Markman, Eyal The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians. (English) Zbl 1523.14011 J. Eur. Math. Soc. (JEMS) 25, No. 1, 231-321 (2023). Summary: We compute the subgroup of the monodromy group of a generalized Kummer variety associated to equivalences of derived categories of abelian surfaces. The result was announced by the author and S. Mehrotra [Math. Nachr. 290, No. 5–6, 876–884 (2017; Zbl 1400.14096)]. G. Mongardi [Algebr. Geom. 3, No. 3, 385–391 (2016; Zbl 1373.14011)] showed that the subgroup constructed here is in fact the whole monodromy group. As an application we prove the Hodge conjecture for the generic abelian fourfold of Weil type with complex multiplication by an arbitrary imaginary quadratic number field \(K\), but with trivial discriminant invariant in \(\mathbb{Q}^\ast / \mathrm{Nm} ( K^\ast )\). The latter result is inspired by a recent observation of O’Grady that the third intermediate Jacobians of smooth projective varieties of generalized Kummer deformation type form complete families of abelian fourfolds of Weil type. Finally, we prove the surjectivity of the Abel-Jacobi map from the Chow group \(\mathrm{CH}^2 ( Y )_0\) of codimension 2 algebraic cycles homologous to zero on every projective irreducible holomorphic symplectic manifold \(Y\) of Kummer type onto the third intermediate Jacobian of \(Y\), as predicted by the generalized Hodge Conjecture. Cited in 2 ReviewsCited in 10 Documents MSC: 14C25 Algebraic cycles 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry 14G05 Rational points 14J42 Holomorphic symplectic varieties, hyper-Kähler varieties Keywords:abelian surfaces and fourfolds; hyperkähler varieties; Hodge conjecture; derived categories contents Citations:Zbl 1400.14096; Zbl 1373.14011 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Acknowledgements. I am grateful to Kieran O’Grady for sharing with me an early draft of his insightful paper [43]; the insight it provided was crucial in the proof of Theorem 1.5. I thank Misha Verbitsky for reference [40]. [2] I am grateful to Claire Voisin for her suggestion of Theorem 1.10. I thank the referees for their insightful comments and suggestions. I thank Bert van Geemen for pointing out a typo in a previous version of equation (4.29). [3] Atiyah, M. F.: K-theory. W. A. Benjamin, New York (1967) Zbl 0159.53302 MR 0224083 · Zbl 0159.53302 [4] Beauville, A.: Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differential Geom. 18, 755-782 (1984) (1983) Zbl 0537.53056 MR 730926 · Zbl 0537.53056 [5] Boissière, S., Nieper-Wißkirchen, M., Sarti, A.: Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties. J. Math. Pures Appl. (9) 95, 553-563 (2011) Zbl 1215.14046 MR 2786223 · Zbl 1215.14046 [6] Buskin, N.: Every rational Hodge isometry between two K3 surfaces is algebraic. J. Reine Angew. Math. 755, 127-150 (2019) Zbl 07121906 MR 4015230 · Zbl 1475.32008 [7] Charles, F., Markman, E.: The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces. Compos. Math. 149, 481-494 (2013) Zbl 1312.14012 MR 3040747 · Zbl 1312.14012 [8] Chevalley, C. C.: The Algebraic Theory of Spinors. Columbia