Logarithmic comparison theorem and D-modules: an overview

This article is based on the 5th Takagi Lectures delivered at the University of Tokyo in 2008. We discuss a hypothetical correspondence between holonomic D-modules on an algebraic variety X defined over a field of zero characteristic, and certain families of Lagrangian subvarieties in the cotangent bundle to X. The correspondence is based on the reduction to positive characteristic.

We write down a new "logarithmic" quasicoherent category $\operatorname{Qcoh}_{log}(U, X, D)$ attached to a smooth open algebraic variety $U$ with toroidal compactification $X$ and boundary divisor $D$. This is a (large) symmetric monoidal Abelian category, which we argue can be thought of as the categorical substrate for logarithmic Hodge theory of $U$. We show that its Hochschild homology theory coincides with the theory of log-forms on $X$ with logarithmic structure induce...

We study an adaptation to the logarithmic case of the Kobayashi-Eisenman pseudo-volume form, or rather an adaptation of its variant defined by Claire Voisin, for which she replaces holomorphic maps by holomorphic K-correspondences. We define an intrinsic logarithmic pseudo-volume form \Phi_{X,D} for every pair (X,D) consisting of a complex manifold X and a normal crossing Weil divisor, the positive part of which is reduced. We then prove that \Phi_{X,D} is generically non-deg...

Let $X$ be a nonsingular complex variety and $D$ a reduced effective divisor in $X$. In this paper we study the conditions under which the formula $c_{SM}(1_U)=c(\textup{Der}_X(-\log D))\cap [X]$ is true. We prove that this formula is equivalent to a Riemann-Roch type of formula. As a corollary, we show that over a surface, the formula is true if and only if the Milnor number equals the Tjurina number at each singularity of $D$. We also show the Rimann-Roch type of formula is...

We introduce the definition of De Rham logarithmic classes. We show that the De Rham class of an algebraic cycle of a smooth algebraic variety over a field of characteristic zero is logarithmic and conversely that a logarithmic class of bidegree $(d,d)$ is the De Rham class of an algebraic cycle (of codimension $d$). We also give for smooth algebraic varieties over a $p$-adic field an analytic version of this result. We deduce from the analytic case the Tate conjecture for sm...

In this note, we prove the logarithmic $p$-adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally $K(\pi,1)$ (in a certain sense) with respect to $\mathbb{F}_p$-local systems and ramified coverings along the divisor. We follow Scholze's method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After intr...

Let $X$ be a compact K\"ahler manifold and $D$ be a simple normal crossing divisor. If $D$ is the support of some effective $k$-ample divisor, we show $$ H^q(X,\Omega^p_X(\log D))=0,\quad \text{for}\quad p+q>n+k.$$

In this paper, we prove a $\partial\bar{\partial}$-type lemma on compact K\"ahler manifolds for logarithmic differential forms valued in the dual of a certain pseudo-effective line bundle, thereby confirming a conjecture proposed by X. Wan. We then derive several applications, including strengthened results by H. Esnault-E. Viehweg on the degeneracy of the spectral sequence at the $E_1$-stage for projective manifolds associated with the logarithmic de Rham complex, as well ...

We study relative and logarithmic characteristic cycles associated to holonomic $\mathscr D$-modules. As applications, we obtain: (1) an alternative proof of Ginsburg's log characteristic cycle formula for lattices of regular holonomic $\mathscr D$-modules following ideas of Sabbah and Briancon-Maisonobe-Merle, and (2) the constructibility of the log de Rham complexes for lattices of holonomic $\mathscr D$-modules, which is a natural generalization of Kashiwara's constructibi...

Estas notas son las memorias del cursillo dictado en el XXII Congreso Colombiano de Matem\'aticas en la Universidad del Cauca en Popay\'an - Colombia. El objetivo de este escrito es brindar un acercamiento a la teor\'ia de m\'odulos sobre el anillo de operadores diferenciales de una variedad algebraica suave. These are the lecture notes of a short course given at the XXII Colombian Congress of Mathematics held at Universidad del Cauca in Popay\'an - Colombia. The aim of thi...