[proofplan] We prove very ampleness by showing that sufficiently high powers of the positive line bundle separate both points and tangent vectors. The analytic input is the Kodaira vanishing theorem, applied on blow-ups of $X$ at one point or at two points. A standard uniform blow-up positivity lemma, equivalently the compactness of the length-two Hilbert-scheme parameter space with uniform local metric estimates, ensures that the relevant auxiliary line bundles on all such blow-ups are positive once the tensor power is large. The resulting cohomology vanishing makes the restriction maps to first-order jets surjective, and the standard jet-separation criterion then identifies this surjectivity with very ampleness and with the Kodaira map being an embedding. [/proofplan]

[step:Choose a uniform tensor power that survives the required blow-ups]Let $n := \dim_{\mathbb{C}} X$. For each point $x \in X$, let \begin{align*} \pi_x: \widetilde X_x &\to X \end{align*} denote the blow-up of $X$ at $x$, and let $E_x \subset \widetilde X_x$ denote its exceptional divisor. For each ordered pair of distinct points $(x,y) \in X \times X$ with $x \neq y$, let \begin{align*} \pi_{x,y}: \widetilde X_{x,y} &\to X \end{align*} denote the blow-up of $X$ at the reduced set $\{x,y\}$, and let $E_{x,y} \subset \widetilde X_{x,y}$ denote the sum of the two exceptional divisors. Since $L \to X$ is positive, choose a Hermitian metric $h$ whose curvature form \begin{align*} \omega:=\frac{i}{2\pi}\Theta_h(L) \end{align*} is positive. We invoke the following external auxiliary theorem (citing a result not yet in the wiki: relative exceptional-twist positivity for compact blow-up families). Let $B\to X$ be any fixed holomorphic line bundle. For the compact complex manifold $X$, the positive line bundle $L$, and the integers $n+1$ and $n$, there exists $m_0\in\mathbb N$ such that for every $m\ge m_0$, every $x\in X$, and every pair $x\ne y$, the line bundles \begin{align*} \pi_x^*(B\otimes L^m)\otimes\mathcal O_{\widetilde X_x}(-(n+1)E_x), \qquad \pi_{x,y}^*(B\otimes L^m)\otimes\mathcal O_{\widetilde X_{x,y}}(-nE_{x,y}) \end{align*} are positive. The cited theorem is precisely the relative compactness and mixed-curvature estimate needed here: it treats the universal one-point blow-up over $X$ and the universal length-two blow-up over the compact Douady parameter space, proves the positivity of the exceptional twists in directions contracted by the blow-down, and uses the positive curvature of $L^m$ to dominate the remaining directions uniformly. Thus the single integer $m_0$ in its conclusion works simultaneously for all points and all distinct pairs. Apply this auxiliary result with $B=K_X^{-1}$. Thus, after replacing $m_0$ by the integer supplied by the auxiliary result, for every $m \ge m_0$, every $x \in X$, and every pair $x\neq y$, the holomorphic line bundles \begin{align*} \mathcal A_{x,m} &:= \pi_x^*(K_X^{-1}\otimes L^m) \otimes \mathcal O_{\widetilde X_x}(-(n+1)E_x),\\ \mathcal A_{x,y,m} &:= \pi_{x,y}^*(K_X^{-1}\otimes L^m) \otimes \mathcal O_{\widetilde X_{x,y}}(-nE_{x,y}) \end{align*} are positive. The cited theorem avoids treating $X\times X\setminus\Delta$ as compact: the universal family over the compact length-two Douady parameter space supplies uniform curvature bounds for reduced pairs near the diagonal, while the one-point universal blow-up supplies the separate first-jet positivity estimate. In particular, the positivity needed below is arranged for the Kodaira-vanishing auxiliary bundles \begin{align*} \mathcal A_{x,m}, \qquad \mathcal A_{x,y,m}. \end{align*}[/step]

