Einstein metrics on rational homology 7-spheres
Annales de l'Institut Fourier, Volume 52 (2002) no. 5, pp. 1569-1584.

In this paper we demonstrate the existence of Sasakian-Einstein structures on certain 2- connected rational homology 7-spheres. These appear to be the first non-regular examples of Sasakian-Einstein metrics on simply connected rational homology spheres. We also briefly describe the rational homology 7-spheres that admit regular positive Sasakian structures.

Dans cet article nous démontrons l'existence de structures d'Einstein sasakiennes sur certaines 7-sphères d'homologie rationnelle, 2-connexes. Elle apparaissent comme étant les premiers exemples non réguliers de métriques d'Einstein sasakiennes sur les sphères d'homologie rationnelle, simplement connexes. Nous décrivons aussi brièvement les 7- sphères d'homologie rationnelle qui admettent des structures sasakiennes positives régulières.

DOI: 10.5802/aif.1925
Classification: 53C25,  53C12
Keywords: Einstein metrics, sasakian structures, homology spheres
Boyer, Charles P. 1; Galicki, Krzysztof 1; Nakamaye, Michael 1

1 University of New Mexico, Department of Mathematics and Statistics, Albuquerque NM 87131 (USA)
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     title = {Einstein metrics on rational homology 7-spheres},
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     pages = {1569--1584},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Boyer, Charles P.; Galicki, Krzysztof; Nakamaye, Michael. Einstein metrics on rational homology 7-spheres. Annales de l'Institut Fourier, Volume 52 (2002) no. 5, pp. 1569-1584. doi : 10.5802/aif.1925. http://www.numdam.org/articles/10.5802/aif.1925/

[BE] R.J. Baston; M.G. Eastwood The Penrose Transform, Oxford University Press, New York, 1989 | MR | Zbl

[BFGK] H. Baum; T. Friedrich; R. Grunewald; I. Kath Twistors and Killing Spinors on Riemannian Manifolds, Teubner-Texte für Mathematik, vol. 124, Teubner, Stuttgart, Leipzig, 1991 | MR | Zbl

[BG1] C. P. Boyer; K. Galicki On Sasakian-Einstein Geometry, Int. J. Math, Volume 11 (2000), pp. 873-909 | DOI | MR | Zbl

[BG2] C. P. Boyer; K. Galicki 3-Sasakian manifolds. Surveys in differential geometry: essays on Einstein manifolds (Surv. Differ. Geom.) (1999), pp. 123-184 | Zbl

[BG3] C. P. Boyer; K. Galicki New Einstein Metrics in Dimension Five, J. Diff. Geom., Volume 57 (2001), pp. 443-463 | MR | Zbl

[BGN1] C. P. Boyer; K. Galicki; M. Nakamaye On the Geometry of Sasakian-Einstein 5-Manifolds (e-print. To appear in Math. Ann., math.DG/0012047) | MR | Zbl

[BGN2] C. P. Boyer; K. Galicki; and M. Nakamaye On Positive Sasakian Geometry (e-print. To appear in Geom. Ded., math.DG/0104126) | MR | Zbl

[BGN3] C. P. Boyer; K. Galicki; M. Nakamaye Sasakian-Einstein Structures on 9#(S 2 ×S 3 ), Trans. Amer. Math. Soc., Volume 354 (2002), pp. 2983-2996 | DOI | MR | Zbl

[BGN3] C. P. Boyer; K. Galicki; M. Nakamaye Sasakian-Einstein Structures on 9#(S 2 ×S 3 ) (e-print, math.DG/0102181)

[BGN4] C. P. Boyer; K. Galicki; M. Nakamaye Sasakian Geometry, Homotopy Spheres and Positive Ricci Curvature (e-print. To appear in Topology, math.DG/0201147) | MR | Zbl

[BGP] C. P. Boyer; K. Galicki; P. Piccinni 3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients, Ann. Global Anal. Geom, Volume 21 (2002), pp. 85-110 | DOI | MR | Zbl

