0 Ju n 20 06 CM Stability And The Generalised Futaki Invariant II Sean

Let (X,L) be a smooth polarized complex variety of dimension n. In this paper we identify the leading terms of the (reduced) K-energy map with the weight F1(λ) provided that (X,L) moves in a good family X f → B. Consequently, we deduce that the properness of the K-Energy map implies the K-Stability of the variety. §0 Resumé of Results Let (X, L) be a compact polarised manifold. Assume L is very ample so that we have an embedding X L →֒ P . In this case ωFS is a Kähler form representing c1(L). We consider λ : C∗ → G := SL(N + 1,C) an algebraic one parameter subgroup. Let φλ(t) be the associated potential. νω denotes the K-Energy map. This paper revolves around the following Question What is the asymptotic behavior of νω(φt) as t →+ 0? Theorem 1. Assume that (X,L) moves in a good family 1 X. Then there is a function ΨX : G → R such that −∞ ≤ ΨX ≤ C and an asymptotic expansion νω(φλ(t))−ΨX(λ(t)) = F1(λ) log(t) +O(1) as t → 0. (∗) Moreover ΨX(λ(t)) → −∞ whenever X has a component of multiplicity greater than one. F1(λ) is the generalised Futaki invariant of the degeneration λ. Moser iteration together with a refined Sobolev inequality (see [3] and [4]) yields the next result. Columbia University and the University of Wisconsin Madison. Massachusetts Institute of Technology and Princeton University. The definition of a good family appears below. See [1] and [2]. 1 Theorem 2. Assume that (X,L) moves in a good family. If νω is proper then X is K-Stable. X is K-Semistable provided νω is bounded from below. Definition 1. A polarised manifold (X,L) is said to move in a good family provided there exists a diagram: