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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Carathéodory measure hyperbolicity and positivity of canonical bundles


Author: Shin Kikuta
Journal: Proc. Amer. Math. Soc. 139 (2011), 1411-1420
MSC (2010): Primary 32Q45; Secondary 32J18, 32J25
Published electronically: September 1, 2010
MathSciNet review: 2748434
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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.


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Additional Information

Shin Kikuta
Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
Email: sa6m15@math.tohoku.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10564-6
PII: S 0002-9939(2010)10564-6
Keywords: Carathéodory pseudo-volume form, curvature, canonical bundle, Bergman kernel form
Received by editor(s): November 4, 2009
Received by editor(s) in revised form: April 20, 2010, and April 28, 2010
Published electronically: September 1, 2010
Communicated by: Franc Forstneric
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.