Let $(X,L)$ be a polarized projective variety

Given an ample line bundle $L\to X$, then a test configuration for the pair $(X,L)$ consists of :

  • a scheme $\mathfrak X$ with a $\mathbb C^*$-action
  • a flat $\mathbb C^*$-equivariant map $\pi:\mathfrak X\to \mathbb C$ with fibres $X_t$;
  • an eqivariant line bundle $\mathfrak L\to \mathfrak X$, ample on all fibres;

  • for some $r>0$, an isomorphism of the pair $(\mathfrak X_1, \mathfrak L_1)$ with the original pair $(X,L^r)$.

Let $U_k=H^0(\mathfrak X_0,\mathfrak L_0^k\mid_{\mathfrak X_1})$ be vector spaces of $\mathbb C^*$-action, and let $A_k:U_k\to U_k$ be the endomorphisms generating those actions, then


$$Tr(A_k)=b_0k^{n+1}+b_1k^n+...$$ ,

Then Donaldson-Futaki invariant of a test configuration $(\mathfrak X,\mathfrak L)$ is

$$Fut(\mathfrak X,\mathfrak L)=\frac{2(a_1b_0-a_0b_1)}{a_0}$$,

Now, let a $\mathbb C^*$-action on $\mathbb CP^n$ then the infinitesimal generator is given by a Hermitian matrix , say $A$. The Hamiltonian function for the $S^1$ action on $\mathbb CP^N$ is

$$H_A(z)=\frac{z^*Az}{\mid z\mid^2}$$ , define

$$\mu(V)=\int_V\frac{zz^*}{\mid z\mid^2}d\mu_{FS}-\frac{Vol(V)}{N+1}I$$

where $V$ is a projective subvariety of $\mathbb CP^N$

and $$CH(V,A)=-\int_VH_Ad\mu_{FS}+\frac{Vol(V)}{N+1}TrA$$, Then , I am looking for a proof for the following identity.

$$lim_{k\to \infty}\frac{CH_k(\mathfrak X_0,-A_k)}{k^n}=Fut(\mathfrak X,\mathfrak L)$$

where here $A_k$ is as follows

In fact a test configuration $(\mathfrak X,\mathfrak L)$ can realize as family of projective schemes in $\mathbb P(H^0(X,L^k)^*)$ with a Hermitian metric and $\mathbb C^*$ action generated by Hermitian matrix $-A_k$


This result was proved at around the same time independently by

Xiaowei Wang in Moment map, Futaki invariant and stability of projective manifolds, Comm. Anal. Geom. 12 (2004), no. 5, 1009–1037 (link), see Theorem 26, and by

Simon Donaldson in Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005), no. 3, 453–472 (link), see Proposition 3.

These two proofs are not exactly the same.

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