Schur-Finiteness (and Bass-Finiteness) Conjecture for Quadric Fibrations and Families of Sextic du Val del Pezzo Surfaces
Doc. Math. 25, 2339-2354 (2020)
DOI: 10.25537/dm.2020v25.2339-2354
Communicated by Max Karoubi
Summary
Let \(Q \to B\) be a quadric fibration and \(T \to B\) a family of sextic du Val del Pezzo surfaces. Making use of the theory of noncommutative mixed motives, we establish a precise relation between the Schur-finiteness conjecture for \(Q\), resp. for \(T\), and the Schur-finiteness conjecture for \(B\). As an application, we prove the Schur-finiteness conjecture for \(Q\), resp. for \(T\), when \(B\) is low-dimensional. Along the way, we obtain a proof of the Schur-finiteness conjecture for smooth complete intersections of two or three quadric hypersurfaces. Finally, we prove similar results for the Bass-finiteness conjecture.
Mathematics Subject Classification
14A22, 14C15, 14D06
Keywords/Phrases
Schur-finiteness conjecture, Bass-finiteness conjecture, quadric fibrations, du Val del Pezzo surfaces, noncommutative algebraic geometry, noncommutative mixed motives
References
1. D. Abramovich, T. Graber and A. Vistoli, Gromov-Witten theory of Deligne-Mumford stacks. Amer. J. Math. 130 (2008), no. 5, 1337-1398. DOI 10.1353/ajm.0.0017; zbl 1193.14070; MR2450211; arxiv math/0603151.
2. A. Auel, M. Bernardara and M. Bolognesi, Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems. J. Math. Pures Appl. (9) 102 (2014), no. 1, 249-291. DOI 10.1016/j.matpur.2013.11.009; zbl 1327.14078; MR3212256; arxiv 1109.6938.
3. J. Ayoub, Motives and algebraic cycles: a selection of conjectures and open questions. Hodge theory and \(L^2\)-analysis, 87-125, Adv. Lect. Math. (ALM), vol. 39. Int. Press, Somerville, MA, 2017. zbl 1379.14003; MR3751289.
4. J. Ayoub, Topologie feuilletée et la conservativité des réalisations classiques en caractéristique nulle. Available at http://user.math.uzh.ch/ayoub.
5. H. Bass, Some problems in classical algebraic \(K\)-theory. Algebraic K-theory, II: ''Classical'' algebraic \(K\)-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), 3-73. LNM 342, 1973. zbl 0384.18008; MR0409606.
6. S. Bloch, A. Kas and D. Lieberman, Zero cycles on surfaces with \(p_g=0\). Compositio Math. 33 (1976), 135-145. zbl 0337.14006; MR0435073.
7. A. Bondal and D. Orlov, Derived categories of coherent sheaves. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 47-56. zbl 0996.18007; MR1957019; arxiv math/0206295.
8. A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties. arxiv alg-geom/9506012.
9. J. Bouali, Motives of quadric bundles. Manuscr. Math. 149 (2016), no. 3-4, 347-368. DOI 10.1007/s00229-015-0783-1; zbl 1350.14007; MR3458173; arxiv 1310.2782.
10. C. Cadman, Using stacks to impose tangency conditions on curves. Amer. J. Math. 12 (2007), no. 2, 405-427. DOI 10.1353/ajm.2007.0007; zbl 1127.14002; MR2306040; arxiv math/0312349.
11. P. Deligne, Catégories tensorielles. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. Mosc. Math. J. 2 (2002), no. 2, 22-248. zbl 1005.18009; MR1944506.
12. V. Guletskii, Finite-dimensional objects in distinguished triangles. J. Number Theory 119 (2006), no. 1, 99-127. DOI 10.1016/j.jnt.2005.10.008; zbl 1102.14003; MR2228952; arxiv math/0306297.
13. V. Guletskii and C. Pedrini, Finite-dimensional motives and the conjectures of Beilinson and Murre. Special issue in honor of Hyman Bass on his seventieth birthday. Part III. \(K\)-Theory 30 (2003), no. 3, 24-263. DOI 10.1023/B:KTHE.0000019787.69435.89; zbl 1060.19001; MR2064241; arxiv math/0303170.
14. D. Grayson, Finite generation of \(K\)-groups of a curve over a finite field (after Daniel Quillen). Algebraic K-theory, Part I (Oberwolfach, 1980), 69-90, LNM 966, 1982. zbl 0502.14004; MR0689367.
15. G. Harder, Die Kohomologie \(S\)-arithmetischer Gruppen über Funktionenkörpern. Invent. Math. 42 (1977), 135-175. DOI 10.1007/BF01389786; zbl 0391.20036; MR0473102.
16. K. Helmsauer, Chow Motives of del Pezzo surfaces of degree \(5\) and \(6\). MSc. thesis (2013). Available at https://search.library.ualberta.ca/catalog/6504220.
17. M. Hovey, Model categories. Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence, RI, 1999. zbl 0909.55001; MR1650134.
18. A. Ishii and K. Ueda, The special McKay correspondence and exceptional collections. Tohoku Math. J. (2) 67 (2015), no. 4, 585-609. DOI 10.2748/tmj/1450798075; zbl 1335.14005; MR3436544; arxiv 1104.2381.
19. B. Kahn, Algebraic \(K\)-theory, algebraic cycles and arithmetic geometry. Handbook of Algebraic \(K\)-theory, 351-428, Berlin, New York. Springer-Verlag, 2005. zbl 1115.19003; MR2181827.
