×

The \(K3\) category of a cubic fourfold. (English) Zbl 1440.14180

Summary: Smooth cubic hypersurfaces \(X\subset \mathbb{P}^{5}\) (over \(\mathbb{C}\)) are linked to \(K3\) surfaces via their Hodge structures, due to the work of B. Hassett [Compos. Math. 120, No. 1, 1–23 (2000; Zbl 0956.14031)], and via a subcategory \({\mathcal{A}}_{X}\subset \text{D}^{\text{b}}(X)\), due to the work of A. Kuznetsov [Prog. Math. 282, 219–243 (2010; Zbl 1202.14012)]. The relation between these two viewpoints has recently been elucidated by N. Addington and R. Thomas [Duke Math. J. 163, No. 10, 1885–1927 (2014; Zbl 1309.14014)]. In this paper, both aspects are studied further and extended to twisted \(K3\) surfaces, which in particular allows us to determine the group of autoequivalences of \({\mathcal{A}}_{X}\) for the general cubic fourfold. Furthermore, we prove finiteness results for cubics with equivalent \(K3\) categories and study periods of cubics in terms of generalized \(K3\) surfaces.

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] N.Addington, On two rationality conjectures for cubic fourfolds, Math. Res. Lett.23 (2016), 1-13.10.4310/MRL.2016.v23.n1.a1 · Zbl 1375.14134
[2] N.Addington and R.Thomas, Hodge theory and derived categories of cubic fourfolds, Duke Math. J.163 (2014), 1885-1927.10.1215/00127094-2738639 · Zbl 1309.14014
[3] A.Bayer and T.Bridgeland, Derived automorphism groups of K3 surfaces of Picard rank 1, Duke Math. J., to appear. Preprint (2013), arXiv:1310.8266. · Zbl 1358.14019
[4] M.Ballard, D.Favero and L.Katzarkov, Orlov spectra: bounds and gaps, Invent. Math.189 (2012), 359-430.10.1007/s00222-011-0367-y · Zbl 1266.14013
[5] A.Beauville and R.Donagi, La variété des droites d’une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris Sér. I Math.301 (1985), 703-706. · Zbl 0602.14041
[6] M.Bernardara, E.Macrì, S.Mehrotra and P.Stellari, A categorical invariant for cubic threefolds, Adv. Math.229 (2012), 770-803.10.1016/j.aim.2011.10.007 · Zbl 1242.14012
[7] T.Bridgeland, Stability conditions on K3 surfaces, Duke Math. J.141 (2008), 241-291.10.1215/S0012-7094-08-14122-5 · Zbl 1138.14022
[8] T.Bridgeland and A.Maciocia, Complex surfaces with equivalent derived categories, Math. Z.236 (2001), 677-697.10.1007/PL00004847 · Zbl 1081.14023
[9] R.-O.Buchweitz and H.Flenner, A semiregularity map for modules and applications to deformations, Compositio Math.137 (2003), 135-210.10.1023/A:1023999012081 · Zbl 1085.14503
[10] R.-O.Buchweitz and H.Flenner, The global decomposition theorem for Hochschild (co-)homology of singular spaces via the Atiyah-Chern character, Adv. Math.217 (2008), 243-281.10.1016/j.aim.2007.06.013 · Zbl 1144.14015
[11] A.Căldăraru, Derived categories of twisted sheaves on Calabi-Yau manifolds, PhD thesis, Cornell (2000).
[12] A.Căldăraru, The Mukai paring II. The Hochschild-Kostant-Rosenberg isomorphism, Adv. Math.194 (2005), 34-66.10.1016/j.aim.2004.05.012 · Zbl 1098.14011
[13] A.Canonaco and P.Stellari, Twisted Fourier-Mukai functors, Adv. Math.212 (2007), 484-503.10.1016/j.aim.2006.10.010 · Zbl 1116.14009
[14] F.Charles, A remark on the Torelli theorem for cubic fourfolds, Preprint (2012), arXiv:1209.4509.
[15] F.Charles, Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for K3 surfaces, and the Tate conjecture, Ann. of Math. (2)184 (2016), 487-526.10.4007/annals.2016.184.2.4 · Zbl 1387.14102
[16] D.Cox, Primes of the form x^2 + ny^2 . Fermat, class field theory and complex multiplication (John Wiley & Sons, New York, 1989). · Zbl 0701.11001
[17] S.Galkin and E.Shinder, The Fano variety of lines and rationality problem for a cubic hypersurface, Preprint (2014), arXiv:1405.5154.
