get that, on $U_{ij}$, $$\eta_i-\eta_j=\frac{dg_{ij}}{g_{ij}},$$ as sections of $\Gamma(U_{ij},\Omega^1_X /{\mathcal{L}})$. Therefore we get a Čech cochain $(\eta_i)_{i\in I}$ of $C^0(\Omega^1_X/{\mathcal{L}})$ whose border is $$\partial (\eta)_{ij}=d\log g_{ij}\in B^1(\Omega^1/{\mathcal{L}}).$$ As the cocycle $(d\log g_{ij})\in Z^1(\Omega^1_X)$ represents $(2\pi i )c_1({\mathcal{L}})$, the existence of the cochain $(\eta_i)$ implies $c_1({\mathcal{L}})$ is in the kernel of the map $H^1(\Omega^1_X)\to H^1(\Omega^1_X/{\mathcal{L}})$ induced by the short exact sequence of sheaves $$0\to{\mathcal{L}}\xrightarrow{\omega}\Omega^1_X\to \Omega^1_X/{\mathcal{L}}\to 0.$$ The hypothesis $H^1({\mathcal{L}})=0$ then implies $c_1({\mathcal{L}})=0$.

Let $X$ be a smooth projective variety over ${\mathbb{C}}$ such that every line bundle ${\mathcal{L}}$ verifies $H^1(X,{\mathcal{L}})=0$ and such that $Pic(X)$ is torsion-free (*e.g.*: $X$ smooth complete intersection). Then every foliation on $X$ have persistent singularities.

From the exponential sequence and the hypothesis $H^1(X,{\mathcal{O}}_X)=0$ it follows that $c_1:Pic(X)\to H^2(X,{\mathbb{Z}})$ is injective. Assume that $\omega:{\mathcal{L}}\to\Omega^1_X$ is a foliation without persistent singularities. Then the above Proposition imply that $c_1({\mathcal{L}})$ is a torsion element in $H^2(X,{\mathbb{Z}})$. But given that $Pic(X)$ is torsion-free, we get ${\mathcal{L}}\cong{\mathcal{O}}_X$.

In particular, $\omega$ is a global differential $1$-form which contradicts the fact that $H^0(X,\Omega^1_X)=H^1(X,{\mathcal{O}}_X)=0$.

\[[domega]{}\] Given a trivializing open set $U$, a choice of a trivialization ${\mathcal{O}}_X|_U\cong {\mathcal{L}}|_U$ and a local generator $\varpi$ of ${\mathcal{L}}(U)$, the mapping $\varpi\mapsto d\varpi$ defines a ${\mathcal{O}}_X$-linear morphism ${\mathcal{L}}\to \Omega^2_X\otimes {\mathcal{O}}_{{\mathrm{Sing}(\omega)}}$. We 