Λ-modules and holomorphic Lie algebroid connections

Pietro Tortella

Open Mathematics (2012)

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

Abstract

top
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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.