[proofplan] The proof has two separate parts. First, the type $(1,1)$ hypothesis and the relative exponential sequence lift the locally constant integral class $c_1(L)$ to a holomorphic line bundle on the total space over a smaller base, normalized so that the central fibre is $L$. Second, ampleness of $L$ gives a positive curvature representative on the central fibre; by smooth local triviality and the elliptic Kodaira-Spencer stability argument, this representative can be corrected fibrewise to nearby positive closed $(1,1)$-forms in the same integral class. The curvature criterion then makes the fibrewise restrictions positive line bundles, and Kodaira's embedding theorem makes the nearby fibres projective. [/proofplan]

[step:Shrink the base and transport the Chern class through the local system] Since $\pi:\mathcal X\to B$ is a proper holomorphic submersion, the fibres $\mathcal X_b:=\pi^{-1}(b)$ are compact complex manifolds. By Ehresmann's fibration theorem, after shrinking $B$ to a connected open neighbourhood $B_1\subset B$ of $0$, there is a $C^\infty$ trivialisation over $B_1$. Equivalently, the local system $R^2\pi_*\mathbb Z$ identifies all groups $H^2(\mathcal X_b,\mathbb Z)$ with $H^2(X,\mathbb Z)$. Let \begin{align*} \alpha_0:=c_1(L)\in H^2(X,\mathbb Z). \end{align*} Let \begin{align*} \alpha_b\in H^2(\mathcal X_b,\mathbb Z) \end{align*} denote the locally constant section of $R^2\pi_*\mathbb Z$ whose value at $0$ is $\alpha_0$. By hypothesis, the image of $\alpha_b$ in $H^2(\mathcal X_b,\mathbb C)$ belongs to \begin{align*} H^{1,1}(\mathcal X_b)\subset H^2(\mathcal X_b,\mathbb C) \end{align*} for every $b\in B_1$. [/step]

[step:Lift the relative integral class to a holomorphic line bundle]Apply the local relative Picard theorem, equivalently the relative exponential sequence for a proper holomorphic submersion, to the integral section $(\alpha_b)_{b\in B_1}$ of $R^2\pi_*\mathbb Z$. Its hypothesis is exactly that the complexification of $\alpha_b$ has type $(1,1)$ on every fibre. Therefore, after shrinking to a connected open neighbourhood $B_2\subset B_1$ of $0$, there is a holomorphic line bundle \begin{align*} \mathcal L_1\to \mathcal X|_{B_2} \end{align*} such that \begin{align*} c_1(\mathcal L_1|_{\mathcal X_b})=\alpha_b \end{align*} for every $b\in B_2$. On the central fibre, $\mathcal L_1|_X$ and $L$ have the same first Chern class. Their quotient \begin{align*} P:=L\otimes(\mathcal L_1|_X)^{-1} \end{align*} lies in the identity component $\operatorname{Pic}^0(X)$ of the Picard group of $X$. By the local relative Picard construction, after possibly shrinking $B_2$ to an open neighbourhood $B_3$ of $0$, the line bundle $P$ extends to a holomorphic line bundle \begin{align*} \mathcal P\to \mathcal X|_{B_3} \end{align*} with $\mathcal P|_X\cong P$. Define \begin{align*} \mathcal L:=\mathcal L_1|_{\mathcal X|_{B_3}}\otimes \mathcal P. \end{align*} Then \begin{align*} \mathcal L|_X\cong \mathcal L_1|_X\otimes L\otimes(\mathcal L_1|_X)^{-1}\cong L. \end{align*} Moreover $c_1(\mathcal P|_{\mathcal X_b})=0$, so \begin{align*} c_1(\mathcal L|_{\mathcal X_b})=\alpha_b \end{align*} for every $b\in B_3$.[/step]

