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Studying $D$-modules, I often see the notation $\Omega_X^{\otimes -1}$ where $X$ is a complex manifold and $\Omega_X$ are the top holomorphic forms on $X$. If I understand it well, this sheaf verifies the property $$\Omega_X \otimes_{\mathcal{O}_X} \Omega_X^{\otimes -1} \simeq \mathcal{O}_X.$$ My question is : What is exactly this sheaf ? How can you describe it explicitely ? And how do you prove that it is an "inverse" of $\Omega_X$ ? I would like to have a more intuitive and explicit approach to this problem of "invertible sheaves". Thank you for any help.

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  • $\begingroup$ A little remark : if $\Omega^1_X$ is the sheaf of holomorphic $1$-forms, then $\Omega_X = \wedge^n \Omega^1_X$ so the dual of $\Omega_X$ is given by $\wedge^n \Theta_X$ where $\Theta_X$ is the sheaf of vector fields on $X$, i.e the top poly-vector fields. $\endgroup$ – Nicolas Hemelsoet Jun 24 '18 at 20:03
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In general, if $\mathcal{L}$ is a line bundle on $X$, then its inverse (for the tensor product) is its dual $\mathcal{L}^{-1}=Hom_{\mathcal{O}_{X}}(\mathcal{L},\mathcal{O}_X)$.

What's remarkable in the case where $\mathcal{L}=\Omega_X$ is that it has a natural $D_X$ action (on the right). Its dual, $\Omega_X^{-1}$, also comes equipped with a $D_X$ action.

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