I am starting to study K-stability of Fano varieties, by reading the C. Xu's paper "K-stability of Fano varieties: an algebro-geometric approach", but I'm having probelms proving a simple identity.
Background and notation
Let $X$ be a Fano variety, $(\mathcal{X},\mathcal{L})$ a test configuration, and consider the induced $\mathbb{C}^*$-action on the fiber $(\mathcal{X}_0,\mathcal{L}_0)$. Notice that we obtain an induced $\mathbb{C}^*$-action on $\text{H}^0(\mathcal{X}_0, r\mathcal{L}_0)$ for any positive integer $r$.
For sufficiently divisible $k\in \mathbb{N}$, let $w_k$ be the total weight of the action, that is $ w_k=\sum_{m\geq 0} m\dim \text{H}^0(\mathcal{X}_0,k\mathcal{L}_0)_m.$ Moreover, let $d_k=\dim \text{H}^0(\text{H}^0(\mathcal{X}_0,k\mathcal{L}_0)$. Then uing asymptotic Riemann-Roch we know that
$$w_k=b_0k^{n+1}+b_1k^n+\mathcal{O}(k^{n-1})$$ $$d_k=a_0k^{n}+a_1k^{n-1}+\mathcal{O}(k^{n-2}),$$
where $n=\dim X$, and $a_0,b_0,a_1,b_1$ are rationals.
We can then consider the quotient
$$\frac{w_k}{kd_k}=F_0+F_1k^{-1}+\mathcal{O}(k^{-2}).$$
Then one define the Donaldson-Futaki invariant as $$DF(\mathcal{X},\mathcal{L})=-F_1=\frac{a_0b_1-a_1b_0}{a_0^2}$$
Question
I'm having trouble proving that $-F_1$ can be described as the fraction. I apologize because I think it's a rather simple computation, but it's driving me crazy. This is what I've done so far:
\begin{equation*} \begin{split} \frac{w_k}{kd_k}& = \frac{b_0k^n+1+b_1k^n+\mathcal{O}(k^{n-1})}{a_0k^{n+1}+a_1k^n+\mathcal{O}(k^{n+1})}\\ & = \frac{b_0k^n+1}{a_0k^{n+1}+a_1k^n+\mathcal{O}(k^{n+1})} + \frac{b_1k^n}{a_0k^{n+1}+a_1k^n+\mathcal{O}(k^{n+1})} +\mathcal{O}(k^{n-1})\\ & = k^n\frac{b_0k}{a_0k+a_1} + k^n\frac{b_1}{a_0k+a_1} +\mathcal{O}(k^{n-1})\\ \end{split} \end{equation*}
but still I don't know how to derive that identity. I've tried also to compute $F_0$, knowing already that $F_1$ had such an indetity, but the formula is messy and nothing interesting seems to pop-up.
I apologize again for the simple question!
It seems there is a mistake in the degrees of the polynomials, in your computation. Notice that $w_k$ is of degree $n+1$, while $d_k$ is of degree $d$. So, the two polynomials in the numerator and denominator are of the same degree. Anyway, using your notation we can get $$ \frac{w_k}{k\,d_k}= \frac{b_0k^{n+1}+b_1k^n+\dots}{a_0k^{n+1}+a_1k^n+\dots}= \frac{b_0+b_1k^{-1}+\dots}{a_0+a_1k^{-1}+\dots}. $$ It is clear that as $k\to\infty$ this approaches $b_0/a_0$; now, what is the next term in the expansion for $k\to\infty$? To find out, compute $$ k \left(\frac{w_k}{k\,d_k}-\frac{b_0}{a_0}\right)=k\left(\frac{b_0+b_1k^{-1}+\dots}{a_0+a_1k^{-1}+\dots}-\frac{b_0}{a_0}\right)=k\frac{a_0b_0+a_0b_1k^{-1}-(a_0b_0+a_1b_0k^{-1})+O(k^{-2})}{a_0^2+O(k^{-1})}= \frac{a_0b_1-b_0a_1+O(k^{-1})}{a_0^2+O(k^{-1})} $$ and the limit is more or less what we wanted: there is a sign difference, which is important when talking about the (Donaldson-)Futaki Invariant! Unfortunately, over the years there have been quite a few different sign conventions. Couple this with some sign mistakes here and there, and the situation is even more confusing.
It seems to me that the correct definition is $$DF=-\frac{a_0b_1-b_0a_1}{a_0^2}=-F_1,$$ you were just mistaken as to what that term of the expansion is. But again, it is quite common to be confused by these things!
A second question: why is that $w_k$ is of degree $n+1$ in $k$? That is a claim usually just addressed by quoting some "equivariant Riemann-Roch". You can find a good discussion of the relevant results in the original paper by Donaldson, "Scalar curvature and stability of toric varieties", in particular the proof of Proposition $2.2.2$.
I find that the intersection-theoretic approach to the Futaki Invariant actually explains a little bit better why you would expect a piece of degree $n+1$ to come up: instead of computing the weight of the $\mathbb{C}^*$-action on $H^0(X_0,L_0)$, you can compactify $(\mathcal{X},\mathcal{L})$ to some $\bar{\mathcal{X}}\to\mathbb{P}^1$ using the $\mathbb{C}^*$-action. It turns out that the weight of the action on $H^0(X_0,L_0)$ can also be computed as the dimension of a $H^0$ on $\mathcal{X}$, which is of dimension $N+1$.