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On hyper Kähler manifolds associated to Lagrangian Kähler submanifolds of $T^*\mathbb{C}^n$

Author: Vicente Cortés
Journal: Trans. Amer. Math. Soc. 350 (1998), 3193-3205
MSC (1991): Primary 53C25
MathSciNet review: 1466946
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Abstract: For any Lagrangian Kähler submanifold $M \subset T^*\mathbb{C}^n$, there exists a canonical hyper Kähler metric on $T^*M$. A Kähler potential for this metric is given by the generalized Calabi Ansatz of the theoretical physicists Cecotti, Ferrara and Girardello. This correspondence provides a method for the construction of (pseudo) hyper Kähler manifolds with large automorphism group. Using it, an interesting class of pseudo hyper Kähler manifolds of complex signature $(2,2n)$ is constructed. For any manifold $N$ in this class a group of automorphisms with a codimension one orbit on $N$ is specified. Finally, it is shown that the bundle of intermediate Jacobians over the moduli space of gauged Calabi Yau 3-folds admits a natural pseudo hyper Kähler metric of complex signature $(2,2n)$.

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  • [A] D.V. Alekseevsky: Classification of quaternionic spaces with a transitive solvable group of motions, Math. USSR Izvestija 9, No. 2 (1975), 297-339.
  • [A-C1] D.V. Alekseevsky, V. Cortés: Isometry groups of homogeneous quaternionic Kähler manifolds (to appear in Journal of Geometric Analysis); available as preprint Erwin Schrödinger Institut 230 (1995).
  • [A-C2] D.V. Alekseevsky, V. Cortés: Classification of stationary compact homogeneous special pseudo Kähler manifolds of semisimple group (to appear); available as preprint SFFB256 no. 519 (1997).
  • [Be] A.L. Besse: Einstein manifolds, Springer, Berlin, 1987. MR 88f:53087
  • [Bo] F.A. Bogomolov: Hamiltonian Kähler manifolds, Dokl. Akad. Nauk. SSSR 245 (1978), 1102-1104. MR 80c:32024
  • [B-G] R.L. Bryant, P.A. Griffiths, Some observations on the infinitesimal period relations for regular threefolds with trivial canonical bundle, in Arithmetic and geometry, Papers dedicated to I.R. Shafarevich, Progress in Mathematics 36, Birkhäuser, 1983. MR 86a:32044
  • [Ce] S. Cecotti: Homogeneous Kähler manifolds and T-Algebras in N=2 supergravity and superstrings, Commun. Math. Phys. 124 (1989), 23-55. MR 90g:53085
  • [C-F-G] S. Cecotti, S. Ferrara, L. Girardello: Geometry of type II superstrings and the moduli of superconformal field theories, Int. J. Mod. Phys. A4 (1989), 2475-2529. MR 90j:81175
  • [C1] V. Cortés: Alekseevskian spaces, Diff.Geom. Appl. 6 (1996), 129-168. MR 97m:53079
  • [C2] V. Cortés: Homogeneous Special Geometry, Transformation Groups 1, No. 4 (1996), 337-373. MR 98a:53064
  • [dW-V-VP] B. de Wit, F. Vanderseypen, A. Van Proeyen: Symmetry structure of special geometries, Nucl. Phys. B400 (1993), 463-521. MR 95a:83095
  • [dW-VP] B. de Wit, A. Van Proeyen: Special geometry, cubic polynomials and homogeneous quaternionic spaces, Commun. Math. Phys. 149 (1992), 307-333. MR 94a:53079
  • [D-M] R. Donagi, E. Markman: Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, Lecture Notes in Math, 1620, Springer, Berlin, 1996. MR 97h:14017
  • [F-S] S. Ferrara, S. Sabharwal: Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces, Nucl. Phys. B332 (1990), 317-332. MR 91g:53051
  • [G] P.A. Griffiths: Periods of integrals on algebraic manifolds I, II, Am. J. Math. 90 (1968), 568-626, 805-865; Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems, Bull. Am. Math. Soc. 76 (1970), 228-296. MR 37:5215; MR 38:2146
  • [K-N2] S. Kobayashi, K. Nomizu: Foundations of differential geometry, Vol. II, Interscience, Wiley, New York, 1969. MR 38:6501
  • [Kn] M. Kontsevich: Mirror symmetry in dimension 3, Séminaire BOURBAKI 47ème année, no. 801 (1994-95), 1-17. MR 98a:14055
  • [L-M] H.B. Lawson, M.-L. Michelsohn: Spin Geometry, Princeton Univ. Press, 1989. MR 91g:53001
  • [M-K] J. Morrow, K. Kodaira: Complex manifolds, Holt, Rinehart and Winston, Inc., New York, 1970. MR 46:2080
  • [Ti] G. Tian: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric, in Mathematical aspects of string theory, ed. S.-T. Yau, World Scientific, Singapore, 1987, 629-646. MR 89m:81001
  • [To] A. Todorov: The Weil-Petersson geometry of the moduli space of $SU(n\ge 3)$ (Calabi-Yau) manifolds I, Commun. Math. Phys. 126 (1989), 325-346. MR 91f:32022
  • [Y] S.-T. Yau (ed.): Essays on mirror manifolds, International Press, Hong Kong (1992). MR 94b:32001

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Additional Information

Vicente Cortés
Affiliation: Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany

Received by editor(s): August 29, 1996
Additional Notes: Supported by the Alexander von Humboldt Foundation and MSRI (Berkeley). Research at MSRI is supported in part by grant DMS-9022140.
Article copyright: © Copyright 1998 American Mathematical Society

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