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 $-A\_k$



An identity for Futaki-Donaldson invariant
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](http://dx.doi.org/10.4310/CAG.2004.v12.n5.a2)), see Theorem 26, and by 


Simon Donaldson in *Lower bounds on the Calabi functional*, J. Differential Geom. 70 (2005), no. 3, 453–472 ([link](http://projecteuclid.org/euclid.jdg/1143642909)), see Proposition 3. 


These two proofs are not exactly the same.