Univ. Press, New York (1954) Zbl 0057.25901 MR 0060497 · Zbl 0057.25901 [9] Deligne, P.: Notes on spinors. In: Quantum Fields and Strings: a Course for Mathematicians, Vols. 1 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI, 99-135 (1999) Zbl 1170.81380 MR 1701598 · Zbl 1170.81380 [10] El Zein, F., Zucker, S.: Extendability of normal functions associated to algebraic cycles. In: Topics in Transcendental Algebraic Geometry (Princeton, NJ, 1981/1982), Ann. of Math. Stud. 106, Princeton Univ. Press, Princeton, NJ, 269-288 (1984) Zbl 0545.14017 MR 756857 · Zbl 0545.14017 [11] Golyshev, V., Lunts, V., Orlov, D.: Mirror symmetry for abelian varieties. J. Algebraic Geom. 10, 433-496 (2001) Zbl 1014.14020 MR 1832329 · Zbl 1014.14020 [12] Göttsche, L.: Hilbert Schemes of Zero-Dimensional Subschemes of Smooth Varieties. Lecture Notes in Math. 1572, Springer, Berlin (1994) Zbl 0814.14004 MR 1312161 · Zbl 0814.14004 [13] Göttsche, L., Soergel, W.: Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces. Math. Ann. 296, 235-245 (1993) Zbl 0789.14002 MR 1219901 · Zbl 0789.14002 [14] Green, M. L.: Infinitesimal methods in Hodge theory. In: Algebraic Cycles and Hodge Theory (Torino, 1993), Lecture Notes in Math. 1594, Springer, Berlin, 1-92 (1994) Zbl 0846.14001 MR 1335239 · Zbl 0846.14001 [15] Hassett, B., Tschinkel, Y.: Hodge theory and Lagrangian planes on generalized Kummer fourfolds. Moscow Math. J. 13, 33-56, 189 (2013) Zbl 1296.14008 MR 3112215 · Zbl 1296.14008 [16] Helgason, S.: Differential Geometry and Symmetric Spaces. Pure Appl. Math. 12, Academic Press, New York (1962) Zbl 0111.18101 MR 0145455 · Zbl 0111.18101 [17] Huybrechts, D.: Compact hyperkähler manifolds: basic results. Invent. Math. 135, 63-113 (1999); · Zbl 0953.53031 [18] Erratum, Invent. Math. 152, 209-212 (2003) Zbl 0953.5301 Zbl 1029.53058(err.) MR 1664696 MR 1965365(err.) [19] Huybrechts, D.: Moduli spaces of hyperkähler manifolds and mirror symmetry. In: Inter-section Theory and Moduli, ICTP Lect. Notes 19, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 185-247 (2004) Zbl 1110.53034 MR 2172498 · Zbl 1110.53034 [20] Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry. Oxford Math. Monogr., Clarendon Press, Oxford Univ. Press, Oxford (2006) Zbl 1095.14002 MR 2244106 · Zbl 1095.14002 [21] Huybrechts, D.: A global Torelli theorem for hyperkähler manifolds [after M. Verbitsky]. In: Séminaire Bourbaki, Volume 2010/2011, Astérisque 348, exp. 1040, x, 375-403 (2012) Zbl 1272.32014 MR 3051203 · Zbl 1272.32014 [22] Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. 2nd ed., Cambridge Math. Library, Cambridge Univ. Press, Cambridge (2010) Zbl 1206.14027 MR 2665168 · Zbl 1206.14027 [23] Kapfer, S., Menet, G.: Integral cohomology of the generalized Kummer fourfold. Algebr. Geom. 5, 523-567 (2018) Zbl 1423.14243 MR 3847205 · Zbl 1423.14243 [24] Karoubi, M.: K-theory. Grundlehren Math. Wiss. 226, Springer, Berlin (1978) Zbl 0382.55002 MR 0488029 · Zbl 0382.55002 [25] Lam, T. Y.: Introduction to Quadratic Forms over Fields. Grad. Stud. Math. 67, Amer. Math. Soc., Providence, RI (2005) Zbl 1068.11023 MR 2104929 · Zbl 