[guided]We repeat the uniform positivity argument in detail. Let $n=\dim_{\mathbb C}X$. For $x\in X$, let \begin{align*} \pi_x: \widetilde X_x &\to X \end{align*} be the blow-up of $X$ at $x$, with exceptional divisor $E_x$. For distinct $x,y\in X$, let \begin{align*} \pi_{x,y}: \widetilde X_{x,y} &\to X \end{align*} be the blow-up at the reduced set $\{x,y\}$, with exceptional divisor $E_{x,y}$ equal to the sum of the two exceptional components. Choose a Hermitian metric $h$ on $L$ with positive curvature \begin{align*} \omega:=\frac{i}{2\pi}\Theta_h(L). \end{align*} The external auxiliary theorem says this: for every fixed holomorphic line bundle $B\to X$, there is $m_0\in\mathbb N$ such that, for all $m\ge m_0$, \begin{align*} \pi_x^*(B\otimes L^m)\otimes\mathcal O_{\widetilde X_x}(-(n+1)E_x) \end{align*} is positive for every $x\in X$, and \begin{align*} \pi_{x,y}^*(B\otimes L^m)\otimes\mathcal O_{\widetilde X_{x,y}}(-nE_{x,y}) \end{align*} is positive for every $x\ne y$. The theorem includes the construction of the universal one-point and length-two blow-up families, the relative exceptional divisors, and the mixed horizontal-vertical curvature estimate. Its hypotheses are satisfied because $X$ is compact complex, $L$ has the positive curvature form $\omega$, and $B$ is fixed. Thus we may choose $m_0 \in \mathbb N$ so that, for all $m \ge m_0$, the line bundles \begin{align*} \mathcal A_{x,m} = \pi_x^*(K_X^{-1}\otimes L^m)\otimes \mathcal O_{\widetilde X_x}(-(n+1)E_x), \qquad \mathcal A_{x,y,m} = \pi_{x,y}^*(K_X^{-1}\otimes L^m)\otimes \mathcal O_{\widetilde X_{x,y}}(-nE_{x,y}) \end{align*} are positive for every allowed $x$ and $(x,y)$. This is precisely the positivity needed to apply Kodaira vanishing in the next step.[/guided]

[step:Use Kodaira vanishing on the blow-ups to remove the first cohomology obstruction]Fix $m \ge m_0$. Let $K_X:=\Lambda^n(T^{1,0}X)^*$ denote the canonical bundle of $X$. For $x \in X$, let $\mathfrak m_x \subset \mathcal O_X$ be the ideal sheaf of holomorphic germs vanishing at $x$. The canonical bundle formula for the blow-up at a point gives \begin{align*} K_{\widetilde X_x} \cong \pi_x^*K_X \otimes \mathcal O_{\widetilde X_x}((n-1)E_x). \end{align*} Define \begin{align*} \mathcal A_{x,m} := \pi_x^*(K_X^{-1}\otimes L^m) \otimes \mathcal O_{\widetilde X_x}(-(n+1)E_x). \end{align*} By the previous step, $\mathcal A_{x,m}$ is positive. The canonical-bundle formula gives the exact identity \begin{align*} K_{\widetilde X_x}\otimes\mathcal A_{x,m} &\cong \pi_x^*K_X\otimes\mathcal O_{\widetilde X_x}((n-1)E_x) \otimes \pi_x^*(K_X^{-1}\otimes L^m) \otimes \mathcal O_{\widetilde X_x}(-(n+1)E_x)\\ &\cong \pi_x^*L^m\otimes\mathcal O_{\widetilde X_x}(-2E_x). \end{align*} Kodaira vanishing applied to the positive bundle $\mathcal A_{x,m}$ on the compact complex manifold $\widetilde X_x$ gives \begin{align*} H^1(\widetilde X_x,\pi_x^*L^m\otimes\mathcal O_{\widetilde X_x}(-2E_x))=0. \end{align*} For the blow-up of a smooth point, \begin{align*} (\pi_x)_*\mathcal O_{\widetilde X_x}(-2E_x)=\mathfrak m_x^2, \qquad R^j(\pi_x)_*\mathcal O_{\widetilde X_x}(-2E_x)=0 \quad (j>0). \end{align*} The projection formula and the Leray spectral sequence therefore identify the preceding group with \begin{align*} H^1(X,L^m\otimes \mathfrak m_x^2)=0. \end{align*} Similarly, set \begin{align*} \mathcal A_{x,y,m} := \pi_{x,y}^*(K_X^{-1}\otimes L^m) \otimes \mathcal O_{\widetilde X_{x,y}}(-nE_{x,y}). \end{align*} The canonical-bundle formula for the blow-up of the two reduced points gives \begin{align*} K_{\widetilde X_{x,y}}\otimes \mathcal A_{x,y,m} \cong \pi_{x,y}^*L^m\otimes\mathcal O_{\widetilde X_{x,y}}(-E_{x,y}). \end{align*} Kodaira vanishing, together with \begin{align*} (\pi_{x,y})_*\mathcal O_{\widetilde X_{x,y}}(-E_{x,y}) = \mathfrak m_x\mathfrak m_y, \qquad R^j(\pi_{x,y})_*\mathcal O_{\widetilde X_{x,y}}(-E_{x,y})=0 \quad (j>0), \end{align*} gives \begin{align*} H^1(X,L^m\otimes \mathfrak m_x\mathfrak m_y)=0 \end{align*} for every pair of distinct points $x,y \in X$.[/step]

