HPD-invariance of the Tate, Beilinson and Parshin conjectures

TL;DR

This paper demonstrates that the Tate, Beilinson, and Parshin conjectures remain invariant under Homological Projective Duality, and applies this to prove these conjectures in new geometric cases, including certain stacks.

Contribution

It establishes the HPD-invariance of key conjectures and extends their validity to new classes of geometric objects like stacks and specific varieties.

Findings

Proved invariance of conjectures under HPD.

Established the conjectures for linear sections of determinantal varieties.

Extended conjectures to certain low-dimensional stacks.

Abstract

We prove that the Tate, Beilinson and Parshin conjectures are invariant under Homological Projective Duality (=HPD). As an application, we obtain a proof of these celebrated conjectures (as well as of the strong form of the Tate conjecture) in the new cases of linear sections of determinantal varieties and complete intersections of quadrics. Furthermore, we extend the original conjectures of Tate, Beilinson and Parshin from schemes to stacks and prove these extended conjectures for certain low-dimensional global orbifolds.