[proofplan] We prove very ampleness by constructing local holomorphic sections with prescribed zero-th or first-order jets, cutting them off, and correcting the resulting smooth global sections by solving a $\bar\partial$-equation. The correction is forced to vanish to the required order by inserting logarithmic singularities in the $L^2$ weight. The positivity of $L$ makes the high tensor powers $L^m$ sufficiently curved to absorb the bounded negative curvature introduced by the cutoffs in those singular weights. Finally, compactness of $X$ and of the relevant finite jet data gives one exponent $m_0$ working uniformly. [/proofplan] [step:Choose positive metrics and a uniform $\bar\partial$ estimate for large powers] Let $n := \dim_{\mathbb{C}} X$. Since $L$ is positive, choose a smooth Hermitian metric $h$ on $L$ whose Chern curvature form \begin{align*} \omega := \frac{i}{2\pi}\Theta(L,h) \end{align*} is a positive $(1,1)$-form on $X$. For $m \in \mathbb{N}$, let $h_m := h^{\otimes m}$ denote the induced Hermitian metric on $L^m$. We use the following standard analytic input: the external [Hörmander-Demailly $L^2$ $\bar\partial$ estimate with singular weights](https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/estimations_l2.pdf). The form $\omega$ is Kähler, because it is a positive closed $(1,1)$-form on the compact complex manifold $X$. Hence $(X,\omega)$ is a complete Kähler manifold, and the compact Kähler version of the Hörmander-Demailly estimate applies to the line bundle $L^m$ with metric $h_m$ and to quasi-plurisubharmonic weights whose curvature is bounded below in the sense of currents. A function $\psi:X\to[-\infty,\infty)$ is quasi-plurisubharmonic here means that locally it is the sum of a plurisubharmonic function and a smooth function. Its curvature inequality is understood in the sense of currents. The weighted norm of a measurable section $u$ of $L^m$ is \begin{align*} \|u\|_{m,\psi}^2 := \int_X |u(x)|_{h_m}^2 e^{-\psi(x)}\,dV_\omega(x), \end{align*} where $dV_\omega := \omega^n/n!$ is the smooth volume measure determined by $\omega$. The weighted norm of an $L^m$-valued $(0,1)$-form is defined using the same metric $h_m$, the Hermitian metric induced by $\omega$ on cotangent vectors, and the same measure $dV_\omega$. In the form needed here, the estimate says that if $\psi$ is quasi-plurisubharmonic and \begin{align*} \frac{i}{2\pi}\Theta(L^m,h_m) + \frac{i}{2\pi}\partial\bar\partial \psi \geq c\,\omega \end{align*} as currents for some constant $c>0$, then every smooth $L^m$-valued $(0,1)$-form $f$ satisfying $\bar\partial f=0$ and $\|f\|_{m,\psi}<\infty$ admits an $L^2_{\mathrm{loc}}$ section $u$ of $L^m$ satisfying \begin{align*} \bar\partial u = f \end{align*} in the distributional sense and $\|u\|_{m,\psi}<\infty$. The hypotheses required by the estimate are verified here as follows: compactness of $X$ gives completeness of the Kähler metric, quasi-plurisubharmonicity and the current lower bound are built into the weights below, and the forms to which the estimate is applied are smooth, $\bar\partial$-closed, and have finite weighted norm because their supports avoid the logarithmic poles. On every [open set](/page/Open%20Set) where $f$ is smooth, the standard local elliptic regularity for the $\bar\partial$-operator makes $u$ smooth; on every open set where $f=0$, the local Weyl lemma for $\bar\partial$ makes $u$ holomorphic. The weights used below have the local form $A\log |z-a|^2$ near their poles, plus smooth cutoff terms away from the poles. Such functions are quasi-plurisubharmonic because $\log |z-a|^2$ is plurisubharmonic and the remaining cutoff contribution is smooth. The final compactness step constructs these weights with a uniform lower bound for their smooth negative curvature part, so after increasing $m_0$ the curvature hypothesis holds for every later application. [guided] The proof needs one analytic theorem: the external [Hörmander-Demailly $L^2$ $\bar\partial$ estimate with singular weights](https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/estimations_l2.pdf). We choose a Hermitian metric $h$ on $L$ whose curvature is positive because positivity of the line bundle means precisely that such a metric exists. The form \begin{align*} \omega := \frac{i}{2\pi}\Theta(L,h) \end{align*} is positive and closed, so $\omega$ is a Kähler form on $X$. Since $X$ is compact, the Kähler metric determined by $\omega$ is complete. Thus the completeness hypothesis in the Hörmander-Demailly estimate is satisfied. For $m\in\mathbb{N}$, the induced metric on $L^m$ is \begin{align*} h_m := h^{\otimes m}, \end{align*} and its curvature is \begin{align*} \frac{i}{2\pi}\Theta(L^m,h_m)=m\omega. \end{align*} A quasi-plurisubharmonic weight is a function $\psi:X\to[-\infty,\infty)$ which is locally the sum of a plurisubharmonic function and a smooth function. Its curvature inequality is interpreted as an inequality of currents. For a measurable section $u$ of $L^m$, the weighted norm is \begin{align*} \|u\|_{m,\psi}^2 := \int_X |u(x)|_{h_m}^2 e^{-\psi(x)}\,dV_\omega(x), \end{align*} where $dV_\omega := \omega^n/n!$. The same metric $h_m$, the Hermitian metric induced by $\omega$ on cotangent vectors, and the same measure $dV_\omega$ define the weighted norm of $L^m$-valued $(0,1)$-forms. The theorem applies when \begin{align*} \frac{i}{2\pi}\Theta(L^m,h_m)+\frac{i}{2\pi}\partial\bar\partial\psi\geq c\omega \end{align*} for some $c>0$. Under this curvature hypothesis, every smooth $L^m$-valued $(0,1)$-form $f$ satisfying $\bar\partial f=0$ and $\|f\|_{m,\psi}<\infty$ has a solution $u\in L^2_{\mathrm{loc}}(X,L^m)$ such that \begin{align*} \bar\partial u=f \end{align*} in the distributional sense and $\|u\|_{m,\psi}<\infty$. The estimate is used as follows. We build a smooth section $\tilde{s}$ that is already holomorphic near the points or jets we want to protect, but not globally holomorphic. Its defect is \begin{align*} f := \bar\partial\tilde{s}. \end{align*} The support of $f$ is placed away from the logarithmic poles, so $\|f\|_{m,\psi}<\infty$. Also $\bar\partial f=\bar\partial^2\tilde{s}=0$. If the compactness step gives the curvature lower bound above, the Hörmander-Demailly estimate gives $u$ satisfying $\bar\partial u=f$. Then \begin{align*} s := \tilde{s}-u \end{align*} is holomorphic. On the neighbourhoods where $f=0$, the local Weyl lemma for $\bar\partial$ makes $u$ holomorphic, and the logarithmic integrability of $u$ forces the required vanishing order. This is why subtracting $u$ corrects global holomorphicity without changing the protected value or first jet. The only uniformity issue is curvature. Cutoff logarithmic weights have a bounded negative curvature contribution away from their logarithmic poles. Since the curvature of $L^m$ grows linearly in $m$, compactness of $X$ gives a single integer $m_0$ such that, for every $m\geq m_0$, the positive curvature of $L^m$ dominates those bounded errors for all one-pole and two-pole weights appearing in the finite-cover construction. [/guided] [/step] [step:Record the vanishing consequence of logarithmic integrability] Let $(U,\varphi)$ be a holomorphic coordinate chart on $X$ with coordinates $z=(z_1,\dots,z_n)$, and let $e:U \to L|_U$ be a nowhere-vanishing holomorphic frame. For a point $p \in U$, write $a := \varphi(p) \in \varphi(U) \subset \mathbb{C}^n$. [claim:Logarithmic weights force vanishing of holomorphic germs] Let $q \in \mathbb{N}$. Suppose $g:\varphi(U) \to \mathbb{C}$ is holomorphic near $a$ and, for some neighbourhood $V \subset \varphi(U)$ of $a$, \begin{align*} \int_V |g(z)|^2 |z-a|^{-2(n+q-1)}\,d\mathcal{L}^{2n}(z) < \infty. \end{align*} Then all partial derivatives $D^\alpha g(a)$ with $|\alpha|\leq q-1$ vanish. Equivalently, $g$ vanishes to order at least $q$ at $a$. [/claim] [proof] Write the Taylor expansion of $g$ at $a$ as \begin{align*} g(z)=P_k(z-a)+O(|z-a|^{k+1}), \end{align*} where $k \in \mathbb{N}\cup\{0\}$ is the vanishing order of $g$ at $a$ and $P_k$ is a nonzero homogeneous holomorphic polynomial of degree $k$, unless $g$ vanishes identically near $a$. If $g$ vanishes identically near $a$, the conclusion follows. Assume $P_k \neq 0$. Since $P_k$ is not identically zero on the