Abstract
In this paper, we give a criterion for the properness of the \(K\)-energy in a general Kähler class of a compact Kähler manifold by using Song–Weinkove’s result in (Commun Pure Appl Math 61(2):210–229, 2008). As applications, we give some Kähler classes on \(\mathbb {C}\mathbb {P}^2\#3\overline{\mathbb {C}\mathbb {P}^2}\) and \(\mathbb {C}\mathbb {P}^2\#8\overline{\mathbb {C}\mathbb {P}^2}\) in which the \(K\)-energy is proper. Finally, we prove Song-Weinkove’s result on the existence of critical points of \(\hat{J}\) functional by the continuity method.
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Notes
In Fulton’s terminology, it is called “convex”. We choose this name according to the usual notation.
We can also prove this by observing that \((DL(f),\eta )_{L^2}=-\int _X \eta <\frac{\sqrt{-1}}{2} \partial \bar{\partial }f, \omega _t>_{\chi _{t_0}} {\chi _{t_0}^n\over n!}= -\int _X \eta \frac{\sqrt{-1}}{2} \partial \bar{\partial }f\wedge \star \omega _t\). Here \(\star \) is the Hodge star operator associated with \(\chi _{t_0}\). Since \(\partial \) and \(\bar{\partial }\) commute with \(\star \) It is obvious that this equals \(-\int _X f \frac{\sqrt{-1}}{2} \partial \bar{\partial }\eta \wedge \star \omega _t= (f,DL(\eta ))_{L^2}\). A good reference for Hodge star operator on Kähler manifold is [32].
The adaptation of the classical Evans–Krylov theorem to the complex case has been carried out by Siu [22] P100–107.
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Acknowledgments
The authors would like to thank Professor Julius Ross for bringing our attention to this problem and many helpful discussions. We are also grateful to the anonymous referee for valuable comments and suggestions.
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H. Li’s research partially supported by NSFC Grant No. 11001080 and No. 11131007.
Y. Shi’s research partially supported by NSFC Grants No. 11101206, No. 11171143, No. 11171144, No. 11331001 and by a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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Li, H., Shi, Y. & Yao, Y. A criterion for the properness of the \(K\)-energy in a general Kähler class. Math. Ann. 361, 135–156 (2015). https://doi.org/10.1007/s00208-014-1073-z
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DOI: https://doi.org/10.1007/s00208-014-1073-z