Uniform $L^{2}$-estimates for flat nontrivial line bundles on compact complex manifolds

The strategy of the proof is to chase the Dolbeault isomorphism between $\bar{\partial}$-cohomology and Čech cohomology, using the Hörmander’s $L^{2}$-method, which allows us to apply Ueda’s lemma for the $\delta$-equation.

We work with a fixed Hermitian metric $g$ on $X$, with all norms and volume forms defined with respect to $g$. We now recall a foundational result by Hörmander (see e.g. [Demailly_book]*Chapter VIII, Theorem 6.9, [Dem82]*5, 4.1 Théorème, or [Hormbook]*Theorem 4.4.2), which guarantees the following: Let $U^{*}_{x}$ be a Stein open neighborhood of a given point $x\in X$, and let $h=e^{-\varphi}$ be a singular Hermitian metric on the trivial line bundle with $\varphi$ psh. Then, for any smooth $(0,1)$-form $v_{x}$ on $U^{*}_{x}$ such that $\bar{\partial}v_{x}=0$ and $\|v_{x}\|_{L^{2}(U^{*}_{x}),h}<\infty$, there exists a smooth function $u_{x}$ on $U^{*}_{x}$ such that

$$\bar{\partial}u_{x}=v_{x}\text{ on }U_{x}^{*}$$

with the $L^{2}$ estimate

$$\|u_{x}\|_{L^{2}(U^{*}_{x}),h}\leq C_{x,g}\|v_{x}\|_{L^{2}(U^{*}_{x}),h},$$

where the constant $C_{x,g}>0$ depends only on $U^{*}_{x}$ and $g$. Note that the solution of $u_{x}$ can be chosen to be smooth if $h=e^{-\varphi}$ and $v_{x}$ are smooth.

We may assume, without loss of generality and by shrinking each $U^{*}_{x}$, that $U^{*}_{x}$ trivializes any flat line bundle on $X$ (see [KH22]*Lemma 2.7). For each $x\in X$, we pick a relatively compact open subset $U^{**}_{x}\Subset U^{*}_{x}$. We then cover $X$ with the open sets $\{U^{**}_{x}\}_{x\in X}$, and by the compactness of $X$, we may select a finite subcover $\{U^{**}_{x_{j}}\}_{j\in I}$. By setting $U^{\prime}_{j}:=U^{**}_{x_{j}}$ and $U_{j}:=U^{*}_{x_{j}}$ for each $j\in I$, we obtain two finite covers $\{U^{\prime}_{j}\}_{j\in I}$ and $\{U_{j}\}_{j\in I}$ of $X$. By construction, we can observe that $U_{j}$ are Stein, $U^{\prime}_{j}\Subset U_{j}$, and $U_{j}$ (and thus $U^{\prime}_{j}$ as well) trivializes any flat line bundle on $X$.

Let $F$ be a flat line bundle endowed with a flat Hermitian metric $h$, which is unique up to a multiplicative constant (see e.g. [KH22]*Lemma 2.4). Let $v$ be a smooth $\bar{\partial}$-closed $(0,1)$-form with values in $F$, whose Dolbeault cohomology class $[v]\in H^{0,1}(X,F)$ is trivial. In what follows, all norms are with respect to the fixed flat Hermitian metric $h$ on $F$ and the fixed Hermitian metric $g$ on $X$. By applying Hörmander’s estimates and trivializing $F$ equipped with the metric $h$, we find that for each $j\in I$ there exists a smooth function $u_{j}$ on $U_{j}$ such that

(1)

$$\bar{\partial}u_{j}=v|_{U_{j}}\text{ and }\|u_{j}\|_{L^{2}(U_{j}),h}\leq C^{(1)}_{j,g}\|v\|_{L^{2}(U_{j}),h}$$

for a constant $C^{(1)}_{j,g}>0$, which depends only on $U_{j}$ and $g$. The key point of this estimate is that it holds uniformly for all flat line bundles $(F,h)$ and for all $v$ representing a trivial Dolbeault class in $H^{0,1}(X,F)$.

