Note on the singular Hermitian metrics with analytic singularities

By the assumption of Stein, we may regard $X$ as a submanifold of $\mathbb{C}^{N}$ for some positive integer $N$. We denote by $i:X\hookrightarrow\mathbb{C}^{N}$ the inclusion of $X$ and according to Siu’s result in [Siu76], there exist an open neighborhood $M$ of $X$ in $\mathbb{C}^{N}$ and a holomorphic retraction $p:M\rightarrow X$ such that $p\circ i=\text{Id}_{X}$. Now $(p^{\ast}E,p^{\ast}h)$ is the trivial vector bundle with Griffiths semi-positive metric $p^{\ast}h$ on $M$. From the Lemma 2.5, one can obtain a sequence of smooth Hermitian metrics $\{g_{\nu}\}$ with Griffiths semi-positive curvature increasing to $p^{\ast}h$ on any relative compact subset in $M$. Taking an exhaustion $\{X_{j}\}_{j=1}^{\infty}$ of $X$, where each $X_{j}$ is a relative compact Stein sub-domain in $X$, satisfying each $X_{j}$ is the relatively compact subset of $X_{j+1}$ and $\bigcup X_{j}=X$. Set $\{h_{\nu}:=i^{\ast}g_{\nu}\}$ and so $\{h_{\nu}\}$ is an approximation sequence with Griffiths semi-positive curvature increasing to $h$ on any relatively compact subset of $X$. Since each $h_{\nu}$ is Griffiths semi-positive, due to Demailly-Skoda’s theorem 2.3, $h_{\nu}\otimes\det h_{\nu}$ is Nakano semi-positive. The curvature of $(E\otimes\det^{m}E,h_{\nu}\otimes\det^{m}h_{\nu}\cdot e^{-\phi})$ can be calculated as

		$\displaystyle\sqrt{-1}\Theta_{h_{\nu}\otimes\det^{m}h_{\nu}\cdot e^{-\phi}}$
	$\displaystyle=$	$\displaystyle\sqrt{-1}\Theta_{h_{\nu}\otimes\det^{m}h_{\nu}}+\sqrt{-1}\partial\overline{\partial}\phi\otimes\text{Id}_{E\otimes\det^{m}E}$
	$\displaystyle=$	$\displaystyle\sqrt{-1}\Theta_{h_{\nu}\otimes\det h_{\nu}}\otimes\text{Id}_{\det^{m-1}E}+(m-1)\sqrt{-1}\partial\overline{\partial}(-\log\det h_{\nu})\otimes\text{Id}_{E\otimes\det E}$
		$\displaystyle+\sqrt{-1}\partial\overline{\partial}\phi\otimes\text{Id}_{E\otimes\det^{m}E}$
	$\displaystyle\geq$	$\displaystyle~{}C\omega\otimes\text{Id}_{E\otimes\det^{m}E}.$

The last inequality is in the sense of Nakano and $C$ is independent to $\nu$. Thus for any $E\otimes\det^{m}E$ valued $(n,q)$-form $u$ with finite norm, we have

$$\langle[\sqrt{-1}\Theta_{h_{\nu}\otimes\det^{m}h_{\nu}\cdot e^{-\phi}},\Lambda_{\omega}]u,u\rangle\geq qC|u|^{2}_{h_{\nu}\otimes\det^{m}h_{\nu}\cdot e^{-\phi}}$$

Now we fix one sub-domain $X_{j}$. By using Theorem 2.2 we get a solution $\alpha_{\nu}$ of the $\overline{\partial}$-equation satisfying

