## Sasakian-Einstein structures on $9\#(S^2\times S^3)$

HTML articles powered by AMS MathViewer

- by Charles P. Boyer, Krzysztof Galicki and Michael Nakamaye PDF
- Trans. Amer. Math. Soc.
**354**(2002), 2983-2996 Request permission

## Abstract:

We show that $\scriptstyle {9\#(S^2\times S^3)}$ 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 $\scriptstyle {b_2(M)\leq 8}$ which holds for any regular Sasakian-Einstein $\scriptstyle {M}$ 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.## References

- Arthur L. Besse,
*Einstein manifolds*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR**867684**, DOI 10.1007/978-3-540-74311-8 - Charles P. Boyer and Krzysztof Galicki,
*On Sasakian-Einstein geometry*, Internat. J. Math.**11**(2000), no. 7, 873–909. MR**1792957**, DOI 10.1142/S0129167X00000477 - Charles Boyer and Krzysztof Galicki,
*3-Sasakian manifolds*, Surveys in differential geometry: essays on Einstein manifolds, Surv. Differ. Geom., vol. 6, Int. Press, Boston, MA, 1999, pp. 123–184. MR**1798609**, DOI 10.4310/SDG.2001.v6.n1.a6 - C. P. Boyer and K. Galicki,
*New Einstein Metrics in Dimension Five*, J. Diff. Geom. 57 (2001), 443–463. math.DG/0003174. - C. P. Boyer, K. Galicki, and M. Nakamaye,
*On the Geometry of Sasakian-Einstein 5-Manifolds*, submitted for publication; math.DG/0012047. - C. P. Boyer, K. Galicki, and M. Nakamaye,
*On Positive Sasakian Geometry*, submitted for publication; math.DG/0104126. - Raimund Blache,
*Chern classes and Hirzebruch-Riemann-Roch theorem for coherent sheaves on complex-projective orbifolds with isolated singularities*, Math. Z.**222**(1996), no. 1, 7–57. MR**1388002**, DOI 10.1007/PL00004527 - Shigetoshi Bando and Toshiki Mabuchi,
*Uniqueness of Einstein Kähler metrics modulo connected group actions*, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 11–40. MR**946233**, DOI 10.2969/aspm/01010011 - 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. - Igor Dolgachev,
*Weighted projective varieties*, Group actions and vector fields (Vancouver, B.C., 1981) Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 34–71. MR**704986**, DOI 10.1007/BFb0101508 - A. R. Iano-Fletcher,
*Working with weighted complete intersections*, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 101–173. MR**1798982** - Nigel Hitchin,
*Compact four-dimensional Einstein manifolds*, J. Differential Geometry**9**(1974), 435–441. MR**350657** - J. M. Johnson and J. Kollár,
*Kähler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-spaces*, Ann. Inst. Fourier (Grenoble)**51**(2001), no. 1, 69–79 (English, with English and French summaries). MR**1821068** - Jennifer M. Johnson and János Kollár,
*Fano hypersurfaces in weighted projective 4-spaces*, Experiment. Math.**10**(2001), no. 1, 151–158. MR**1822861** - T. Kawasaki,
*The signature theorem for $V$-manifolds*, Topology 17 (1978), 75–83. - János Kollár and Shigefumi 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**1658959**, DOI 10.1017/CBO9780511662560 - Claude LeBrun,
*Four-dimensional Einstein manifolds, and beyond*, Surveys in differential geometry: essays on Einstein manifolds, Surv. Differ. Geom., vol. 6, Int. Press, Boston, MA, 1999, pp. 247–285. MR**1798613**, DOI 10.4310/SDG.2001.v6.n1.a10 - John Milnor,
*Singular points of complex hypersurfaces*, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR**0239612** - John Milnor and Peter Orlik,
*Isolated singularities defined by weighted homogeneous polynomials*, Topology**9**(1970), 385–393. MR**293680**, DOI 10.1016/0040-9383(70)90061-3 - M. Bershadsky,
*On measure over supermoduli space and holomorphic factorization*, Strings ’88 (College Park, MD, 1988) World Sci. Publ., Teaneck, NJ, 1989, pp. 69–76. MR**1119603** - Richard C. Randell,
*The homology of generalized Brieskorn manifolds*, Topology**14**(1975), no. 4, 347–355. MR**413149**, DOI 10.1016/0040-9383(75)90019-1 - Ichirô Satake,
*The Gauss-Bonnet theorem for $V$-manifolds*, J. Math. Soc. Japan**9**(1957), 464–492. MR**95520**, DOI 10.2969/jmsj/00940464 - Stephen Smale,
*On the structure of $5$-manifolds*, Ann. of Math. (2)**75**(1962), 38–46. MR**141133**, DOI 10.2307/1970417 - 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). - Kentaro Yano and Masahiro Kon,
*Structures on manifolds*, Series in Pure Mathematics, vol. 3, World Scientific Publishing Co., Singapore, 1984. MR**794310**

## 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
- MR Author ID: 40590
- Email: galicki@math.unm.edu
**Michael Nakamaye**- Affiliation: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico
- MR Author ID: 364646
- Email: nakamaye@math.unm.edu
- 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
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1897386