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Simplicial Toric Varieties as Leaf Spaces

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Special Metrics and Group Actions in Geometry

Part of the book series: Springer INdAM Series ((SINDAMS,volume 23))

Abstract

We present a summary of some results from our article (Battaglia and Zaffran, Int. Math. Res. Not. IMRN 2015 no. 22 (2015), 11785–11815) and other recent results on the so-called LVMB manifolds. We emphasize some features by taking a different point of view. We present a simple variant of the Delzant construction, in which the group that is used to perform the symplectic reduction can be chosen of arbitrarily high dimension, and is always connected.

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References

  1. M. Atiyah, Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 14, 1–15 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Audin, The Topology of Torus Actions on Symplectic Manifolds. Progress in Mathematics, vol. 93 (Birkhäuser, Basel, 1991)

    Google Scholar 

  3. F. Battaglia, Betti numbers of the geometric spaces associated to nonrational simple convex polytopes. Proc. Am. Math. Soc. 139, 2309–2315 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Battaglia, E. Prato, Generalized toric varieties for simple nonrational convex polytopes. Intern. Math. Res. Not. 24, 1315–1337 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. F. Battaglia, E. Prato, The symplectic geometry of Penrose rhombus tilings. J. Symplectic Geom. 6, 139–158 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. F. Battaglia, E. Prato, The symplectic Penrose kite. Commun. Math. Phys. 299, 577–601 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. F. Battaglia, E. Prato, Nonrational symplectic toric cuts. Preprint (2016). arXiv:1606.00610 [math.SG]

    Google Scholar 

  8. F. Battaglia, E. Prato, Nonrational symplectic toric reduction, in preparation.

    Google Scholar 

  9. F. Battaglia, D. Zaffran, Foliations modeling nonrational simplicial toric varieties. Int. Math. Res. Not. IMRN 2015(22), 11785–11815 (2015)

    MathSciNet  MATH  Google Scholar 

  10. F. Battaglia, D. Zaffran, LVMB-manifolds as equivariant group compatifications, in preparation

    Google Scholar 

  11. F. Battaglia, L. Godinho, A. Mandini, Contact-symplectic toric spaces, in preparation

    Google Scholar 

  12. L. Battisti, LVMB manifolds and quotients of toric varieties. Math. Z. 275(1–2), 549–568 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Bosio, Variétés complexes compactes: une généralisation de la construction de Meersseman et de López de Medrano-Verjovsky. Ann. Inst. Fourier 51(5), 1259–1297 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Cox, The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4(1), 17–50 (1995)

    MathSciNet  MATH  Google Scholar 

  15. D.A. Cox, J.B. Little, H.K. Schenck, Toric Varieties, Graduate Studies in Mathematics, vol. 124 (American Mathematical Society, Providence, 2011)

    Google Scholar 

  16. S. Cupit, D. Zaffran, Non-Kähler manifolds and GIT-quotients. Math. Z. 257(4), 783–797 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. J.A. De Loera, J. Rambau, F. Santos, Triangulations: Structures for Algorithms and Applications. Algorithms and Computation in Mathematics, vol. 25 (Springer, Berlin, 2010), 539 pp.

    Google Scholar 

  18. T. Delzant, Hamiltoniens periodiques et images convexes de l’application moment. Bull. Soc. Math. France 116, 315–339 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  19. A. El Kacimi Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications. Compos. Math. 73, 57–106 (1990)

    MATH  Google Scholar 

  20. W. Fulton, Introduction to Toric Varieties (Princeton University Press, Princeton, 1993)

    MATH  Google Scholar 

  21. V. Guillemin, S. Sternberg, Convexity properties of the moment mapping. Invent. Math. 67, 491–513 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  22. V. Guillemin, S. Sternberg, Birational equivalence in the symplectic category. Invent. Math. 97, 485–522 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Ishida, Torus invariant transverse Kähler foliations. Trans. Am. Math. Soc. 369, 5137–515 (2017)

