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

Abstract

Let $ \mathcal{D} = \{D_{1}, \ldots, D_{\ell}\} $ be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space $ \mathbb{P}^{n} $ and let $ \Omega^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ be the logarithmic bundle attached to it. Our aim is to study the injectivity of the correspondence $ \mathcal{D} \longrightarrow \Omega^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $. In order to do that, we first show that $ \Omega^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ admits a resolution of 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 $, by recovering them as unstable smooth irreducible degree-$d$ hypersurfaces of $ \Omega^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $. The cases of one quadric and a pair of quadrics in $ \mathbb{P}^{n} $ are not Torelli; in particular, through a duality argument, we prove that the isomorphism class of the logarithmic bundle attached to a pair of quadrics is determined by the tangent hyperplanes to the pair. Finally, by describing the moduli spaces containing $ \Omega^{1}_{\mathbb{P}^{2}}(log \mathcal{D}) $, we show that some line-conic arrangements are not of Torelli type.