ScalingStacks

This section is included for motivational purposes. On any compact complex manifold $X$ with a positive line bundle $L$, any fixed Kähler metric $\omega$ in the class $2\pi c_{1}(L)$ is the curvature form of a Hermitian metric $h$ on $L$. Consider the projective embedding $\iota_{k}:X\hookrightarrow\mathbb{P}(H^{0}(X,L^{k})^{*})$ for $k\gg 1$. The $L^{2}$ norms on sections induce Euclidean metrics on the vector spaces $H^{0}(X,L^{k})$, hence Fubini-Study metrics $\omega_{FS,k}$ on $\mathbb{P}(H^{0}(X,L^{k})^{*})$. A famous result of Tian says that $\omega$ is approximated by the algebraic metrics $k^{-1}\iota_{k}^{*}\omega_{FS,k}$ as $k\to\infty$; this idea has been much exploited in regularization theorems.

This construction is particularly transparent in the toric case, as explained in [13]. Let $(X,L)$ be an $n$-dimensional polarised toric manifold with moment polytope $P$, so a $T^{n}$-invariant basis $\{s_{m}\}$ of $H^{0}(X,L^{k})$ corresponds to $kP\cap\mathbb{Z}^{n}$, or equivalently $P\cap k^{-1}\mathbb{Z}^{m}$ after rescaling. The $L^{2}$-metric on $H^{0}(X,L^{k})$ is diagonal in the basis; i.e. the toric assumption reduces the unitary group acting on $H^{0}(X,L^{k})$ to its maximal torus. Concretely, let $\phi$ denote the torus invariant Kähler potential on $X\cap(\mathbb{C}^{*})^{n}$, equivalently thought as some convex function of $\vec{t}\in\mathbb{R}^{n}$ via the logarithm map $(\mathbb{C}^{*})^{n}\to\mathbb{R}^{n}$. Then

$$I_{m}(k)=\left\lVert s_{m}\right\rVert_{L^{2}}^{2}=\int_{X}|s_{m}|^{2}d\text{Vol}=\text{const}\int_{\mathbb{R}^{n}}e^{-k(\phi-\vec{t}\cdot m)}d\vec{t},\quad m\in P\cap k^{-1}\mathbb{Z}^{n},$$

(4)

and the Fubini-Study potentials are

$$k^{-1}\iota_{k}^{*}\phi_{FS,k}=k^{-1}\log\left(\sum_{m\in P\cap k^{-1}\mathbb{Z}^{n}}I_{m}(k)^{-1}|s_{m}|^{2}\right).$$

(5)

Now the RHS of (4) is a Laplace type integral, and its dominant contribution comes from the neighbourhood of the point $\vec{t}_{0}$ where $\vec{t}\cdot m-\phi(\vec{t})$ is maximized among $\vec{t}\in\mathbb{R}^{n}$. The maximum is the value of the Legendre transform of $\phi$:

$$u(m)=\sup_{\vec{t}}(\vec{t}\cdot m-\phi(t)).$$

The steepest descent method yields the asymptote

$$k^{-1}\log I_{m}(k)=u(m)+O(k^{-1}\log k),\quad k\to\infty.$$

In the ‘continuum limit’ $k\to\infty$, the discrete sum $\sum_{m\in P\cap k^{-1}\mathbb{Z}^{n}}$ is replaced by an integral. Now the RHS of (5) is to leading order

$$k^{-1}\log\int_{P}e^{k(-u(m)+\vec{t}\cdot m)}dm,\quad\vec{t}\in\mathbb{R}^{n}.$$

This is another Laplace type integral, and its limit as $k\to\infty$ is the Legendre transform of $u$, which gives back the function $\phi$.

The moral is that in the presence of toric symmetry, algebraic approximation of Kähler metrics is related to Legendre transforms.