Let $\pi:\mathcal X\to B$ be a proper holomorphic submersion over a complex manifold $B$, let $0\in B$, and let $X=\mathcal X_0$ be the central fibre. Let $L\to X$ be an ample holomorphic line bundle. After shrinking $B$ around $0$, identify the groups $H^2(\mathcal X_b,\mathbb Z)$ with $H^2(X,\mathbb Z)$ by the local system $R^2\pi_*\mathbb Z$, equivalently by a differentiable trivialisation of $\pi$. Suppose that the locally constant class determined by $c_1(L)\in H^2(X,\mathbb Z)$ maps to a class of Hodge type $(1,1)$ in $H^2(\mathcal X_b,\mathbb C)$ for every $b$ in this neighbourhood. Then, after possibly shrinking to an open neighbourhood $U\subset B$ of $0$, there exists a holomorphic line bundle $\mathcal L\to \mathcal X|_U$ such that $\mathcal L|_X\cong L$, and every fibre $\mathcal X_b$ with $b\in U$ is projective.