Abstract
In this article we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980s. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this invariant coincides with the former one. We also show that the invariant vanishes for any compact Vaisman manifold.
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References
Bando S.: An obstruction for Chern class forms to be harmonic. Kodai Math. J. 29, 337–345 (2006)
Calabi, E.: Extremal Kähler metrics II. In: Chavel, I., Farkas, H.M. (eds.) Differential geometry and complex analysis, pp. 95–114. Springer-Verlag, Berlin (1985)
Donaldson S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62, 289–349 (2002)
Donaldson, S.K.: Kähler metrics with cone singularities along a divisor, preprint. arXiv:1102.1196
DragomirS. , Ornea L.: Locally conformal Kähler geometry, progress in mathematics, vol. 155. Birkhäuser, Boston (1998)
Futaki A.: An obstruction to the existence of Einstein Kähler metrics. Invent. Math. 73, 437–443 (1983)
Futaki A.: On compact Kähler manifolds of constant scalar curvature. Proc. Jpn. Acad. A 59, 401–402 (1983)
Futaki, A.: Kähler–Einstein metrics and integral invariants. Lecture notes in mathematics, vol. 1314. Springer-Verlag, Berlin (1988)
Futaki A.: Asymptotic Chow semi-stability and integral invariants. Int. J. Math. 15, 967–979 (2004)
Futaki A., Morita S.: Invariant polynomials of the automorphism group of a compact complex manifold. J. Differ. Geom. 21, 135–142 (1985)
Futaki A., Ono H., Sano Y.: Hilbert series and obstructions to asymptotic semistability. Adv. Math. 226, 254–284 (2011)
Futaki A., Ono H., Wang G.: Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. J. Differ. Geom. 83, 585–636 (2009)
Kamishima Y., Ornea L.: Geometric flow on compact locally conformally Kähler manifolds. Tohoku Math. J. 57(2), 201–221 (2005)
Li, C.: Remarks on logarithmic K-stability. Preprint. arXiv:1104.0428v1
Nill, B., Paffenholz, A.: Examples of non-symmetric Kähler-Einstein toric Fano manifolds. Preprint. arXiv:0905.2054
Ono, H., Sano, Y., Yotsutani, N.: An example of asymptotically Chow unstable manifolds with constant scalar curvature. To appear in Annales de L’Institut Fourier. arXiv:0906.3836
Ross J., Thomas R.P.: An obstruction to the existence of constant scalar curvature Kähler metrics. J. Differ. Geom. 72(3), 429–466 (2006)
Tricerri F.: Some examples of locally conformal Kähler manifolds. Rend. Sem. Mat. Univ. Politec. Torino 40, 81–92 (1982)
Vuletescu V.: Blowing-up points on l.c.K. manifolds. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52(100), 387–390 (2009)
Yau S.-T.: On Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. USA 74, 1798–1799 (1977)
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Futaki, A., Hattori, K. & Ornea, L. An integral invariant from the view point of locally conformally Kähler geometry. manuscripta math. 140, 1–12 (2013). https://doi.org/10.1007/s00229-011-0527-9
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DOI: https://doi.org/10.1007/s00229-011-0527-9