	$\displaystyle\int_{X_{j}}|\alpha_{\nu}|^{2}_{\omega,h_{\nu}\otimes\det^{m}h_{\nu}}e^{-\phi}dV_{\omega}$	$\displaystyle\leq\frac{1}{qC}\int_{X_{j}}|u|^{2}_{\omega,h_{\nu}\otimes\det^{m}h_{\nu}}e^{-\phi}dV_{\omega}$
		$\displaystyle\leq\frac{1}{qC}\int_{X_{j}}|u|^{2}_{\omega,h\otimes\det^{m}h}e^{-\phi}dV_{\omega}$
		$\displaystyle\leq\frac{1}{qC}\int_{X}|u|^{2}_{\omega,h\otimes\det^{m}h}e^{-\phi}dV_{\omega}<+\infty$

for sufficiently large $\nu$. Fix sufficiently large $\nu_{0}$. We have that for $\nu\geq\nu_{0}$

	$\displaystyle\int_{X_{j}}|\alpha_{\nu}|^{2}_{\omega,h_{\nu_{0}}\otimes\det^{m}h_{\nu_{0}}}e^{-\phi}dV_{\omega}$	$\displaystyle\leq\int_{X_{j}}|\alpha_{\nu}|^{2}_{\omega,h_{\nu}\otimes\det^{m}h_{\nu}}e^{-\phi}dV_{\omega}$
		$\displaystyle\leq\frac{1}{qC}\int_{X}|u|^{2}_{\omega,h\otimes\det^{m}h}e^{-\phi}dV_{\omega}<+\infty.$

Then $\{\alpha_{\nu}\}_{\nu\geq\nu_{0}}$ forms a bounded sequence on $X_{j}$ with respect to the norm

$$\int_{X_{j}}|~{}\cdot~{}|^{2}_{\omega,h_{\nu_{0}}\otimes\det^{m}h_{\nu_{0}}}e^{-\phi}dV_{\omega}.$$

We can get a weakly convergent subsequence $\{\alpha_{\nu_{0},k}\}_{k}$. Thus, the weak limit $\alpha_{j}$ satisfies

$$\int_{X_{j}}|\alpha_{j}|^{2}_{\omega,h_{\nu_{0}}\otimes\det^{m}h_{\nu_{0}}}e^{-\phi}dV_{\omega}\leq\frac{1}{qC}\int_{X}|u|^{2}_{\omega,h\otimes\det^{m}h}e^{-\phi}dV_{\omega}<+\infty.$$

Next, we fix $\nu_{1}>\nu_{0}$. Repeating the above argument, we can choose a weakly convergent subsequence $\{\alpha_{\nu_{1},k}\}_{k}\subset\{\alpha_{\nu_{0},k}\}_{k}$ with respect to $\int_{X_{j}}|~{}\cdot~{}|^{2}_{\omega,h_{\nu_{1}}\otimes\det^{m}h_{\nu_{1}}}e^{-\phi}dV_{\omega}$. Then by taking a sequence $\{\nu_{n}\}_{n}$ increasing to $+\infty$ and a diagonal sequence, we obtain a weakly convergent sequence $\{\alpha_{\nu_{k},k}\}_{k}$ with respect to $\int_{X_{j}}|~{}\cdot~{}|^{2}_{\omega,h_{\nu_{\ell}}\otimes\det^{m}h_{\nu_{\ell}}}e^{-\phi}dV_{\omega}$ for all $\ell$. Hence, $\alpha_{j}$ satisfies

$$\int_{X_{j}}|\alpha_{j}|^{2}_{\omega,h\otimes\det^{m}h}e^{-\phi}dV_{\omega}\leq\frac{1}{qC}\int_{X}|u|^{2}_{\omega,h\otimes\det^{m}h}e^{-\phi}dV_{\omega}$$

thanks to the monotone convergence theorem. Since the right-hand side of the above inequality is independent of $j$, by using the exactly same argument, we can get an $E\otimes\det^{m}E$-valued $(n,q-1)$-form $\alpha$ satisfying $\overline{\partial}\alpha=u$ and

$$\int_{X}|\alpha|^{2}_{\omega,h\otimes\det^{m}h}e^{-\phi}dV_{\omega}\leq\frac{1}{qC}\int_{X}|u|^{2}_{\omega,h\otimes\det^{m}h}e^{-\phi}dV_{\omega},$$

which completes the proof. ∎

We divide the proof into several steps.

step $1$.  The sheaf $\mathcal{E}(E,h)$ is coherent if and only if $K_{X}\otimes\mathcal{E}(E,h)$ is coherent. We thus consider the coherence of $K_{X}\otimes\mathcal{E}(E,h)$ since it has the nice functorial property under the proper modification. Indeed, if $\pi:X^{\prime}\rightarrow X$ be a proper modification, then due to previous Lemma 2.6, we have

$$\pi_{\ast}(K_{X}\otimes\mathcal{E}(\pi^{\ast}E,\pi^{\ast}h))=K_{X}\otimes\mathcal{E}(E,h).$$

According to Grauert’s coherence theorem, if $K_{X}\otimes\mathcal{E}(\pi^{\ast}E,\pi^{\ast}h)$ is coherent, then its direct image $K_{X}\otimes\mathcal{E}(E,h)$ is coherent too. Note that $(\pi^{\ast}E,\pi^{\ast}h)$ is also singular Griffiths semi-positive, see for example [PT18, Lemma 2.3.2]. By the assumption, the weight of induced metric $\det h$ on the determinant line bundle $\det E$ has algebraic singularities, i.e., locally can be written as

$$-\log\det h=\gamma\log(\sum_{j}|h_{j}|^{2})+\xi.$$

Here $\gamma$ is a positive rational number and $\xi$ is bounded function. Therefore after blow-up or modification, we can assume that

$$-\log\det h=\sum_{j}\gamma_{j}\log|h_{j}|^{2}+\xi.$$

All $\gamma_{j}$ are positive rational number, we can write $\gamma_{j}=\frac{n_{j}}{m}$, where $n_{j},m\in\mathbb{Z}$ and $m\neq 0$. So we have

(3.1)

$\displaystyle-m\log\det h=\sum_{j}n_{j}\log|h_{j}|^{2}+m\xi$

be the local weight of $\det^{m}h$, the metric of line bundle $\det^{m}E$. Now locally the metric

(3.2)

$\displaystyle det^{m}h=\frac{1}{\prod_{j}|h_{j}|^{2n_{j}}}e^{-m\xi}.$

Let $H=\prod h_{j}^{n_{j}}$ be the corresponded holomorphic function. Locally $|H|^{2}\det^{m}h=e^{-m\psi}$ is positive and bounded function.

step $2$.  As we have said, since the coherence is a local property, we may assume that $X:=\Delta:=\Delta^{n}$ is a small polydisc in $\mathbb{C}^{n}$. Let $H^{0}_{2,h\otimes\det^{m}h}(\Delta,E\otimes\det^{m}E)$ be the space of holomorphic section $s$ of $E\otimes\det^{m}E$ on $\Delta$ such that

$$\int_{\Delta}|s|^{2}_{h\otimes\det^{m}h}dV_{\omega}<+\infty.$$

We consider the evaluation map

$$ev:H^{0}_{2,h\otimes\det^{m}h}(\Delta,E\otimes det^{m}E)\times\mathcal{O}_{\Delta}\rightarrow\mathcal{O}(E\otimes det^{m}E).$$

We denote by $e\otimes\det^{m}e$ the image of $ev$. Every coherent $\mathcal{O}_{\Delta}$-sheaf enjoys the Noether property, it is obvious that $e\otimes\det^{m}e$ is coherent, and by the Remark 3.2 we have the equality

(3.3)

$\displaystyle(e\otimes det^{m}e)_{x}=\mathcal{E}(E\otimes det^{m}E,h\otimes det^{m}h)_{x}.$

Similarly, for the line bundle $(\det^{m}E,\det^{m}h)$, we denote by $\det^{m}e$ the image of square integrable sections under the evaluation map, and according to the coherence of $\mathcal{E}(\det^{m}E,\det^{m}h)$, we have

(3.4)

