Sasakian-Einstein structures on

Authors:
Charles P. Boyer, Krzysztof Galicki and Michael Nakamaye

Journal:
Trans. Amer. Math. Soc. **354** (2002), 2983-2996

MSC (2000):
Primary 53C25, 53C12, 14E30

DOI:
https://doi.org/10.1090/S0002-9947-02-03015-5

Published electronically:
April 1, 2002

MathSciNet review:
1897386

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that admits an 8-dimensional complex family of inequivalent non-regular Sasakian-Einstein structures. These are the first known Einstein metrics on this 5-manifold. In particular, the bound which holds for any regular Sasakian-Einstein does not apply to the non-regular case. We also discuss the failure of the Hitchin-Thorpe inequality in the case of 4-orbifolds and describe the orbifold version.

**[Be]**A. Besse,*Einstein Manifolds*, Springer-Verlag, Berlin and New York, 1987. MR**88f:53087****[BG1]**C. P. Boyer and K. Galicki,*On Sasakian-Einstein Geometry*, Internat. J. of Math. 11 (2000), 873-909. MR**2001k:53081****[BG2]**C. P. Boyer and K. Galicki,*3-Sasakian Manifolds*, Surveys in Differential Geometry VI:*Essays on Einstein Manifolds*A supplement to the Journal of Differential Geometry, pp. 123-184, (eds. C. LeBrun, M. Wang); International Press, Cambridge 1999. MR**2001m:53076****[BG3]**C. P. Boyer and K. Galicki,*New Einstein Metrics in Dimension Five*, J. Diff. Geom. 57 (2001), 443-463. math.DG/0003174.**[BGN1]**C. P. Boyer, K. Galicki, and M. Nakamaye,*On the Geometry of Sasakian-Einstein 5-Manifolds*, submitted for publication; math.DG/0012047.**[BGN2]**C. P. Boyer, K. Galicki, and M. Nakamaye,*On Positive Sasakian Geometry*, submitted for publication; math.DG/0104126.**[Bla]**R. Blache,*Chern classes and Hirzebruch-Riemann-Roch theorem for coherent sheaves on complex projective orbifolds with isolated singularities*, Math. Z. 222 (1996), 7-57. MR**97d:14015****[BM]**S. Bando and T. Mabuchi,*Uniqueness of Einstein Kähler Metrics Modulo Connected Group Actions*, Adv. Stud. Pure Math. 10 (1987), 11-40. MR**89c:53029****[DK]**J.-P. Demailly and J. Kollár,*Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds*, preprint AG/9910118, to appear in Ann. Scient. Ec. Norm. Sup. Paris (4) 34 (2001), 525-556.**[Do1]**I. Dolgachev,*Weighted projective varieties*, in Proceedings, Group Actions and Vector Fields, Vancouver (1981) LNM 956, 34-71. MR**85g:14060****[Fle]**A.R. Iano-Fletcher,*Working with weighted complete intersections*, Preprint MPI/89-95, revised version in*Explicit birational geometry of 3-folds*, A. Corti and M. Reid, eds., Cambridge Univ. Press, 2000, pp 101-173. MR**2001k:14089****[Hit]**N. Hitchin,*On compact four-dimensional Einstein manifolds*, J. Diff. Geom. 9 (1974), 435-442. MR**50:3149****[JK1]**J.M. Johnson and J. Kollár,*Kähler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-space*, Ann. Inst. Fourier 51(1) (2001) 69-79. MR**2002b:32041****[JK2]**J.M. Johnson and J. Kollár,*Fano hypersurfaces in weighted projective 4-spaces*, Experimental Math. 10(1) (2001) 151-158. MR**2002a:14048****[Kaw]**T. Kawasaki,*The signature theorem for -manifolds*, Topology 17 (1978), 75-83.**[KM]**J. Kollár, and S. Mori,*Birational Geometry of Algebraic Varieties*, Cambridge University Press, 1998. MR**2000b:14018****[LeB]**C. LeBrun,*Four-Dimensional Einstein Manifolds, and Beyond, Surveys in Differential Geometry VI*:*Essays on Einstein Manifolds*; A supplement to the Journal of Differential Geometry, pp. 247-285, (eds. C. LeBrun, M. Wang); International Press, Cambridge (1999). MR**2001m:53072****[Mil]**J. Milnor,*Singular Points of Complex Hypersurfaces*, Ann. of Math. Stud. 61, Princeton Univ. Press, 1968. MR**39:969****[MO]**J. Milnor and P. Orlik,*Isolated singularities defined by weighted homogeneous polynomials*, Topology 9 (1970), 385-393. MR**45:2757****[Na]**A.M. Nadel,*Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature*, Ann. Math. 138 (1990), 549-596. MR**92d:32028****[Ran]**R.C. Randell,*The homology of generalized Brieskorn manifolds*, Topology 14 (1975), 347-355. MR**54:1270****[Sat]**I. Satake,*The Gauss-Bonnet theorem for -manifolds*, J. Math. Soc. Japan 9 4. (1957), 464-476. MR**20:2022****[Sm]**S. Smale,*On the structure of 5-manifolds*, Ann. Math. 75 (1962), 38-46. MR**25:4544****[Y]**S. -T. Yau,*Einstein manifolds with zero Ricci curvature*, Surveys in Differential Geometry VI:*Essays on Einstein Manifolds*; A supplement to the Journal of Differential Geometry, pp.1-14, (eds. C. LeBrun, M. Wang); International Press, Cambridge (1999).**[YK]**K. Yano and M. Kon,*Structures on manifolds*, Series in Pure Mathematics 3, World Scientific Pub. Co., Singapore, 1984. MR**86g:53001**

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Additional Information

**Charles P. Boyer**

Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico

Email:
cboyer@math.unm.edu

**Krzysztof Galicki**

Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico

Email:
galicki@math.unm.edu

**Michael Nakamaye**

Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico

Email:
nakamaye@math.unm.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-03015-5

Received by editor(s):
November 7, 2001

Published electronically:
April 1, 2002

Additional Notes:
During the preparation of this work the first two authors were partially supported by NSF grant DMS-9970904, and third author by NSF grant DMS-0070190

Article copyright:
© Copyright 2002
American Mathematical Society