[DK] J.-P. Demailly; J. Kollár Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Scient. Ec. Norm. Sup. Paris, Volume 34 (2001), pp. 525-556 | Numdam | MR | Zbl

[DK] J.-P. Demailly; J. Kollar Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds (e-print, AG/9910118)

[Dol] I. Dolgachev Weighted projective varieties, Proceedings, Group Actions and Vector Fields, Vancouver (LNM), Volume 956 (1981), pp. 34-71 | Zbl

[Fle] A.R. Fletcher; A. Corti and M. Reid, eds. Working with weighted complete intersections, revised version in Explicit birational geometry of 3-folds (Preprint MPI) (2000), pp. 101-173

[GS] K. Galicki; S. Salamon On Betti numbers of 3-Sasakian manifolds, Geom. Ded., Volume 63 (1996), pp. 45-68 | MR | Zbl

[HZ] F. Hirzebruch; D. Zagier The Atiyah-Singer Theorem and Elementary Number Theory, Publish or Perish, Inc., Berkeley, 1974 | MR | Zbl

[Isk] V. A. Iskovskikh Anticanonical Models of Three-dimensional Algebraic Varieties, J. Soviet Math., Volume 13 (1980), pp. 745-814 | DOI | Zbl

[IsPr] V. A. Iskovskikh; Yu.G. Prokhorov; A.N. Parshin and I.R. Shaferevich, Eds. Fano Varieties, Algebraic Geometry V (Enc. Math. Sci), Volume Vol 47 (1999) | Zbl

[JK1] J. M. Johnson; J. Kollár Kähler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-space, Ann. Inst. Fourier, Volume 51 (2001) no. 1, pp. 69-79 | DOI | Numdam | MR | Zbl

[JK2] J. M. Johnson; J. Kollár Fano hypersurfaces in weighted projective 4-spaces, Experimental Math, Volume 10(1) (2001), pp. 151-158 | MR | Zbl

[Mil] J. Milnor Singular Points of Complex Hypersurface, Ann. of Math. Stud, 61, Princeton Univ. Press, 1968 | MR | Zbl

[MO] J. Milnor; P. Orlik Isolated singularities defined by weighted homogeneous polynomials, Topology, Volume 9 (1970), pp. 385-393 | DOI | MR | Zbl

[Mo] S. Morita A Topological Classification of Complex Structures on S 1 ×Σ 2n-1 , Topology, Volume 14 (1975), pp. 13-22 | MR | Zbl

[MU] S. Mukai; H. Umemura; M. Raynaud and T. Shioda Eds Minimal Rational Threefolds, Algebraic Geometry (LNM), Volume 1016 (1983), pp. 490-518 | Zbl

[Sa] H. Sato Remarks Concerning Contact Manifolds, Tôhoku Math. J, Volume 29 (1977), pp. 577-584 | DOI | MR | Zbl

[TaYu] S. Tachibana; W.N. Yu On a Riemannian space admitting more than one Sasakian structure, Tôhoku Math. J, Volume 22 (1970), pp. 536-540 | DOI | MR | Zbl

[Ti1] G. Tian Kähler-Einstein metrics with positive scalar curvature, Invent. Math, Volume 137 (1997), pp. 1-37 | DOI | MR | Zbl

[Ti2] G. Tian Canonical Metrics in Kähler Geometry, Birkhäuser, Boston, 2000 | MR | Zbl

[Us1] I. Ustilovsky Infinitely Many Contact Structures on S 4m+1 , Int. Math. Res. Notices, Volume 14 (1999), pp. 781-791 | DOI | MR | Zbl

[Us2] I. Ustilovsky Contact Homology and Contact Structures on S 4m+1 (2000) (Ph.d. thesis, Stanford Univ)

[YK] K. Yano; M. Kon Structures on manifolds, Series in Pure Mathematics, 3, World Scientific Pub. Co., Singapore, 1984 | MR | Zbl

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