Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


A class of torus manifolds with nonconvex orbit space

Authors: Mainak Poddar and Soumen Sarkar
Journal: Proc. Amer. Math. Soc. 143 (2015), 1797-1811
MSC (2010): Primary 57R17, 57R91
Published electronically: November 24, 2014
MathSciNet review: 3314091
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study a class of smooth torus manifolds whose orbit space has the combinatorial structure of a simple polytope with holes. We construct moment angle manifolds for such polytopes with holes and use them to prove that the associated torus manifolds admit stable almost complex structure. We give a combinatorial formula for the Hirzebruch $ \chi _y$ genus of these torus manifolds. We show that they have (invariant) almost complex structure if they admit positive omniorientation. We give examples of almost complex manifolds that do not admit a complex structure. When the dimension is four, we calculate the homology groups and describe a method for computing the cohomology ring.

References [Enhancements On Off] (What's this?)

  • [At57] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181-207. MR 0086359 (19,172c)
  • [BP02] Victor M. Buchstaber and Taras E. Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series, vol. 24, American Mathematical Society, Providence, RI, 2002. MR 1897064 (2003e:57039)
  • [BPV84] W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574 (86c:32026)
  • [BR01] Victor M. Buchstaber and Nigel Ray, Tangential structures on toric manifolds, and connected sums of polytopes, Internat. Math. Res. Notices 4 (2001), 193-219. MR 1813798 (2002b:57043),
  • [Dav78] Michael Davis, Smooth $ G$-manifolds as collections of fiber bundles, Pacific J. Math. 77 (1978), no. 2, 315-363. MR 510928 (80b:57034)
  • [DJ91] Michael W. Davis and Tadeusz Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417-451. MR 1104531 (92i:52012),
  • [GH94] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523 (95d:14001)
  • [GK98] Michael D. Grossberg and Yael Karshon, Equivariant index and the moment map for completely integrable torus actions, Adv. Math. 133 (1998), no. 2, 185-223. MR 1604738 (2000f:53112),
  • [Gom95] Robert E. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2) 142 (1995), no. 3, 527-595. MR 1356781 (96j:57025),
  • [Gro86] Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505 (90a:58201)
  • [HM03] Akio Hattori and Mikiya Masuda, Theory of multi-fans, Osaka J. Math. 40 (2003), no. 1, 1-68. MR 1955796 (2004d:53103)
  • [IK12] Hiroaki Ishida and Yael Karshon, Completely integrable torus actions on complex manifolds with fixed points, Math. Res. Lett. 19 (2012), no. 6, 1283-1295. MR 3091608,
  • [Kus09] A. A. Kustarev, Equivariant almost complex structures on quasitoric manifolds, Tr. Mat. Inst. Steklova 266 (2009), no. Geometriya, Topologiya i Matematicheskaya Fizika. II, 140-148 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 266 (2009), no. 1, 133-141. MR 2603265 (2011b:57031),
  • [McD88] Dusa McDuff, The moment map for circle actions on symplectic manifolds, J. Geom. Phys. 5 (1988), no. 2, 149-160. MR 1029424 (91c:58042),
  • [Mas99] Mikiya Masuda, Unitary toric manifolds, multi-fans and equivariant index, Tohoku Math. J. (2) 51 (1999), no. 2, 237-265. MR 1689995 (2000e:57058),
  • [MP06] Mikiya Masuda and Taras Panov, On the cohomology of torus manifolds, Osaka J. Math. 43 (2006), no. 3, 711-746. MR 2283418 (2007j:57039)
  • [Pan01] T. E. Panov, Hirzebruch genera of manifolds with torus action, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001), no. 3, 123-138 (Russian, with Russian summary); English transl., Izv. Math. 65 (2001), no. 3, 543-556. MR 1853368 (2002i:57047),
  • [Sil01] Ana Cannas da Silva, Symplectic toric manifolds, Symplectic geometry of integrable Hamiltonian systems (Barcelona, 2001), Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2003, pp. 85-173. MR 2000746
  • [Tho67] Emery Thomas, Complex structures on real vector bundles, Amer. J. Math. 89 (1967), 887-908. MR 0220310 (36 #3375)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 57R17, 57R91

Retrieve articles in all journals with MSC (2010): 57R17, 57R91

Additional Information

Mainak Poddar
Affiliation: Departamento de Matemáticas, Universidad de los Andes, Bogota, Colombia

Soumen Sarkar
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea

Keywords: Almost complex, symplectic, Hirzebruch genus, moment angle complex, torus action
Received by editor(s): September 28, 2011
Received by editor(s) in revised form: July 11, 2012, and July 18, 2013
Published electronically: November 24, 2014
Additional Notes: The first author was partially supported by the Proyecto de investigaciones grant from the Universidad de los Andes
The second author was partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (No. 2012-0000795)
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society