Categorical joins

Kuznetsov, Alexander; Perry, Alexander

doi:10.1090/jams/963

Categorical joins
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by Alexander Kuznetsov and Alexander Perry

J. Amer. Math. Soc. 34 (2021), 505-564

DOI: https://doi.org/10.1090/jams/963

Published electronically: February 18, 2021

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Abstract:

We introduce the notion of a categorical join, which can be thought of as a categorification of the classical join of two projective varieties. This notion is in the spirit of homological projective duality, which categorifies classical projective duality. Our main theorem says that the homological projective dual category of the categorical join is naturally equivalent to the categorical join of the homological projective dual categories. This categorifies the classical version of this assertion and has many applications, including a nonlinear version of the main theorem of homological projective duality.

References

Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. MR 2450211, DOI 10.1353/ajm.0.0017
Asher Auel, Marcello Bernardara, and Michele 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 (English, with English and French summaries). MR 3212256, DOI 10.1016/j.matpur.2013.11.009
David Ben-Zvi, John Francis, and David Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc. 23 (2010), no. 4, 909–966. MR 2669705, DOI 10.1090/S0894-0347-10-00669-7
David Ben-Zvi, David Nadler, and Anatoly Preygel, Integral transforms for coherent sheaves, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 12, 3763–3812. MR 3730514, DOI 10.4171/JEMS/753
Lev A. Borisov, Andrei Căldăraru, and Alexander Perry, Intersections of two Grassmannians in $\Bbb {P}^9$, J. Reine Angew. Math. 760 (2020), 133–162. MR 4069885, DOI 10.1515/crelle-2018-0014
Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427. MR 2306040, DOI 10.1353/ajm.2007.0007
Francesca Carocci and Zak Turčinović, Homological projective duality for linear systems with base locus, Int. Math. Res. Not. IMRN 21 (2020), 7829–7856. MR 4176839, DOI 10.1093/imrn/rny222
Dennis Gaitsgory and Nick Rozenblyum, A study in derived algebraic geometry. Vol. I. Correspondences and duality, Mathematical Surveys and Monographs, vol. 221, American Mathematical Society, Providence, RI, 2017. MR 3701352, DOI 10.1090/surv/221.1
Daisuke Inoue, Calabi–Yau 3-folds from projective joins of del Pezzo manifolds, arXiv:1902.10040, 2019.
Qingyuan Jiang and Conan Naichung Leung, Categorical duality between joins and intersections, arXiv:1811.05135, 2018.
Qingyuan Jiang, Conan Naichung Leung, and Ying Xie, Categorical Plücker formula and homological projective duality, arXiv:1704.01050, 2017.
A. G. Kuznetsov, Hyperplane sections and derived categories, Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006), no. 3, 23–128 (Russian, with Russian summary); English transl., Izv. Math. 70 (2006), no. 3, 447–547. MR 2238172, DOI 10.1070/IM2006v070n03ABEH002318
Alexander Kuznetsov, Homological projective duality, Publ. Math. Inst. Hautes Études Sci. 105 (2007), 157–220. MR 2354207, DOI 10.1007/s10240-007-0006-8
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, habilitation thesis (in Russian), unpublished (2008).
Alexander Kuznetsov, Lefschetz decompositions and categorical resolutions of singularities, Selecta Math. (N.S.) 13 (2008), no. 4, 661–696. MR 2403307, DOI 10.1007/s00029-008-0052-1
Alexander Kuznetsov, Derived categories of cubic fourfolds, Cohomological and geometric approaches to rationality problems, Progr. Math., vol. 282, Birkhäuser Boston, Boston, MA, 2010, pp. 219–243. MR 2605171, DOI 10.1007/978-0-8176-4934-0_{9}
Alexander Kuznetsov, Base change for semiorthogonal decompositions, Compos. Math. 147 (2011), no. 3, 852–876. MR 2801403, DOI 10.1112/S0010437X10005166
Alexander Kuznetsov, Semiorthogonal decompositions in algebraic geometry, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 635–660. MR 3728631
A. G. Kuznetsov, Embedding derived categories of an Enriques surface into derived categories of Fano varieties, Izv. Ross. Akad. Nauk Ser. Mat. 83 (2019), no. 3, 127–132 (Russian, with Russian summary); English transl., Izv. Math. 83 (2019), no. 3, 534–539. MR 3954308, DOI 10.4213/im8825
Alexander Kuznetsov and Valery A. Lunts, Categorical resolutions of irrational singularities, Int. Math. Res. Not. IMRN 13 (2015), 4536–4625. MR 3439086, DOI 10.1093/imrn/rnu072
Alexander Kuznetsov and Alexander Perry, Derived categories of Gushel-Mukai varieties, Compos. Math. 154 (2018), no. 7, 1362–1406. MR 3826460, DOI 10.1112/s0010437x18007091
Alexander Kuznetsov and Alexander Perry, Categorical cones and quadratic homological projective duality, arXiv:1902.09824, 2019.
Alexander Kuznetsov and Alexander Perry, Homological projective duality for quadrics, J. Algebraic Geom., DOI 10.1090/jag/767, arXiv:1902.09832, 2021.
Alexander Kuznetsov and Maxim Smirnov, On residual categories for Grassmannians, Proc. Lond. Math. Soc. (3) 120 (2020), no. 5, 617–641. MR 4017162, DOI 10.1112/plms.12294
Joseph Lipman and Amnon Neeman, Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor, Illinois J. Math. 51 (2007), no. 1, 209–236. MR 2346195
Jacob Lurie, Higher algebra, available at https://www.math.ias.edu/~lurie/.
Jacob Lurie, Spectral algebraic geometry, available at https://www.math.ias.edu/~lurie/.
Laurent Manivel, Double spinor Calabi-Yau varieties, Épijournal Géom. Algébrique 3 (2019), Art. 2, 14 (English, with English and French summaries). MR 3936623, DOI 10.46298/epiga.2019.volume3.3965
D. O. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 852–862 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 133–141. MR 1208153, DOI 10.1070/IM1993v041n01ABEH002182
John Christian Ottem and Jørgen Vold Rennemo, A counterexample to the birational Torelli problem for Calabi-Yau threefolds, J. Lond. Math. Soc. (2) 97 (2018), no. 3, 427–440. MR 3816394, DOI 10.1112/jlms.12111
Alexander Perry, Noncommutative homological projective duality, Adv. Math. 350 (2019), 877–972. MR 3948688, DOI 10.1016/j.aim.2019.04.052
Richard P. Thomas, Notes on homological projective duality, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 585–609. MR 3821163, DOI 10.1090/pspum/097.1/01686
Bertrand Toën, Derived algebraic geometry, EMS Surv. Math. Sci. 1 (2014), no. 2, 153–240. MR 3285853, DOI 10.4171/EMSS/4
Bertrand Toën and Gabriele Vezzosi, Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008), no. 902, x+224. MR 2394633, DOI 10.1090/memo/0902
Michel van den Bergh, Non-commutative crepant resolutions, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 749–770. MR 2077594
Michel Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423–455. MR 2057015, DOI 10.1215/S0012-7094-04-12231-6