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

HTML articles powered by AMS MathViewer

- by Vicente Cortés PDF
- Trans. Amer. Math. Soc.
**350**(1998), 3193-3205 Request permission

## 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 $(2,2n)$.## References

- D.V. Alekseevsky:
*Classification of quaternionic spaces with a transitive solvable group of motions*, Math. USSR Izvestija**9**, No. 2 (1975), 297-339. - 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). - 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). - 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 - F. A. Bogomolov,
*Hamiltonian Kählerian manifolds*, Dokl. Akad. Nauk SSSR**243**(1978), no. 5, 1101–1104 (Russian). MR**514769** - Robert L. Bryant and Phillip A. Griffiths,
*Some observations on the infinitesimal period relations for regular threefolds with trivial canonical bundle*, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 77–102. MR**717607** - S. Cecotti,
*Homogeneous Kähler manifolds and $T$-algebras in $N=2$ supergravity and superstrings*, Comm. Math. Phys.**124**(1989), no. 1, 23–55. MR**1012857** - S. Cecotti, S. Ferrara, and L. Girardello,
*Geometry of type II superstrings and the moduli of superconformal field theories*, Internat. J. Modern Phys. A**4**(1989), no. 10, 2475–2529. MR**1017548**, DOI 10.1142/S0217751X89000972 - Vicente Cortés,
*Alekseevskian spaces*, Differential Geom. Appl.**6**(1996), no. 2, 129–168. MR**1395026**, DOI 10.1016/0926-2245(96)89146-7 - V. Cortés,
*Homogeneous special geometry*, Transform. Groups**1**(1996), no. 4, 337–373. MR**1424448**, DOI 10.1007/BF02549212 - B. de Wit, F. Vanderseypen, and A. Van Proeyen,
*Symmetry structure of special geometries*, Nuclear Phys. B**400**(1993), no. 1-3, 463–521. MR**1227264**, DOI 10.1016/0550-3213(93)90413-J - B. de Wit and A. Van Proeyen,
*Special geometry, cubic polynomials and homogeneous quaternionic spaces*, Comm. Math. Phys.**149**(1992), no. 2, 307–333. MR**1186031** - Ron Donagi and Eyal Markman,
*Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles*, Integrable systems and quantum groups (Montecatini Terme, 1993) Lecture Notes in Math., vol. 1620, Springer, Berlin, 1996, pp. 1–119. MR**1397273**, DOI 10.1007/BFb0094792 - S. Ferrara and S. Sabharwal,
*Quaternionic manifolds for type $\textrm {II}$ superstring vacua of Calabi-Yau spaces*, Nuclear Phys. B**332**(1990), no. 2, 317–332. MR**1046353**, DOI 10.1016/0550-3213(90)90097-W - Phillip A. Griffiths,
*Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties*, Amer. J. Math.**90**(1968), 568–626. MR**229641**, DOI 10.2307/2373545 - Shoshichi Kobayashi and Katsumi Nomizu,
*Foundations of differential geometry. Vol. II*, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR**0238225** - Maxim Kontsevich,
*Mirror symmetry in dimension $3$*, Astérisque**237**(1996), Exp. No. 801, 5, 275–293. Séminaire Bourbaki, Vol. 1994/95. MR**1423628** - H. Blaine Lawson Jr. and Marie-Louise Michelsohn,
*Spin geometry*, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR**1031992** - James Morrow and Kunihiko Kodaira,
*Complex manifolds*, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1971. MR**0302937** - S.-T. Yau (ed.),
*Mathematical aspects of string theory*, Advanced Series in Mathematical Physics, vol. 1, World Scientific Publishing Co., Singapore, 1987. MR**915812**, DOI 10.1142/0383 - Andrey N. Todorov,
*The Weil-Petersson geometry of the moduli space of $\textrm {SU}(n\geq 3)$ (Calabi-Yau) manifolds. I*, Comm. Math. Phys.**126**(1989), no. 2, 325–346. MR**1027500** - Shing-Tung Yau (ed.),
*Essays on mirror manifolds*, International Press, Hong Kong, 1992. MR**1191418**

## Additional Information

**Vicente Cortés**- Affiliation: Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany
- Email: vicente@math.uni-bonn.de
- 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.
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**350**(1998), 3193-3205 - MSC (1991): Primary 53C25
- DOI: https://doi.org/10.1090/S0002-9947-98-02156-4
- MathSciNet review: 1466946