Positive Sasakian Structures on 5-Manifolds

  • János Kollár
Part of the Progress in Mathematics book series (PM, volume 271)


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.


Modulus Space Fundamental Group Einstein Metrics Pezzo Surface Quotient Singularity 
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  1. 1.
    D. Barden, Simply connected five-manifolds, Ann. Math. (2) 82 (1965), pp. 365–385. MR 0184241 (32 #1714).CrossRefMathSciNetGoogle Scholar
  2. 2.
    C.P. Boyer & K. Galicki, On Sasakian-Einstein geometry, Int. J. Math. 11 (2000), 7, pp. 873– 909. MR 2001k:53081.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    C.P. Boyer, K. Galicki & J. Koll á r, Einstein metrics on spheres, Ann. Math. 162 (2005), pp. 1–24.CrossRefGoogle Scholar
  4. 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.MATHGoogle Scholar
  5. 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.MATHMathSciNetGoogle Scholar
  6. 6.
    S. Kobayashi, Topology of positively pinched Kähler manifolds, Tôhoku Math. J. (2) 15 (1963), pp. 121–139. MR 27 #4185.MATHCrossRefGoogle Scholar
  7. 7.
    J. Koll á r, Einstein metrics on five-dimensional Seifert bundles, J. Geom. Anal. 15 (2005), 3, pp. 445–476. MR 2190241.MathSciNetGoogle Scholar
  8. 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. 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.CrossRefGoogle Scholar
  10. 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. 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. 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.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    —— Gorenstein log del Pezzo surfaces. II, J. Algebra, 156 (1993), 1, pp. 183–193. MR 94m:14045.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    S. Smale, On the structure of 5-manifolds, Ann. Math. (2) 75 (1962), pp. 38–46. MR 25 #4544.CrossRefMathSciNetGoogle Scholar
  15. 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. 16.
    C. Xu, Notes on π1 of Smooth Loci of Log Del Pezzo Surfaces, arXiv:math.AG/0706.3957, 2007.Google Scholar
  17. 17.
    Q. Ye, On Gorenstein log del Pezzo surfaces, Japan. J. Math. (N.S.) 28 (2002), 1, pp. 87–136. MR 1933881 (2003h:14063).MATHMathSciNetGoogle Scholar

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

Authors and Affiliations

  • János Kollár
    • 1
  1. 1.Princeton UniversityPrincetonUSA

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