Logarithmic comparison theorem and D-modules: an overview

Let $X$ be a normal variety. Assume that for some reduced divisor $D \subset X$, logarithmic 1-forms defined on the snc locus of $(X, D)$ extend to a log resolution $\tilde X \to X$ as logarithmic differential forms. We prove that then the Lipman-Zariski conjecture holds for $X$. This result applies in particular if $X$ has log canonical singularities. Furthermore, we give an example of a 2-form defined on the smooth locus of a three-dimensional log canonical pair $(X, \emp...

The purpose of this paper is to study singular holomorphic foliations of arbitrary codimension defined by logarithmic forms on projective spaces.

Let Z be the germ of a complex hypersurface isolated singularity of equation f, with Z at least of dimension 2. We consider the family of analytic D-modules generated by the powers of 1/f and describe it in terms of the pole order filtration on the de Rham cohomology of the complement of {f=0} in the neighborhood of the singularity.

In a beautiful paper Deligne and Illusie proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. In a recent paper Arinkin, C\u{a}ld\u{a}raru and the author of this paper gave a geometric interpretation of the problem of Deligne-Illusie showing that the triviality of a certain line bundle on a derived scheme implies the the Deligne-Illusie result. In the present paper we generalize these ideas to logarithmic schemes and using t...

In the first part we deepen the six-functor theory of (holonomic) logarithmic D-modules, in particular with respect to duality and pushforward along projective morphisms. Then, inspired by work of Ogus, we define a logarithmic analogue of the de Rham functor, sending logarithmic D-modules to certain graded sheaves on the so-called Kato-Nakayama space. For holonomic modules we show that the associated sheaves have finitely generated stalks and that the de Rham functor intertwi...

We study divisors in a complex manifold in view of the property that the algebra of logarithmic differential operators along the divisor is generated by logarithmic vector fields. We give a sufficient criterion for the property, a simple proof of F.J. Calderon-Moreno's theorem that free divisors have the property, a proof that divisors in dimension 3 with only isolated quasi-homogeneous singularities have the property, an example of a non-free divisor with non-isolated singul...

The Galois theory of logarithmic differential equations with respect to relative D-groups in partial differential-algebraic geometry is developed.

Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$. They are called {\it $D$-equivalent} if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, while {\it $K$-equivalent} if they are birationally equivalent and the pull-backs of their canonical divisors to a common resolution coincide. We expect that the two equivalences coincide at least for birationally equivalent varieties. We shall provide a partial an...

We consider a version of the Lipman-Zariski conjecture for logarithmic vector fields and logarithmic $1$-forms on pairs. Let $(X,D)$ be a pair consisting of a normal complex variety $X$ and an effective Weil divisor $D$ such that the sheaf of logarithmic vector fields (or dually the sheaf of reflexive logarithmic $1$-forms) is locally free. We prove that in this case the following holds: If $(X,D)$ is dlt, then $X$ is necessarily smooth and $\lfloor D\rfloor $ is snc. If $(X,...

We provide a new formalism of de Rham--Witt complexes in the logarithmic setting. This construction generalizes a result of Bhatt--Lurie--Mathew, and agrees with those of Hyodo--Kato and Matsuue for log-smooth schemes of log-Cartier type. We then apply our formalism to obtain a more direct proof of the log crystalline comparison of A_inf-cohomology in the case of semistable reduction, which is established by Cesnavicius--Koshiwara.