A note on Grothendieck's standard conjectures of type C and D

TL;DR

This paper extends Grothendieck's standard conjectures of type C and D from smooth projective schemes to smooth proper dg categories, demonstrating their invariance under homological projective duality and proving them in several geometric cases.

Contribution

It generalizes Grothendieck's conjectures to dg categories and establishes their invariance under duality, providing new proofs in specific geometric contexts.

Findings

Grothendieck's conjectures are invariant under homological projective duality.

Confirmed the conjectures for intersections of quadrics and related varieties.

Proved the conjectures for quadric fibrations.

Abstract

Grothendieck conjectured in the sixties that the even Kunneth projector (with respect to a Weil cohomology theory) is algebraic and that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note we extend these celebrated conjectures from smooth projective schemes to the broad setting of smooth proper dg categories. As an application, we prove that Grothendieck's original conjectures are invariant under homological projective duality. This leads to a proof of Grothendieck's conjectures in the case of intersections of quadrics, linear sections of determinantal varieties, and intersections of bilinear divisors. Along the way, we prove also the case of quadric fibrations.