Mixed Hodge structures and Sullivan’s minimal models of Sasakian manifolds
Annales de l'Institut Fourier, Volume 67 (2017) no. 6, p. 2533-2546
We show that the Malčev Lie algebra of the fundamental group of a compact 2n+1-dimensional Sasakian manifold with n2 admits a quadratic presentation by using Morgan’s bigradings of minimal models of mixed-Hodge diagrams. By using bigradings of minimal models, we also simplify the proof of the result of Cappelletti–Montano, De Nicola, Marrero and Yudin on Sasakian nilmanifolds.
Nous montrons, en utilisant les bigraduations de Morgan de modèles minimaux de diagrammes de Hodge, que l’algèbre de Lie de Malčev du groupe fondamental d’une variété sasakienne compacte de dimension 2n+1 admet une présentation quadratique pour n2. A l’aide de bigraduations de modèles minimaux, nous simplifions également la démonstration du résultat de Cappelletti–Montano, De Nicola, Marrero et Yudin sur les nilvariétés sasakiennes.
Received : 2015-11-23
Revised : 2016-12-09
Accepted : 2017-03-15
Published online : 2017-12-14
DOI : https://doi.org/10.5802/aif.3142
Classification:  53C25,  55P62,  58A14
Keywords: Sasakian structure, Sullivan’s minimal model, Morgan’s mixed Hodge diagram, formality
@article{AIF_2017__67_6_2533_0,
     author = {Kasuya, Hisashi},
     title = {Mixed Hodge structures and Sullivan's minimal models of Sasakian manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {6},
     year = {2017},
     pages = {2533-2546},
     doi = {10.5802/aif.3142},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_6_2533_0}
}
Kasuya, Hisashi. Mixed Hodge structures and Sullivan’s minimal models of Sasakian manifolds. Annales de l'Institut Fourier, Volume 67 (2017) no. 6, pp. 2533-2546. doi : 10.5802/aif.3142. https://aif.centre-mersenne.org/item/AIF_2017__67_6_2533_0/

[1] Amorós, Jaume; Burger, Marc; Corlette, A.; Kotschick, Dieter; Toledo, Domingo Fundamental groups of compact Kähler manifolds, American Mathematical Society, Mathematical Surveys and Monographs, Tome 44 (1996), xi+140 pages | Zbl 0849.32006

[2] Biswas, Indranil; Fernández, Marisa; Muñoz, Vicente; Tralle, Aleksy Formality of Sasakian manifoldss, J. Topol., Tome 9 (2016) no. 1, pp. 161-180 | Article | Zbl 1339.55014

[3] Bourgeois, Frédéric Odd dimensional tori are contact manifolds, Int. Math. Res. Not., Tome 2002 (2002) no. 30, pp. 1571-1574 | Article | Zbl 1021.53055

[4] Bowden, Jonathan; Crowley, Diarmuid; Stipsicz, András I. Contact structures on M×S 2 , Math. Ann., Tome 358 (2014) no. 1-2, pp. 351-359 | Article | Zbl 1286.57022

[5] Boyer, Charles P.; Galicki, Krzysztof Sasakian geometry, Oxford University Press, Oxford Mathematical Monographs (2008), xi+613 pages | Zbl 1155.53002

[6] Cappelletti-Montano, Beniamino; De Nicola, Antonio; Marrero, Juan Carlos; Yudin, Ivan Sasakian nilmanifolds, Int. Math. Res. Not., Tome 2015 (2015) no. 15, pp. 6648-6660 | Article | Zbl 1330.53064

[7] Carlson, James A.; Toledo, Domingo Quadratic presentations and nilpotent Kähler groups, J. Geom. Anal., Tome 5 (1995) no. 3, pp. 351-359 (erratum in ibid. 7 (1997), no. 3, p. 511-514) | Article | Zbl 0835.57014

[8] Deligne, Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis Real homotopy theory of Kähler manifolds, Invent. Math., Tome 29 (1975) no. 3, pp. 245-274 | Article | Zbl 0312.55011

[9] Fernández, Marisa; Muñoz, Vicente Formality of Donaldson submanifolds, Math. Z., Tome 250 (2005) no. 1, pp. 149-175 (erratum in ibid. 257 (2007), no. 2, p. 465-466) | Article | Zbl 1071.57024

[10] Hain, Richard M. Nil-manifolds as links of isolated singularities, Compos. Math., Tome 84 (1992) no. 1, pp. 91-99 | Zbl 0767.57010

[11] Hasegawa, Keizo Minimal models of nilmanifolds, Proc. Am. Math. Soc., Tome 106 (1989) no. 1, pp. 65-71 | Article | Zbl 0691.53040

[12] Malčev, Anatoliĭ Ivanovich On a class of homogeneous spaces, Am. Math. Soc. Transl., Tome 39 (1951), pp. 1-33 | Zbl 0034.01701

[13] Morgan, John W. The algebraic topology of smooth algebraic varieties, Publ. Math., Inst. Hautes Étud. Sci., Tome 48 (1978), pp. 137-204 (erratum in ibid. 64 (1986), p. 185) | Article | Zbl 0401.14003

[14] Muñoz, Vicente; Tralle, Aleksy Simply connected K-contact and Sasakian manifolds of dimension 7, Math. Z., Tome 281 (2015) no. 1-2, pp. 457-470 | Article | Zbl 06481993

[15] Nomizu, Katsumi On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. Math., Tome 59 (1954), pp. 531-538 | Article | Zbl 0058.02202

[16] Sullivan, Dennis Infinitesimal computations in topology, Publ. Math., Inst. Hautes Étud. Sci., Tome 47 (1977), pp. 269-331 | Article | Zbl 0374.57002

[17] Tievsky, Aaron M Analogues of Kähler geometry on Sasakian manifolds, Massachusetts Institute of Technology (USA) (1954) (Ph. D. Thesis)