Quantisation of Extremal Kähler Metrics

Abstract

Suppose that a polarised Kähler manifold (X, L) admits an extremal metric $\omega$. We prove that there exists a sequence of Kähler metrics $\{ \omega _k \}_k$, converging to $\omega$ as $k \rightarrow \infty$, each of which satisfies ${\bar{\partial }} \text {grad}^{1,0}_{\omega _k} \rho _k (\omega _k)=0$; the (1, 0)-part of the gradient of the Bergman function is a holomorphic vector field.

Appendix A: Some Results on the Lichnerowicz Operator Used in Sect. 3.2

Lemma 26

For any $F \in C^{\infty } (X , {\mathbb {R}})$, there exists $F_1 \in C^{\infty } (X , {\mathbb {R}})$, $F_2 \in C^{\infty } (X , {\mathbb {R}})$ such that ${\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega } F_1 = F + F_2$ with ${\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega } F_2=0$. Moreover, writing $pr _{\omega }: C^{\infty } (X , {\mathbb {R}}) \twoheadrightarrow \ker {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega }$ for the projection associated to the $L^2$-orthogonal direct sum decomposition $C^{\infty } (X, {\mathbb {R}}) \cong im {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega } \oplus \ker {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega }$, $F_2$ is in fact $F_2 = - pr _{\omega } (F)$.

Proof

This is a well-known result, which follows from the self-adjointness and the elliptic regularity of ${\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega }$. $\square$

Lemma 27

Let $\{ F_{k} \}$ be a family of smooth functions parametrised by k, converging to a smooth function $F_{\infty }$ in $C^{\infty }$ as $k \rightarrow \infty$, and $(\phi _{1,k} , \dots , \phi _{m,k})$ be smooth functions, each of which converges to a smooth function $\phi _{i ,\infty }$ as $k \rightarrow \infty$. Write $\omega _{(m)} := \omega + \sqrt{-1}\partial {\bar{\partial }}(\sum _{i=1}^m \phi _{i,k} / k^i)$. Let $pr _{\omega } : C^{\infty } (X , {\mathbb {R}}) \rightarrow \ker {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega }$ and $pr _{(m)} : C^{\infty } (X , {\mathbb {R}}) \rightarrow \ker {\mathcal {D}}_{(m)}^* {\mathcal {D}}_{(m)}$ be the projection to $\ker {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega }$ and $\ker {\mathcal {D}}_{(m)}^* {\mathcal {D}}_{(m)}$, respectively. Then, $pr _{(m)} F_k$ converges to $pr _{\omega } F_{\infty }$ in $C^{\infty }$.

Proof

Note that we can write ${\mathcal {D}}_{(m)}^* {\mathcal {D}}_{(m)} = {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega } + D/k$ for some differential operator D of order at most 4, which depends on $\omega$ and $(\phi _{1,k} , \dots , \phi _{m,k})$. Since we know that each $\phi _{i,k}$ converges to a smooth function $\phi _{i ,\infty }$ in $C^{\infty }$, the operator norm of D can be controlled by a constant which depends only on $\omega$ and $(\phi _{1,\infty } , \dots , \phi _{m,\infty })$ but not on k. Thus, $|| pr _{(m)} F - pr _{\omega }F ||_{C^{\infty }} \rightarrow 0$ for any fixed $F \in C^{\infty } (X, {\mathbb {R}})$ as $k \rightarrow \infty$. On the other hand, $|| pr _{(m)} F_k - pr _{(m)} F_{\infty } ||_{C^{\infty }} \rightarrow 0$ since $F_k$ converges to $F_{\infty }$ in $C^{\infty }$. Combining these estimates,

$$\begin{aligned} || pr _{(m)} F_k - pr _{\omega } F_{\infty } ||_{C^{\infty }} \le || pr _{(m)} ( F_k - F_{\infty } ) ||_{C^{\infty }} + || pr _{(m)} F_{\infty } - pr _{\omega } F_{\infty } ||_{C^{\infty }} \rightarrow 0 \end{aligned}$$

as $k \rightarrow \infty$. $\square$

Lemma 28

Suppose that the following four conditions hold for an arbitrary but fixed $m \ge 1$.

