## Bochner-Kähler metrics

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- by Robert L. Bryant
- J. Amer. Math. Soc.
**14**(2001), 623-715 - DOI: https://doi.org/10.1090/S0894-0347-01-00366-6
- Published electronically: March 20, 2001
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## Abstract:

A Kähler metric is said to be*Bochner-Kähler*if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least $2$. In this article it will be shown that, in a certain well-defined sense, the space of Bochner-Kähler metrics in complex dimension $n$ has real dimension $n{+}1$ and a recipe for an explicit formula for any Bochner-Kähler metric will be given. It is shown that any Bochner-Kähler metric in complex dimension $n$ has local (real) cohomogeneity at most $n$. The Bochner-Kähler metrics that can be ‘analytically continued’ to a complete metric, free of singularities, are identified. In particular, it is shown that the only compact Bochner-Kähler manifolds are the discrete quotients of the known symmetric examples. However, there are compact Bochner-Kähler orbifolds that are not locally symmetric. In fact, every weighted projective space carries a Bochner-Kähler metric. The fundamental technique is to construct a canonical infinitesimal torus action on a Bochner-Kähler metric whose associated momentum mapping has the orbits of its symmetry pseudo-groupoid as fibers.

## References

- Miguel Abreu,
*Kähler geometry of toric varieties and extremal metrics*, Internat. J. Math.**9**(1998), no. 6, 641–651. MR**1644291**, DOI 10.1142/S0129167X98000282
AG V. Apostolov and P. Gauduchon, - 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 - Morgan Ward,
*Ring homomorphisms which are also lattice homomorphisms*, Amer. J. Math.**61**(1939), 783–787. MR**10**, DOI 10.2307/2371336 - Sam Perlis,
*Maximal orders in rational cyclic algebras of composite degree*, Trans. Amer. Math. Soc.**46**(1939), 82–96. MR**15**, DOI 10.1090/S0002-9947-1939-0000015-X
Ca É. Cartan, - Bang-Yen Chen,
*Some topological obstructions to Bochner-Kaehler metrics and their applications*, J. Differential Geometry**13**(1978), no. 4, 547–558 (1979). MR**570217** - Johan Deprez, Kouei Sekigawa, and Leopold Verstraelen,
*Classifications of Kaehler manifolds satisfying some curvature conditions*, Sci. Rep. Niigata Univ. Ser. A**24**(1988), 1–12. MR**929631** - Andrzej Derdziński,
*Self-dual Kähler manifolds and Einstein manifolds of dimension four*, Compositio Math.**49**(1983), no. 3, 405–433. MR**707181** - Norio Ejiri,
*Bochner Kähler metrics*, Bull. Sci. Math. (2)**108**(1984), no. 4, 423–436 (English, with French summary). MR**784677** - William Fulton and Joe Harris,
*Representation theory*, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR**1153249**, DOI 10.1007/978-1-4612-0979-9 - Victor Guillemin,
*Kaehler structures on toric varieties*, J. Differential Geom.**40**(1994), no. 2, 285–309. MR**1293656** - Sigurdur Helgason,
*Differential geometry, Lie groups, and symmetric spaces*, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**514561** - Yoshinobu Kamishima,
*Uniformization of Kähler manifolds with vanishing Bochner tensor*, Acta Math.**172**(1994), no. 2, 299–308. MR**1278113**, DOI 10.1007/BF02392648 - U-Hang Ki and Byung Hak Kim,
*Manifolds with Kaehler-Bochner metric*, Kyungpook Math. J.**32**(1992), no. 2, 285–290. MR**1203945** - J. Leysen, M. Petrović-Torgašev, and L. Verstraelen,