[18] B.Hassett, Special cubic fourfolds, Compositio Math.120 (2000), 1-23.10.1023/A:1001706324425 · Zbl 0956.14031
[19] L.Hille and M.van den Bergh, Fourier-Mukai transforms, in Handbook of tilting theory, London Mathematical Society Lecture Note Series, vol. 332 (Cambridge University Press, Cambridge, 2007), 147-177.10.1017/CBO9780511735134.007
[20] S.Hosono, B.Lian, K.Oguiso and S.-T.Yau, Fourier-Mukai number of a K3 surface, CRM Proc. Lecture Notes38 (2004), 177-192. · Zbl 1076.14045
[21] D.Huybrechts, Generalized Calabi-Yau structures, K3 surfaces, and B-fields, Int. J. Math.19 (2005), 13-36.10.1142/S0129167X05002734 · Zbl 1120.14027
[22] D.Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs (Oxford University Press, Oxford, 2006). · Zbl 1095.14002
[23] D.Huybrechts, The global Torelli theorem: classical, derived, twisted, in Algebraic geometry-Seattle 2005. Part 1, Proceedings of Symposia in Pure Mathematics, vol. 80 (American Mathematical Society, Providence, RI, 2009), 235-258. · Zbl 1179.14015
[24] D.Huybrechts, Introduction to stability conditions, in Moduli spaces, London Mathematical Society Lecture Notes Series, vol. 411 (Cambridge University Press, Cambridge, 2014), 179-229.10.1017/CBO9781107279544.005 · Zbl 1316.14003
[25] D.Huybrechts, Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 2016), http://www.math.uni-bonn.de/people/huybrech/K3.html.10.1017/CBO9781316594193 · Zbl 1360.14099
[26] D.Huybrechts, E.Macrì and P.Stellari, Stability conditions for generic K3 categories, Compositio Math.144 (2008), 134-162.10.1112/S0010437X07003065 · Zbl 1152.14037
[27] D.Huybrechts, E.Macrì and P.Stellari, Derived equivalences of K3 surfaces and orientation, Duke Math. J.149 (2009), 461-507.10.1215/00127094-2009-043 · Zbl 1237.18008
[28] D.Huybrechts and P.Stellari, Equivalences of twisted K3 surfaces, Math. Ann.332 (2005), 901-936.10.1007/s00208-005-0662-2 · Zbl 1092.14047
[29] D.Huybrechts and P.Stellari, Proof of Căldăraru’s conjecture, in Moduli spaces and arithmetic geometry, Advanced Studies in Pure Mathematics, vol. 45, (2006), 31-42. · Zbl 1118.14049
[30] D.Huybrechts and R.Thomas, Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes, Math. Ann.346 (2010), 545-569.10.1007/s00208-009-0397-6 · Zbl 1186.14014
[31] K.Kawatani, A hyperbolic metric and stability conditions on K3 surfaces with<![CDATA \([\unicode[STIX]{x1D70C}=1]]\)>, Preprint (2012), arXiv:1204.1128.
[32] M.Kneser, Quadratische formen (Springer, 2002).10.1007/978-3-642-56380-5 · Zbl 1001.11014
[33] A.Kuznetsov, Derived categories of cubic and V_14 threefolds, Proc. Steklov Inst. Math.3 (2004), 171-194; arXiv:math/0303037. · Zbl 1107.14028
[34] A.Kuznetsov, Homological projective duality for Grassmannians of lines, Preprint (2006),arXiv:math.AG/0610957.
[35] A.Kuznetsov, Hochschild homology and semiorthogonal decompositions, Preprint (2009),arXiv:0904.4330.