• 1. $(\phi _{1,k} , \dots , \phi _{m,k})$ are smooth functions parametrised by k such that each $\phi _{i,k}$ converges to a smooth function $\phi _{i ,\infty }$ in $C^{\infty }$ as $k \rightarrow \infty$, so that the Kähler metrics $\omega _{(m)}:= \omega + \sqrt{-1}\partial {\bar{\partial }}( \sum _{i=1}^m \phi _{i,k}/k^i)$ converges to $\omega$ in $C^{\infty }$,

• 2. $\{ G_k \}$ is a family of smooth functions on X parametrised by k such that it converges to a smooth function $G_{\infty }$ in $C^{\infty }$ as $k \rightarrow \infty$,

• 3. $\{ F_k \}$ is another family of smooth functions on X parametrised by k, each of which is the solution to the equation

  $$\begin{aligned} {\mathcal {D}}_{(m)}^* {\mathcal {D}}_{(m)} F_k = G_k , \end{aligned}$$

  with the minimum $L^2$-norm,

• 4. there exists a smooth function $F_{\infty }$ which is the solution to the equation

  $$\begin{aligned} {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega } F_{\infty } = G_{\infty } \end{aligned}$$

  with the minimum $L^2$-norm.

Then $F_{k}$ converges to $F_{\infty }$ in ${C^{\infty }}$ as $k \rightarrow \infty$.

Proof

Consider the equation

$$\begin{aligned} {\mathcal {D}}_{(m)}^* {\mathcal {D}}_{(m)} (F_{\infty } - F_k) = {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega } F_{\infty } + O(1/k) - G_k = G_{\infty } - G_k +O(1/k) \end{aligned}$$

in $C^{\infty } (X, {\mathbb {R}})$. As before, we use the $L^2$-orthogonal direct sum decomposition $C^{\infty } (X , {\mathbb {R}}) = \ker {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega } \oplus im {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega }$ (and hence $im {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega } = \ker {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega }^{\perp }$), to write $(F_{\infty } - F_k)^{\perp }$ for the $\ker {\mathcal {D}}_{(m)}^* {\mathcal {D}}_{(m)}^{\perp }$-component of $F_{\infty } - F_k$. By the standard elliptic estimate, we have

$$\begin{aligned} || (F_{\infty } - F_k)^{\perp } ||_{L^2_{p+4}} \le C_{1,p}(\omega , \{ \phi _{i,k} \}) || G_{\infty } - G_k + O(1/k)||_{L^2_{p}} \rightarrow 0 \end{aligned}$$

Recalling also $im {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega } = \ker {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega }^{\perp }$, the hypothesis 4 implies $F_{\infty } \in \mathrm {im} {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega }$, and hence there exists a function $F' \in C^{\infty } (X , {\mathbb {R}})$ such that $F_{\infty } = {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega } F'$ with the estimate

$$\begin{aligned} ||F'||_{L^2_{p}} \le C_{2,p} (\omega ) ||F_{\infty }||_{L^2_{p-4}} \end{aligned}$$

following from the standard elliptic regularity. On the other hand,

$$\begin{aligned} F_{\infty } = {\mathcal {D}}_{\omega }^* {\mathcal {D}}_{\omega } F' = {\mathcal {D}}_{(m)}^* {\mathcal {D}}_{(m)} F' + \frac{1}{k}D(F'), \end{aligned}$$

with some differential operator D of order at most 4 which depends on $\omega$ and $(\phi _{1,k} , \dots , \phi _{m,k})$. This means that $F_{\infty } - D(F')/k \in \ker {\mathcal {D}}_{(m)}^* {\mathcal {D}}_{(m)}^{\perp }$, and hence

$$\begin{aligned} || F_{\infty } - (F_{\infty })^{\perp } ||_{L^2_{p+4}}< ||D(F')||_{L^2_{p+4}}/k&< C_{3,p} (\omega , \{ \phi _{i,k} \} ) ||F'||_{L^2_{p}} /k \\&< C_{4,p} (\omega , \{ \phi _{i,k} \} ) ||F_{\infty }||_{L^2_{p-4}} /k \rightarrow 0 \end{aligned}$$

as $k \rightarrow \infty$, where we used the fact that $\phi _{i,k}$ converges to some smooth function as $k \rightarrow \infty$, so that $C_4 (\omega , \{ \phi _{i,k} \})$ stays bounded when k goes to infinity. Thus, recalling that $F_k$ is the solution to ${\mathcal {D}}_{(m)}^* {\mathcal {D}}_{(m)} F_k = G_k$ with the minimum $L^2$-norm (implying $(F_k)^{\perp } = F_k$), we have

$$\begin{aligned} || F_{\infty } -F_k||_{L^2_{p+4}} \le || (F_{\infty } -F_k)^{\perp }||_{L^2_{p+4}} + || F_{\infty } - (F_{\infty })^{\perp }||_{L^2_{p+4}} \rightarrow 0 \end{aligned}$$

as $k \rightarrow \infty$.

Since the above argument holds for all large enough p, we see that $F_k$ converges to $F_{\infty }$ in $C^{\infty }$. $\square$