TITLE: What is a Futaki invariant, what is the intuition behind it, and why is it important?
QUESTION [16 upvotes]: As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?

REPLY [9 votes]: 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.