What is a Futaki invariant, what is the intuition behind it, and why is it important?
The best reference is the nice survey paper of Tian about Futaki invariant and CM polarization http://bicmr.pku.edu.cn/~tian/?page_id=31 and another paper of Tian with Ding http://www.maths.ed.ac.uk/cheltsov/cambridge/pdf/tian92.pdf
The Futaki invariant is a Lie algebra character on the space of holomorphic vector fields. Its vanishing is a necessary (but not sufficient) condition for the existence of a Kahler-Einstein metric on Fano varieties. The Futaki invariant is related to the Chow weight in GIT. In fact, on any Fano variety, we have an obstruction to finding a Kaehler-Einstein metric: if the Futaki invariant doesn't vanishes then there is no Kahler Einstein metric.
Donaldson generalized the Futaki invariant; his generalization is known as the Donaldson-Futaki invariant. It can be rewritten as a CM-line bundle due to Tian.
It is worth mentioning that for arithmetic varieties, the Donaldson-Tian-Futaki invariant is a generalized version of the Faltings height. See http://arxiv.org/pdf/1508.07716.pdf
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 equivariant 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 with $\mathbb C^*$-action, and let $A_k \colon U_k\to U_k$ be the endomorphisms generating those actions. Then
$$\operatorname{dim} U_k=a_0k^n+a_1k^{n-1}+\dots$$
$$\operatorname{Tr}\left(A_k\right)=b_0k^{n+1}+b_1k^n+\dots$$
Then the Donaldson-Futaki invariant of a test configuration $(\mathfrak X,\mathfrak L)$ is
$$Fut(\mathfrak X,\mathfrak L)=\frac{2(a_1b_0-a_0b_1)}{a_0}.$$
See http://www.math.purdue.edu/~li2285/notes/Futaki.pdf
http://math.newark.rutgers.edu/~xiaowwan/Teaching/Math744/futaki.pdf
for more information
Moreover, if you use the notion of slope stability of holomorphic vector bundles and use CM bundle of Tian then you see how the Futaki Donaldson invariant appear ;)
The Futaki invariant $F(X,[\omega])$ is a quantity that needs two pieces of information on a compact complex manifold $M$.
1) A Kahler class $[\omega]$
2) A holomorphic vector field $X$.
It is an obstruction for the existence of a constant scalar curvature Kahler (cscK) metric in the Kahler class $[\omega]$. If it does not vanish then there cannot exist a cscK metric in that class. (The converse is not true in general.)
Its definition is as follows. Let $\hat{S}$ be the average scalar curvature with respect to any Kahler metric. This is a topological quantity. Let $S-\hat{S} = \Delta f$. Then $F(X,[\omega])=-\displaystyle \int _M X(f) \omega^n$. It is a miraculous fact that this quantity depends solely on the Kahler class (as opposed to the specific Kahler metric chosen).
Now, one question could be "How the heck could one have come up with such an invariant by oneself?" The deeper idea is the following simple one, originally due to Bourgougnon. If $G$ is a lie group acting on a manifold $M$, and $\alpha$ is a $G$-invariant closed $1$-form, then $\alpha(X)$ is a constant for every $X$ arising from the Lie algebra of $G$. When one applies this philosophy for appropriately chosen $G, M,$ and $\alpha$ (where infinite dimensional things are allowed) then one gets a wealth of invariants including the Futaki invariant.