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Let $ \mathcal{D} = \{D_{1}, \ldots, D_{\ell}\} $ be an arrangement of smoothhypersurfaces with normal crossings on the complex projective space $\mathbb{P}^{n} $ and let $ \Omega^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ bethe logarithmic bundle attached to it. Our aim is to study the injectivity ofthe correspondence $ \mathcal{D} \longrightarrow\Omega^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $. In order to do that, we firstshow that $ \Omega^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ admits a resolutionof length $ 1 $ depending on the degrees and on the equations of $ D_{1},\ldots, D_{\ell} $. Then, we prove a Torelli type theorem when $ \mathcal{D} $has a sufficiently large number of components of the same degree $ d $, byrecovering them as unstable smooth irreducible degree-$d$ hypersurfaces of $\Omega^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $. The cases of one quadric and apair of quadrics in $ \mathbb{P}^{n} $ are not Torelli; in particular, througha duality argument, we prove that the isomorphism class of the logarithmicbundle attached to a pair of quadrics is determined by the tangent hyperplanesto the pair. Finally, by describing the moduli spaces containing $\Omega^{1}_{\mathbb{P}^{2}}(log \mathcal{D}) $, we show that some line-conicarrangements are not of Torelli type.

The Torelli problem for Logarithmic bundles of hypersurface arrangements in the projective space