## Journal of the Mathematical Society of Japan

### Relative stability associated to quantised extremal Kähler metrics

Yoshinori HASHIMOTO

#### Abstract

We study algebro-geometric consequences of the quantised extremal Kähler metrics, introduced in the previous work of the author. We prove that the existence of quantised extremal metrics implies weak relative Chow polystability. As a consequence, we obtain asymptotic weak relative Chow polystability and relative $K$-semistability of extremal manifolds by using quantised extremal metrics; this gives an alternative proof of the results of Mabuchi and Stoppa–Székelyhidi. In proving them, we further provide an explicit local density formula for the equivariant Riemann–Roch theorem.

#### Article information

Source
J. Math. Soc. Japan, Volume 71, Number 3 (2019), 861-880.

Dates
Received: 23 February 2018
First available in Project Euclid: 25 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1556179398

Digital Object Identifier
doi:10.2969/jmsj/79947994

Mathematical Reviews number (MathSciNet)
MR3984245

Subjects
Primary: 32Q26: Notions of stability
Secondary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]

#### Citation

HASHIMOTO, Yoshinori. Relative stability associated to quantised extremal Kähler metrics. J. Math. Soc. Japan 71 (2019), no. 3, 861--880. doi:10.2969/jmsj/79947994. https://projecteuclid.org/euclid.jmsj/1556179398

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