Holomorphic Line Bundles over a Tower of Coverings

Abstract

We study a tower of normal coverings over a compact Kähler manifold with holomorphic line bundles. When the line bundle is sufficiently positive, we obtain an effective estimate, which implies the Bergman stability. As a consequence, we deduce the equidistribution for zero currents of random holomorphic sections. Furthermore, we obtain a variance estimate for those random zero currents, which yields the almost sure convergence under some geometric condition.

Appendix

We include a proof of the Bergman stability as stated in Theorem 1.3 by using the standard Hörmander–Demailly type $L^2$ estimate, for a slightly more general setup (complete noncompact base manifold with bounded geometry). The proof is well known to the experts, which we record here for its independent interest.

Proposition 5.3

Assume the Kähler manifold $(M,\omega _h)$ is complete (not necessarily compact), and satisfies the following geometric finite conditions:

• (a) The sectional curvature of $(M, \omega _h)$ is uniformly bounded;

• (b) The injectivity radius of $(M, \omega _h)$ is uniformly bounded from below by $R>0$.

Then there exists some $N_4=N_4(M,L,h)>0$ such that any tower of normal coverings with line bundles $\{(M_j,L^N)\}$ is Bergman stable whenever $N\ge N_4$.

Proof

We essentially follow the argument of [31] (see also [6, 35]) to break the argument into two parts:

• (i) $\displaystyle \limsup _{j\rightarrow \infty }|\Pi _{j,L^N}\left( p_j(z),p_j(z)\right) |_{h^N}\le |\tilde{\Pi }_{L^N}(z,z)|_{h^N}$ for any $z\in \tilde{M}$ and any $N\ge 1$;

• (ii) $\displaystyle \liminf _{j\rightarrow \infty }|\Pi _{j,L^N}\left( p_j(z),p_j(z)\right) |_{h^N}\ge |\tilde{\Pi }_{L^N}(z,z)|_{h^N}$ for any $z\in \tilde{M}$ and any $N\ge N_4$.

Part (i) follows by a straightforward normal family argument (cf. [6, 31, 35]) which we will omit here, while part (ii) is a combination of Hörmander’s $L^2$-estimate and Agmon estimates. For any $z\in \tilde{M}$, define $\tau _j(z)=\inf \left\{ dist(z,\gamma _j z):\ \gamma _j\in \Gamma _j\setminus \{1\}\right\}$. Then $p_j|_{B(z,{\textstyle \frac{1}{2}}\tau _j(z))}$ is one-to-one and $p_j|_{B(z,{\textstyle \frac{1}{2}}\tau _j(z))}:B(z,{\textstyle \frac{1}{2}}\tau _j(z))\rightarrow p_j\left( B(z,{\textstyle \frac{1}{2}}\tau _j(z))\right)$ is a biholomorphism. It is proved in [8] that $\tau _j(z) \rightarrow \infty$ uniformly on compact subsets of $\tilde{M}$, as $j \rightarrow \infty$.

Now fix a point $x \in \tilde{M}$. We only need to show the case that $\tilde{\Pi }_{L^N}(x,x)\ne 0$. Let $\rho (\cdot )=dist(\cdot , x)\in \mathcal {C}^0(\tilde{M})$ and $x_j=p_j(x) \in M_j$ for any $j \ge 0$.

Step 1 Define sections $\{T_j\in \Gamma (M_j,L^N)\}$.

Consider the coherent state

$$\begin{aligned} S_{x}(y):=\frac{\tilde{\Pi }_{L^N}(y, x)}{\sqrt{\tilde{\Pi }_{L^N}(x, x)}}. \end{aligned}$$

Then $S_{x} \in SH^0(\tilde{M},L^N)$ and $|S_{x}(x)|_{h^N}^2=|\tilde{\Pi }_{L^N}(x, x)|_{h^N}$. For any $j\ge 0$, let $\tilde{T}_j(y)=\chi _j(\rho (y))S_{x}(y)\in \Gamma (\tilde{M},L^N)$, where the nonincreasing function $\chi _j\in \mathcal {C}_c^{\infty }([0,\infty ),{\mathbb R}^+)$ satisfies $\chi _j(r)=1$ for $0\le r\le \frac{1}{4}\tau _j(x)$, $\chi _j(r)=0$ for $r\ge \frac{1}{3}\tau _j(x)$ and $\Vert \chi '_j\Vert _{\infty }=O(\tau _j(x)^{-1})$. Since $p_j|_{B(x,{\textstyle \frac{1}{2}}\tau _j(x))}$ is one-to-one, the sections $\{T_j\in \Gamma (M_j,L^N)\}$ are defined as follows:

$$\begin{aligned} T_j(z)= {\left\{ \begin{array}{ll} \tilde{T}_j\left( \Big (p_j|_{B(x,{\textstyle \frac{1}{2}}\tau _j(x))}\Big )^{-1}(z)\right) &{}\quad \text {if} ~z\in p_j(B(x,{\textstyle \frac{1}{2}}\tau _j(x))), \\ 0 &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

Step 2 Construct potential functions $\{\phi _j\}$ following [34].

