so again shrinking $U$ if necessary we have that $h$ is a unit, hence $1\in {\mathscr{I}}_{\mathfrak{p}}$.

Most of the known families of foliations on algebraic varieties present persistent singularities, see [@gmln; @omegar; @celn; @pullback; @fji; @fmi; @fj; @mmq]. As it happens the absence of persistent singularities impose some restrictions on the line bundle ${\mathcal{L}}$. To explain this we have to make explicit use of a result that is implied in the proof of Lefschetz Theorem on $(1,1)$ classes as is proved in [@GH Chapter 1.1 p.:  141].

Let ${\mathcal{L}}$ be a line bundle. Choose a trivialization $(U_i, \phi_i)_{i\in I}$ of ${\mathcal{L}}$ with gluing data $g_{ij}\in {\mathcal{O}}^*_X(U_{ij})$. The Čech cocycle $\frac{1}{2\pi i }[d\log g_{ij}]\in Z^1(\Omega^1_X)$ represents the Chern class $c_1({\mathcal{L}})$ of ${\mathcal{L}}$ in $H^1(X,\Omega^1_X)$.

The claim follows from a careful reading of the proof of the Proposition in page 141 of [@GH Chern classes of line bundles, Chapter 1.1, p. 141], as we will show next. There is shown that de Rham’s Theorem for $\mathcal{C}^\infty$-forms with complex coefficients give us exact sequence of sheaves $$0\to {\mathbb{C}}\to \mathcal{A}^0\to \mathcal{Z}^1_d\to 0,\qquad 0\to\mathcal{Z}^1_d\to\mathcal{A}^1\to \mathcal{Z}^2_d\to 0,$$ where $\mathcal{A}^i$ are $\mathcal{C}^\infty$ $i$-forms with complex coefficients and $\mathcal{Z}^i_d$ are the cycles of the de Rham complex. These 