Given a smooth projective variety X and a divisor D on it, the sheaf of logarithmic differential forms $\Omega^1_X(log D)$ is defined as the sheaf of one-forms having logarithmic poles along D. The sheaf was studied extensively by Deligne and Saito in connection with Hodge theory. On the other hand, this sheaf, or equivalently its dual sheaf $\mathcal{T}_X(- log D)$ of vector fields tangent along D, has been investigated from the point of view of stability theory, as well as related to Torelli problems.
The goal of my talk will be twofold: firstly, I am going to resume the state-of-the-art regarding the aforementioned problems. Secondly, I am going to introduce a generalization of $\Omega^1_X(log D)$ for $X$ a smooth surface that takes into account certain vanishings along a fixed set of points $Z$ on $D$. This is joint work with S.Marchesi, S.Huh and J. Vallès.