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

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

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

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\pages 123--138
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\jour Izv. Math.
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\vol 65
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• http://mi.mathnet.ru/eng/izv338
• https://doi.org/10.4213/im338
• http://mi.mathnet.ru/eng/izv/v65/i3/p123

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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