0 M ay 2 00 6 CM Stability And The Generalised Futaki Invariant I Sean

Based on the Cayley, Grothendieck, Knudsen Mumford theory of determinants we extend the CM polarisation to the Hilbert scheme. The Baum Fulton Macpherson GRR Theorem enables us to show that on any flat, proper, local complete intersection family the restriction of the extension and the original CM sheaf are isomorphic (under a mild hypothesis on the base). As a consequence the CM stability implies the K Stability in the sense of Donaldson. When the CM polarisation is ample on the base of the family, CM stability and K stability are equivalent. §0 Introduction In order to establish a relationship between Kähler Einstein metrics on a fixed Fano manifold X and the stability of an associated projective model the second author , in [2], was led to consider a family of projective varieties parametrized by a base B: