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

Tabuada, Gonçalo

doi:10.1090/proc/13955

A note on Grothendieck’s standard conjectures of type $C^+$ and $D$
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Proc. Amer. Math. Soc. 146 (2018), 1389-1399 Request permission

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.

References

Yves André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses [Panoramas and Syntheses], vol. 17, Société Mathématique de France, Paris, 2004 (French, with English and French summaries). MR 2115000
M. Bernardara, M. Marcolli, and G. Tabuada Some remarks concerning Voevodsky’s nilpotence conjecture. To appear in J. Reine Angew. Math. Available at arXiv:1403.0876.
Marcello Bernardara, Michele Bolognesi, and Daniele Faenzi, Homological projective duality for determinantal varieties, Adv. Math. 296 (2016), 181–209. MR 3490767, DOI 10.1016/j.aim.2016.04.003
A. Grothendieck, Standard conjectures on algebraic cycles, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) Oxford Univ. Press, London, 1969, pp. 193–199. MR 0268189
Bernhard Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 151–190. MR 2275593
Steven L. Kleiman, The standard conjectures, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 3–20. MR 1265519, DOI 10.1090/pspum/055.1/1265519
S. L. Kleiman, Algebraic cycles and the Weil conjectures, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 359–386. MR 292838
M. Kontsevich, Mixed noncommutative motives. Talk at the Workshop on Homological Mirror Symmetry, Miami, 2010. Notes available at www-math.mit.edu/auroux/frg/miami10-notes.
Maxim Kontsevich, Notes on motives in finite characteristic, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Progr. Math., vol. 270, Birkhäuser Boston, Boston, MA, 2009, pp. 213–247. MR 2641191, DOI 10.1007/978-0-8176-4747-6_{7}
Maxim Kontsevich, Noncommutative motives. Talk at the IAS on the occasion of the $61^{\mathrm {st}}$ birthday of Pierre Deligne (2005). Available at http://video.ias.edu/Geometry-and-Arithmetic.
A. Kuznetsov, Semiorthogonal decompositions in algebraic geometry. To appear in Proceedings of the ICM 2014. Available at 1404.3143.
Alexander Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008), no. 5, 1340–1369. MR 2419925, DOI 10.1016/j.aim.2008.03.007
Alexander Kuznetsov, Homological projective duality. Publ. Math. IHÉS (2007), no. 105, 157–220.
Alexander Kuznetsov, Homological projective duality for Grassmannians of lines. Available at arXiv:math/0610957.
David I. Lieberman, Numerical and homological equivalence of algebraic cycles on Hodge manifolds, Amer. J. Math. 90 (1968), 366–374. MR 230336, DOI 10.2307/2373533
Valery A. Lunts and Dmitri O. Orlov, Uniqueness of enhancement for triangulated categories, J. Amer. Math. Soc. 23 (2010), no. 3, 853–908. MR 2629991, DOI 10.1090/S0894-0347-10-00664-8
Matilde Marcolli and Gonçalo Tabuada, Noncommutative numerical motives, Tannakian structures, and motivic Galois groups, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 3, 623–655. MR 3463419, DOI 10.4171/JEMS/598
J. V. Rennemo, The homological projective dual of $\mathrm {Sym}^2\mathbb {P}(V)$. Available at arXiv:1509.04107.
Gonçalo Tabuada, Noncommutative motives, University Lecture Series, vol. 63, American Mathematical Society, Providence, RI, 2015. With a preface by Yuri I. Manin. MR 3379910, DOI 10.1090/ulect/063
Gonçalo Tabuada and Michel Van den Bergh, Noncommutative motives of Azumaya algebras, J. Inst. Math. Jussieu 14 (2015), no. 2, 379–403. MR 3315059, DOI 10.1017/S147474801400005X
Charles Vial, Algebraic cycles and fibrations, Doc. Math. 18 (2013), 1521–1553. MR 3158241