    Article  MATH  Google Scholar 

  24. L. Katzarkov, E. Lupercio, L. Meersseman, A. Verjovsky, The definition of a non-commutative toric variety. Contemp. Math. 620, 223–250 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. G. Kempf, L. Ness, The length of vectors in representation spaces, in Algebraic Geometry, Summer Meeting, Copenhagen, August 7–12, 1978. Lecture Notes in Mathematics, vol. 732 (1979), pp. 233–243

    Google Scholar 

  26. F. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry. Mathematical Notes, vol. 31 (Princeton University Press, Princeton, 1984)

    Google Scholar 

  27. E. Lerman, S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties. Trans. Am. Math. Soc. 349(10), 4201–4230 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Loeb, M. Nicolau, On the complex geometry of a class of non Kählerian manifolds. Isr. J. Math. 110, 371–379 (1999)

    Article  MATH  Google Scholar 

  29. S. López de Medrano, A. Verjovsky, A new family of complex, compact, non symplectic manifolds. Bull. Braz. Math. Soc. 28, 253–269 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. L. Meersseman, A new geometric construction of compact complex manifolds in any dimension. Math. Ann. 317, 79–115 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. L. Meerssemann, A. Verjovsky, Holomorphic principal bundles over projective toric varieties. J. Reine Angew. Math. 572, 57–96 (2004)

    MathSciNet  MATH  Google Scholar 

  32. T.Z. Nguyen, T. Ratiu, Presymplectic convexity and (ir)rational polytopes. Preprint arXiv:1705.11110 [math.SG]

    Google Scholar 

  33. T. Oda, Convex Bodies and Algebraic Geometry (Springer, Berlin, 1988)

    MATH  Google Scholar 

  34. T. Panov, Y. Ustinovsky, Complex-analytic structures on moment-angle manifolds. Mosc. Math. J. 12(1), 149–172 (2012)

    MathSciNet  MATH  Google Scholar 

  35. T. Panov, Y. Ustinovski, M. Verbitsky, Complex geometry of moment-angle manifolds. Math. Z. 284(1), 309–333 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. E. Prato, Simple non-rational convex polytopes via symplectic geometry. Topology 40, 961–975 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. J. Tambour, LVMB manifolds and simplicial spheres. Ann. Inst. Fourier 62(4), 1289–1317 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Y. Ustinovsky, Geometry of compact complex manifolds with maximal torus action. Proc. Steklov Inst. Math. 286(1), 198–208 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Leonor Godinho for pointing out the relevance of [22].

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica U. Dini, Università di Firenze, Via S. Marta 3, 50139, Firenze, Italy

    Fiammetta Battaglia

  2. College of Marin, 835 College Ave, Kentfield, CA, 94904, USA

    Dan Zaffran

Authors
  1. Fiammetta Battaglia
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  2. Dan Zaffran
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Corresponding author

Correspondence to Fiammetta Battaglia .

Editor information

Editors and Affiliations

  1. GMA - IME, Universidade Federal Fluminense GMA - IME, Niteroi, Rio de Janeiro, Brazil

    Simon G. Chiossi

  2. Dipartimento di Mathematica "G. Peano", Università di Torino, Torino, Italy

    Anna Fino

  3. Dipartimento di Matematica e Informatica, Università di Firenze, Firenze, Firenze, Italy

    Emilio Musso

  4. Dipartimento di Scienze Matematiche, Politecnico di Torino, Torino, Italy

    Fabio Podestà

  5. Dipart. di Matematica, Giuseppe Peano, Università di Torino, Torino, Italy

    Luigi Vezzoni

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Battaglia, F., Zaffran, D. (2017). Simplicial Toric Varieties as Leaf Spaces. In: Chiossi, S., Fino, A., Musso, E., Podestà, F., Vezzoni, L. (eds) Special Metrics and Group Actions in Geometry. Springer INdAM Series, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-319-67519-0_1

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