LVMB manifolds and simplicial spheres
Annales de l'Institut Fourier, Volume 62 (2012) no. 4, p. 1289-1317
LVM and LVMB manifolds are a large family of non kähler manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be seen as LVMB manifolds. The LVM manifolds have a natural action of a real torus and the quotient of this action is a polytope. This quotient allows us to relate closely LVM manifolds to the moment-angle manifolds studied by Buchstaber and Panov. Our aim is to generalize the polytope associated to a LVM manifold to the LVMB case and study the properties of this generalization. In particular, we show that the object we obtain belongs to a very large class of simplicial spheres. Moreover, we show that for every sphere belonging to this class, we can construct a LVMB manifold whose associated sphere is the given sphere. We use this latter result to show that many moment-angle complexes can be endowed with a complex structure (up to product with circles).
Les variétés LVM et LVMB constituent une grande famille de variétés complexes non kählériennes. Par exemple, les variétés de Hopf ou de Calabi-Eckmann peuvent être vues comme des cas particuliers de variétés LVMB. Les variétés LVM sont munies d’une action naturelle du tore compact et le quotient de cette action est un polytope simple. Ce quotient permet de nouer des liens profonds entre variétés LVM et les complexes moment-angle (étudiés par Buchstaber et Panov). Notre but est de généraliser le polytope associé à une variété LVM au cas des variétés LVMB et d’étudier les propriétés de cette généralisation. En particulier, nous montrons que l’objet obtenu appartient à une grande classe de sphères simpliciales. De plus, pour toute sphère appartenant à cette classe, on peut construire une variété LVMB ayant cette sphère pour complexe associé. On utilise ce dernier résultat pour munir une grande famille de complexe moment-angle d’une structure complexe.
DOI : https://doi.org/10.5802/aif.2723
Classification:  05E45,  32Q99,  32M05,  55U10
Keywords: non Kähler compact complex manifolds, simplicial spheres, toric varieties, complex structure on some moment-angle complexes
@article{AIF_2012__62_4_1289_0,
     author = {Tambour, J\'er\^ome},
     title = {LVMB manifolds and simplicial spheres},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {4},
     year = {2012},
     pages = {1289-1317},
     doi = {10.5802/aif.2723},
     mrnumber = {3025744},
     zbl = {1253.05151},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2012__62_4_1289_0}
}
Tambour, Jérôme. LVMB manifolds and simplicial spheres. Annales de l'Institut Fourier, Volume 62 (2012) no. 4, pp. 1289-1317. doi : 10.5802/aif.2723. http://www.numdam.org/item/AIF_2012__62_4_1289_0/

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