Asian Journal of Mathematics

Volume 22 (2018)

Number 4

Special issue in honor of Ngaiming Mok (3 of 3)

Guest Editors: Jun-Muk Hwang, Korea Institute for Advanced Study; Yum-Tong Siu, Harvard University; Wing-Keung To, National University of Singapore; Stephen S.-T. Yau, Tsinghua University; Sai-Kee Yeung, Purdue University

Asymptotic polybalanced kernels on extremal Kähler manifolds

Pages: 647 – 664



Toshiki Mabuchi (Department of Mathematics, Osaka University, Toyonaka, Osaka, Japan)


In this paper, improving a result in “Stability of extremal Kähler manifolds” [Osaka J. Math., 41 (2004), pp. 563–582], we obtain asymptotic polybalanced kernels associated to extremal Kähler metrics on polarized algebraic manifolds. As a corollary, we strengthen a result in “An energy-theoretic approach to the Hitchin–Kobayashi correspondence for manifolds, II” [Osaka J. Math., 46 (2009), pp. 115–139] on asymptotic relative Chow-polystability for extremal Kähler polarized algebraic manifolds. Finally, related to the Yau–Tian–Donaldson Conjecture for extremal Kähler metrics, we shall discuss the difference between strong relative $K$-stability and relative $K$-stability.


asymptotic polybalanced kernels, extremal Kähler metrics, asymptotic relative Chow-stability

2010 Mathematics Subject Classification

Primary 53C25. Secondary 32Q15, 32Q26.

Supported by JSPS Grant-in-Aid for Scientific Research (B) No. 25287010.

Received 31 October 2016

Accepted 8 June 2017

Published 20 September 2018