$\displaystyle(det^{m}e)_{x}=\mathcal{E}(det^{m}E,det^{m}h)_{x}.$

For the vector bundle $(E,h)$, we denote by $e$ the image of square integrable sections under the evaluation map as above. We want to show that $e_{x}=\mathcal{E}(E,h)_{x}$ for any point $x\in\Delta$. Since $e$ is coherent and $e_{x}\subseteq\mathcal{E}(E,h)_{x}$, one just need to check

(3.5)

$\displaystyle e_{x}+\mathcal{E}(E,h)_{x}\cap m_{x}^{k}\cdot E_{(x)}=\mathcal{E}(E,h)_{x}$

for any positive integer $k$, here $E_{(x)}:=\underrightarrow{\lim}_{x\in U}H^{0}(U,E)$. Indeed, if this is the case, by the Artin–Rees lemma, there exists a positive integer $l$ such that

$$\mathcal{E}(E,h)_{x}\cap m_{x}^{k}\cdot E_{(x)}=m_{x}^{k-l}\cdot(\mathcal{E}(E,h)_{x}\cap m_{x}^{l}\cdot E_{(x)})$$

holds for any $k>l$. Therefore according to the above equality (3.5), one have

$$\mathcal{E}(E,h)_{x}=e_{x}+\mathcal{E}(E,h)_{x}\cap m_{x}^{k}\cdot E_{(x)}\subset e_{x}+m_{x}\cdot\mathcal{E}(E,h)_{x}\subset\mathcal{E}(E,h)_{x}.$$

By Nakayama’s lemma, one can obtain $e_{x}=\mathcal{E}(E,h)_{x}$, which is the desired result. Now we begin to prove the equality (3.5), one inclusion is trivial, we just need to check that $e_{x}+\mathcal{E}(E,h)_{x}\cap m_{x}^{k}\cdot E_{(x)}\supseteq\mathcal{E}(E,h)_{x}$

step $3$. Let $f_{x}\in\mathcal{(}E,h)_{x}$, according to our assumption

$$det^{m}h=\frac{1}{\prod_{j}|h_{j}|^{2n_{j}}}\cdot e^{-m\xi}$$

on $\Delta$. So we can choose $g=H=\prod_{j}h_{j}^{n_{j}}\in H^{0}_{2,\det^{m}h}(\Delta,\det^{m}E)$ and

(3.6)

$\displaystyle|g|^{2}_{\det^{m}h}=|g|^{2}\cdot det^{m}h=e^{-m\xi}\in(C_{1},C_{2}).$

Here $C_{1},C_{2}$ are two strictly positive bounded real numbers. In this situation, one have

(3.7)

$\displaystyle\int_{\Delta^{\prime}}|f|^{2}_{h}\cdot|g|^{2}_{(\det h)^{m}}dV_{\omega}\leq C_{2}\int_{\Delta^{\prime}}|f|^{2}_{h}dV_{\omega}<+\infty.$

Here $\Delta^{\prime}$ is the very small neighborhood of $x$. Therefore we have $f_{x}\cdot g_{x}\in\mathcal{E}(E\otimes\det^{m}E,h\otimes\det^{m}h)_{x}$. According to equality (3.3), $(e\otimes\det^{m}e)_{x}=\mathcal{E}(E\otimes\det^{m}E,h\otimes\det^{m}h)_{x}$, there exist a global section $r\in H^{0}_{2,h\otimes\det^{m}h}(\Delta,E\otimes\det^{m}E)$ such that

$$r_{x}=f_{x}\cdot g_{x}.$$

step $4$. We choose a sufficient small neighbour $U$ of $x$ and a cut off function $\rho$ such that supp $\rho\subseteq U$ and $\rho=1$ on a small neighbour of $x$.

