Example for which the Donaldson-Futaki invariant is $0$ Background:
Let $X$ be a Fano variety, and consider a test configuration for $(X,-K_X)$, tha is a pair $(\mathcal{X}, \mathcal{L})$ where:

*

*$\mathcal{X}$ is a normal variety, endowed with a $\mathbb{C}^*$-action;

*there is a flat $\mathbb{C}^*$-equivariant morphism $f:X\to \mathbb{P}^1$, with $\mathbb{C}^*$ acts on $\mathbb{P}^1$ as $[tx:y]$;

*$\mathcal{L}$ is an $f$-ample line bundle on $\mathcal{X}$, and there is a $\mathbb{C}^*$-equivariant isomorphism $$(\mathcal{X}\setminus \mathcal{X}_0, \mathcal{L}|_{\mathcal{X}\setminus \mathcal{X}_0})\simeq (X\times (\mathbb{P}^1\setminus 0), \text{pr}_1^\star(-K_X)),$$
where $0=[0:1]$ and $\text{pr}_1 : X\times \mathbb{P}^1\to X$.

With this setting, one defines the Donaldson-Futaki invariant as
$$DF(\mathcal{X},\mathcal{L})=\frac{1}{(-K_X)^n}(\mathcal{L}^n\cdot K_{\mathcal{X}|\mathbb{P}^1}+\frac{n}{n+1}\mathcal{L}^{n+1}),$$
with $n=\dim X$, and one says that the Fano variety is

*

*semistable if for every test configuration one has $DF(\mathcal{X};\mathcal{L})\geq 0$;

*polystable if it is semistable, and $DF(\mathcal{X},\mathcal{L})=0 \iff (\mathcal{X},\mathcal{L})$ is of product type, i.e. it holds $\mathcal{X}\setminus \mathcal{X}_\infty\simeq X\times (\mathbb{P}^1\setminus \infty)$, with $\infty=[1:0]$.

Question:
From this definition it seems easy to see that, if a test configuration is of product type, then its Donaldson-Futaki invariant is $0$. However, I am having troubles proving it, or at least even convincing myself it should be true. Seeing the expression, $DF=0$ should correspond to having $\mathcal{L}=-K_{\mathcal{X}\mid \mathbb{P}^1}$, but still I don't see why we have the $\frac{n}{n+1}$ factor. Moreover, so far most of the proofs of stability of Fano varieties I've seen use different approachs, instead of computing the DF-invariant.
Any help would be much appreciated!
 A: I can comment on the case when $X$ is smooth. In this case we are looking at an anti-canonically polarized (hence Kähler) manifold, and this is the setting in which K-stability was first studied.
Assume that $(\mathcal{X},\mathcal{L})$ is a product test configuration for $(X,-K_X)$; in particular the central fibre $(X_0,L_0)$ equals $(X,L)$, and as the $\mathbb{C}^*$-action on $\mathcal{X}$ fixes the central fibre, it induces a holomorphic vector field $V$ on $X_0\,(\cong X)$. It turns out that the Donladson-Futaki invariant of this test configuration is the Futaki invariant of $\mathrm{c}_1(L)$ evaluated on $V$. This was proven by Donaldson, who later defined an algebraic generalization of Futaki's invariant.
This Futaki invariant $\mathcal{F}_{\mathrm{c}_1(L)}(V)$ was introduced by A. Futaki in the $90$s as a differential-geometric notion related to the (non-)existence of cscK metrics. The condition $\mathcal{F}_{\mathrm{c}_1(L)}(V)=0$ is not always satisfied. There are some famous examples on complex surfaces: I guess that the simplest is $\mathrm{Bl}_p{\mathbb{CP}^2}$. So, the condition that the Donaldson-Futaki invariant vanishes on product test configurations is not trivial, it really needs to be checked whenever the $\mathbb{C}^*$-action is not trivial.
Anyway, I hope this was helpful!