[guided]The type $(1,1)$ hypothesis is needed precisely at the point where one passes from topology to holomorphic geometry. A topological complex line bundle on $\mathcal X_b$ is controlled by an integral class in $H^2(\mathcal X_b,\mathbb Z)$, but it is holomorphic only when its Chern class is compatible with the [Hodge decomposition](/theorems/2745), namely when the complexified class has type $(1,1)$. We use the local relative Picard theorem, also described as the relative exponential sequence for a proper holomorphic submersion. The theorem says that a locally constant integral cohomology class \begin{align*} b\mapsto \alpha_b\in H^2(\mathcal X_b,\mathbb Z) \end{align*} whose complexification has type $(1,1)$ on each fibre is locally the first Chern class of a holomorphic line bundle on the total family. The hypotheses are verified as follows: $\pi$ is a proper holomorphic submersion by assumption, the base has been shrunk so that the relevant local system is trivialised, and the class $\alpha_b$ has type $(1,1)$ by the theorem's hypothesis. Hence, after shrinking the base to some neighbourhood $B_2$ of $0$, there is a holomorphic line bundle \begin{align*} \mathcal L_1\to \mathcal X|_{B_2} \end{align*} with \begin{align*} c_1(\mathcal L_1|_{\mathcal X_b})=\alpha_b \end{align*} for every $b\in B_2$. This line bundle has the correct Chern class on the central fibre, but it need not literally restrict to the prescribed line bundle $L$. The discrepancy is the degree-zero line bundle \begin{align*} P:=L\otimes(\mathcal L_1|_X)^{-1}. \end{align*} Because $c_1(\mathcal L_1|_X)=c_1(L)$, additivity of the first Chern class gives \begin{align*} c_1(P)=c_1(L)-c_1(\mathcal L_1|_X)=0. \end{align*} Thus $P\in \operatorname{Pic}^0(X)$. The same local relative Picard construction extends this degree-zero central fibre line bundle to a holomorphic line bundle \begin{align*} \mathcal P\to \mathcal X|_{B_3} \end{align*} after shrinking to a smaller neighbourhood $B_3\subset B_2$. Now tensor the first lift by this correction: \begin{align*} \mathcal L:=\mathcal L_1|_{\mathcal X|_{B_3}}\otimes \mathcal P. \end{align*} On the central fibre, \begin{align*} \mathcal L|_X\cong \mathcal L_1|_X\otimes P\cong \mathcal L_1|_X\otimes L\otimes(\mathcal L_1|_X)^{-1}\cong L. \end{align*} Since $\mathcal P$ has fibrewise first Chern class zero, tensoring by $\mathcal P$ does not change the transported Chern class. Therefore \begin{align*} c_1(\mathcal L|_{\mathcal X_b})=\alpha_b \end{align*} for every $b\in B_3$.[/guided]

[step:Choose a positive curvature representative on the central fibre] Since $L$ is ample, it is positive. By [citetheorem:9099], there exists a Hermitian metric $h_0$ on $L$ whose Chern curvature form \begin{align*} \omega_0:=i\Theta(L,h_0) \end{align*} is a positive real closed $(1,1)$-form on $X$ representing $c_1(L)$ in real cohomology. Here $\Theta(L,h_0)$ denotes the Chern curvature of the Hermitian holomorphic line bundle $(L,h_0)$. Thus $\omega_0$ is a Kähler form on the compact complex manifold $X$. [/step]

[step:Correct the transported forms to fibrewise positive $(1,1)$ representatives] Use the chosen differentiable trivialisation to regard nearby fibres as the same smooth compact manifold $M$, with complex structures \begin{align*} J_b:TM\to TM \end{align*} depending smoothly on $b$ and satisfying $J_0=J_X$. Let $A^2(M;\mathbb R)$ denote the [vector space](/page/Vector%20Space) of smooth real-valued two-forms on $M$. Let \begin{align*} \eta_b\in A^2(M;\mathbb R) \end{align*} be a smooth family of real closed two-forms representing the transported real class $\alpha_b\otimes 1$ and satisfying $\eta_0=\omega_0$. The Kodaira-Spencer elliptic deformation argument for the $\partial\bar\partial$-lemma says the following: because $\omega_0$ is a real closed $(1,1)$-form for $J_0$ and the transported class $\alpha_b$ remains of Hodge type $(1,1)$ for $J_b$, after shrinking to an open neighbourhood $B_4\subset B_3$ of $0$, there are real one-forms \begin{align*} \gamma_b\in A^1(M;\mathbb R) \end{align*} depending smoothly on $b$, with $\gamma_0=0$, such that \begin{align*} \omega_b:=\eta_b+d\gamma_b \end{align*} is a real closed $(1,1)$-form with respect to $J_b$ for every $b\in B_4$. Moreover $\omega_b\to \omega_0$ in the $C^\infty$ topology as $b\to0$. Since positivity of a real $(1,1)$-form is an open condition in the $C^0$ topology on the compact manifold $M$, shrink once more to an open neighbourhood $U\subset B_4$ of $0$ so that $\omega_b$ is positive for every $b\in U$. Hence $\omega_b$ is a positive real closed $(1,1)$-form on $\mathcal X_b$ representing \begin{align*} c_1(\mathcal L|_{\mathcal X_b})\in H^2(\mathcal X_b,\mathbb R). \end{align*} [/step]

[step:Conclude that the relative bundle is fibrewise positive and the fibres are projective] For each $b\in U$, the line bundle $\mathcal L|_{\mathcal X_b}$ has first Chern class represented by the positive real closed $(1,1)$-form $\omega_b$. By [citetheorem:9099], $\mathcal L|_{\mathcal X_b}$ is positive. The [Kodaira embedding theorem](/theorems/3836), applied to the compact complex manifold $\mathcal X_b$ and the positive holomorphic line bundle $\mathcal L|_{\mathcal X_b}$, implies that $\mathcal X_b$ is projective. The construction also gives \begin{align*} \mathcal L|_X\cong L. \end{align*} Thus, after shrinking to the neighbourhood $U\subset B$ of $0$, there is a holomorphic line bundle $\mathcal L\to\mathcal X|_U$ extending $L$, and every fibre $\mathcal X_b$ for $b\in U$ is projective. This is the desired conclusion. [/step]