unit sphere $S^{2n-1}\subset \mathbb{C}^n$, there is a relatively open subset $A \subset S^{2n-1}$ and a constant $c_A>0$ such that \begin{align*} |P_k(\theta)| \geq c_A \end{align*} for every $\theta \in A$. Hence, for sufficiently small $\rho>0$ and every $\theta \in A$, \begin{align*} |g(a+\rho\theta)|^2 \geq \frac{c_A^2}{4}\rho^{2k}. \end{align*} Let $\mathcal{H}^{2n-1}$ denote $(2n-1)$-dimensional [Hausdorff measure](/page/Hausdorff%20Measure) on the Euclidean unit sphere $S^{2n-1}\subset\mathbb{C}^n\cong\mathbb{R}^{2n}$. Using polar coordinates $z=a+\rho\theta$, where the polar-coordinate formula for Lebesgue measure gives \begin{align*} d\mathcal{L}^{2n}(z)=\rho^{2n-1}\,d\mathcal{L}^1(\rho)\,d\mathcal{H}^{2n-1}(\theta), \end{align*} the weighted integral over the cone $\{a+\rho\theta:0<\rho<\varepsilon,\ \theta\in A\}$ is bounded below by a positive constant times \begin{align*} \int_0^\varepsilon \rho^{2k}\rho^{-2(n+q-1)}\rho^{2n-1}\,d\mathcal{L}^1(\rho) = \int_0^\varepsilon \rho^{2k-2q+1}\,d\mathcal{L}^1(\rho). \end{align*} This one-dimensional integral is finite only if $2k-2q+1>-1$, equivalently $k\geq q$. Therefore $g$ vanishes to order at least $q$ at $a$. [/proof] For $q=1$, the weight $|z-a|^{-2n}$ forces a holomorphic correction to vanish at $p$. For $q=2$, the weight $|z-a|^{-2(n+1)}$ forces the correction to vanish to second order at $p$. [guided] We prove the local vanishing statement directly. Let $q\in\mathbb{N}$, let $g:\varphi(U)\to\mathbb{C}$ be holomorphic near $a$, and suppose there is a neighbourhood $V\subset\varphi(U)$ of $a$ such that \begin{align*} \int_V |g(z)|^2 |z-a|^{-2(n+q-1)}\,d\mathcal{L}^{2n}(z) < \infty. \end{align*} The goal is to show that every partial derivative $D^\alpha g(a)$ with $|\alpha|\leq q-1$ vanishes. Write the Taylor expansion of $g$ at $a$. If $g$ vanishes identically near $a$, then all derivatives at $a$ vanish and there is nothing more to prove. Otherwise, let $k\in\mathbb{N}\cup\{0\}$ be the first degree for which the Taylor expansion has a nonzero homogeneous term, and denote that term by a nonzero homogeneous holomorphic polynomial $P_k:\mathbb{C}^n\to\mathbb{C}$. Then \begin{align*} g(z)=P_k(z-a)+O(|z-a|^{k+1}) \end{align*} as $z\to a$. Why does one look at directions on the sphere? Because homogeneity converts the leading term into a power of the radius. The polynomial $P_k$ is not identically zero on the unit sphere $S^{2n-1}\subset\mathbb{C}^n\cong\mathbb{R}^{2n}$. Hence there are a relatively open set $A\subset S^{2n-1}$ and a constant $c_A>0$ such that \begin{align*} |P_k(\theta)|\geq c_A \end{align*} for every $\theta\in A$. Since the remainder is $O(|z-a|^{k+1})$, after decreasing $\varepsilon>0$ if necessary, for every $0<\rho<\varepsilon$ and every $\theta\in A$ we have \begin{align*} |g(a+\rho\theta)|^2\geq \frac{c_A^2}{4}\rho^{2k}. \end{align*} Let $\mathcal{H}^{2n-1}$ denote $(2n-1)$-dimensional Hausdorff measure on $S^{2n-1}$. The polar-coordinate formula for the substitution $z=a+\rho\theta$ gives \begin{align*} d\mathcal{L}^{2n}(z)=\rho^{2n-1}\,d\mathcal{L}^1(\rho)\,d\mathcal{H}^{2n-1}(\theta). \end{align*} Restricting the original finite integral to the cone $\{a+\rho\theta:0<\rho<\varepsilon,\ \theta\in A\}$ is legitimate because the integrand is nonnegative. On this cone, the previous lower bound gives a positive constant $C_A := c_A^2\mathcal{H}^{2n-1}(A)/4$ such that the weighted integral is bounded below by \begin{align*} C_A\int_0^\varepsilon \rho^{2k}\rho^{-2(n+q-1)}\rho^{2n-1}\,d\mathcal{L}^1(\rho) = C_A\int_0^\varepsilon \rho^{2k-2q+1}\,d\mathcal{L}^1(\rho). \end{align*} The one-dimensional integral $\int_0^\varepsilon \rho^\beta\,d\mathcal{L}^1(\rho)$ is finite exactly when $\beta>-1$. Therefore finiteness forces \begin{align*} 2k-2q+1>-1, \end{align*} which is equivalent to $k\geq q$. Thus every Taylor coefficient of degree at most $q-1$ is zero, meaning $D^\alpha g(a)=0$ for all multi-indices $\alpha$ with $|\alpha|\leq q-1$. For $q=1$, the weight $|z-a|^{-2n}$ forces a holomorphic correction to vanish at the point. For $q=2$, the weight $|z-a|^{-2(n+1)}$ forces both