[guided]We now convert positivity into a cohomology vanishing statement. The obstruction to prescribing a first-order jet at $x$ is the group $H^1(X,L^m\otimes \mathfrak m_x^2)$, where $\mathfrak m_x \subset \mathcal O_X$ is the ideal sheaf of germs vanishing at $x$. The obstruction to prescribing independent values at two distinct points $x$ and $y$ is the group $H^1(X,L^m\otimes \mathfrak m_x\mathfrak m_y)$. The analytic theorem being used is the Kodaira vanishing theorem: if $A$ is a positive holomorphic line bundle on a compact complex manifold $Y$, then \begin{align*} H^q(Y,K_Y\otimes A)=0 \end{align*} for every $q>0$, where $K_Y$ is the canonical bundle of $Y$. We apply it with $Y=\widetilde X_x$ and with a positive line bundle obtained from $\pi_x^*L^m$ by subtracting a fixed exceptional divisor. The hypotheses are satisfied because $\widetilde X_x$ is compact and complex, and positivity was arranged in the previous step. More precisely, the positive bundle used for the one-point blow-up is \begin{align*} \mathcal A_{x,m} = \pi_x^*(K_X^{-1}\otimes L^m)\otimes\mathcal O_{\widetilde X_x}(-(n+1)E_x). \end{align*} The canonical-bundle formula gives \begin{align*} K_{\widetilde X_x}\otimes\mathcal A_{x,m} \cong \pi_x^*L^m\otimes\mathcal O_{\widetilde X_x}(-2E_x). \end{align*} The blow-up calculation relates this sheaf on $\widetilde X_x$ back to the ideal sheaf on $X$. The exceptional divisor $E_x$ records vanishing order at $x$, and the direct image of $\mathcal O_{\widetilde X_x}(-2E_x)$ is $\mathfrak m_x^2$, with no higher direct images. The projection formula and Leray therefore descend the vanishing on the blow-up to \begin{align*} H^1(X,L^m\otimes \mathfrak m_x^2)=0. \end{align*} The same argument on the blow-up of the two-point set $\{x,y\}$ uses \begin{align*} \mathcal A_{x,y,m} = \pi_{x,y}^*(K_X^{-1}\otimes L^m)\otimes\mathcal O_{\widetilde X_{x,y}}(-nE_{x,y}), \end{align*} for which \begin{align*} K_{\widetilde X_{x,y}}\otimes\mathcal A_{x,y,m} \cong \pi_{x,y}^*L^m\otimes\mathcal O_{\widetilde X_{x,y}}(-E_{x,y}). \end{align*} Since the direct image of $\mathcal O_{\widetilde X_{x,y}}(-E_{x,y})$ is $\mathfrak m_x\mathfrak m_y$, again with no higher direct images, we get \begin{align*} H^1(X,L^m\otimes \mathfrak m_x\mathfrak m_y)=0. \end{align*} These two vanishings are the exact cohomological forms of tangent-vector separation and point separation.[/guided]

