Kontsevich's conjecture on an algebraic formula for vanishing cycles of local systems

TL;DR

This paper proves Kontsevich's conjecture on a formula for vanishing cycles of local systems using twisted de Rham complexes, extending previous results to more general D-module contexts.

Contribution

It provides a proof of Kontsevich's conjecture on vanishing cycles formulas and generalizes the result to regular holonomic D-modules.

Findings

Confirmed Kontsevich's conjecture for local systems.

Extended the formula to regular holonomic D-modules.

Connected the result to Brieskorn's work on isolated singularities.

Abstract

For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the corresponding locally free sheaf with integrable connection having regular singularity at infinity. We also prove its local version, which may be viewed as a natural generalization of a result of E. Brieskorn in the isolated singularity case. We then generalize these to the case of the de Rham complexes of regular holonomic D-modules where we have to use the tensor product with a certain sheaf of formal microlocal differential operators instead of the formal completion.