Now we want to solve the $\overline{\partial}$-equation $\overline{\partial}(\rho r)=\overline{\partial}u$ with the weighted $L^{2}$-estimate. We first define two weighted psh functions

	$\displaystyle\varphi_{k}$	$\displaystyle=(n+k+k^{\prime})\log|z-x|^{2}+|z|^{2};$
	$\displaystyle\varphi_{k,\delta}$	$\displaystyle=(n+k+k^{\prime})\log(|z-k|^{2}+\delta^{2})+|z|^{2}.$

Here $k^{\prime}$ represent the order of $g$ at the point $x$, i.e., $g_{x}\in m_{x}^{k^{\prime}}$ but $g_{x}\notin m_{x}^{k^{\prime}+1}$. The previous Theorem 3.1 shows that there exists an $E\otimes\det^{m}E$ valued $(0,0)$ form $u$ such that $\overline{\partial}u=\overline{\partial}(\rho r)$ and

$$\int_{\Delta}|u|^{2}_{h\otimes\det^{m}h}e^{-\varphi_{k,\delta}}dV_{\omega}\leq\int_{\Delta}|\overline{\partial}(\rho r)|^{2}_{h\otimes\det^{m}h}e^{-\varphi_{k,\delta}}dV_{\omega}<+\infty.$$

Taking some subsequence and taking the limit when $\delta\rightarrow 0$, we can get

$$\int_{\Delta}|u|^{2}_{h\otimes\det^{m}h}e^{-\varphi_{k}}dV_{\omega}\leq\int_{\Delta}|\overline{\partial}(\rho r)|^{2}_{h\otimes\det^{m}h}e^{-\varphi_{k}}dV_{\omega}<+\infty.$$

By the definition of $\varphi_{k}$, we know that $u_{x}\in m_{x}^{k+k^{\prime}+1}\cdot(E\otimes\det^{m}E)_{(x)}$, this is actually not obvious. Indeed, by Lemma $2.3$ in [Iwa21], if we write $u=\sum u_{i}\delta_{i}$, here $\delta_{i}$ are the holomorphic frame of vector bundle $E\otimes\det^{m}E$, we then have each $\delta_{i}\in m_{x}^{k+k^{\prime}+1}$. Since $\overline{\partial}(\rho\cdot r-u)=0$, this means $(\rho\cdot r-u)$ is holomorphic section of $E\otimes\det^{m}E$. Then $(\frac{\rho\cdot r}{g}-\frac{u}{g})$ is a meromorphic section of $E$ on the whole $\Delta$ and holomorphic on $\Delta\backslash V(g)$, where $V(g):=\{z\in\Delta:g(z)=0\}$. On one hand, due to the fact

$$(\frac{\rho\cdot r}{g})_{x}=\frac{r_{x}}{g_{x}}=f_{x}$$

and $\rho$ is a cut off function near $x$ with small enough support, the first term $\frac{\rho\cdot r}{g}$ is smooth on the whole $\Delta$ and satisfying

$$\int_{\Delta}|\frac{\rho\cdot r}{g}|^{2}_{h}dV_{\omega}<+\infty.$$

On the other hand, if we have $\int_{\Delta}|\frac{u}{g}|^{2}_{h}dV_{\omega}<+\infty$, then $(\frac{\rho\cdot r}{g}-\frac{u}{g})$ is square integrable with respect to metric $h$, hence it can be extended to the whole $\Delta$ as a holomorphic section of $E$. Grant this for the time being, let $s=\frac{\rho\cdot r}{g}-\frac{u}{g}$, take the germ at $x$, we have $s_{x}=f_{x}-\frac{u_{x}}{g_{x}}$ and

$$f_{x}=s_{x}+\frac{u_{x}}{g_{x}}\in e_{x}+\mathcal{E}(E,h)_{x}\cap m_{x}^{k}\cdot E_{(x)}.$$

This is to say, $f_{x}\subseteq e_{x}+\mathcal{E}(E,h)_{x}\cap m_{x}^{k}\cdot E_{(x)}$ and therefore equality (3.5) is obtained.We now prove the desired result that $\mathcal{E}(E,h)$ is coherent.

