Λ-modules and holomorphic Lie algebroid connections

Pietro Tortella

Open Mathematics (2012)

  • Volume: 10, Issue: 4, page 1422-1441
  • ISSN: 2391-5455

Abstract

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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 ≡ : G r Λ → S y m • 𝒪 X 𝒢 is an isomorphism, and pairs (L, Σ), where L is a holomorphic Lie algebroid structure on 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.

How to cite

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Pietro Tortella. "Λ-modules and holomorphic Lie algebroid connections." Open Mathematics 10.4 (2012): 1422-1441. <http://eudml.org/doc/269694>.

@article{PietroTortella2012,
abstract = {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 \rightarrow 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.},
author = {Pietro Tortella},
journal = {Open Mathematics},
keywords = {Holomorphic Lie algebroids; Filtered algebras; Universal enveloping algebra; Lie algebroid connections; Moduli spaces of flat connections; Generalized holomorphic bundles; holomorphic Lie algebroids; filtered algebras; flat connections; generalized holomorphic bundles; Hodge filtration; characteristic ring; characteristic classes; anchor; matched pairs of Lie algebroids; twilled pairs; twilled sum; sheaves of filtered algebras; holomorphic Poisson structures},
language = {eng},
number = {4},
pages = {1422-1441},
title = {Λ-modules and holomorphic Lie algebroid connections},
url = {http://eudml.org/doc/269694},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Pietro Tortella
TI - Λ-modules and holomorphic Lie algebroid connections
JO - Open Mathematics
PY - 2012
VL - 10
IS - 4
SP - 1422
EP - 1441
AB - 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 \rightarrow 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.
LA - eng
KW - Holomorphic Lie algebroids; Filtered algebras; Universal enveloping algebra; Lie algebroid connections; Moduli spaces of flat connections; Generalized holomorphic bundles; holomorphic Lie algebroids; filtered algebras; flat connections; generalized holomorphic bundles; Hodge filtration; characteristic ring; characteristic classes; anchor; matched pairs of Lie algebroids; twilled pairs; twilled sum; sheaves of filtered algebras; holomorphic Poisson structures
UR - http://eudml.org/doc/269694
ER -

References

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  1. [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 Zbl0078.16002
  2. [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 Zbl0790.22007
  3. [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 Zbl1252.32034
  4. [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 Zbl1141.53084
  5. [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 Zbl1197.14017
  6. [6] Deligne P., Équations Différentielles à Points Singuliers Réguliers, Lecture Notes in Math., 163, Springer, Berlin-New York, 1970 Zbl0244.14004
  7. [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 Zbl0603.32006
  8. [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 Zbl1007.22007
  9. [9] Griffiths P., Harris J., Principles of Algebraic Geometry, Pure Appl. Math., John Wiley & Sons, New York, 1978 Zbl0408.14001
  10. [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 Zbl1235.32020
  11. [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 Zbl1210.53079
  12. [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. [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 Zbl1015.17023
  14. [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 Zbl1206.14027
  15. [15] Laurent-Gengoux C., Stiénon M., Xu P., Holomorphic Poisson manifolds and holomorphic Lie algebroids, Int. Math. Res. Not. IMRN, 2008, #088 Zbl1188.53098
  16. [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 Zbl1078.58011
  17. [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 Zbl1023.53060
  18. [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. [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 Zbl0914.14003
  20. [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 Zbl0099.02301

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