A note on Grothendieck's standard conjectures of type C and D
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.
Keywords
Cite
@article{arxiv.1605.05307,
title = {A note on Grothendieck's standard conjectures of type C and D},
author = {Goncalo Tabuada},
journal= {arXiv preprint arXiv:1605.05307},
year = {2016}
}
Comments
Revised version. New result: we extend Grothendieck's standard conjecture of type C from smooth projective schemes to the broad setting of smooth proper dg categories. This leads to a proof of Grothendieck's original conjecture in several cases