Note
This article below is only for showcasing the features of the thm-environments in progress over hugo-notice. The math presented here is segmented from several math papers and as a whole it should make no sense from the end of current sentence on.
Introduction
With these definitions in place we can state the main definition of this section.
Definition
A polarised variety $(X,L)$ is K-semistable if for any test configuration ${\cal X}$ we have ${\rm Fut}({\cal X})\geq 0$. It is $K$-stable if equality holds only when ${\cal X}$ is a product $X\times {\bf C}$.
Note that in the last clause we allow the ${\bf C}^{*}$-action on $X\times {\bf C}$ to be induced from a non-trivial action1 on $X$. What we have called K-stability is often called K-polystability in the literature. The precise statement of the result mentioned in the previous section, verifying Yau’s conjecture, is
Theorem
A Fano manifold $X$ admits a Kahler-Einstein metric if and only if $(X,K_{X}^{-1})$ is K-stable.
Here the “only if” is usually regarded as the easier direction and is due to Berman, following related results of various authors The uniqueness of the metric, modulo holomorphic automorphisms, is a relatively old result of Bando and Mabuchi. We will not say more about these results here but focus on the ‘‘if” direction.
Background to the technical aspects
To give some background to the technical aspects of the proofs sketched in Section 4 below we will now try to explain why Theorem 1 is plausible.
Proposition
Consider the map $\mathbf{G}_z:\mathbf{Q}_* \ni [(V,E,L,W, \beta, U,\mathbf{x_0}) ] \mapsto G_z(\mathbf{x_0},\mathbf{x_0})\in \mathbb C$. Let $M>m>0$, $D\in \mathbb N$. For all $z\in \mathbb C\setminus \mathbb R$, $\mathbf{G}_z$ is continuous on $\mathbf{Q}_*^{D,m,M}$.
First we go back to the definition of the Futaki invariant of $(Z,\Lambda)$ in the case when $Z$ is a manifold, which was in fact the original context for Futaki’s definition. Choose a Kähler metric $\omega$ on $Z$ in the class $c_{1}(\Lambda)$ preserved by the action of $S^{1}\subset {\bf C}^{*}$.
Lemma
Suppose $f: R → R$ is twice continuously differentiable and $\mathrm dX = a_t \mathrm dt + b_t \mathrm dW$. Then $f (X)$ is the Ito process, moreover, we have
$$ \begin{aligned} f (X_t) = & f (X_0) + \int_0^t f'(X_s) a_s \mathrm d s \\ + & \int_0^t f'(Xs) b_s \mathrm dW + \frac 1 2 \int_0^t f^{''}(X_s) b_s^2 \mathrm d s,\\ \end{aligned} $$for $t ≥ 0$.
Viewing $\omega$ as a symplectic structure, this action is generated by a Hamiltonian function $H$ on $Z$. Then the Futaki invariant can be given by a differential geometric formula
$$\int_{Z} (R-\hat{R}) H \frac{\omega^{n}}{n!},$$where $R$ is the scalar curvature of $\omega$ and $\hat{R}$ is the average value of $R$ over $Z$.
Remark
For chaotic billiards, Ingremeau recently and independently formulated a conjecture of the same nature. Using results of Bourgain, Buckley and Wigman he also proved that certain deterministic families of eigenfunctions on the $2$-torus satisfy the conclusion of Berry’s conjecture. Note that in this case, the curvature is 0 and no chaotic dynamics are present.
The Futaki invariant
This formula can be derived from the equivariant Riemann-Roch theorem and can also be understood in terms of the asymptotic geometry of sections of $L^{k}$ as $k\rightarrow \infty$, in the vein of quasi-classical asymptotics in quantisation theory. What this formula shows immediately is that if $\omega$ can be chosen to have constant scalar curvature—in particular if it is a Kähler-Einstein metric—then the Futaki invariant vanishes2. This given another way, different from the Matshusima theorem, of ruling out Kähler-Einstein metrics on 1 or 2 point blow-ups of ${\bf C}{\bf P}^{2}$. The definition of K-stability employs the Futaki invariant in a more subtle way; it is not just the automorphisms of $X$ which need to be considered but of the degenerations. The Mabuchi functional gives a way to understand this phenomenon.
Corollary
Let $D\in\mathbb N$, $M>m>0$, and let $\mathcal Q _N=(V_N,E_N,L_N,W_N, \beta_N, U_N)$ be a sequence of quantum graphs satisfying cool equation for all $N\in \mathbb N$. Then there is a subsequence $\mathcal Q_{N_k}$ which converges in the sense of Benjamini-Schramm (i.e. there exists $\mathbb{P}\in \mathcal{P}(\mathbf{Q}_*^{D,m,M})$ such that $\nu_{{Q}_{N_k}}\xrightarrow{w^*} \mathbb{P}$).
This is a functional ${\cal F}$ on the space ${\cal H}$ of Kähler metrics in a given cohomology class on a manifold $X$ defined via its first variation
$$\delta {\cal F} = \int_{X} (R-\hat{R}) \delta \phi \frac{\omega_{\phi}^{n}}{n!}.$$Here $\delta \phi$ is an infinitesimal variation in the Kähler potential and one shows that such a functional ${\cal F}$ is well-defined, up to the addition of an arbitrary constant.
Conjecture
The rank of an elliptic curve is equal to the order to which the associated $L$-function $L(s)$ vanishes at $s = 1$. In particular, the rank is $0$ (so there are only finitely many rational solutions) if the graph of $L(s)$ does not meet the $s$-axis at $s = 1$ ; the rank is $1$ if the graph meets the $s$-axis at $s = 1$ with non-zero slope; and the rank is $≥ 2$ if the graph is tangent to the $s$-axis at $s = 1$.
The three situations are illustrated for the $L$-functions of the Sylvester curves $x^3 + y^3 = p$ for $p = 5, 13$ and $19$ $(r = 0, 1 \text{ and } 2)$.
The conjecture of Birch and Swinnerton-Dyer is still open, and the million dollars have not been claimed. All that one knows in general is that $r = 0$ if $L(1) \not = 0$ (as explained in the last section) and that $r = 1$ if $L(s)$ vanishes simply at $s = 1$ (Gross-Zagier (1983) and Kolyvagin (1988)). The solution of the problem, one of the deepest and most beautiful in all of number theory, will constitute a huge step forward in our understanding of the mysteries of numbers.
By construction a critical point of ${\cal F}$ is exactly a constant scalar curvature metrics which, in the setting of Theorem 1 can be shown to be Kähler-Einstein. (We mention here that there is another functional, the Ding functional which has many similar properties to the Mabuchi functional and plays an important part in many developments. This is only defined for manifolds polarised by $K^{\pm 1}$.)
There are three possibilities:
${\cal F}$ is bounded below on ${\cal H}$ and attains its infimum;
${\cal F}$ is bounded below but does not attain its infimum;
${\cal F}$ is not bounded below.
An extension of Theorem 1 is the statement that these three possibilities correspond to $X$ being respectively $K$-stable, $K$-semistable (but not $K$-stable) and not $K$-semistable.