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L2-Cohomology of Spaces with Nonisolated Conical Singularities and Nonmultiplicativity of the Signature

  • Jeff Cheeger
  • Xianzhe Dai
Part of the Progress in Mathematics book series (PM, volume 271)

Abstract

We study from a mostly topological standpoint the L 2-signature of certain spaces with nonisolated conical singularities. The contribution from the singularities is identified with a topological invariant of the link fibration of the singularities. This invariant measures the failure of the signature to behave multiplicatively for fibrations for which the boundary of the fiber is nonempty. The result extends easily to cusp singularities and can be used to compute the L 2-cohomology of certain noncompact hyperkähler manifolds that admit geometrically fibered end structures.

Keywords

Vector Bundle Spectral Sequence Ahler Manifold Conical Singularity Adiabatic Limit 
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References

  1. 1.
    M.F. Atiyah. The signature of fibre-bundles. In Global Analysis. Princeton University Press, 1969, pp. 73–84.Google Scholar
  2. 2.
    M.F. Atiyah, V.K. Patodi, and I.M. Singer. Spectral asymmetry and riemannian geometry. I,II,III. Math. Proc. Cambridge Philos. Soc., 77(1975):43–69, 78(1975):405–432, 79(1976):71–99.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J.-M. Bismut and J. Cheeger. Families index for manifolds with boundary, superconnections, and cones. I,II. J. Funct. Anal., 89:313–363, 90:306–354, 1990.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J.-M. Bismut and J. Cheeger. Remarks on famlies index theorem for manifolds with boundary. eds. Blaine Lawson and Kitti Tenanbaum. Differential Geometry. Pitman Monogr. Surveys Pure Appl. Math., 52, Longman Sci. Tech., Harlow, 1991, pp. 59–83.Google Scholar
  5. 5.
    J.-M. Bismut and J. Cheeger. η-invariants and their adiabatic limits. J. Am. Math. Soc., 2:33– 70, 1989.MATHMathSciNetGoogle Scholar
  6. 6.
    J.-M. Bismut and D.S. Freed. The analysis of elliptic families I,II. Commun. Math. Phys., 106:159–167, 107:103–163, 1986.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    R. Bott and L. Tu. Differential Forms in Algebraic Topology. Graduate Text in Mathematics, 82. Springer-Verlag, New York Berlin, 1982.MATHGoogle Scholar
  8. 8.
    J. Cao and X. Frederico. Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature. Math. Ann., 319(3):483–491, 2001.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    J. Cheeger. On the Hodge theory of Riemannian pseudomanifolds. Geometry of the Laplace Operator, Proc. Sympos. Pure Math., XXXVI, Am. Math. Soc., Providence, RI, 1980, pp. 91–146.Google Scholar
  10. 10.
    J. Cheeger. Spectral geometry of singular Riemannian spaces. J. Diff. Geom., 18:575–657, 1983.MATHMathSciNetGoogle Scholar
  11. 11.
    J. Cheeger. Eta invariants, the adiabatic approximation and conical singularities. J. Diff. Geom., 26:175–211, 1987.MATHMathSciNetGoogle Scholar
  12. 12.
    J. Cheeger, M. Goresky, and R. MacPherson. L 2-cohomology and intersection homology of singular algebraic varieties. Seminar on Differential Geometry, Ann. of Math. Stud., 102, Princeton Univ. Press, Princeton, N.J, 1982, pp. 303–340.Google Scholar
  13. 13.
    X. Dai. Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence. J. Am. Math. Soc., 4:265–321, 1991.MATHGoogle Scholar
  14. 14.
    M. Goresky and R. MacPherson. Intersection homology theory. Topology, 19:135–162, 1980.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    M. Goresky and R. MacPherson. Intersection homology II. Invent. Math., 71:77–129, 1983.CrossRefMathSciNetGoogle Scholar
  16. 16.
    T. Hausal, E. Hunsicker, and R. Mazzeo. The hodge cohomology of gravitational instantons. Duke Math. J. 122, no. 3:485–548, 2004.CrossRefMathSciNetGoogle Scholar
  17. 17.
    N. Hitchin. L 2-cohomology of hyperkähler quotients. Comm. Math. Phys., 211(1):153–165, 2000.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    E. Hunsicker. Hodge and signature theorems for a family of manifolds with fibration boundary. Geometry and Topology, 11:1581–1622, 2007.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    E. Hunsicker, R. Mazzeo. Harmonic forms on manifolds with edges. Int. Math. Res. Not. no. 52, 3229–3272, 2005.Google Scholar
  20. 20.
    J. Jost and K. Zuo. Vanishing theorems for L 2-cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry. Comm. Anal. Geom., 8(1):1–30, 2000.MATHMathSciNetGoogle Scholar
  21. 21.
    P. Kronheimer. A Torelli-type theorem for gravitational instantons. J. Diff. Geom., 29(3):685– 697, 1989.MATHMathSciNetGoogle Scholar
  22. 22.
    E. Looijenga. L 2-cohomology of locally symmetric varieties. Compositio Math., 67:3–20, 1988.MATHMathSciNetGoogle Scholar
  23. 23.
    R. Mazzeo. The Hodge cohomology of a conformally compact metric. J. Diff. Geom., 28:309– 339, 1988.MATHMathSciNetGoogle Scholar
  24. 24.
    R. Mazzeo and R. Melrose. The adiabatic limit, Hodge cohomology and Leray’s spectral sequence for a fibration. J. Diff. Geom., 31, no. 1, 185–213, 1990.MATHMathSciNetGoogle Scholar
  25. 25.
    R. Mazzeo and R. Phillips. Hodge theory on hyperbolic manifolds. Duke Math. J., 60:509– 559, 1990.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    W. Pardon and M. Stern. L 2-∂¯-cohomology of complex projective varieties. J. Am. Math. Soc., 4(3):603–621, 1991.MATHMathSciNetGoogle Scholar
  27. 27.
    J. Serre, S. Chern, F. Hirzebruch. On the index of a fibered manifold. Proc. AMS, 8:587–596, 1957.MATHMathSciNetGoogle Scholar
  28. 28.
    L. Saper and M. Stern. L 2-cohomology of arithmetic varieties. Ann. Math., 132(2):1–69, 1990.CrossRefMathSciNetGoogle Scholar
  29. 29.
    L. Saper and S. Zucker. An introduction to L 2-cohomology. Several Complex Variables and Complex Geometry, Proc. Sympos. Pure Math., 52, Part 2, Am. Math. Soc., Providence, RI, 519–534, 1991.Google Scholar
  30. 30.
    E. Witten. Global gravitational anomalies. Commun. Math. Phys., 100:197–229, 1985.MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    S. Zucker. Hodge theory with degenerating coefficients. L 2 cohomology in the Poincará metric.Ann. Math. (2), 109:415–476, 1979.CrossRefMathSciNetGoogle Scholar
  32. 32.
    S. Zucker. L 2 cohomology of warped products and arithmetic groups. Invent. Math., 70:169– 218, 1982.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Jeff Cheeger
    • 1
  • Xianzhe Dai
    • 2
  1. 1.Courant InstituteNew YorkUSA
  2. 2.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA

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