As the question title suggests, 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 KahlerEinstein 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 KaehlerEinstein 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 DonaldsonFutaki invariant. It can be rewritten as a CMline bundle due to Tian. It is worth mentioning that for arithmetic varieties, the DonaldsonTianFutaki 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:
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 $$\operatorname{dim} U_k=a_0k^n+a_1k^{n1}+\dots$$ $$\operatorname{Tr}\left(A_k\right)=b_0k^{n+1}+b_1k^n+\dots$$ Then the DonaldsonFutaki invariant of a test configuration $(\mathfrak X,\mathfrak L)$ is $$Fut(\mathfrak X,\mathfrak L)=\frac{2(a_1b_0a_0b_1)}{a_0}.$$ 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 ;) 