The construction is due to [34]. Since the injectivity radius of the base manifold $M=M_0$ is bounded from below by $R>0$, then the injectivity radius of $M_j$ at $x_j$ is at least $R$ since injectivity radius is nondecreasing along the tower of coverings. Let $\delta \in \mathcal {C}^{\infty }_c([0,\infty ),{\mathbb R}^+)$ (fixed and independent of $j$) be a nonincreasing cut-off function satisfying $\delta (r)=1$ for $0\le r\le \frac{1}{2}R$ and $\delta (r)=0$ for $r\ge R$. In addition, one can pick up $\delta (r)$ so that $-\frac{2+1}{R} \le \delta '(r) \le 0$ and $\left| \delta ''(r) \right| \le \frac{4(2+1)}{r^2}$. As $\tau _j(x) \rightarrow \infty$ as $j \rightarrow \infty$, by choosing $j$ sufficiently large, we can always assume that $\tau _j(x) > 4R$. Define a function on $\tilde{M}$ by

$$\begin{aligned} \phi (y)= \log \left( \frac{4 \rho ^2(y)}{R^2} \right) \times \delta (\rho (y)). \end{aligned}$$

Then the potential function $\phi _j$ on $M_j$ is defined by

$$\begin{aligned} \phi _j(z)= {\left\{ \begin{array}{ll} n \phi \left( \Big (p_j|_{B(x,{\textstyle \frac{1}{2}}\tau _j(x))}\Big )^{-1}(z)\right) &{}\quad \text {if} ~z\in p_j(B(x,{\textstyle \frac{1}{2}}\tau _j(x))), \\ 0 &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

As the sectional curvature of $M_j$ is uniformly bounded independent of $j$, by the Hessian comparison theorem [13], as shown in [34], one can control the complex Hessian of $\phi _j$ to have

$$\begin{aligned} \frac{\sqrt{-1}}{2} \partial \bar{\partial }\phi _j \ge -K \omega _h, \end{aligned}$$

where the positive constant $K=K(M,L,h)$ is independent of $j$ and the base point $x \in \tilde{M}$.

Step 3 Apply Hörmander’s theorem to solve $\bar{\partial }$-equation $\bar{\partial }T'_j=\bar{\partial }T_j$.

There exists $N_4'=N_4'(M,L,h) >0$, such that

$$\begin{aligned} NRic(h)+\frac{\sqrt{-1}}{2}\partial \bar{\partial }\phi _j+Ric(\omega _h) \ge \omega _h ~\text { for}~ N \ge N_4' . \end{aligned}$$

(6.1)

For $N\ge N_4'$ and sufficiently large $j$, we consider the line bundle $(L^N,h^Ne^{-\phi _j})\rightarrow (M_j,dV_h)$. By (6.1), we apply Hörmander’s $L^2$-estimate for the $\bar{\partial }$-equation (cf. [10] Theorem 5.1). There exists $T'_j\in L^2(M_j,(L^N,h^Ne^{-\phi _j}))$, such that $\bar{\partial }T'_j=\bar{\partial }T_j$ with

$$\begin{aligned} \Vert T'_j\Vert ^2_{L^2(h^Ne^{-\phi _j})}\!=\!\!\int _{M_j} |T'_j|^2_{h^N}e^{-\phi _j}dV_h \!\le \!\int _{M_j}|\bar{\partial }T_j|^2_{(h^N,\omega _h)}e^{-\phi _j}dV_h\!=\! \Vert \bar{\partial }T_j\Vert ^2_{L^2(h^Ne^{-\phi _j})}. \end{aligned}$$

(6.2)

Note that $\bar{\partial } T_j$ is supported in $p_j\left( \bar{B}(x,\frac{1}{3}\tau _j(x))\setminus B(x,\frac{1}{4}\tau _j(x))\right) =p_j(\bar{B}(x,\frac{1}{3}\tau _j(x)))\setminus p_j(B(x,\frac{1}{4}\tau _j(x)))$. For any $z\in p_j(B(x,{\textstyle \frac{1}{2}}\tau _j(x)))$,

$$\begin{aligned} \bar{\partial }T_j(z)= & {} \bar{\partial }\left[ \chi _j\left( \rho \circ (p_j|_{B(x,{\textstyle \frac{1}{2}}\tau _j(x))})^{-1}(z) \right) S_x \left( (p_j|_{B(x,{\textstyle \frac{1}{2}}\tau _j(x))})^{-1}(z)\right) \right] \\= & {} \chi '_j\left( \rho \circ (p_j|_{B(x,{\textstyle \frac{1}{2}}\tau _j(x))})^{-1}(z) \right) \bar{\partial }\rho \left( (p_j|_{B(x,{\textstyle \frac{1}{2}}\tau _j(x))})^{-1}(z) \right) \\&\quad \times S_{x}\left( (p_j|_{B(x,{\textstyle \frac{1}{2}}\tau _j(x))})^{-1}(z)\right) . \end{aligned}$$

