# Characteristic cycles and the conductor of direct image

## TL;DR

This paper proves the functoriality of characteristic cycles under proper push-forward for constructible complexes on smooth projective schemes, establishing a conductor formula related to Bloch's conjecture.

## Contribution

It establishes the functoriality of characteristic cycles under proper push-forward and derives a conductor formula for morphisms to curves, confirming a conjecture by Bloch.

## Key findings

- Proves functoriality of characteristic cycles for proper push-forward.
- Derives a conductor formula for morphisms to curves.
- Confirms a special case of Bloch's conjecture.

## Abstract

We prove the functoriality for proper push-forward of the characteristic cycles of constructible complexes by morphisms of smooth projective schemes over a perfect field, under the assumption that the direct image of the singular support has the dimension at most that of the target of the morphism. The functoriality is deduced from a conductor formula which is a special case for morphisms to curves. The conductor formula in the constant coefficient case gives the geometric case of a formula conjectured by Bloch.