We then find that $\{u_{i}-u_{j}\}_{i,j\in I}$ defines a holomorphic section of $F$ on $U_{ij}:=U_{i}\cap U_{j}$. The isomorphism between Čech cohomology and Dolbeault cohomology implies that $\{u_{i}-u_{j}\}_{i,j\in I}$ represents the Čech cohomology class corresponding to $[v]$. Since $[v]$ is trivial, we find that $\{u_{i}-u_{j}\}_{i,j\in I}$ is a Čech $1$-coboundary (after replacing $U^{*}_{x}$ with a smaller polydisk if necessary), meaning that there exist local holomorphic sections $\{f_{j}\}_{j\in I}$ of $F$ such that

$$f_{i}-f_{j}=u_{i}-u_{j}$$

on $U_{ij}$ for all $i,j\in I$. We take a partition of unity $\{\rho_{j}\}_{j\in I}$ subordinate to $\{U_{j}\}_{j\in I}$. The equation $f_{i}-f_{j}=u_{i}-u_{j}$ on $U_{ij}$ for all $i,j\in I$ then implies

$$\sum_{j\in I}\rho_{j}(u_{i}-u_{j})=\sum_{j\in I}\rho_{j}(f_{i}-f_{j})$$

on $U_{i}$. Taking the $\bar{\partial}$-operator of both sides, we have

$$\bar{\partial}u_{i}-\bar{\partial}\left(\sum_{j\in I}\rho_{j}u_{j}\right)=-\bar{\partial}\left(\sum_{j\in I}\rho_{j}f_{j}\right)$$

on $U_{i}$, as $\sum_{j\in I}\rho_{j}\equiv 1$. Since $\bar{\partial}u_{i}=v|_{U_{i}}$, we get

$$v|_{U_{i}}=\bar{\partial}\left(\sum_{j\in I}\rho_{j}(u_{j}-f_{j})|_{U_{i}}\right).$$

Noting that $\sum_{j\in I}\rho_{j}(u_{j}-f_{j})$ is defined globally on $X$, we have

$$v=\bar{\partial}\left(\sum_{j\in I}\rho_{j}(u_{j}-f_{j})\right),$$

and hence $u:=\sum_{j\in I}\rho_{j}(u_{j}-f_{j})$ gives the solution to the equation $\bar{\partial}u=v$, which is necessarily unique since $F$ is flat and nontrivial (a well-known result, see e.g. [KH22]*Lemma 2.3 for the proof).

The argument above (except for the uniqueness of solutions) is valid for any $(0,q)$-forms $v$ with values in $F\otimes\mathcal{I}(h)$, where $h$ is a singular Hermitian metric on $F$ with positive curvature current (see [Mat]*Subsection 5.3 for details). In the following discussion, we essentially use the flatness property.

We apply Ueda’s lemma (Theorem 1.2) to the Čech $0$-cochain $\mathfrak{f}:=\{(U^{\prime}_{j},f_{j}|_{U^{\prime}_{j}})\}$ corresponding to the smaller cover $\mathcal{U}^{\prime}:=\{U^{\prime}_{j}\}_{j\in I}$, to obtain

(2)

$$\max_{j\in I}\sup_{U^{\prime}_{j}}|f_{j}|_{h}\leq\frac{K_{\mathcal{U}^{\prime}}}{\mathsf{d}(\mathbb{I}_{X},F)}\max_{j,k\in I}\sup_{U^{\prime}_{jk}}|f_{jk}|_{h}$$

uniformly for all $F\in\mathcal{P}(X)\setminus\{\mathbb{I}_{X}\}$, for some constant $K_{\mathcal{U}^{\prime}}>0$. Note that we have

(3)

$$\|f_{j}\|_{L^{2}(U^{\prime}_{j})}\leq\mathrm{Vol}(U^{\prime}_{j})^{1/2}\max_{j\in I}\sup_{U^{\prime}_{j}}|f_{j}|_{h}$$

for each $j\in I$, and

$$\max_{j,k\in I}\sup_{U^{\prime}_{jk}}|f_{jk}|_{h}=\max_{j,k\in I}\sup_{U^{\prime}_{jk}}|f_{j}-f_{k}|_{h}=\max_{j,k\in I}\sup_{U^{\prime}_{jk}}|u_{j}-u_{k}|_{h}$$