The distance function $\rho$ is differentiable almost everywhere (away from the cut-locus). Moreover, we have $|\bar{\partial }\rho |_{\omega _h}^2={\textstyle \frac{1}{2}}|d\rho |_{\omega _h}^2={\textstyle \frac{1}{2}}$ almost everywhere. Hence for almost every $z\in p_j(B(x,{\textstyle \frac{1}{2}}\tau _j(x)))$,

$$\begin{aligned} |\bar{\partial }T_j(z)|^2_{(h^N,\omega _h)}\!= {\textstyle \frac{1}{2}}\left| \chi '_j\left( \!\Big (p_j|_{B(x,{\textstyle \frac{1}{2}}\tau _j(x))}\Big )^{-1}(z)\!\right) \right| ^2\ \left| S_{x}\left( \!\Big (p_j|_{B(x,{\textstyle \frac{1}{2}}\tau _j(x))}\Big )^{-1}(z)\!\right) \right| ^2_{h^N}. \end{aligned}$$

(6.3)

From the definition of $\chi _j$,

$$\begin{aligned} \left| \chi '_j\left( \Big (p_j|_{B(x,{\textstyle \frac{1}{2}}\tau _j(x))}\Big )^{-1}(z)\right) \right| ^2\lesssim \tau _j(x)^{-2}. \end{aligned}$$

(6.4)

Applying Agmon estimates on the support of $\bar{\partial }T_j$, when $N\ge N_0$,

$$\begin{aligned} \left| S_x\left( \Big (p_j|_{B(x,{\textstyle \frac{1}{2}}\tau _j(x))}\Big )^{-1}(z)\right) \right| ^2_{h^N}\lesssim e^{-2\beta \sqrt{N}\frac{1}{4}\tau _j(x)}=e^{-{\textstyle \frac{1}{2}}\beta \sqrt{N}\tau _j(x)}, \end{aligned}$$

(6.5)

provided that $j\ge 0$ is large enough to satisfy $\frac{1}{4}\tau _j(x) \ge 1$. Combining (6.3), (6.4) and (6.5), we have that for $j$ large enough, the following holds almost everywhere in $p_j(B(x,{\textstyle \frac{1}{2}}\tau _j(x)))$:

$$\begin{aligned} |\bar{\partial }T_j|^2_{(h^N,\omega _h)}\lesssim \tau _j(x)^{-2}e^{-{\textstyle \frac{1}{2}}\beta \sqrt{N}\tau _j(x)}. \end{aligned}$$

(6.6)

As $\bar{\partial }T_j$ is supported in $p_j(\bar{B}(x,\frac{1}{3}\tau _j(x)))\setminus p_j(B(x,\frac{1}{4}\tau _j(x)))$ and $\phi _j$ is supported in $p_j(B(x,R))$, $\phi _j=0$ in the support of $\bar{\partial }T_j$ for $j$ large enough. Therefore, for such $j$, by (6.6),

$$\begin{aligned} \Vert \bar{\partial }T_j\Vert ^2_{L^2(h^Ne^{-\phi _j})}= & {} \int _{M_j} |\bar{\partial }T_j|^2_{(h^N,\omega _h)}e^{-\phi _j}dV_h\\\lesssim & {} \tau _j(x)^{-2}e^{-{\textstyle \frac{1}{2}}\beta \sqrt{N}\tau _j(x)} \int _{p_j(B(x,{\textstyle \frac{1}{2}}\tau _j(x)))}\ dV_h \\= & {} \tau _j(x)^{-2}e^{-{\textstyle \frac{1}{2}}\beta \sqrt{N}\tau _j(x)}V({B(x,{\textstyle \frac{1}{2}}\tau _j(x))}). \end{aligned}$$

Since the Ricci curvature of $\tilde{M}$ has a lower bound, by the Bishop volume comparison theorem, $V(B(x,{\textstyle \frac{1}{2}}\tau _j(x)))\ dV_h$ grows at most exponentially. In other words, there exists $C=C(M,L,h)>0$ such that $V(B(x,{\textstyle \frac{1}{2}}\tau _j(x)))\le e^{\frac{C}{2}\tau _j(x)}$. Hence

$$\begin{aligned} \Vert \bar{\partial }T_j\Vert ^2_{L^2(h^Ne^{-\phi _j})} \lesssim \tau _j(x)^{-2}e^{-{\textstyle \frac{1}{2}}\beta \sqrt{N}\tau _j(x)}e^{\frac{C}{2} \tau _j(x)}=\tau _j(x)^{-2}e^{-{\textstyle \frac{1}{2}}(\beta \sqrt{N}-C)\tau _j(x)}. \end{aligned}$$