*Some curvature conditions in Bochner-Kaehler manifolds*, Atti Accad. Peloritana Pericolanti. Cl. Sci. Fis. Mat. Natur.**65**(1987), 85–94 (1988). MR**996511** - 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** - Kirill C. H. Mackenzie,
*Lie algebroids and Lie pseudoalgebras*, Bull. London Math. Soc.**27**(1995), no. 2, 97–147. MR**1325261**, DOI 10.1112/blms/27.2.97 - Masatsune Matsumoto,
*On Kählerian spaces with parallel or vanishing Bochner curvature tensor*, Tensor (N.S.)**20**(1969), 25–28. MR**242099** - Masatsune Matsumoto and Shûkichi Tanno,
*Kählerian spaces with parallel or vanishing Bochner curvature tensor*, Tensor (N.S.)**27**(1973), 291–294. MR**343199** - Jean Pradines,
*Théorie de Lie pour les groupoïdes différentiables. Calcul différenetiel dans la catégorie des groupoïdes infinitésimaux*, C. R. Acad. Sci. Paris Sér. A-B**264**(1967), A245–A248 (French). MR**216409** - Jean Pradines,
*Troisième théorème de Lie les groupoïdes différentiables*, C. R. Acad. Sci. Paris Sér. A-B**267**(1968), A21–A23 (French). MR**231414** - C. Procesi,
*The invariant theory of $n\times n$ matrices*, Advances in Math.**19**(1976), no. 3, 306–381. MR**419491**, DOI 10.1016/0001-8708(76)90027-X - Nevena Pušić,
*On an invariant tensor of a conformal transformation of a hyperbolic Kaehlerian manifold*, Zb. Rad.**4**(1990), 55–64 (English, with Serbo-Croatian summary). MR**1141611** - Nevena Pušić,
*On HB-flat hyperbolic Kaehlerian spaces*, Mat. Vesnik**49**(1997), no. 1, 35–44. 11th Yugoslav Geometrical Seminar (Divčibare, 1996). MR**1491945** - M. S. Robertson,
*The variation of the sign of $V$ for an analytic function $U+iV$*, Duke Math. J.**5**(1939), 512–519. MR**51**, DOI 10.1215/S0012-7094-39-00542-9
SW A. Cannas da Silva and A. Weinstein, - Shun-ichi Tachibana and Richard Chieng Liu,
*Notes on Kählerian metrics with vanishing Bochner curvature tensor*, K\B{o}dai Math. Sem. Rep.**22**(1970), 313–321. MR**266121**, DOI 10.2996/kmj/1138846167 - Hitoshi Takagi and Yoshiyuki Watanabe,
*Kählerian manifolds with vanishing Bochner curvature tensor satisfying $R(X,\,Y)\cdot R_{1}=0$*, Hokkaido Math. J.**3**(1974), 129–132. MR**338973**, DOI 10.14492/hokmj/1381758951 - Dirk Van Lindt and Leopold Verstraelen,
*A survey on axioms of submanifolds in Riemannian and Kaehlerian geometry*, Colloq. Math.**54**(1987), no. 2, 193–213. MR**948513**, DOI 10.4064/cm-54-2-193-213

*Self-dual Einstein Hermitian four-manifolds*, preprint, 2000, arXiv:math.DG/0003162.

*Sur la structure des groupes inifinis de transformations*, Ann. Éc. Norm.

**3**(1904), 153–206. (Reprinted in Cartan’s Collected Works, Part II.)

*Geometric Models for Noncommutative Algebras*, University of California at Berkeley Lecture Notes, American Mathematical Society, 1999.

## Bibliographic Information

**Robert L. Bryant**- Affiliation: Department of Mathematics, Duke University, P.O. Box 90320, Durham, North Carolina 27708-0320
- MR Author ID: 42675
- Email: bryant@math.duke.edu
- Received by editor(s): July 6, 2000
- Received by editor(s) in revised form: December 19, 2000
- Published electronically: March 20, 2001
- Additional Notes: The research for this article was made possible by support from the National Science Foundation through grant DMS-9870164 and from Duke University.
- © Copyright 2001 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**14**(2001), 623-715 - MSC (2000): Primary 53B35; Secondary 53C55
- DOI: https://doi.org/10.1090/S0894-0347-01-00366-6
- MathSciNet review: 1824987