the value and all first derivatives to vanish there. [/guided] [/step] [step:Construct sections separating a fixed pair of points] Fix distinct points $p,r\in X$. Choose a holomorphic coordinate chart $(U,\varphi)$ containing $p$. If $r$ is sufficiently close to $p$, choose $U$ so that $r\in U$ as well. If $r\notin U$, choose a relatively compact open set $U'\Subset U$ with $p\in U'$ and $r\notin \overline{U'}$. Choose a nowhere-vanishing holomorphic frame $e:U\to L|_U$. Choose a smooth cutoff function $\eta:X\to[0,1]$ such that $\operatorname{supp}\eta\subset U$; in the case $r\in U$ require $\eta=1$ on neighbourhoods of both $p$ and $r$, while in the case $r\notin U$ require $\eta=1$ on a neighbourhood of $p$ and $\operatorname{supp}\eta\subset U'$. If $r\in U$, choose a coordinate component $z_j$ such that $z_j(p)\neq z_j(r)$ and define the local [holomorphic function](/page/Holomorphic%20Function) \begin{align*} g_{p,r}:\varphi(U)&\to\mathbb{C}\\ z&\mapsto z_j-z_j(r). \end{align*} If $r\notin U$, define \begin{align*} g_{p,r}:\varphi(U)&\to\mathbb{C}\\ z&\mapsto 1. \end{align*} In both cases, the smooth global section $\tilde{s}_{p,r}$ of $L^m$ is defined by \begin{align*} \tilde{s}_{p,r}(x) := \begin{cases} \eta(x)\,g_{p,r}(\varphi(x))\,e(x)^{\otimes m}, & x\in U,\\ 0, & x\in X\setminus \operatorname{supp}\eta. \end{cases} \end{align*} Then $\tilde{s}_{p,r}$ is holomorphic near $p$ and $r$, satisfies $\tilde{s}_{p,r}(p)\neq 0$, and satisfies $\tilde{s}_{p,r}(r)=0$. Choose coordinate balls around $p$ and $r$ with smooth logarithmic cutoffs as in the compactness step below, and define a singular weight $\psi_{p,r}$ which equals \begin{align*} n\log |\varphi_p(x)-\varphi_p(p)|^2 \end{align*} near $p$ in a chart $(U_p,\varphi_p)$ and equals \begin{align*} n\log |\varphi_r(x)-\varphi_r(r)|^2 \end{align*} near $r$ in a chart $(U_r,\varphi_r)$. Away from those two neighbourhoods, $\psi_{p,r}$ is extended by smooth cutoff terms. This weight is quasi-plurisubharmonic because each logarithmic term is plurisubharmonic in its local coordinates and the remaining terms are smooth. Since $f_{p,r}:=\bar\partial\tilde{s}_{p,r}$ is supported where the cutoff $\eta$ is nonconstant, it is supported away from $p$ and $r$; hence its weighted $L^2$ norm is finite. Also \begin{align*} \bar\partial f_{p,r}=\bar\partial^2\tilde{s}_{p,r}=0, \end{align*} so the $\bar\partial$-closedness hypothesis in the weighted estimate is satisfied. For $m\geq m_0$, the curvature lower bound verified in the compactness step gives the remaining hypothesis of the weighted $\bar\partial$ estimate. Hence there exists an $L^2_{\mathrm{loc}}$ section $u_{p,r}$ of $L^m$ such that \begin{align*} \bar\partial u_{p,r}=f_{p,r} \end{align*} and $u_{p,r}$ has finite weighted norm with respect to $h_m e^{-\psi_{p,r}}$. On neighbourhoods of $p$ and $r$ the form $f_{p,r}$ is zero, so the local Weyl lemma for $\bar\partial$ implies that $u_{p,r}$ is holomorphic there. The previous step with $q=1$ implies \begin{align*} u_{p,r}(p)=0, \qquad u_{p,r}(r)=0. \end{align*} Therefore \begin{align*} s_{p,r}:=\tilde{s}_{p,r}-u_{p,r} \end{align*} is a holomorphic global section of $L^m$ satisfying \begin{align*} s_{p,r}(p)=\tilde{s}_{p,r}(p)\neq 0, \qquad s_{p,r}(r)=\tilde{s}_{p,r}(r)=0. \end{align*} Thus $H^0(X,L^m)$ separates the fixed pair $p,r$. [guided] Fix distinct points $p,r\in X$. Choose a holomorphic coordinate chart $(U,\varphi)$ containing $p$. If $r$ is sufficiently close to $p$, choose $U$ so that $r\in U$. If $r\notin U$, choose a relatively compact open set $U'\Subset U$ with $p\in U'$ and $r\notin\overline{U'}$. Choose a nowhere-vanishing holomorphic frame \begin{align*} e:U\to L|_U. \end{align*} Choose a smooth cutoff function $\eta:X\to[0,1]$ with $\operatorname{supp}\eta\subset U$. In the case $r\in U$, require $\eta=1$ on neighbourhoods of both $p$ and $r$. In the case $r\notin U$, require $\eta=1$ on a neighbourhood of $p$ and $\operatorname{supp}\eta\subset U'$. If $r\in U$, the coordinate map separates the two distinct coordinate points. Choose an index $j\in\{1,\dots,n\}$ such that $z_j(p)\neq