Abstract

The original GIT -stability notion for a polarized variety is “asymptotic (Chow or Hilbert) stability”, studied by Mumford and Gieseker in the 1970s, and some moduli spaces were constructed as consequences. Recently a version was introduced with a differential geometric motivation, so-called “K-stability”, by Tian (1997) and reformulated by Donaldson (2002), with the goal of establishing an equivalence between “stability” and the existence of “canonical” metrics. The notion is subtly different from the original “asymptotic stability”.

We give an applicable formula of Donaldson’s Futaki invariants, which defines K-stability as (a sort of) “GIT weight”, after Donaldson, Ross-Thomas and X.Wang.

Based on it, we show that: (1) (K-)semistability implies ``semi-log-canonicity” (partially observed in the 1970s).

(2) The converse holds in the canonically polarized case (among others).

This yields a natural expectation on the construction of Moduli, which can be seen as an algebraized version of the Fujiki-Donaldson picture.