Relative stability associated to quantised extremal Kähler metrics

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 $K$-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.

18 pages. Section 3 previously part of arXiv:1508.02643v1, separated as an independent paper combined with new results

References in corpus (2)

• A moment map picture of relative balanced metrics on extremal Kähler manifolds
• Asymptotic polybalanced kernels on extremal Kaehler manifolds

Cited by in corpus (3)

• Quantisation of extremal Kähler metrics
• Asymptotic Chow semistability implies Ding polystability for Gorenstein toric Fano varieties
• Balanced metrics for extremal Kähler metrics and Fano manifolds