1068.11023 [26] Looijenga, E., Lunts, V. A.: A Lie algebra attached to a projective variety. Invent. Math. 129, 361-412 (1997) Zbl 0890.53030 MR 1465328 · Zbl 0890.53030 [27] Markman, E.: Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces. J. Reine Angew. Math. 544, 61-82 (2002) Zbl 0988.14019 MR 1887889 · Zbl 0988.14019 [28] Markman, E.: Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces. Adv. Math. 208, 622-646 (2007) Zbl 1115.14036 MR 2304330 · Zbl 1115.14036 [29] Markman, E.: On the monodromy of moduli spaces of sheaves on K3 surfaces. J. Algebraic Geom. 17, 29-99 (2008) Zbl 1185.14015 MR 2357680 · Zbl 1185.14015 [30] Markman, E.: Integral constraints on the monodromy group of the hyperKähler resolution of a symmetric product of a K3 surface. Int. J. Math. 21, 169-223 (2010) Zbl 1184.14074 MR 2650367 · Zbl 1184.14074 [31] Markman, E.: A survey of Torelli and monodromy results for holomorphic-symplectic vari-eties. In: Complex and Differential Geometry, Springer Proc. Math. 8, Springer, Heidelberg, 257-322 (2011) Zbl 1229.14009 MR 2964480 · Zbl 1229.14009 [32] Markman, E.: The Beauville-Bogomolov class as a characteristic class. J. Algebraic Geom. 29, 199-245 (2020) Zbl 1439.14123 MR 4069649 · Zbl 1439.14123 [33] Markman, E.: On the existence of universal families of marked irreducible holomorphic symplectic manifolds. Kyoto J. Math. 61, 207-223 (2021) Zbl 1465.53065 MR 4260432 · Zbl 1465.53065 [34] Markman, E., Mehrotra, S.: Hilbert schemes of K3 surfaces are dense in moduli. Math. Nachr. 290, 876-884 (2017) Zbl 1400.14096 MR 3636385 · Zbl 1400.14096 [35] Markman, E., Mehrotra, S.: Integral transforms and deformations of K3 surfaces. arXiv:1507.03108v1 (2015) [36] Mongardi, G.: On the monodromy of irreducible symplectic manifolds. Algebr. Geom. 3, 385-391 (2016) Zbl 1373.14011 MR 3504537 · Zbl 1373.14011 [37] Moonen, B. J. J., Zarhin, Yu. G.: Hodge classes and Tate classes on simple abelian fourfolds. Duke Math. J. 77, 553-581 (1995) Zbl 0874.14034 MR 1324634 · Zbl 0874.14034 [38] Mukai, S.: Duality between D.X/ and D. y X/ with its application to Picard sheaves. Nagoya Math. J. 81, 153-175 (1981) Zbl 0417.14036 MR 607081 · Zbl 0417.14036 [39] Mukai, S.: Symplectic structure of the moduli space of sheaves on an abelian or K3 surface. Invent. Math. 77, 101-116 (1984) Zbl 0565.14002 MR 751133 · Zbl 0565.14002 [40] Mukai, S.: Fourier functor and its application to the moduli of bundles on an abelian variety. In: Algebraic Geometry (Sendai, 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 515-550 (1987) Zbl 0672.14025 MR 946249 · Zbl 0672.14025 [41] Mukai, S.: On the moduli space of bundles on K3 surfaces. I. In: Vector Bundles on Algebraic Varieties (Bombay, 1984), Tata Inst. Fund. Res. Stud. Math. 11, Tata Inst. Fund. Res., Bombay, 341-413 (1987) Zbl 0674.14023 MR 893604 · Zbl 0674.14023 [42] Mukai, S.: Abelian varieties and spin representations. Preprint Warwick Univ. (1998) (English translation from Proceedings of the symposium “Hodge Theory and Algebraic Geometry” (Sapporo, 1994), 110-135) [43] Muñoz, V.: Spin.7/-instantons, stable bundles and the Bogomolov