Carathéodory measure hyperbolicity and positivity of canonical bundles

Kikuta, Shin

doi:10.1090/S0002-9939-2010-10564-6

Carathéodory measure hyperbolicity and positivity of canonical bundles
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by Shin Kikuta PDF

Proc. Amer. Math. Soc. 139 (2011), 1411-1420 Request permission

Abstract:

In this paper, we prove that the curvature of the Carathéodory pseudo-volume form is bounded above by $-1$. On the set where the pseudo-volume form is non-degenerate, the curvature current of the singular Hermitian metric associated with the Carathéodory pseudo-volume form is proved to be strictly positive. Due to these curvature properties, we obtain an explicit relation between the Carathéodory measure hyperbolicity and the positivity of the canonical bundle. Moreover, we show a relation between the Carathéodory measure hyperbolicity, the existence of the Bergman kernel form and the existence of the Bergman metric.

References

Sébastien Boucksom, On the volume of a line bundle, Internat. J. Math. 13 (2002), no. 10, 1043–1063. MR 1945706, DOI 10.1142/S0129167X02001575
Jacob Burbea, On the Hessian of the Carathéodory metric, Rocky Mountain J. Math. 8 (1978), no. 3, 555–559. MR 481122, DOI 10.1216/RMJ-1978-8-3-555
Bo-Yong Chen, The Bergman metric on complete Kähler manifolds, Math. Ann. 327 (2003), no. 2, 339–349. MR 2015074, DOI 10.1007/s00208-003-0456-3
J.-P. Demailly, Complex analytic and differential geometry, www.agbook.ps.gz.
Jean-Pierre Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), no. 3, 361–409. MR 1158622
Shoshichi Kobayashi, Hyperbolic complex spaces, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 318, Springer-Verlag, Berlin, 1998. MR 1635983, DOI 10.1007/978-3-662-03582-5
Xiaonan Ma and George Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, vol. 254, Birkhäuser Verlag, Basel, 2007. MR 2339952, DOI 10.1007/978-3-7643-8115-8
Dan Popovici, Regularization of currents with mass control and singular Morse inequalities, J. Differential Geom. 80 (2008), no. 2, 281–326. MR 2454895
Masaaki Suzuki, The intrinsic metrics on the domains in $\textbf {C}^{n}$, Math. Rep. Toyama Univ. 6 (1983), 143–177. MR 719152
B. Wong, On the holomorphic curvature of some intrinsic metrics, Proc. Amer. Math. Soc. 65 (1977), no. 1, 57–61. MR 454081, DOI 10.1090/S0002-9939-1977-0454081-7
H. Wu, Old and new invariant metrics on complex manifolds, Several complex variables (Stockholm, 1987/1988) Math. Notes, vol. 38, Princeton Univ. Press, Princeton, NJ, 1993, pp. 640–682. MR 1207887