step $5$. The last step is to show that $\int_{\Delta}|\frac{u}{g}|^{2}_{h}dV_{\omega}<+\infty$. But this is easy by the choose of the $g$, see (3.6), indeed, we have

	$\displaystyle\int_{\Delta}\mid\frac{u}{g}\mid^{2}_{h}dV_{\omega}$	$\displaystyle=\int_{\Delta}\frac{|u|^{2}_{h\otimes\det^{m}h}}{|g|^{2}_{\det^{m}h}}dV_{\omega}$
		$\displaystyle\leq\frac{1}{C_{1}}\int_{\Delta}|u|^{2}_{h\otimes\det^{m}h}dV_{\omega}<+\infty$

as desired. ∎

Like the proof in the previous Theorem 3.1, we first regularize the metric $h$ and the psh function $\theta$ on the smaller polydisk as Lemma 2.5. The approximation sequence of smooth metrics $h_{\nu}$ increasing to $h$ and the approximation sequence of smooth psh functions $\theta_{\nu}$ decreasing to $\theta$. We then calculate the curvature of trivial vector bundle $(E\otimes\det^{m}E,h_{\nu}\otimes e^{-\theta_{\nu}}\cdot e^{-\phi})$. Firstly we write $h_{\nu}\otimes e^{-\theta_{\nu}}=h_{\nu}\otimes\det^{m}h_{\nu}\cdot(\det^{m}h_{\nu})^{-1}\cdot e^{-\theta_{\nu}}$.

Due to Demailly-Skoda theorem, $h_{\nu}\otimes\det^{m}h_{\nu}$ is Nakano semi-positive. At the same time, we can arrange the sequence smooth psh function $\theta_{\nu}$ such that

$$\sqrt{-1}\partial\overline{\partial}\theta_{\nu}-m\sqrt{-1}\partial\overline{\partial}(-\log(\det h_{\nu}))\geq-C\omega,$$

here $C$ are some positive constant independent to $\nu$. Hence the curvature of vector bundle $(E\otimes\det^{m}E,h_{\nu}\otimes e^{-\theta_{\nu}}\cdot e^{-\phi})$ can be calculated as follow

	$\displaystyle\sqrt{-1}\Theta_{h_{\nu}\otimes e^{-\theta_{\nu}}\cdot e^{-\phi}}=$	$\displaystyle\sqrt{-1}\Theta_{h_{\nu}\otimes\det^{m}h_{\nu}}+\sqrt{-1}\partial\overline{\partial}(\theta_{\nu}+\log(\det h_{\nu})^{m})\otimes\text{Id}_{E\otimes\det^{m}E}$
		$\displaystyle+\sqrt{-1}\partial\overline{\partial}\phi\otimes\text{Id}_{E\otimes\det^{m}E}.$

Now if we choose the very positive psh function $\phi$, we can make the curvature of vector bundle $(E\otimes\det^{m}E,h_{\nu}\otimes e^{-\theta_{\nu}}\cdot e^{-\phi})$ is strictly Nakano positive, i.e.,

	$\displaystyle\sqrt{-1}\Theta_{h_{\nu}\otimes\det^{m}h_{\nu}}+\sqrt{-1}\partial\overline{\partial}(\theta_{\nu}+\log(\det h_{\nu})^{m})\otimes\text{Id}_{E\otimes\det^{m}E}+\sqrt{-1}\partial\overline{\partial}\phi\otimes\text{Id}_{E\otimes\det^{m}E}$
	$\displaystyle\geq\varepsilon\omega\otimes Id_{E\otimes\det^{m}E}$

Thus for any $E\otimes\det^{m}E$ valued $(n,q)$-form $u$, we have

$$\langle[\sqrt{-1}\Theta_{h_{\nu}\otimes e^{-\theta_{\nu}}\cdot e^{-\phi}},\Lambda_{\omega}]u,u\rangle\geq q\varepsilon|u|^{2}.$$

The rest proof basically the same as the previous Theorem 3.1 and we omit it. ∎