Embedding theorems for quasi-toric manifolds given by combinatorial data

V. M. Buchstaber ; A. A. Kustarev

Izvestiya. Mathematics , Tome 79 (2015) no. 6, pp. 1157-1183.

Résumé

This paper is devoted to problems on equivariant embeddings of quasi-toric manifolds in Euclidean and projective spaces. We construct explicit embeddings and give bounds for the dimensions of the embeddings in terms of combinatorial data that determine such manifolds. We show how familiar results on complex projective varieties in toric geometry can be obtained under additional restrictions on the combinatorial data.

Détail

Keywords: equivariant embedding, moment-angle manifold, characteristic function.

Bibliographie

[1] G. D. Mostow, “Equivariant embeddings in Euclidean space”, Ann. of Math. (2), 65:3 (1957), 432–446 | DOI | MR | Zbl

[2] R. S. Palais, “Imbedding of compact, differentiable transformation groups in orthogonal representations”, J. Math. Mech., 6 (1957), 673–678 | MR | Zbl

[3] G. Bredon, Vvedenie v teoriyu kompaktnykh grupp preobrazovanii, Nauka, M., 1980, 440 pp. ; G. E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, 46, Academic Press, New York–London, 1972, xiii+459 pp. | MR | Zbl | MR | Zbl

[4] K. Kodaira, “On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties)”, Ann. of Math. (2), 60:1 (1954), 28–48 | DOI | MR | Zbl

[5] M. L. Gromov, “A topological technique for the construction of solutions of differential equations and inequalities”, Actes du Congrès International des Mathématiciens (Nice, 1970), v. 2, Gauthier-Villars, Paris, 1971, 221–225 | MR | Zbl

[6] D. Tischler, “Closed 2-forms and an embedding theorem for symplectic manifolds”, J. Differential Geometry, 12:2 (1977), 229–235 | MR | Zbl

[7] V. M. Buchstaber, T. E. Panov, Torus actions and their applications in topology and combinatorics, Univ. Lecture Ser., 24, Amer. Math. Soc., Providence, RI, 2002, viii+144 pp. | MR | Zbl

[8] V. M. Bukhshtaber, T. E. Panov, Toricheskie deistviya v topologii i kombinatorike, MTsNMO, M., 2004, 272 pp. | MR | Zbl

[9] V. Buchstaber, T. Panov, Toric topology, Math. Surveys Monogr., 204, Amer. Math. Soc., Providence, RI, 2015, xiv+518 pp. ; arXiv: 1210.2368 | MR | Zbl

[10] V. I. Danilov, “Geometriya toricheskikh mnogoobrazii”, UMN, 33:2(200) (1978), 85–134 | MR | Zbl

[11] G. Ewald, Combinatorial convexity and algebraic geometry, Grad. Texts in Math., 168, Springer-Verlag, New York, 1996, xiv+372 pp. | DOI | MR | Zbl

[12] T. Oda, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, Ergeb. Math. Grenzgeb. (3), 15, Springer-Verlag, Berlin, 1988, viii+212 pp. | MR | Zbl

[13] V. M. Buchstaber, S. Terzic, “$(2n, k)$-manifolds and applications”, Report No. 27/2014, Mathematisches Forschungsinstitut Oberwolfach, 2014, 11–14

[14] V. M. Buchstaber, S. Terzić, Towards the theory of $(2n,k)$-manifolds, 2014, arXiv: 1410.2.2482

[15] V. M. Buchstaber, T. E. Panov, N. Ray, “Spaces of polytopes and cobordism of quasitoric manifolds”, Moscow Math. J., 7:2 (2007), 219–242 | MR | Zbl

[16] M. W. Davis, T. Januszkiewicz, “Convex polytopes, Coxeter orbifolds and torus actions”, Duke Math. J., 62:2 (1991), 417–451 | DOI | MR | Zbl

[17] V. M. Bukhshtaber, T. E. Panov, “Deistviya torov, kombinatornaya topologiya i gomologicheskaya algebra”, UMN, 55:5(335) (2000), 3–106 | DOI | MR | Zbl

[18] V. M. Buchstaber, N. Ray, “An invitation to toric topology: vertex four of a remarkable tetrahedron”, Toric topology, Contemp. Math., 460, Amer. Math. Soc., Providence, RI, 2008, 1–27 ; MIMS EPrint: 2008.31 http://eprints.ma.man.ac.uk/1055 | DOI | MR | Zbl

[19] M. W. Davis, “When are two Coxeter orbifolds diffeomorphic?”, Michigan Math. J., 63:2 (2014), 401–421 | DOI | MR | Zbl

[20] N. Yu. Erokhovets, “Teoriya invarianta Bukhshtabera simplitsialnykh kompleksov i vypuklykh mnogogrannikov”, Algebraicheskaya topologiya, vypuklye mnogogranniki i smezhnye voprosy, Sbornik statei. K 70-letiyu so dnya rozhdeniya chlena-korrespondenta RAN Viktora Matveevicha Bukhshtabera, Tr. MIAN, 286, MAIK, M., 2014, 144–206 | DOI | Zbl

[21] N. Robertson, D. Sanders, P. Seymour, R. Thomas, “The four colour theorem”, J. Combin. Theory Ser. B, 70:1 (1997), 2–44 | DOI | MR | Zbl

[22] M. F. Atiyah, “Convexity and commuting Hamiltonians”, Bull. London Math. Soc., 14:1 (1982), 1–15 | DOI | MR | Zbl

[23] Th. Delzant, “Hamiltoniens périodiques et images convexes de l'application moment”, Bull. Soc. Math. France, 116:3 (1988), 315–339 | MR | Zbl

[24] V. Guillemin, V. Ginzburg, Y. Karshon, Moment maps, cobordisms, and Hamiltonian group actions, Math. Surveys Monogr., 98, Amer. Math. Soc., Providence, RI, 2002, viii+350 pp. | DOI | MR | Zbl

[25] A. Wasserman, “Equivariant differential topology”, Topology, 8:2 (1969), 127–150 | DOI | MR | Zbl

[26] W. Fulton, Introduction to toric varieties, The W. H. Roever lectures in geometry, Ann. of Math. Stud., 131, Princeton Univ. Press, Princeton, NJ, 1993, xii+157 pp. | MR | Zbl

[27] Yu. Suyama, “Examples of toric manifolds which are not quasitoric manifolds”, Algebr. Geom. Topol. (to appear)

[28] V. M. Buchstaber, V. D. Volodin, “Combinatorial 2-truncated cubes and applications”, Associahedra, Tamari lattices and related structures, Progr. Math., 299, Birkhäuser/Springer, Basel, 2012, 161–186 | DOI | MR | Zbl