[step:Separate first-order jets at every point]For $x \in X$, define the first jet space of $L^m$ at $x$ by \begin{align*} J_x^1(L^m) := H^0(X,L^m\otimes \mathcal O_X/(\mathfrak m_x^2)). \end{align*} The quotient sequence of sheaves \begin{align*} 0 \to L^m\otimes \mathfrak m_x^2 \to L^m \to L^m\otimes \mathcal O_X/(\mathfrak m_x^2) \to 0 \end{align*} induces a restriction map \begin{align*} \rho_x: H^0(X,L^m) &\to J_x^1(L^m). \end{align*} Since $H^1(X,L^m\otimes \mathfrak m_x^2)=0$, the associated [long exact cohomology sequence](/theorems/3471) shows that $\rho_x$ is surjective. Therefore global sections of $L^m$ prescribe arbitrary first-order jets at every point $x \in X$.[/step]

[guided]We need tangent-vector separation, and tangent-vector separation is exactly first-order jet separation. For a point $x \in X$, let $\mathfrak m_x \subset \mathcal O_X$ be the ideal sheaf of germs vanishing at $x$. The quotient $\mathcal O_X/(\mathfrak m_x^2)$ remembers the value and the first derivatives at $x$. Accordingly, we define \begin{align*} J_x^1(L^m) := H^0(X,L^m\otimes \mathcal O_X/(\mathfrak m_x^2)). \end{align*} The sheaf sequence \begin{align*} 0 \to L^m\otimes \mathfrak m_x^2 \to L^m \to L^m\otimes \mathcal O_X/(\mathfrak m_x^2) \to 0 \end{align*} is exact by definition of the quotient sheaf. Taking global sections gives the beginning of a long exact sequence \begin{align*} H^0(X,L^m) \xrightarrow{\rho_x} J_x^1(L^m) \to H^1(X,L^m\otimes \mathfrak m_x^2). \end{align*} The last group is zero by the vanishing proved above. Exactness therefore forces the connecting map out of $J_x^1(L^m)$ to be the zero map, and hence the restriction map \begin{align*} \rho_x: H^0(X,L^m) \to J_x^1(L^m) \end{align*} is surjective. This means that for every prescribed value and every prescribed cotangent-linear first derivative at $x$, there is a global holomorphic section of $L^m$ realizing that first-order data.[/guided]

[step:Separate distinct points]For distinct $x,y \in X$, define the two-point value space by \begin{align*} V_{x,y}(L^m) := H^0(X,L^m\otimes \mathcal O_X/(\mathfrak m_x\mathfrak m_y)). \end{align*} Because $\mathfrak m_x+\mathfrak m_y=\mathcal O_X$ for $x\ne y$, the [Chinese remainder theorem](/theorems/734) gives \begin{align*} \mathcal O_X/(\mathfrak m_x\mathfrak m_y) \cong \mathcal O_X/\mathfrak m_x \oplus \mathcal O_X/\mathfrak m_y. \end{align*} After tensoring with $L^m$, the quotient records exactly the two fiber values at $x$ and $y$. The exact sequence \begin{align*} 0 \to L^m\otimes \mathfrak m_x\mathfrak m_y \to L^m \to L^m\otimes \mathcal O_X/(\mathfrak m_x\mathfrak m_y) \to 0 \end{align*} induces \begin{align*} \rho_{x,y}: H^0(X,L^m) &\to V_{x,y}(L^m). \end{align*} Since $H^1(X,L^m\otimes \mathfrak m_x\mathfrak m_y)=0$, the long exact cohomology sequence shows that $\rho_{x,y}$ is surjective. Hence there exists a section $s \in H^0(X,L^m)$ with $s(x)=0$ and $s(y)\neq 0$, and global sections of $L^m$ separate distinct points.[/step]

