytuł artykułu

Λ-modules and holomorphic Lie algebroid connections

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EN
Abstrakty
EN
Let X be a complex smooth projective variety, and G a locally free sheaf on X. We show that there is a one-to-one correspondence between pairs (Λ, Ξ), where Λ is a sheaf of almost polynomial filtered algebras over X satisfying Simpson’s axioms and $ \equiv :Gr\Lambda \to Sym \bullet _{\mathcal{O}_X } \mathcal{G}$ is an isomorphism, and pairs (L, Σ), where L is a holomorphic Lie algebroid structure on $\mathcal{G}$ and Σ is a class in F 1 H 2(L, ℂ), the first Hodge filtration piece of the second cohomology of L. As an application, we construct moduli spaces of semistable flat L-connections for any holomorphic Lie algebroid L. Particular examples of these are given by generalized holomorphic bundles for any generalized complex structure associated to a holomorphic Poisson manifold.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
4
Strony
1422-1441
Opis fizyczny
Daty
wydano
2012-08-01
online
2012-05-31
Bibliografia
  • [1] Atiyah M.F., Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., 1957, 85, 181–207 http://dx.doi.org/10.1090/S0002-9947-1957-0086359-5
  • [2] Beĭlinson A., Bernstein J., A proof of Jantzen conjectures, In: I.M. Gel’fand Seminar, Adv. Soviet Math., 16(1), American Mathematical Society, Providence, 1993, 1–50
  • [3] Bruzzo U., Rubtsov V.N., Cohomology of skew-holomorphic Lie algebroids, Teoret. Mat. Fiz., 2010, 165(3), 426–439 (in Russian) http://dx.doi.org/10.4213/tmf6586
  • [4] Calaque D., Dolgushev V., Halbout G., Formality theorems for Hochschild chains in the Lie algebroid setting, J. Reine Angew. Math., 2007, 612, 81–127
  • [5] Calaque D., Van den Bergh M., Hochschild cohomology and Atiyah classes, Adv. Math., 2010, 224(5), 1839–1889 http://dx.doi.org/10.1016/j.aim.2010.01.012
  • [6] Deligne P., Équations Différentielles à Points Singuliers Réguliers, Lecture Notes in Math., 163, Springer, Berlin-New York, 1970
  • [7] Esnault H., Viehweg E., Logarithmic de Rham complexes and vanishing theorems, Invent. Math., 1986, 86(1), 161–194 http://dx.doi.org/10.1007/BF01391499
  • [8] Fernandes R.L., Lie algebroids, holonomy and characteristic classes, Adv. Math., 2002, 170(1), 119–179 http://dx.doi.org/10.1006/aima.2001.2070
  • [9] Griffiths P., Harris J., Principles of Algebraic Geometry, Pure Appl. Math., John Wiley & Sons, New York, 1978
  • [10] Gualtieri M., Generalized complex geometry, Ann. of Math., 2011, 174(1), 75–123 http://dx.doi.org/10.4007/annals.2011.174.1.3
  • [11] Hitchin N., Generalized holomorphic bundles and the B-field action, J. Geom. Phys., 2011, 61(1), 352–362 http://dx.doi.org/10.1016/j.geomphys.2010.10.014
  • [12] Huebschmann J., Extensions of Lie-Rinehart algebras and the Chern-Weil construction, In: Higher Homotopy Structures in Topology and Mathematical Physics, Poughkeepsie, June 13–15, 1996, Contemp. Math., 227, Amerrican Mathematical Society, Providence, 1999, 145–176 http://dx.doi.org/10.1090/conm/227/03255
  • [13] Huebschmann J., Differential Batalin-Vilkovisky algebras arising from twilled Lie-Rinehart algebras, In: Poisson Geometry, Warsaw, August 3–15, 1998, Banach Center Publ., 51, Polish Acadamy of Sciences, Warsaw, 2000, 87–102
  • [14] Huybrechts D., Lehn M., The Geometry of Moduli Spaces of Sheaves, 2nd ed., Cambridge Math. Lib., Cambridge University Press, Cambridge, 2010 http://dx.doi.org/10.1017/CBO9780511711985
  • [15] Laurent-Gengoux C., Stiénon M., Xu P., Holomorphic Poisson manifolds and holomorphic Lie algebroids, Int. Math. Res. Not. IMRN, 2008, #088
  • [16] Mackenzie K.C.H., General Theory of Lie Groupoids and Lie Algebroids, London Math. Soc. Lecture Note Ser., 213, Cambridge University Press, Cambridge, 2005
  • [17] Nest R., Tsygan B., Deformation of symplectic Lie algebroids, deformation of holomorphic symplectic structures, and index theorems, Asian J. Math., 2001, 5(4), 599–635
  • [18] Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math., 1994, 79, 47–129 http://dx.doi.org/10.1007/BF02698887
  • [19] Simpson C., The Hodge filtration on nonabelian cohomology, In: Algebraic Geometry, Santa Cruz, July 9–29, 1995, Proc. Sympos. Pure Math., 62(2), American Mathematical, Society, Providence, 1997, 217–281
  • [20] Sridharan R., Filtered algebras and representations of Lie algebras, Trans. Amer. Math. Soc., 1961, 100, 530–550 1 http://dx.doi.org/10.1090/S0002-9947-1961-0130900-1
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