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

$$\text{dim}U_k=a_0k^n+a_1k^{n-1}+...$$

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

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 $-A_k$