Primitive cohomology and the tube mapping

TL;DR

This paper demonstrates that primitive cohomology in the middle degree of a smooth complex projective variety is generated by specific tube classes derived from hyperplane section monodromy, using a novel group cohomology result.

Contribution

It introduces a new geometric construction of primitive cohomology classes via tube classes linked to monodromy, supported by a group cohomology result.

Findings

Primitive cohomology generated by tube classes

Construction of classes from hyperplane section monodromy

Group cohomology result of independent interest

Abstract

Let X be a smooth complex projective variety of dimension d. We show that its primitive cohomology in degree d is generated by certain "tube classes," constructed from the monodromy of the family of smooth hyperplane sections on X. The proof makes use of a result about the group cohomology of certain representations that may be of independent interest.