Logarithmic differential forms on singular varieties

The concept of logarithmic differential forms with poles along a reduced (i.e., without multiple components) divisor $D$ defined on a complex smooth manifold $M$, appeared in the early 1960s in relation with the study of Hodge structures and the Gauss-Manin connection in the cohomology of singular varieties. More precisely, a meromorphic differential form $\omega$ with poles along $D$ is called logarithmic if $\omega$ and the total differential $d\omega$ have at worst simple poles only along the divisor $D$. The corresponding sheaves are usually denoted by $\Omega^p_M(\log D)$, $p\geq 0.$ First, P. Deligne, N. Katz, Ph. Griffiths and others considered this notion for a union of smooth subvarieties with normal crossings, then K. Saito and his successors analyzed the case of divisors with other types of singularities, and so on.
We are developing a different approach to the study of logarithmic differential forms; it is based on an original interpretation of the classical de Rham lemma, adapted to the study of differential forms defined on hypersurfaces with arbitrary singularities. This approach allows us to extend the concept of logarithmic differential forms to the case of effective Cartier divisors defined on singular varieties. Some useful applications and further generalizations, as well as connections with the theory of multi-logarithmic differential forms and their residues, will be also discussed.