will denote by $\{d\omega\}$ this morphism or equivalently the global section of $\Omega^2_X\otimes {\mathcal{O}}_{{\mathrm{Sing}(\omega)}}\otimes{\mathcal{L}}^{-1}$ it defines.

\[KK2\]The subscheme of *Kupka singularities* of $\omega$ is the one defined by the ideal sheaf ${\mathcal{K}}(\omega):={\mathrm{ann}}(\{d\omega\})\in\Omega^2_X\otimes {\mathcal{O}}_{{\mathrm{Sing}(\omega)}}\otimes{\mathcal{L}}^{-1}$. We will denote it just as ${\mathcal{K}}$ if no confusion arises.

\[JJcIIcKK\] Let ${\mathscr{J}}$ be the ideal sheaf of the singular set of $\omega$, ${\mathcal{K}}$ the ideal of the Kupka singularities of $\omega$ and ${\mathscr{I}}$ the ideal of persistent singularities. Then the following inclusions hold, $${\mathscr{J}}\subseteq {\mathscr{I}}\subseteq {\mathcal{K}}.$$

Let $U\subseteq X$ be an open subscheme such that ${\mathcal{L}}|_U\simeq{\mathcal{O}}_X$, and $\varpi$ a local generator of ${\mathcal{L}}(U)$.

Suppose $h\in{\mathscr{J}}(U)\subseteq{\mathcal{O}}_X(U)$ is a local section. By shrinking $U$ if necessary we may assume that there is a vector field $v\in T_X(U)$ such that $h= i_v(\omega)$. Then we have $$0=i_v(\varpi\wedge d\varpi)=i_v(\varpi)d\varpi - \varpi \wedge i_v(d\varpi).$$ So, calling $\eta=i_v(d\varpi)$, we get $hd\varpi=\varpi\wedge \eta$. Hence $h$ is in ${\mathscr{I}}(U)$, which proves the first inclusion.

Now assume $h\in {\mathscr{I}}(U)$, then again by shrinking $U$ if necessary, we may assume that there is a $\eta\in \Omega^1_X/{\mathcal{L}}(U)$ such that $hd\varpi= \varpi\wedge\eta$. By definition we have $\varpi \in {\mathscr{J}}(U)\cdot \Omega^1_X(U)$, then $hd\varpi\in {\mathscr{J}}(U)\cdot \Omega^2_X(U)$ so $h$ is in the annihilator of $\{d\omega\}$ in $\Omega^2_X\otimes {\mathcal{O}}_{{\mathrm{Sing}(\omega)}}$. Then $h\in {\mathcal{K}}(U)$, 