Cleanliness and log-characteristic cycles for vector bundles with flat connections

TL;DR

This paper introduces a cleanliness condition for flat connections with irregular singularities on vector bundles over algebraic varieties, enabling explicit computation of log-characteristic cycles and linking them to irregularities, with applications to de Rham cohomology.

Contribution

It defines a new cleanliness condition controlling singularities of flat connections and computes their log-characteristic cycles explicitly, relating them to refined irregularities.

Findings

Explicit formula for log-characteristic cycle under cleanliness

Relation between characteristic cycle and refined irregularities

Euler characteristic of de Rham cohomology derived from log-variant Kashiwara-Dubson formula

Abstract

Let $X$ be a proper smooth algebraic variety over a field $k$ of characteristic zero and let $D$ be a divisor with simple normal crossings. Let $M$ be a vector bundle over $X-D$ equipped with a flat connection with possible irregular singularities along $D$. We define a cleanliness condition which roughly says that the singularities of the connection are controlled by the singularities at the generic points of $D$. When this condition is satisfied, we compute explicitly the associated log-characteristic cycle, and relate it to the so-called refined irregularities. As a corollary of a log-variant of Kashiwara-Dubson formula, we obtain the Euler characteristic of the de Rham cohomology of the vector bundle, under a mild technical hypothesis on $M$.