Or, equivalently, a (non trivial) global section of $H^2(\omega)\otimes{\mathcal{L}}$. We will denote the global section or the morphism indistinctly by $[d\omega]$. By Theorem \[koszul\] above, we conclude that $codim({\mathrm{Sing}(\omega)})\geq 2$.

\[II2\] The subscheme of *persistent singularities* of $\omega$ is the one defined by the ideal sheaf ${\mathscr{I}}(\omega):={\mathrm{ann}}([d\omega])$, for $[d\omega]\in H^0\left(X,H^2(\omega)\otimes{\mathcal{L}}\right)$. We will denote it just as ${\mathscr{I}}$ if no confusion arises.

\[remarkI\] Let $\varpi\in \Omega^1_X(U)$ be a local generator of the image of ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$, then the local sections of ${\mathscr{I}}(\omega)$ in $U$ are given by $${\mathscr{I}}(U)=\{h\in {\mathcal{O}}_X(U): \text{there is a section } \eta\in\Gamma(U,\Omega^1_X/{\mathcal{L}})\text{ s.t. }  hd \varpi=\varpi\wedge\eta\}.$$

\[formalremark\] For a regular local ring $(R,\mathfrak{m})$, an $R$-module $M$ and an element $m\in M$, let us denote $\widehat{R}$ the $\mathfrak{m}$-adic completion of $R$ and $\widehat{M}=M\otimes\widehat{R}$. The element $m\otimes 1\in\widehat{M}$ has as annihilator the ideal ${\mathrm{ann}}(m)\otimes \widehat{R}$. Setting $R={\mathcal{O}}_{X,{\mathfrak{p}}}$, $M= \left(H^2(\omega)\otimes{\mathcal{L}}\right)_{\mathfrak{p}}$ and $m=[d\omega]_{\mathfrak{p}}$, and following the notation of Remark \[notation\], we have that $${\mathrm{ann}}([d\omega]_{\mathfrak{p}}\otimes 1)=\{h\in \widehat{{\mathcal{O}}_{X,{\mathfrak{p}}}}: \text{there is a formal $1$-form } \eta \ \text{ s.t. } hd \varpi=\varpi\wedge\eta\}.$$

Let ${\mathfrak{p}}\in X$ be a point in ${\mathrm{Sing}(\omega)}$, ${\mathcal{O}}_{X,{\mathfrak{p}}}$ the local ring around ${\mathfrak{p}}$, and $X_{\mathfrak{p}}=\mathrm{Spec}({\mathcal{O}}_{X,{\mathfrak{p}}})$. Then ${\mathfrak{p}}$ is in the subscheme of persistent singularities if and only if for any infinitesimal first order 