[36] A.Kuznetsov, Derived categories of cubic fourfolds, in Cohomological and geometric approaches to rationality problems, Progress in Mathematics, vol. 282 (Springer, Berlin, 2010), 219-243.10.1007/978-0-8176-4934-0_9 · Zbl 1202.14012
[37] A.Kuznetsov, Calabi-Yau and fractional Calabi-Yau categories, Preprint (2015),arXiv:1509.07657. · Zbl 1440.14092
[38] A.Kuznetsov and D.Markushevich, Symplectic structures on moduli spaces of sheaves via the Atiyah class, J. Geom. Phys.59 (2009), 843-860.10.1016/j.geomphys.2009.03.008 · Zbl 1181.14049
[39] R.Laza, The moduli space of cubic fourfolds via the period map, Ann. of Math. (2)172 (2010), 673-711.10.4007/annals.2010.172.673 · Zbl 1201.14026
[40] M.Lieblich, Moduli of complexes on a proper morphism, J. Algebraic Geom.15 (2006), 175-206.10.1090/S1056-3911-05-00418-2 · Zbl 1085.14015
[41] M.Lieblich, D.Maulik and A.Snowden, Finiteness of K3 surfaces and the Tate conjecture, Ann. Sci. Éc. Norm. Supér.47 (2014), 285-308. · Zbl 1329.14078
[42] E.Looijenga, The period map for cubic fourfolds, Invent. Math.177 (2009), 213-233.10.1007/s00222-009-0178-6 · Zbl 1177.32010
[43] E.Markman, A survey of Torelli and monodromy results for holomorphic-symplectic varieties, in Complex and differential geometry, Proceedings in Mathematics, vol. 8 (Springer, Berlin, 2011), 257-322.10.1007/978-3-642-20300-8_15 · Zbl 1229.14009
[44] E.Macrì and P.Stellari, Infinitesimal derived Torelli theorem for K3 surfaces, (Appendix by S. Mehrotra), Int. Math. Res. Not. IMRN2009 (2009), 3190-3220. · Zbl 1174.14018
[45] E.Macrì and P.Stellari, Fano varieties of cubic fourfolds containing a plane, Math. Ann.354 (2012), 1147-1176.10.1007/s00208-011-0776-7 · Zbl 1266.18016
[46] E.Markman and S.Mehrotra, Integral transforms and deformations of K3 surfaces, Preprint (2015), arXiv:1507.03108.
[47] S.Mukai, On the moduli space of bundles on K3 surfaces. I. Vector bundles on algebraic varieties (Bombay, 1984), Tata Inst. Fund. Res. Stud. Math.11 (1987), 341-413.
[48] V.Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat.43 (1979), 111-177. · Zbl 0408.10011
[49] K.Oguiso, K3 surfaces via almost-primes, Math. Res. Lett.9 (2002), 47-63.10.4310/MRL.2002.v9.n1.a4 · Zbl 1043.14010
[50] D.Orlov, Equivalences of derived categories and K3 surfaces, J. Math. Sci.84 (1997), 1361-1381.10.1007/BF02399195 · Zbl 0938.14019
[51] D.Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, Vol. II, Progress in Mathematics, vol. 270 (Springer, Berlin, 2009), 503-531.10.1007/978-0-8176-4747-6_16 · Zbl 1200.18007
[52] E.Reinecke, Autoequivalences of twisted K3 surfaces, Master thesis, Bonn (2014),http://www.math.uni-bonn.de/people/huybrech/ReineckeMA.pdf. · Zbl 1498.14103
[53] P.Stellari, Some remarks about the FM-partners of K3 surfaces with Picard numbers 1 and 2, Geom. Dedicata108 (2004), 1-13.10.1007/s10711-004-9291-7 · Zbl 1072.14043
[54] Y.Toda, Deformations and Fourier-Mukai transforms, J. Differential Geom.81 (2009), 197-224. · Zbl 1165.14019
[55] Y.Toda, Gepner type stability condition via Orlov/Kuznetsov equivalence, Preprint (2013), arXiv:1308.3791. · Zbl 1334.14011
[56] Y.Toda, Gepner type stability conditions on graded matrix factorizations, Algebr. Geom.1 (2014), 613-665.10.14231/AG-2014-026 · Zbl 1322.14042
[57] C.Voisin, Théorème de Torelli pour les cubiques de ℙ^5, Invent. Math.6 (1986), 577-601.10.1007/BF01389270 · Zbl 0622.14009
[58] C.Voisin, Correction à : ‘Théorème de Torelli pour les cubiques de ℙ^5 ’, Invent. Math.172 (2008), 455-458.10.1007/s00222-008-0116-z · Zbl 1133.14310
[59] K.Yoshioka, Moduli spaces of twisted sheaves on a projective variety, in Moduli Spaces and Arithmetic Geometry, Advanced Studies in Pure Mathematics, vol. 45 (World Scientific, Singapore, 2006), 1-30. · Zbl 1118.14013
[60] K.Yoshioka, Stability and the Fourier-Mukai transform. II, Compositio Math.145 (2009), 112-142.10.1112/S0010437X08003758 · Zbl 1165.14033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.