z_j(r)$, and define \begin{align*} g_{p,r}:\varphi(U)&\to\mathbb{C}\\ z&\mapsto z_j-z_j(r). \end{align*} Then $g_{p,r}(\varphi(p))\neq0$ and $g_{p,r}(\varphi(r))=0$. If $r\notin U$, define instead \begin{align*} g_{p,r}:\varphi(U)&\to\mathbb{C}\\ z&\mapsto 1. \end{align*} The cutoff-supported section will vanish near $r$ in this second case because its support avoids $r$. Define the smooth global section $\tilde{s}_{p,r}$ of $L^m$ by \begin{align*} \tilde{s}_{p,r}(x) := \begin{cases} \eta(x)\,g_{p,r}(\varphi(x))\,e(x)^{\otimes m}, & x\in U,\\ 0, & x\in X\setminus \operatorname{supp}\eta. \end{cases} \end{align*} This section is holomorphic near $p$ and $r$, because on those neighbourhoods the cutoff is constant and the local coefficient is holomorphic. It satisfies \begin{align*} \tilde{s}_{p,r}(p)\neq0, \qquad \tilde{s}_{p,r}(r)=0. \end{align*} Let $\psi_{p,r}$ be a two-pole weight equal near $p$ to \begin{align*} n\log |\varphi_p(x)-\varphi_p(p)|^2 \end{align*} in a chart $(U_p,\varphi_p)$ and equal near $r$ to \begin{align*} n\log |\varphi_r(x)-\varphi_r(r)|^2 \end{align*} in a chart $(U_r,\varphi_r)$. Away from the pole neighbourhoods it is obtained by the smooth cutoff construction in the compactness step. This is a quasi-plurisubharmonic weight: the logarithmic terms are plurisubharmonic in their local coordinates, and the remaining cutoff terms are smooth. Set \begin{align*} f_{p,r}:=\bar\partial\tilde{s}_{p,r}. \end{align*} Because $\eta$ is constant near $p$ and $r$, the support of $f_{p,r}$ is disjoint from neighbourhoods of both logarithmic poles. Hence $\|f_{p,r}\|_{m,\psi_{p,r}}<\infty$. Also \begin{align*} \bar\partial f_{p,r}=\bar\partial^2\tilde{s}_{p,r}=0, \end{align*} so the $\bar\partial$-closedness hypothesis in the weighted estimate is satisfied. For $m\geq m_0$, the compactness step gives \begin{align*} \frac{i}{2\pi}\Theta(L^m,h_m)+\frac{i}{2\pi}\partial\bar\partial\psi_{p,r}\geq c_m\omega \end{align*} for a constant $c_m>0$. The Hörmander-Demailly estimate therefore gives an $L^2_{\mathrm{loc}}$ section $u_{p,r}$ of $L^m$ satisfying \begin{align*} \bar\partial u_{p,r}=f_{p,r} \end{align*} and $\|u_{p,r}\|_{m,\psi_{p,r}}<\infty$. On neighbourhoods of $p$ and $r$, the form $f_{p,r}$ is zero. The local Weyl lemma for $\bar\partial$ implies that $u_{p,r}$ is holomorphic there. In local frames near $p$ and $r$, the finite weighted norm gives exactly the integrability condition in the logarithmic vanishing lemma with $q=1$. Therefore \begin{align*} u_{p,r}(p)=0, \qquad u_{p,r}(r)=0. \end{align*} Define \begin{align*} s_{p,r}:=\tilde{s}_{p,r}-u_{p,r}. \end{align*} Then $\bar\partial s_{p,r}=0$, so $s_{p,r}\in H^0(X,L^m)$, and the protected values are unchanged: \begin{align*} s_{p,r}(p)=\tilde{s}_{p,r}(p)\neq0, \qquad s_{p,r}(r)=\tilde{s}_{p,r}(r)=0. \end{align*} Thus $H^0(X,L^m)$ separates the fixed pair $p,r$. [/guided] [/step] [step:Construct sections separating a fixed tangent vector] Fix $p\in X$ and a nonzero vector $v\in T_pX$. Choose a holomorphic coordinate chart $(U,\varphi)$ around $p$ with coordinates $z=(z_1,\dots,z_n)$ and choose a nowhere-vanishing holomorphic frame $e:U\to L|_U$. Since $d\varphi_p(v)\neq 0$ in $T_{\varphi(p)}\mathbb{C}^n$, there is a complex linear functional $\lambda:T_{\varphi(p)}\mathbb{C}^n\to\mathbb{C}$ such that \begin{align*} \lambda(d\varphi_p(v))\neq 0. \end{align*} Define the local holomorphic function \begin{align*} g_{p,v}:\varphi(U)&\to\mathbb{C}\\ z&\mapsto \lambda(z-\varphi(p)). \end{align*} Choose a smooth cutoff function $\eta:X\to[0,1]$ with $\eta=1$ near $p$ and $\operatorname{supp}\eta\subset U$, and define the smooth global section \begin{align*} \tilde{s}_{p,v}(x):= \begin{cases} \eta(x)\,g_{p,v}(\varphi(x))\,e(x)^{\otimes m}, & x\in U,\\ 0, & x\in X\setminus \operatorname{supp}\eta. \end{cases} \end{align*} Then $\tilde{s}_{p,v}$ is holomorphic near $p$, satisfies $\tilde{s}_{p,v}(p)=0$, and has first derivative \begin{align*} d(g_{p,v}\circ \varphi)_p(v)=\lambda(d\varphi_p(v))\neq 0. \end{align*} Choose a singular weight $\psi_{p,v}$ equal near $p$ to \begin{align*} (n+1)\log |\varphi(x)-\varphi(p)|^2 \end{align*} with the smooth logarithmic cutoff constructed in the compactness step below. This weight is quasi-plurisubharmonic because the logarithmic term is plurisubharmonic in the chart and the cutoff contribution is smooth. Since $f_{p,v}:=\bar\partial\tilde{s}_{p,v}$ is supported where $\eta$ is nonconstant, it is supported away from $p$, so its weighted norm is finite. Also \begin{align*} \bar\partial f_{p,v}=\bar\partial^2\tilde{s}_{p,v}=0, \end{align*} so the $\bar\partial$-closedness hypothesis is satisfied. For $m\geq m_0$, the curvature lower bound verified in the compactness step gives the curvature hypothesis, and the weighted $\bar\partial$ estimate gives an $L^2_{\mathrm{loc}}$ section $u_{p,v}$ satisfying \begin{align*} \bar\partial u_{p,v}=f_{p,v} \end{align*} with $u_{p,v}$ finite in the weighted norm. Since $f_{p,v}=0$ near $p$, the local Weyl lemma for $\bar\partial$ implies that $u_{p,v}$ is holomorphic near $p$. The vanishing lemma with $q=2$ implies that $u_{p,v}$ vanishes to order at least two at $p$. Hence, in the frame $e^{\otimes m}$, the value and first derivative of $u_{p,v}$ at $p$ are both zero. Define \begin{align*} s_{p,v}:=\tilde{s}_{p,v}-u_{p,v}. \end{align*} Write $s_{p,v}=\sigma_{p,v}\,e^{\otimes m}$ and $\tilde{s}_{p,v}=\tilde{\sigma}_{p,v}\,e^{\otimes m}$ near $p$, where $\sigma_{p,v}$ and $\tilde{\sigma}_{p,v}$ are holomorphic scalar functions on a neighbourhood of $p$. Then $s_{p,v}\in H^0(X,L^m)$, $s_{p,v}(p)=0$, and \begin{align*} d\sigma_{p,v,p}(v)=d\tilde{\sigma}_{p,v,p}(v)=\lambda(d\varphi_p(v))\neq 0. \end{align*} Thus $H^0(X,L^m)$ separates the tangent vector $v$ at $p$. [guided] Fix $p\in X$ and a nonzero tangent vector $v\in T_pX$. Choose a holomorphic coordinate chart $(U,\varphi)$ around $p$ with coordinates $z=(z_1,\dots,z_n)$ and choose a nowhere-vanishing holomorphic frame \begin{align*} e:U\to L|_U. \end{align*} The differential of the chart is the complex-linear isomorphism \begin{align*} d\varphi_p:T_pX\to T_{\varphi(p)}\mathbb{C}^n. \end{align*} Because $v\neq0$, the vector $d\varphi_p(v)$ is nonzero. Choose a complex linear functional \begin{align*} \lambda:T_{\varphi(p)}\mathbb{C}^n\to\mathbb{C} \end{align*} with \begin{align*} \lambda(d\varphi_p(v))\neq0. \end{align*} Define the local holomorphic function \begin{align*} g_{p,v}:\varphi(U)&\to\mathbb{C}\\ z&\mapsto \lambda(z-\varphi(p)). \end{align*} Then $g_{p,v}(\varphi(p))=0$, while \begin{align*} d(g_{p,v}\circ\varphi)_p(v)=\lambda(d\varphi_p(v))\neq0. \end{align*} Choose a smooth cutoff function $\eta:X\to[0,1]$ such that $\eta=1$ near $p$ and $\operatorname{supp}\eta\subset U$. Define \begin{align*} \tilde{s}_{p,v}(x):= \begin{cases} \eta(x)\,g_{p,v}(\varphi(x))\,e(x)^{\otimes m}, & x\in U,\\ 0, & x\in X\setminus\operatorname{supp}\eta. \end{cases} \end{align*} This smooth section is holomorphic near $p$, satisfies $\tilde{s}_{p,v}(p)=0$, and has nonzero scalar first derivative in the direction $v$ in the frame $e^{\otimes m}$. Set \begin{align*} f_{p,v}:=\bar\partial\tilde{s}_{p,v}. \end{align*} The support of $f_{p,v}$ is contained where $\eta$ is nonconstant, so it is disjoint from a neighbourhood of $p$. Use the singular weight $\psi_{p,v}$ equal near $p$ to \begin{align*} (n+1)\log |\varphi(x)-\varphi(p)|^2 \end{align*} with the smooth logarithmic cutoff from the compactness step. This weight is quasi-plurisubharmonic because the logarithmic term is plurisubharmonic in the chart and the cutoff contribution is smooth. Since $f_{p,v}$ is supported away from the pole, $\|f_{p,v}\|_{m,\psi_{p,v}}<\infty$, and \begin{align*} \bar\partial f_{p,v}=\bar\partial^2\tilde{s}_{p,v}=0. \end{align*} For $m\geq m_0$, the compactness step gives the curvature lower bound required by the Hörmander-Demailly estimate. Hence there is an $L^2_{\mathrm{loc}}$ section $u_{p,v}$ of $L^m$ satisfying \begin{align*} \bar\partial u_{p,v}=f_{p,v} \end{align*} and $\|u_{p,v}\|_{m,\psi_{p,v}}<\infty$. Because $f_{p,v}=0$ near $p$, the local Weyl lemma for $\bar\partial$ implies that $u_{p,v}$ is holomorphic near $p$. In the frame $e^{\otimes m}$, the