Skip to main content

Part of the book series: Progress in Mathematics ((PM,volume 271))

Abstract

The aim of this paper is to study 5-manifolds that carry a positive Sasakian structure. Strong restrictions are derived for the integral hemology groups. In some cases, all positive sasakian structures are classified. A key step is the study of log Del Pezzo surfaces whose boundary divisor contains positive genus curves.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Barden, Simply connected five-manifolds, Ann. Math. (2) 82 (1965), pp. 365–385. MR 0184241 (32 #1714).

    Article  MathSciNet  Google Scholar 

  2. C.P. Boyer & K. Galicki, On Sasakian-Einstein geometry, Int. J. Math. 11 (2000), 7, pp. 873– 909. MR 2001k:53081.

    Article  MATH  MathSciNet  Google Scholar 

  3. C.P. Boyer, K. Galicki & J. Koll á r, Einstein metrics on spheres, Ann. Math. 162 (2005), pp. 1–24.

    Article  Google Scholar 

  4. J-P. Demailly & J. Koll á r, Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. (4) 34 (2001), 4, pp. 525–556. MR 2002e:32032.

    MATH  Google Scholar 

  5. M. Furushima, Singular del Pezzo surfaces and analytic compactifications of 3-dimensional complex affine space C3, Nagoya Math. J. 104 (1986), pp. 1–28. MR 88m:32051.

    MATH  MathSciNet  Google Scholar 

  6. S. Kobayashi, Topology of positively pinched Kähler manifolds, Tôhoku Math. J. (2) 15 (1963), pp. 121–139. MR 27 #4185.

    Article  MATH  Google Scholar 

  7. J. Koll á r, Einstein metrics on five-dimensional Seifert bundles, J. Geom. Anal. 15 (2005), 3, pp. 445–476. MR 2190241.

    MathSciNet  Google Scholar 

  8. ——, Flips and abundance for algebraic threefolds, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque 211 (1992), Société Mathématique de France, Paris. MR 94f:14013.

    Google Scholar 

  9. J. Koll á r & S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. MR 2000b:14018.

    Book  Google Scholar 

  10. J. Koll á r, K.E. Smith & A. Corti, Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics, vol. 92, Cambridge University Press, Cambridge, 2004. MR 2062787 (2005i:14063).

    Google Scholar 

  11. Y.I. Manin, Cubic forms, North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986. MR 833513 (87d:11037).

    Google Scholar 

  12. M. Miyanishi & D.-Q Zhang, Gorenstein log del Pezzo surfaces of rank one, J. Algebra, 118 (1988), 1, pp. 63–84. MR 89i:14033.

    Article  MATH  MathSciNet  Google Scholar 

  13. —— Gorenstein log del Pezzo surfaces. II, J. Algebra, 156 (1993), 1, pp. 183–193. MR 94m:14045.

    Article  MATH  MathSciNet  Google Scholar 

  14. S. Smale, On the structure of 5-manifolds, Ann. Math. (2) 75 (1962), pp. 38–46. MR 25 #4544.

    Article  MathSciNet  Google Scholar 

  15. A.N. Varčenko, Newton polyhedra and estimates of oscillatory integrals, Funkcional. Anal. i Priložen. 10 (1976), 3, pp. 13–38. MR 0422257 (54 #10248).

    Google Scholar 

  16. C. Xu, Notes on π1 of Smooth Loci of Log Del Pezzo Surfaces, arXiv:math.AG/0706.3957, 2007.

    Google Scholar 

  17. Q. Ye, On Gorenstein log del Pezzo surfaces, Japan. J. Math. (N.S.) 28 (2002), 1, pp. 87–136. MR 1933881 (2003h:14063).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Birkhäuser Boston, a part of Springer Science+Business Media LLC

About this chapter

Cite this chapter

Kollár, J. (2009). Positive Sasakian Structures on 5-Manifolds. In: Galicki, K., Simanca, S.R. (eds) Riemannian Topology and Geometric Structures on Manifolds. Progress in Mathematics, vol 271. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4743-8_5

Download citation

Publish with us

Policies and ethics