Duke Mathematical Journal

Curvature and characteristic numbers of hyper-Kähler manifolds

Nigel Hitchin and Justin Sawon

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Characteristic numbers of compact hyper-Kähler manifolds are expressed in graph-theoretical form, considering them as a special case of the curvature invariants introduced by L. Rozansky and E. Witten. The appropriate graphs are generated by "wheels," and the recently proved Wheeling theorem is used to give a formula for the $\mathscr{L}$2-norm of the curvature of an irreducible hyper-Kähler manifold in terms of the volume and Pontryagin numbers. The formula involves the multiplicative sequence that is the square root of the Â-polynomial.

Article information

Source
Duke Math. J., Volume 106, Number 3 (2001), 599-615.

Dates
First available in Project Euclid: 13 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1092403944

Digital Object Identifier
doi:10.1215/S0012-7094-01-10637-6

Mathematical Reviews number (MathSciNet)
MR1813238

Zentralblatt MATH identifier
1024.53032

Subjects
Primary: 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry
Secondary: 32J27: Compact Kähler manifolds: generalizations, classification 57M27: Invariants of knots and 3-manifolds 57R20: Characteristic classes and numbers

Citation

Hitchin, Nigel; Sawon, Justin. Curvature and characteristic numbers of hyper-Kähler manifolds. Duke Math. J. 106 (2001), no. 3, 599--615. doi:10.1215/S0012-7094-01-10637-6. https://projecteuclid.org/euclid.dmj/1092403944


Export citation

References

  • \lccD. Bar-Natan, T. Le, and D. Thurston, preprint.
  • \lccD. Bar-Natan, S. Garoufalidis, L. Rozansky, and D. P. Thurston, Wheels, wheeling and the Kontsevich integral of the unknot, http://www.arXiv.org/abs/q-alg/9703025, to appear in Israel J. Math.
  • \lccA. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle,J. Differential Geometry 18 (1983), 755–782.
  • ––––, private communication, 1999.
  • \lccA. L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb. (3) 10, Springer, Berlin, 1987.
  • \lccF. A. Bogomolov, Hamiltonian Kählerian manifolds, Soviet Math. Dokl. 19 (1978), 1462–1465.
  • \lccP. Deligne, Letter to D. Bar-Natan, 1996, http://www.ma.huji.ac.il/~drorbn/Deligne/.
  • \lccD. B. Fuks, Cohomology of Infinite-Dimensional Lie Algebras, trans. A. B. Sosinskii, Contemp. Soviet Math., Consultants Bureau, New York, 1986.
  • \lccN. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535–589.
  • \lccM. Kapranov, Rozansky-Witten invariants via Atiyah classes, Compositio Math. 115 (1999), 71–113.
  • \lccM. Kontsevich, Deformation quantization of Poisson manifolds, I, preprint, http://www.arXiv.org/abs/q-alg/9709040.
  • \lccL. Rozansky and E. Witten, Hyper-Kähler geometry and invariants of three-manifolds, Selecta Math. (N.S.) 3 (1997), 401–458.
  • \lccS. Salamon, Riemannian Geometry and Holonomy Groups, Pitman Res. Notes Math. Ser. 201, Longman, Harlow; Wiley, New York, 1989.
  • ––––, On the cohomology of Kähler and hyper-Kähler manifolds, Topology 35 (1996), 137–155.
  • \lccD. P. Thurston, Wheeling: A diagrammatic analogue of the Duflo isomorphism, Ph.D. dissertation, University of California at Berkeley, http://www.arXiv.org/abs/math.QA/0006083.
  • \lccG. Tian, “Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric” in Mathematical Aspects of String Theory (San Diego, 1986), Adv. Ser. Math. Phys. 1, World Sci., Singapore, 1987, 629–646.
  • \lccA. N. Todorov, The Weil-Petersson geometry of the moduli space of $\SU(n\ge 3)$ (Calabi-Yau) manifolds, I, Comm. Math. Phys. 126 (1989), 325–346.