finite weighted norm and the logarithmic vanishing lemma with $q=2$ imply that the scalar coefficient of $u_{p,v}$ vanishes to order at least two at $p$. Therefore both its value and first derivative at $p$ are zero. Define \begin{align*} s_{p,v}:=\tilde{s}_{p,v}-u_{p,v}. \end{align*} Then $\bar\partial s_{p,v}=0$, so $s_{p,v}\in H^0(X,L^m)$, and $s_{p,v}(p)=0$. Writing $s_{p,v}=\sigma_{p,v}e^{\otimes m}$ near $p$, we obtain \begin{align*} d\sigma_{p,v,p}(v)=\lambda(d\varphi_p(v))\neq0. \end{align*} Thus $H^0(X,L^m)$ separates the tangent vector $v$ at $p$. [/guided] [/step] [step:Use compactness to make the exponent independent of the point and vector] Choose finitely many holomorphic coordinate charts $(U_i,\varphi_i)$ and relatively compact open sets $U_i'\Subset U_i$ such that the sets $U_i'$ cover $X$. Shrink the $U_i'$ if necessary so that there is a number $\rho_i>0$ with the following property: for every $a\in\varphi_i(U_i')$, the Euclidean ball $B(a,4\rho_i)$ is contained in $\varphi_i(U_i)$. Choose holomorphic frames $e_i:U_i\to L|_{U_i}$. Fix, for each $i$, a smooth function $\beta_i:[0,\infty)\to[0,1]$ such that $\beta_i(t)=1$ for $0\leq t\leq \rho_i^2$ and $\beta_i(t)=0$ for $t\geq 4\rho_i^2$. For a pole $p\in U_i'$ and an integer $A\in\{n,n+1\}$, define the local singular model \begin{align*} \Psi_{i,p,A}(x):=A\,\beta_i(|\varphi_i(x)-\varphi_i(p)|^2)\log |\varphi_i(x)-\varphi_i(p)|^2 \end{align*} on $U_i$, and extend it by $0$ outside the support of the cutoff. Near $p$ this equals $A\log |\varphi_i(x)-\varphi_i(p)|^2$. On the annulus where the cutoff varies, the point $\varphi_i(p)$ stays a distance between $\rho_i$ and $2\rho_i$ from $\varphi_i(x)$, so all first and second coordinate derivatives of the smooth cutoff part are bounded by a constant depending only on $i$, $A$, $\rho_i$, and the fixed cutoff $\beta_i$, not on $p$. Convert this coordinate bound into a global form bound as follows. On each compact set $\overline{U_i}$ supporting the cutoff, the Hermitian form $\omega$ is uniformly comparable with the Euclidean Hermitian form in the chart $(U_i,\varphi_i)$. Hence any smooth real $(1,1)$-form whose coordinate coefficients are bounded below by a fixed Euclidean constant is bounded below by $-C_i\omega$ for some $C_i>0$. Taking the maximum over finitely many $i$ and over $A\in\{n,n+1\}$ gives a constant $C_1>0$ such that every one-pole model satisfies \begin{align*} \frac{i}{2\pi}\partial\bar\partial\Psi_{i,p,A}\geq -C_1\omega \end{align*} as currents. The logarithmic part contributes a positive current, and $-C_1\omega$ bounds only the smooth cutoff contribution. The tangent-vector weights are exactly one-pole weights with $A=n+1$. The point-separation weights are sums of at most two one-pole weights with $A=n$: if the two cutoff supports overlap, the current lower bounds add; if they are disjoint, the same inequality holds on each support and hence globally. Therefore every two-pole point-separation weight $\psi_{p,r}$ satisfies \begin{align*} \frac{i}{2\pi}\partial\bar\partial\psi_{p,r}\geq -2C_1\omega, \end{align*} and every one-pole tangent weight satisfies the stronger bound with $-C_1\omega$. Set $C_0:=2C_1$. Since \begin{align*} \frac{i}{2\pi}\Theta(L^m,h_m)=m\omega, \end{align*} choosing $m_0>C_0$ gives, for every $m\geq m_0$ and every weight $\psi$ used above, \begin{align*} \frac{i}{2\pi}\Theta(L^m,h_m)+\frac{i}{2\pi}\partial\bar\partial\psi\geq (m-C_0)\omega. \end{align*} Thus the curvature hypothesis in the weighted $\bar\partial$ estimate holds uniformly for every point-separation and tangent-separation construction. It remains to make the point-pair cutoff choice uniform. Because the sets $U_i'$ form a finite open cover of the compact [metric space](/page/Metric%20Space) $X$, choose a Lebesgue number $\delta>0$ for this cover with respect to the distance induced by $\omega$. Also shrink the local coordinate balls used in each $U_i$ so that their closures remain in $U_i$. If $p\neq r$ and $d_\omega(p,r)<\delta$, then $p$ and $r$ lie in some common $U_i'\Subset U_i$, so the construction uses that one chart and a coordinate