Logarithmic comparison theorem and D-modules: an overview

The relationship between ${\cal D}$-modules and free divisors has been studied in a general setting by L. Narv\'aez and F.J. Calder\'on. Using the ideas of these works we prove in this article a duality formula between two ${\cal D}$-modules associated to a class of free divisors on ${\bf C}^n$ and we give some applications

In this paper we study topics related to one of Kato-Nakayama's comparison theorems using analytic log etale topoi.

In this article, we prove the comparison theorem between the relative log de Rham-Witt cohomology and the relative log crystalline cohomology for a log smooth saturated morphism of fs log schemes satisfying certain condition. Our result covers the case where the base fs log scheme is etale locally log smooth over a scheme with trivial log structure or the case where the base fs log scheme is hollow, and so it generalizes the previously known results of Matsuue. In Appendix, w...

We survey nearby and vanishing cycles for both perverse sheaves and D-modules under analytic setting. Following ideas of A. Beilinson, M. Kashiwara and M. Saito, we explain in detail the proof of the comparison theorem between them in the sense of Riemann-Hilbert correspondence.

Let $X$ be a nonsingular complex projective variety and $D$ a locally quasi-homogeneous free divisor in $X$. In this paper we study a numerical relation between the Chern class of the sheaf of logarithmic derivations on $X$ with respect to $D$, and the Chern-Schwartz-MacPherson class of the complement of $D$ in $X$. Our result confirms a conjectural formula for these classes, at least after push-forward to projective space; it proves the full form of the conjecture for locall...

We characterize Kaehler manifolds with trivial logarithmic tangent bundle (with respect to a divisor D) as a class of certain compatifications of complex semi-tori.

Let $\mathcal{V}$ be a complete discrete valuation ring of unequal characteristic with perfect residue field, $\mathcal{P}$ be a smooth, quasi-compact, separated formal scheme over $\mathcal{V}$, $\mathcal{Z}$ be a strict normal crossing divisor of $\mathcal{P}$ and $\mathcal{P}^\sharp := (\mathcal{P}, \mathcal{Z})$ the induced smooth formal log-scheme over $\mathcal{V}$. In Berthelot's theory of arithmetic $\mathcal{D}$-modules, we work with the inductive system of sheaves o...

We consider the Hodge filtration on the sheaf of meromorphic functions along free divisors for which the logarithmic comparison theorem holds. We describe the Hodge filtration steps as submodules of the order filtration on a cyclic presentation in terms of a special factor of the Bernstein-Sato polynomial of the divisor and we conjecture a bound for the generating level of the Hodge filtration. Finally, we develop an algorithm to compute Hodge ideals of such divisors and we a...

These lecture notes provide an introduction to logarithmic geometry with a view towards recent applications in the desingularization theory.

A complex hypersurface D in complex affine n-space C^n is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for n at most 4. Analogous to Grothendieck's comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic differential forms computes the complex cohomology of the complement of D in C^n. We develop a general criter...