inequality for complex 4-tori. J. Math. Pures Appl. (9) 102, 124-152 (2014) Zbl 1326.14107 MR 3212251 · Zbl 1326.14107 [44] Namikawa, Y.: Counter-example to global Torelli problem for irreducible symplectic manifolds. Math. Ann. 324, 841-845 (2002) Zbl 1028.53081 MR 1942252 · Zbl 1028.53081 [45] Nikulin, V. V.: Integer symmetric bilinear forms and some of their geometric applications. Izv. Akad. Nauk SSSR Ser. Mat. 43, 111-177, 238 (1979) (in Russian) Zbl 0427.10014 MR 525944 · Zbl 0408.10011 [46] O’Grady, K. G.: Compact tori associated to hyperkähler manifolds of Kummer type. Int. Math. Res. Notices 2021, 12356-12419 Zbl 07471389 MR 4300229 · Zbl 1490.53068 [47] Oguiso, K.: No cohomologically trivial nontrivial automorphism of generalized Kummer manifolds. Nagoya Math. J. 239, 110-122 (2020) Zbl 1440.14197 MR 4138897 · Zbl 1440.14197 [48] Orlov, D. O.: Derived categories of coherent sheaves on abelian varieties and equivalences between them. Izv. Ross. Akad. Nauk Ser. Mat. 66, 131-158 (2002) (in Russian) Zbl 1031.18007 MR 1921811 · Zbl 1031.18007 [49] Ramón Marí, J. J.: On the Hodge conjecture for products of certain surfaces. Collect. Math. 59, 1-26 (2008) Zbl 1188.14004 MR 2384535 · Zbl 1188.14004 [50] Schoen, C.: Hodge classes on self-products of a variety with an automorphism. Compos. Math. 65, 3-32 (1988) Zbl 0663.14006 MR 930145 · Zbl 0663.14006 [51] Schoen, C.: Addendum to: “Hodge classes on self-products of a variety with an automor-phism”. Compos. Math. 114, 329-336 (1998) Zbl 0926.14002 MR 1665776 · Zbl 0926.14002 [52] van Geemen, B.: An introduction to the Hodge conjecture for abelian varieties. In: Algebraic Cycles and Hodge Theory (Torino, 1993), Lecture Notes in Math. 1594, Springer, Berlin, 233-252 (1994) Zbl 828.14004 MR 1335243 · Zbl 0828.14004 [53] Verbitsky, M.: Cohomology of compact hyper-Kähler manifolds and its applications. Geom. Funct. Anal. 6, 601-611 (1996) Zbl 0861.53069 MR 1406664 · Zbl 0861.53069 [54] Verbitsky, M.: Mirror symmetry for hyper-Kähler manifolds. In: Mirror Symmetry, III (Montreal, PQ, 1995), AMS/IP Stud. Adv. Math. 10, Amer. Math. Soc., Providence, RI, 115-156 (1999) Zbl 0926.32036 MR 1673084 · Zbl 0926.32036 [55] Verbitsky, M.: Mapping class group and a global Torelli theorem for hyperkähler manifolds. Duke Math. J. 162, 2929-2986 (2013); · Zbl 1295.53042 [56] Errata, Duke Math. J. 169, 1037-1038 (2020) Zbl 1295.53042 MR 3161308 MR 4079420(err.) [57] Verbitsky, M.: Hyperholomorphic sheaves and new examples of hyperkähler manifolds. In: Hyperkähler Manifolds, by D. Kaledin and M. Verbitsky, Math. Phys. (Somerville) 12, Int. Press, Somerville, MA (1999) Zbl 0990.53048 MR 1815021 · Zbl 0990.53048 [58] Wall, C. T. C.: On the orthogonal groups of unimodular quadratic forms. II. J. Reine Angew. Math. 213, 122-136 (1963/64) Zbl 0135.08802 MR 155798 · Zbl 0135.08802 [59] Weil, A.: Abelian varieties and the Hodge ring. In: Collected Papers, Vol. III, Springer, 421-429 (1980) [60] Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321, 817-884 (2001) Zbl 1066.14013 MR 1872531 · Zbl 1066.14013 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.