20. B. Keller, On differential graded categories. International Congress of Mathematicians (Madrid), Vol. II, 151-190. Eur. Math. Soc., Zürich, 2006. zbl 1140.18008; MR2275593; arxiv math/0601185.
21. S.-I. Kimura, Chow groups are finite dimensional, in some sense. Math. Ann. 33 (2005), no. 1, 173-201. DOI 10.1007/s00208-004-0577-3; zbl 1067.14006; MR2107443.
22. M. Kontsevich, Mixed noncommutative motives. Talk at the Workshop on Homological Mirror Symmetry, Miami, 2010. Available at www-math.mit.edu/auroux/frg/miami10-notes.
23. M. Kontsevich, Notes on motives in finite characteristic. Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 213-247, Progr. Math., vol. 270, Birkhäuser Boston, Inc., Boston, MA, 2009. DOI 10.1007/978-0-8176-4747-6_7; zbl 1279.11065; MR2641191; arxiv math/0702206.
24. M. Kontsevich, Noncommutative motives. Talk at the IAS on the occasion of the 61. birthday of Pierre Deligne (2005). Available at http://video.ias.edu/Geometry-and-Arithmetic.
25. A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics. Adv. Math. 218 (2008), no. 5, 1340-1369. DOI 10.1016/j.aim.2008.03.007; zbl 1168.14012; MR2419925; arxiv math/0510670.
26. A. Kuznetsov, Derived categories of families of sextic del Pezzo surfaces. To appear in IMRN. arxiv 1708.00522.
27. V. Lunts and D. Orlov, Uniqueness of enhancement for triangulated categories. J. Amer. Math. Soc. 23 (2010), no. 3, 853-908. DOI 10.1090/S0894-0347-10-00664-8; zbl 1197.14014; MR2629991; arxiv 0908.4187.
28. C. Mazza, Schur functors and motives. \(K\)-Theory 33 (2004), no. 2, 89-106. DOI 10.1007/s10977-004-6468-2; zbl 1071.14026; MR2131746; arxiv 1010.3932.
29. D. Quillen, Finite generation of the groups \(K_i\) of rings of algebraic integers. Cohomology of groups and algebraic \(K\)-theory, 479-488, Adv. Lect. Math. (ALM), 12 (2010). zbl 1197.19002; MR2655185.
30. D. Quillen, On the cohomology and \(K\)-theory of the general linear groups over a finite field. Ann. of Math. (2) 96 (1972), 552-586. DOI 10.2307/1970825; zbl 0249.18022; MR0315016.
31. A. Shermenev, The motive of an abelian variety. Funct. Anal. 8 (1974), 47-53. DOI 10.1007/BF02028307; zbl 0294.14003; MR0335523.
32. G. Tabuada, Recent developments on noncommutative motives. New Directions in Homotopy Theory, Contemporary Mathematics 707 (2018), 143-173. DOI 10.1090/conm/707/14258; zbl 1397.14013; MR3807746; arxiv 1611.05439.
33. G. Tabuada, Noncommutative Motives. With a preface by Yuri I. Manin. University Lecture Series, 63. American Mathematical Society, Providence, RI, 2015. DOI 10.1090/ulect/063; zbl 1333.14002; MR3379910.
34. G. Tabuada, Voevodsky's mixed motives versus Kontsevich's noncommutative mixed motives. Advances in Mathematics 264 (2014), 506-545. DOI 10.1016/j.aim.2014.07.022; zbl 1349.14025; MR3250292; arxiv 1402.4438.
35. G. Tabuada, Higher \(K\)-theory via universal invariants. Duke Math. J. 145 (2008), no. 1, 121-206. DOI 10.1215/00127094-2008-049; zbl 1166.18007; MR2451292; arxiv 0706.2420.
36. G. Tabuada and M. Van den Bergh, Additive invariants of orbifolds. Geometry and Topology 22 (2018), 3003-3048. DOI 10.2140/gt.2018.22.3003; zbl 1397.14005; MR3811776; arxiv 1612.03162.
37. G. Tabuada and M. Van den Bergh, Noncommutative motives of Azumaya algebras. J. Inst. Math. Jussieu 14 (2015), no. 2, 379-403. DOI 10.1017/S147474801400005X; zbl 1356.14007; MR3315059; arxiv 1307.7946.
38. C. Vial, Algebraic cycles and fibrations. Doc. Math. 18 (2013), 1521-1553. https://elibm.org/article/10000253; zbl 1349.14027; MR3158241; arxiv 1203.2650.
39. V. Voevodsky, Triangulated categories of motives over a field. Cycles, transfers, and motivic homology theories, 188-238, Ann. of Math. Stud., 143, Princeton, NJ, 2000. zbl 1019.14009; MR1764202.
40. C. Voisin, Bloch's conjecture for Catanese and Barlow surfaces. J. Differential Geom. 97 (2014), no. 1, 149-175. DOI 10.4310/jdg/1404912107; zbl 1386.14145; MR3229054; arxiv 1210.3935.
41. C. Voisin, Sur les zéro-cycles de certaines hypersurfaces munies d'un automorphisme. Ann. Scuola Norm. Sup. Pisa 19 (1992), 473-492. zbl 0786.14006; MR1205880.
Affiliation
Tabuada, Gonçalo
Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK, and Departamento de Matemática and Centro de Matemática e Aplicações (CMA), FCT, UNL, Quinta da Torre, 2829-516 Caparica, Portugal