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Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 3, Pages 123–138 (Mi izv338)  

This article is cited in 12 scientific papers (total in 12 papers)

Hirzebruch genera of manifolds with torus action

T. E. Panov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: A quasitoric manifold is a smooth $2n$-manifold $M^{2n}$ with an action of the compact torus $T^n$ such that the action is locally isomorphic to the standard action of $T^n$ on $\mathbb C^n$ and the orbit space is diffeomorphic, as a manifold with corners, to a simple polytope $P^n$. The name refers to the fact that topological and combinatorial properties of quasitoric manifolds are similar to those of non-singular algebraic toric varieties (or toric manifolds). Unlike toric varieties, quasitoric manifolds may fail to be complex. However, they always admit a stably (or weakly almost) complex structure, and their cobordism classes generate the complex cobordism ring. Buchstaber and Ray have recently shown that the stably complex structure on a quasitoric manifold is determined in purely combinatorial terms, namely, by an orientation of the polytope and a function from the set of codimension-one faces of the polytope to primitive vectors of the integer lattice. We calculate the $\chi_y$-genus of a quasitoric manifold with a fixed stably complex structure in terms of the corresponding combinatorial data. In particular, this gives explicit formulae for the classical Todd genus and the signature. We also compare our results with well-known facts in the theory of toric varieties.

DOI: https://doi.org/10.4213/im338

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English version:
Izvestiya: Mathematics, 2001, 65:3, 543–556

Bibliographic databases:

MSC: Primary 57R20, 57S25; Secondary 14M25, 58G10
Received: 25.02.2000

Citation: T. E. Panov, “Hirzebruch genera of manifolds with torus action”, Izv. RAN. Ser. Mat., 65:3 (2001), 123–138; Izv. Math., 65:3 (2001), 543–556

Citation in format AMSBIB
\by T.~E.~Panov
\paper Hirzebruch genera of manifolds with torus action
\jour Izv. RAN. Ser. Mat.
\yr 2001
\vol 65
\issue 3
\pages 123--138
\jour Izv. Math.
\yr 2001
\vol 65
\issue 3
\pages 543--556

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    This publication is cited in the following articles:
    1. V. M. Buchstaber, T. E. Panov, N. Ray, “Spaces of polytopes and cobordism of quasitoric manifolds”, Mosc. Math. J., 7:2 (2007), 219–242  mathnet  crossref  mathscinet  zmath
    2. Maeda H., Masuda M., Panov T., “Torus graphs and simplicial posets”, Advances in Mathematics, 212:2 (2007), 458–483  crossref  mathscinet  zmath  isi  elib  scopus
    3. M. Masuda, T. E. Panov, “Semifree circle actions, Bott towers and quasitoric manifolds”, Sb. Math., 199:8 (2008), 1201–1223  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    4. Feldman K.E., “Miraculous cancellation and Pick's theorem”, Toric Topology, Contemporary Mathematics Series, 460, 2008, 71–86  crossref  mathscinet  zmath  isi
    5. A. A. Kustarev, “Equivariant almost complex structures on quasi-toric manifolds”, Russian Math. Surveys, 64:1 (2009), 156–158  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. A. A. Kustarev, “Equivariant Almost Complex Structures on Quasitoric Manifolds”, Proc. Steklov Inst. Math., 266 (2009), 133–141  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    7. Buchstaber V., Panov T., Ray N., “Toric Genera”, International Mathematics Research Notices, 2010, no. 16, 3207–3262  mathscinet  zmath  isi  elib
    8. Poddar M., Sarkar S., “On Quasitoric Orbifolds”, Osaka J Math, 47:4 (2010), 1055–1076  mathscinet  zmath  isi  elib
    9. V. M. Buchstaber, “Complex cobordism and formal groups”, Russian Math. Surveys, 67:5 (2012), 891–950  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. Hiroaki Ishida, Yukiko Fukukawa, Mikiya Masuda, “Topological toric manifolds”, Mosc. Math. J., 13:1 (2013), 57–98  mathnet  crossref  mathscinet
    11. Buchstaber V.M. Terzic S., “Toric Genera of Homogeneous Spaces and their Fibrations”, Int. Math. Res. Notices, 2013, no. 6, 1324–1403  crossref  mathscinet  zmath  isi  elib  scopus
    12. Poddar M., Sarkar S., “a Class of Torus Manifolds With Nonconvex Orbit Space”, Proc. Amer. Math. Soc., 143:4 (2015), PII S0002-9939(2014)12075-2, 1797–1811  crossref  mathscinet  zmath  isi  elib  scopus
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