Abstract
For any regular Lie algebroid A, the kernel K and the image F of its anchor map ρA together with A itself fit into a short exact sequence, called the Atiyah sequence, of Lie algebroids. We prove that the Atiyah and Todd classes of dg manifolds arising from a regular Lie algebroid respect the Atiyah sequence, i.e., the Atiyah and Todd classes of A restrict to the Atiyah and Todd classes of the bundle K of Lie algebras on the one hand, and project onto the Atiyah and Todd classes of the integrable distribution F ⊆ TM on the other hand.
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References
Abad C A, Crainic M. Representations up to homotopy of Lie algebroids. J Reine Angew Math, 2012, 663: 91–126
Atiyah M F. Complex analytic connections in fibre bundles. Trans Amer Math Soc, 1957, 85: 181–207
Chen Z, Stiénon M, Xu P. From Atiyah classes to homotopy Leibniz algebras. Comm Math Phys, 2016, 341: 309–349
Chen Z, Xiang M S, Xu P. Atiyah and Todd classes arising from integrable distributions. J Geom Phys, 2019, 136: 52–67
Cueca M, Mehta R A. Courant cohomology, Cartan calculus, connections, curvature, characteristic classes. Comm Math Phys, 2021, 381: 1091–1113
Gracia-Saz A, Mehta R A. Lie algebroid structures on double vector bundles and representation theory of Lie algebroids. Adv Math, 2010, 223: 1236–1275
Hirzebruch F, Berger T, Jung R. Manifolds and Modular Forms. Aspects of Mathematics, vol. E20. Wiesbaden: Vieweg+Teubner Verlag, 1992
Kapranov M M. Rozansky-Witten invariants via Atiyah classes. Compos Math, 1999, 115: 71–113
Kontsevich M. Deformation quantization of Poisson manifolds. Lett Math Phys, 2003, 66: 157–216
Kotov A, Strobl T. Characteristic classes associated to Q-bundles. Int J Geom Methods Mod Phys, 2015, 12: 1550006
Kubarski J. Bott’s vanishing theorem for regular Lie algebroids. Trans Amer Math Soc, 1996, 348: 2151–2167
Liao H-Y, Stiénon M, Xu P. Formality theorem for differential graded manifolds. C R Math Acad Sci Paris, 2018, 356: 27–43
Lyakhovich S L, Mosman E A, Sharapov A A. Characteristic classes of Q-manifolds: Classification and applications. J Geom Phys, 2010, 60: 729–759
Manetti M. A relative version of the ordinary perturbation lemma. Rend Mat Appl (7), 2010, 30: 221–238
Manin Y I. Gauge Field Theory and Complex Geometry, 2nd ed. Grundlehren der mathematischen Wissenschaften, vol. 289. Berlin: Springer-Verlag, 1997
Mehta R A. Lie algebroid modules and representations up to homotopy. Indag Math (NS), 2014, 25: 1122–1134
Mehta R A, Stiénon M, Xu P. The Atiyah class of a dg-vector bundle. C R Math Acad Sci Paris, 2015, 353: 357–362
Molino P. Classe d’Atiyah d’un feuilletage et connexions transverses projetables. C R Acad Sci Paris Sér A, 1971, 272: A779–A781
Seol S, Stiénon M, Xu P. Dg manifolds, formal exponential maps and homotopy Lie algebras. Comm Math Phys, 2022, 391: 33–76
Stiénon M, Xu P. Atiyah classes and Kontsevich-Duflo type theorem for dg manifolds. In: Homotopy Algebras, Deformation Theory and Quantization. Banach Center Publications, vol. 123. Warsaw: Polish Acad Sci Inst Math, 2021, 63–110
Vaǐntrob A. Lie algebroids and homological vector fields. Russian Math Surveys, 1997, 52: 428–429
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11901221). The author thanks Zhuo Chen, Yu Qiao and Ping Xu for helpful discussions and comments. The author is also grateful to the referees for helpful suggestions and comments to improve the presentation of the manuscript.
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Xiang, M. Atiyah and Todd classes of regular Lie algebroids. Sci. China Math. 66, 1569–1592 (2023). https://doi.org/10.1007/s11425-021-2017-7
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DOI: https://doi.org/10.1007/s11425-021-2017-7