Singular support and Characteristic cycle of a rank one sheaf in codimension two

TL;DR

This paper computes the singular support and characteristic cycle of a rank 1 sheaf in codimension two on smooth varieties, introducing a new partially logarithmic ramification theory to handle the algebraic cycles involved.

Contribution

It develops a general theory of partially logarithmic ramification and defines algebraic cycles on logarithmic cotangent bundles, linking them to singular support and characteristic cycle.

Findings

Computed singular support and characteristic cycle in codimension two

Introduced partially logarithmic ramification theory

Established equality of cycles outside codimension > 2

Abstract

We compute the singular support and the characteristic cycle of a rank 1 sheaf on a smooth variety in codimension 2 using ramification theory, when the ramification of the sheaf is clean. We develop a general theory, called the partially logarithmic ramification theory, and define an algebraic cycle on a logarithmic cotangent bundle with partial logarithmic poles along the boundary. We prove that the inverse image of the support of the cycle and the pull-back of the cycle to the cotangent bundle are equal to the singular support and the characteristic cycle, respectively, outside a closed subset of the variety of codimension greater than 2 under a mild assumption.