## Osaka Journal of Mathematics

### Locally conformal Kähler structures on compact solvmanifolds

Hiroshi Sawai

#### Abstract

Let $(M, g, J)$ be a compact Hermitian manifold and $\Omega$ the fundamental 2-form of $(g, J)$. A Hermitian manifold $(M, g, J)$ is said to be locally conformal Kähler if there exists a closed 1-form $\omega$ such that $d\Omega=\omega \wedge \Omega$. The purpose of this paper is to investigate a relation between a locally conformal Kähler structure and the adapted differential operator on compact solvmanifolds.

#### Article information

Source
Osaka J. Math., Volume 49, Number 4 (2012), 1087-1102.

Dates
First available in Project Euclid: 19 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1355926888

Mathematical Reviews number (MathSciNet)
MR3007955

Zentralblatt MATH identifier
1275.53066

Subjects
Primary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]
Secondary: 17B30: Solvable, nilpotent (super)algebras

#### Citation

Sawai, Hiroshi. Locally conformal Kähler structures on compact solvmanifolds. Osaka J. Math. 49 (2012), no. 4, 1087--1102. https://projecteuclid.org/euclid.ojm/1355926888

#### References

• E. Abbena and A. Grassi: Hermitian left invariant metrics on complex Lie groups and cosymplectic Hermitian manifolds, Boll. Un. Mat. Ital. A (6) 5 (1986), 371–379.
• D. Arapura and M. Nori: Solvable fundamental groups of algebraic varieties and Kähler manifolds, Compositio Math. 116 (1999), 173–188.
• F.A. Belgun: On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), 1–40.
• C. Benson and C.S. Gordon: Kähler structures on compact solvmanifolds, Proc. Amer. Math. Soc. 108 (1990), 971–980.
• L.A. Cordero, M. Fernández and M. de León: Compact locally conformal Kähler nilmanifolds, Geom. Dedicata 21 (1986), 187–192.
• S. Dragomir and L. Ornea: Locally Conformal Kähler Geometry, Birkhäuser Boston, Boston, MA, 1998.
• K. Hasegawa: Complex and Kähler structures on compact solvmanifolds, J. Symplectic Geom. 3 (2005), 749–767.
• K. Hasegawa: A note on compact solvmanifolds with Kähler structures, Osaka J. Math. 43 (2006), 131–135.
• A. Hattori: Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 289–331.
• M. Inoue: On surfaces of class $\mathrm{VII}_{0}$, Invent. Math. 24 (1974), 269–310.
• \begingroup Y. Kamishima: Note on locally conformal Kähler surfaces, Geom. Dedicata 84 (2001), 115–124. \endgroup
• H. Sawai: Locally conformal Kähler structures on compact nilmanifolds with left-invariant complex structures, Geom. Dedicata 125 (2007), 93–101.
• H. Sawai: A construction of lattices on certain solvable Lie groups, Topology Appl. 154 (2007), 3125–3134.
• F. Tricerri: Some examples of locally conformal Kähler manifolds, Rend. Sem. Mat. Univ. Politec. Torino 40 (1982), 81–92.
• I. Vaisman: Locally conformal Kähler manifolds with parallel Lee form, Rend. Mat. (6) 12 (1979), 263–284.
• H.-C. Wang: Complex parallisable manifolds, Proc. Amer. Math. Soc. 5 (1954), 771–776.