component separating $\varphi_i(p)$ from $\varphi_i(r)$. If $d_\omega(p,r)\geq\delta$, choose $i$ with $p\in U_i'$ and choose a cutoff supported in a coordinate ball around $p$ of $\omega$-radius less than $\delta/3$; this support avoids $r$, so the constant local function $1$ gives point separation. These choices use only the finite family of charts, fixed radii, and fixed cutoff profiles already controlled above. Hence, for all $m\geq m_0$, the estimates produce separating sections for every pair of distinct points and every nonzero tangent vector. Consequently $L^m$ separates points and tangent vectors for every $m\geq m_0$. [guided] We must justify that the lower bound for the singular weights does not deteriorate as the pole moves, and that the same finite construction covers every pair of distinct points. Choose finitely many charts $(U_i,\varphi_i)$ with compactly contained subcharts $U_i'\Subset U_i$ covering $X$. Since each $\varphi_i(U_i')$ is compactly contained in $\varphi_i(U_i)$, choose $\rho_i>0$ so that \begin{align*} B(a,4\rho_i)\subset\varphi_i(U_i) \end{align*} for every $a\in\varphi_i(U_i')$. For $A\in\{n,n+1\}$ and $p\in U_i'$, define \begin{align*} \Psi_{i,p,A}(x)=A\,\beta_i(|\varphi_i(x)-\varphi_i(p)|^2)\log |\varphi_i(x)-\varphi_i(p)|^2. \end{align*} Near $p$, where $\beta_i=1$, this is exactly the logarithmic pole needed for the vanishing lemma. Where $\beta_i$ changes, the distance $|\varphi_i(x)-\varphi_i(p)|$ is bounded below by $\rho_i$ and above by $2\rho_i$. Therefore the derivatives of $\log |\varphi_i(x)-\varphi_i(p)|^2$ and of the cutoff factor are bounded uniformly in $p$. This coordinate derivative bound must be compared with $\omega$. On each compact support of a cutoff inside $U_i$, the Euclidean Hermitian form in the chart and the Hermitian form $\omega$ are uniformly comparable. Thus a lower bound for the coordinate coefficients of the smooth cutoff curvature gives a lower bound by $-C_i\omega$. Taking the maximum over finitely many charts and over $A=n,n+1$ gives $C_1>0$ with \begin{align*} \frac{i}{2\pi}\partial\bar\partial\Psi_{i,p,A}\geq -C_1\omega \end{align*} as currents for every allowed pole. The singular logarithmic part is plurisubharmonic and contributes a positive current; only the smooth cutoff annulus can contribute negatively. The tangent-vector construction uses one pole, so it is covered by this bound. The point-separation construction may use two poles. In that case define the weight as the sum of the two one-pole models, one centered at $p$ and one centered at $r$, each with coefficient $A=n$. If their cutoff annuli overlap, current inequalities still add; if they do not overlap, the same sum inequality holds support by support. Hence every two-pole weight satisfies \begin{align*} \frac{i}{2\pi}\partial\bar\partial\psi_{p,r}\geq -2C_1\omega. \end{align*} Set $C_0:=2C_1$. Since \begin{align*} \frac{i}{2\pi}\Theta(L^m,h_m)=m\omega, \end{align*} we get, for every weight $\psi$ used in the proof, \begin{align*} \frac{i}{2\pi}\Theta(L^m,h_m)+\frac{i}{2\pi}\partial\bar\partial\psi \geq (m-C_0)\omega. \end{align*} Choose $m_0>C_0$. Then for every $m\geq m_0$ the right-hand side is a positive multiple of $\omega$, so the Hörmander-Demailly estimate applies uniformly. Finally, we verify that every pair of distinct points falls into a controlled local construction. The finite cover $(U_i')$ of the compact metric space $(X,d_\omega)$ has a Lebesgue number $\delta>0$. If $d_\omega(p,r)<\delta$, then $p$ and $r$ lie in a common $U_i'\Subset U_i$, so one chart contains both points and some coordinate component separates $\varphi_i(p)$ from $\varphi_i(r)$. If $d_\omega(p,r)\geq\delta$, choose $i$ with $p\in U_i'$ and take the cutoff support inside a coordinate ball around $p$ of $\omega$-radius less than $\delta/3$; this support cannot contain $r$, so the local constant function $1$ separates the pair after cutoff. The charts, radii, cutoff profiles, and curvature constants all come from finite data. This proves that one exponent threshold works for all points, pairs of points, and tangent vectors. [/guided] [/step]