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

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 $$M$$ .