# What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?

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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

$$Fut(\mathfrak X,\mathfrak L)=\frac{2(a_1b_0-a_0b_1)}{a_0}.$$

http://math.newark.rutgers.edu/~xiaowwan/Teaching/Math744/futaki.pdf