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

Author: Gonçalo Tabuada
Journal: Proc. Amer. Math. Soc. 146 (2018), 1389-1399
MSC (2010): Primary 14A22, 14C15, 14M12, 18D20, 18E30
Published electronically: January 12, 2018
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Abstract: Grothendieck conjectured in the sixties that the even Künneth 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 conjectures are invariant under homological projective duality. This leads to a proof of Grothendieck's original conjectures in the case of intersections of quadrics and linear sections of determinantal varieties. Along the way, we also prove the case of quadric fibrations and intersections of bilinear divisors.

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Additional Information

Gonçalo Tabuada
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Portugal; Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Portugal

Received by editor(s): May 20, 2016
Received by editor(s) in revised form: September 24, 2016, and January 23, 2017
Published electronically: January 12, 2018
Additional Notes: The author was partially supported by the NSF CAREER Award #1350472 and by the Portuguese Foundation for Science and Technology grant PEst-OE/MAT/UI0297/2014.
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2018 American Mathematical Society

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