since $u_{j}-u_{k}=f_{j}-f_{k}$. Consider a weight $\varphi_{j}$ of $h$ on $U_{j}$ such that $h=e^{-\varphi_{j}}$. Since $\varphi_{j}$ is pluriharmonic, there exists a holomorphic function $w_{j}$ on $U_{j}$ such that $\varphi_{j}$ is the real part of $w_{j}$, which implies that $h=e^{-\varphi_{j}}=|e^{-w_{j}}|$. Given that $(u_{j}-u_{k})e^{-w_{j}}$ is holomorphic on $U_{jk}$ (which contains $U^{\prime}_{jk}$), we have

$$\sup_{U^{\prime}_{jk}}|u_{j}-u_{k}|_{h}\leq C^{(2)}_{j,g}\|u_{j}-u_{k}\|_{L^{2}(U_{jk}),h}\leq 2C^{(2)}_{j,g}\|u_{j}\|_{L^{2}(U_{jk}),h}$$

for some constant $C^{(2)}_{j,g}>0$, depending only on $U^{\prime}_{j}$, $U_{j}$, and $g$, by the mean value inequality and the Cauchy–Schwarz inequality. Hence

	$\displaystyle\max_{j,k\in I}\sup_{U^{\prime}_{jk}}|f_{jk}|_{h}=\max_{j,k\in I}\sup_{U^{\prime}_{jk}}|u_{j}-u_{k}|_{h}$	$\displaystyle\leq 2\max_{j,k\in I}C^{(2)}_{j,g}\|u_{j}\|_{L^{2}(U_{jk}),h}$
		$\displaystyle\leq 2\max_{j\in I}C^{(2)}_{j,g}\|u_{j}\|_{L^{2}(U_{j}),h}$
(4)			$\displaystyle\leq 2\max_{j\in I}C^{(2)}_{j,g}C^{(1)}_{j,g}\|v\|_{L^{2}(U_{j}),h}\leq C^{(3)}_{g}\|v\|_{L^{2}(X),h}$

by the Hörmander estimate (1) on $U_{j}$, where we set $C^{(3)}_{g}:=2\max_{j}C^{(2)}_{j,g}C^{(1)}_{j,g}>0$. With (2) and (3), the estimate (4) gives

(5)

$$\|f_{j}\|_{L^{2}(U_{j}),h}\leq\sum_{k\in I}\|f_{k}\|_{L^{2}(U^{\prime}_{k}),h}\leq\frac{K_{\mathcal{U}^{\prime}}C^{(3)}_{g}}{\mathsf{d}(\mathbb{I}_{X},F)}\left(\sum_{k\in I}\mathrm{Vol}(U^{\prime}_{k})^{1/2}\right)\|v\|_{L^{2}(X),h}$$

for any $j\in I$.

We now recall $u=\sum_{j\in I}\rho_{j}(u_{j}-f_{j})$ and evaluate

$$\|u\|_{L^{2}(X),h}\leq C^{(4)}_{g}\sum_{j\in I}\left(\|u_{j}\|_{L^{2}(U_{j}),h}+\|f_{j}\|_{L^{2}(U_{j}),h}\right)\leq C^{(4)}_{g}\sum_{j\in I}\left(C^{(1)}_{j,g}\|v\|_{L^{2}(X),h}+\|f_{j}\|_{L^{2}(U_{j}),h}\right)$$

again by the Hörmander estimate (1) on $U_{j}$, where $C^{(4)}_{g}>0$ is a constant which depends only on $g$. Combined with (5), we thus get

$$\|u\|_{L^{2}(X),h}\leq\frac{C_{g}^{(4)}}{\mathsf{d}(\mathbb{I}_{X},F)}\left(\sum_{j\in I}\left(C^{(1)}_{j,g}\mathsf{d}(\mathbb{I}_{X},F)+K_{\mathcal{U}^{\prime}}C^{(3)}_{g}\sum_{k\in I}\mathrm{Vol}(U^{\prime}_{k})^{1/2}\right)\right)\|v\|_{L^{2}(X),h}.$$

Noting that $\sup_{F\in\mathcal{P}(X)}\mathsf{d}(\mathbb{I}_{X},F)$ is finite and $I$ is a finite set, we get the required estimate. ∎