LVMB manifolds and simplicial spheres

Tambour, Jérôme

Annales de l'Institut Fourier, Tome 62 (2012) no. 4, pp. 1289-1317.

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.

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).

Classification : 05E45, 32Q99, 32M05, 55U10
Keywords: non Kähler compact complex manifolds, simplicial spheres, toric varieties, complex structure on some moment-angle complexes
Mot clés : Variétés complexes compactes non kählériennes, sphères simpliciales, variétés toriques, structure complexe des complexes moment-angle.

[1] Białynicki-Birula, A.; Carrell, J. B.; McGovern, W. M. Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action, Encyclopaedia of Mathematical Sciences, 131, 2002 (Invariant Theory and Algebraic Transformation Groups, II) | Zbl

[2] Białynicki-Birula, A.; Święcicka, J. Open subsets of projective spaces with a good quotient by an action of a reductive group, Transform. Groups, Volume 1 (1996) no. 3, pp. 153-185 | DOI | MR | Zbl

[3] Bosio, F. Variétés complexes compactes: une généralisation de la construction de Meersseman et López de Medrano-Verjovsky, Ann. Inst. Fourier (Grenoble), Volume 51 (2001) no. 5, pp. 1259-1297 | DOI | Numdam | MR | Zbl

[4] Bosio, F.; Meersseman, L. Real quadrics in ${ℂ}^n$, complex manifolds and convex polytopes, Acta Math., Volume 197 (2006) no. 1, pp. 53-127 | DOI | MR | Zbl

[5] Bredon, G.E. Topology and geometry, Graduate Texts in Mathematics, 139, Springer-Verlag, New York, 1997 (Corrected third printing of the 1993 original) | MR | Zbl

[6] Buchstaber, V.M.; Panov, T.E. Torus actions and their applications in topology and combinatorics, University Lecture Series, 24, American Mathematical Society, Providence, RI, 2002 | MR | Zbl

[7] Calabi, E.; Eckmann, B. A class of compact, complex manifolds which are not algebraic, Ann. of Math. (2), Volume 58 (1953), pp. 494-500 | DOI | MR | Zbl

[8] Cox, D.; Little, J.; Schenk, H. Toric Varieties, available on Cox’s website, 2009

[9] Cupit-Foutou, S.; Zaffran, D. Non-Kähler manifolds and GIT-quotients, Math. Z., Volume 257 (2007) no. 4, pp. 783-797 | DOI | MR | Zbl

[10] Ewald, G Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics, 168, Springer-Verlag, 1996 | MR | Zbl

[11] Hamm, H. A Very good quotients of toric varieties, Real and complex singularities (São Carlos, 1998) (Chapman & Hall/CRC Res. Notes Math.), Volume 412, 2000, pp. 61-75 | MR | Zbl

[12] Hopf, H. Zur Topologie der komplexen Mannigfaltigkeiten, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, 1948, pp. 167-185 | MR | Zbl

[13] Huybrechts, D. Complex geometry, Universitext, Springer-Verlag, 2005 | MR | Zbl

[14] Lee, D.H. The structure of complex Lie groups, Chapman & Hall/CRC Research Notes in Mathematics, 429, Chapman & Hall/CRC, Boca Raton, FL, 2002 | MR | Zbl

[15] López de Medrano, S. Topology of the intersection of quadrics in ${\mathbf{R}}^n$, Algebraic topology (Arcata, CA, 1986) (Lecture Notes in Math.), Volume 1370, Springer, Berlin, 1989, pp. 280-292 | MR | Zbl

[16] López de Medrano, S.; Verjovsky, A. A new family of complex, compact, non-symplectic manifolds, Bol. Soc. Brasil. Mat. (N.S.), Volume 28 (1997) no. 2, pp. 253-269 | DOI | MR | Zbl

[17] Meersseman, L. A new geometric construction of compact complex manifolds in any dimension, Math. Ann., Volume 317 (2000) no. 1, pp. 79-115 | DOI | MR | Zbl

[18] Meersseman, L.; Verjovsky, A. Holomorphic principal bundles over projective toric varieties, J. Reine Angew. Math., Volume 572 (2004), pp. 57-96 | MR | Zbl

[19] Mihalisin, J.; Williams, G. Nonconvex embeddings of the exceptional simplicial 3-spheres with 8 vertices, J. Combin. Theory Ser. A, Volume 98 (2002) no. 1, pp. 74-86 | DOI | MR | Zbl

[20] Orlik, P. Seifert manifolds, Chapman & Hall/CRC Research Notes in Mathematics, 429, Chapman & Hall/CRC, Boca Raton, FL, 2002

[21] Panov, T.; Ustinovsky, Y. Complex-analytic structures on moment-angle manifolds (http://arxiv.org/abs/1008.4764v1)