[guided]For distinct points, the quotient by $\mathfrak m_x\mathfrak m_y$ remembers the two independent values of a section. The ideals $\mathfrak m_x$ and $\mathfrak m_y$ are comaximal, so the Chinese [remainder theorem](/theorems/1707) gives \begin{align*} \mathcal O_X/(\mathfrak m_x\mathfrak m_y) \cong \mathcal O_X/\mathfrak m_x \oplus \mathcal O_X/\mathfrak m_y. \end{align*} Thus $V_{x,y}(L^m)$ is the space of possible pairs of values in the fibers of $L^m$ at $x$ and $y$. The exact sequence \begin{align*} 0 \to L^m\otimes \mathfrak m_x\mathfrak m_y \to L^m \to L^m\otimes \mathcal O_X/(\mathfrak m_x\mathfrak m_y) \to 0 \end{align*} has connecting map into $H^1(X,L^m\otimes\mathfrak m_x\mathfrak m_y)$. That group vanished in the blow-up step, so every pair of fiber values is realized by a global section. Choosing the pair of values $(0,\ell_y)$ with $\ell_y\ne0$ gives a section vanishing at $x$ and not vanishing at $y$. Therefore global sections separate distinct points.[/guided]

[step:Apply the jet-separation criterion for very ampleness]The jet-separation criterion for holomorphic line bundles states that a holomorphic line bundle $A \to X$ is very ample if its global sections separate distinct points and separate tangent vectors at every point. Applying this criterion to $A=L^m$, the previous two steps show that $L^m$ is very ample for every $m\ge m_0$.[/step]

[guided]The remaining step is a geometric translation. A holomorphic line bundle $A \to X$ is very ample precisely when the complete linear system of its global sections defines a holomorphic embedding into projective space. The jet-separation criterion gives a practical sufficient and necessary condition: global sections must separate distinct points, and at each point their first-order jets must separate tangent directions. For $A=L^m$, point separation was proved by the surjectivity of \begin{align*} \rho_{x,y}: H^0(X,L^m) \to V_{x,y}(L^m) \end{align*} for every $x\neq y$. Tangent-vector separation was proved by the surjectivity of \begin{align*} \rho_x: H^0(X,L^m) \to J_x^1(L^m) \end{align*} for every $x\in X$. Therefore the hypotheses of the jet-separation criterion are verified for $L^m$ whenever $m\ge m_0$, and so $L^m$ is very ample for every such $m$.[/guided]

[step:Identify the resulting Kodaira map as an embedding]For $m\ge m_0$, first-jet surjectivity includes arbitrary value data at each point, so $L^m$ is basepoint-free and the Kodaira map is defined everywhere. Define this map by \begin{align*} \Phi_m: X &\to \mathbb P(H^0(X,L^m)^*) \end{align*} by sending $x\in X$ to the hyperplane of sections in $H^0(X,L^m)$ vanishing at $x$. Since $L^m$ is very ample, this map is holomorphic, injective, and has injective differential at every point. Because $X$ is compact and projective space is Hausdorff, a holomorphic injective immersion \begin{align*} \Phi_m: X &\to \mathbb P(H^0(X,L^m)^*) \end{align*} is a holomorphic embedding onto its image. Thus $\Phi_m:X\hookrightarrow\mathbb P(H^0(X,L^m)^*)$ is a holomorphic embedding for every $m\ge m_0$, as claimed.[/step]

[guided]The first-jet surjectivity proved earlier includes value data, so for every point $x$ there is a section of $L^m$ not vanishing at $x$. Hence the hyperplane of sections vanishing at $x$ is a genuine hyperplane, and the Kodaira map \begin{align*} \Phi_m: X &\to \mathbb P(H^0(X,L^m)^*) \end{align*} is defined for all $x\in X$. Point separation says that if $x\ne y$, then some section distinguishes their values; therefore the hyperplanes of sections vanishing at $x$ and at $y$ are different, so $\Phi_m$ is injective. First-order jet separation says that for every nonzero tangent vector $v\in T_xX$, some section vanishing at $x$ has nonzero derivative in the direction $v$; this is exactly injectivity of the differential $d\Phi_m$ at $x$. Thus $\Phi_m$ is a holomorphic injective immersion. Since $X$ is compact and projective space is Hausdorff, $\Phi_m$ is a homeomorphism onto its image; a holomorphic injective immersion with this topological property is a holomorphic embedding. This proves the embedding assertion.[/guided]