Denote $N_4''=N_4''(M,L,h)=\max \big \{\big \lfloor \big (\frac{C+2}{\beta }\big )^2\big \rfloor +1,N_0\big \}$. Then for $N\ge N_4''$, $\beta \sqrt{N}-C>2$,

$$\begin{aligned} \Vert \bar{\partial }T_j\Vert ^2_{L^2(h^Ne^{-\phi _j})} \lesssim \tau _j(x)^{-2}e^{-\tau _j(x)}. \end{aligned}$$

(6.7)

Take $N_4=\max \{N_4',N_4''\}$. By the $L^2$-estimate (6.2), for $N\ge N_4$ and $j$ large enough, then

$$\begin{aligned} \Vert T'_j\Vert ^2_{L^2(h^N e^{-\phi _j})}=\int _{M_j}|T'_j|^2_{h^N}e^{-\phi _j}dV_h \lesssim \tau _j(x)^{-2}e^{-\tau _j(x)} < \infty . \end{aligned}$$

As $\phi _j\le \log 4$, $e^{-\phi _j}\ge \frac{1}{4}$, then

$$\begin{aligned} \Vert T'_j\Vert ^2_{L^2(h^N)}\lesssim \Vert T'_j\Vert ^2_{L^2(h^Ne^{-\phi _j})} \lesssim \tau _j(x)^{-2}e^{-\tau _j(x)}\rightarrow 0, ~\text {as}~j \rightarrow \infty . \end{aligned}$$

(6.8)

Step 4 Conclusion.

Let $S_j:=T_j-T'_j$. Then $S_j$ satisfies following properties for $N \ge N_4$ and for $j$ sufficiently large:

• (1) $\bar{\partial }S_j=\bar{\partial }T_j-\bar{\partial }T'_j=0$. This implies $S_j\in H^0(M_j,L^N)$ and thus $T'_j\in \Gamma (M_j,L^N)$.

• (2) Since $e^{-\phi _j(z)}\sim \Big (\rho \circ \Big (p_j|_{B(x,{\textstyle \frac{1}{2}}\tau _j(x))}\Big )^{-1}(z)\Big )^{-2n}=dist(z,x_j)^{-2n}$ near $x_j$, $|T'_j|^2_{h^N}e^{-\phi _j}$ is not locally integrable unless we have $T'_j(x_j)=0$. Therefore $S_j(x_j)=T_j(x_j)-T'_j(x_j)=T_j(x_j)$, which implies that $|S_j(x_j)|^2_{h^N}=|T_j(x_j)|^2_{h^N}=|S_{x}(x)|^2_{h^N}=|\tilde{\Pi }_{L^N}(x, x)|_{h^N}>0$.

• (3)
  $$\begin{aligned} 0<\Vert S_j\Vert _{L^2(h^N)}= & {} \Vert T_j-T'_j\Vert _{L^2(h^N)}\le \Vert T_j\Vert _{L^2(h^N)}+\Vert T'_j\Vert _{L^2(h^N)} \\\le & {} \Vert S_{x}\Vert _{L^2(h^N)}+\Vert T'_j\Vert _{L^2(h^N)} \\= & {} 1+\Vert T'_j\Vert _{L^2(h^N)}. \end{aligned}$$

Define $F_j=\frac{S_j}{\Vert S_j\Vert _{L^2(h^N)}}\in SH^0(M_j,L^N)$. Therefore, by the extremal property of the Bergman kernel,

$$\begin{aligned}&|\Pi _{j,L^N}\left( p_j(x),p_j(x)\right) |_{h^N}=|\Pi _{j,L^N}(x_j, x_j)|_{h^N}\ge |F_j(x_j)|^2_{h^N}\\&\quad =\frac{|S_j(x_j)|^2_{h^N}}{\Vert S_j\Vert ^2_{L^2(h^N)}}\ge \frac{|\tilde{\Pi }_{L^N}(x, x)|_{h^N}}{(1+\Vert T'_j\Vert _{L^2(h^N)})^2}. \end{aligned}$$

By (6.8), for $N\ge N_4$,

$$\begin{aligned} \liminf _{j\rightarrow \infty }|\Pi _{j,L^N}\left( p_j(x),p_j(x) \right) |_{h^N}\ge |\tilde{\Pi }_{L^N}(x,x)|_{h^N}. \end{aligned}$$

Hence part (ii) is proved as $x \in \tilde{M}$ is arbitrary. $\square$