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Einstein metrics on rational homology 7-spheres. (English) Zbl 1023.53029

In former work, the authors studied Sasakian-Einstein manifolds and provided a lot of new examples of compact regular and non-regular 5-dimensional Sasakian-Einstein spaces. This work is based on and extends some work of Demailly-Kolář and Johnson-Kolář on the existence of Kähler-Einstein metrics on compact Fano orbifolds.
In the present paper they continue their construction of examples of compact Einstein spaces along the same lines and focus on 7-dimensional Sasakian-Einstein spaces. In particular they are interested in the case of rational homology 7-spheres. Based on the same work, they provide 1936 distinct Sasakian-Einstein structures on certain 2-connected 7-manifolds \(M^7\) realized as links of weighted homogeneous polynomials in \(\mathbb{C}^5\). Among them they discovered 184 2-connected rational homology spheres and discuss some of these examples in more detail. Here it should be noted that before, \(S^{2n+1}\), the Stiefel manifold \(V_2(\mathbb{R}^{2n+1})\) of 2-frames in \(\mathbb{R}^{2n+1}\) and the 3-Sasakian homogeneous 11-manifold \(G_2/\text{Sp}(1)\) were the only examples, known to the authors, of simply-connected rational homology spheres that admit Sasakian-Einstein structures.
Furthermore, the authors discuss 5- and 7-dimensional homology spheres which can be equipped with a regular positive Sasakian structure and finally, determine all rational homology spheres \(M^{2n+1}\) admitting a homogeneous Sasakian-Einstein structure or in particular, a 3-Sasakian homogeneous structure.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C12 Foliations (differential geometric aspects)
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References:

[1] The Penrose Transform (1989) · Zbl 0726.58004
[2] Twistors and Killing Spinors on Riemannian Manifolds, vol. 124 (1991) · Zbl 0734.53003
[3] On Sasakian-Einstein Geometry, Int. J. Math, 11, 873-909 (2000) · Zbl 1022.53038 · doi:10.1142/S0129167X00000477
[4] 3-Sasakian manifolds. Surveys in differential geometry: essays on Einstein manifolds, 123-184 (1999) · Zbl 1008.53047
[5] New Einstein Metrics in Dimension Five, J. Diff. Geom., 57, 443-463 (2001) · Zbl 1039.53048
[6] On the Geometry of Sasakian-Einstein 5-Manifolds · Zbl 1046.53028
[7] On Positive Sasakian Geometry · Zbl 1046.53029
[8] Sasakian-Einstein Structures on \(9\#(S^2\times S^3)\), Trans. Amer. Math. Soc., 354, 2983-2996 (2002) · Zbl 1003.53038 · doi:10.1090/S0002-9947-02-03015-5
[9] Sasakian-Einstein Structures on \(9\#(S^2\times S^3)\)
[10] Sasakian Geometry, Homotopy Spheres and Positive Ricci Curvature · Zbl 1066.53089
[11] 3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients, Ann. Global Anal. Geom, 21, 85-110 (2002) · Zbl 1003.53032 · doi:10.1023/A:1014261219517
[12] Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Scient. Ec. Norm. Sup. Paris, 34, 525-556 (2001) · Zbl 0994.32021
[13] Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds · Zbl 0994.32021
[14] Weighted projective varieties, Proceedings, Group Actions and Vector Fields, Vancouver, 956, 34-71 (1981) · Zbl 0516.14014
[15] Working with weighted complete intersections, revised version in Explicit birational geometry of 3-folds, 101-173 (2000) · Zbl 0960.14027
[16] On Betti numbers of 3-Sasakian manifolds, Geom. Ded., 63, 45-68 (1996) · Zbl 0859.53031
[17] The Atiyah-Singer Theorem and Elementary Number Theory (1974) · Zbl 0288.10001
[18] Anticanonical Models of Three-dimensional Algebraic Varieties, J. Soviet Math., 13, 745-814 (1980) · Zbl 0428.14016 · doi:10.1007/BF01084563
[19] Fano Varieties, Algebraic Geometry V, Vol 47 (1999) · Zbl 0912.14013
[20] Kähler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-space, Ann. Inst. Fourier, 51, 1, 69-79 (2001) · Zbl 0974.14023 · doi:10.5802/aif.1815
[21] Fano hypersurfaces in weighted projective 4-spaces, Experimental Math, 10(1), 151-158 (2001) · Zbl 0972.14034
[22] Singular Points of Complex Hypersurface, 61 (1968) · Zbl 0184.48405
[23] Isolated singularities defined by weighted homogeneous polynomials, Topology, 9, 385-393 (1970) · Zbl 0204.56503 · doi:10.1016/0040-9383(70)90061-3
[24] A Topological Classification of Complex Structures on \(S^1\times \Sigma^{2n-1}\), Topology, 14, 13-22 (1975) · Zbl 0301.57010
[25] Minimal Rational Threefolds, Algebraic Geometry, 1016, 490-518 (1983) · Zbl 0526.14006
[26] Remarks Concerning Contact Manifolds, Tôhoku Math. J, 29, 577-584 (1977) · Zbl 0382.53031 · doi:10.2748/tmj/1178240494
[27] On a Riemannian space admitting more than one Sasakian structure, Tôhoku Math. J, 22, 536-540 (1970) · Zbl 0213.48301 · doi:10.2748/tmj/1178242720
[28] Kähler-Einstein metrics with positive scalar curvature, Invent. Math, 137, 1-37 (1997) · Zbl 0892.53027 · doi:10.1007/s002220050176
[29] Canonical Metrics in Kähler Geometry (2000) · Zbl 0978.53002
[30] Infinitely Many Contact Structures on \(S^{4m+1}\), Int. Math. Res. Notices, 14, 781-791 (1999) · Zbl 1034.53080 · doi:10.1155/S1073792899000392
[31] Contact Homology and Contact Structures on \(S^{4m+1} (2000)\)
[32] Structures on manifolds, 3 (1984) · Zbl 0557.53001
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