A well-known theorem of Donaldson [1] states that a polarized algebraic manifold (X,L) with constant scalar curvature Kähler metric in c1 (L) is asymptotically Chow stable if the group Aut(X,L) is discrete. The following generalization of this theorem to extremal Kähler cases has recently been obtained by many authors [2], [3], [4]:

A polarized algebraic manifold (X,L) with extremal Kähler metric in c1 (L) is asymptotically Chow stable relative to a suitable algebraic torus of Aut(X,L).

In a series of lectures, we discuss such a generalization from various points of view. The main method of the proof is to perturb extremal Kähler metrics to suitable polybalanced metrics. However, there is another way to obtain asymptotic relative Chow stability via strong relative K-stability. Both methods will be discussed in detail.

Some basic knowledge of Kähler (or algebraic) geometry and algebraic groups

[1] S.K. Donaldson: Scalar curvature and projective embeddings I, J. Differential Geom. 59 (2001), 479-522.

[2] R. Seyyedali: Relative Chow stability and extremal metrics, Adv. in Math. 316 (2017), 770-805.

[3] Y. Sano and C. Tipler: A moment map picture of relative balanced metrics on extremal Kähler manifolds, arXiv. math. 1703.09458.

[4] T. Mabuchi: Asymptotic polybalanced kernels on extremal Kähler manifolds, Asian J. Math. 22 (2018), 647-664.