Chern numbers Chern numbers are the integrals of monomials in the Chern classes of $X$ living in top degrees. These are integers that can be used to control various other numerical invariants of varieties.
For a hyperkähler manifold the odd Chern classes vanish, so in the table below we only list monomials using even Chern classes.
All Chern numbers dimension d (2–20) :
monomial K3[3] -type Kum3 -type OG6 $\mathrm{c}_{ 2 }^{ 3 }$ 36800 30208 30720 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }$ 14720 6784 7680 $\mathrm{c}_{ 6 }^{ }$ 3200 448 1920
monomial K3[4] -type Kum4 -type $\mathrm{c}_{ 2 }^{ 4 }$ 1992240 1470000 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }$ 813240 405000 $\mathrm{c}_{ 4 }^{ 2 }$ 332730 111750 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ }$ 182340 37500 $\mathrm{c}_{ 8 }^{ }$ 25650 750
monomial K3[5] -type Kum5 -type OG10 $\mathrm{c}_{ 2 }^{ 5 }$ 126867456 84478464 127370880 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ }$ 52697088 26220672 53071200 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 2 }$ 21921408 8141472 22113000 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 6 }^{ }$ 12168576 3141504 12383280 $\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$ 5075424 979776 5159700 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 8 }^{ }$ 1774080 142560 1791720 $\mathrm{c}_{ 10 }^{ }$ 176256 2592 176904
monomial K3[6] -type Kum6 -type $\mathrm{c}_{ 2 }^{ 6 }$ 9277276480 5603050432 $\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ }$ 3910848640 1881462016 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 2 }$ 1650311720 631808744 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 6 }^{ }$ 927397840 268796752 $\mathrm{c}_{ 4 }^{ 3 }$ 697106648 212190776 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$ 392090040 90412056 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 8 }^{ }$ 139942280 17075912 $\mathrm{c}_{ 6 }^{ 2 }$ 93495320 12976376 $\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$ 59314272 5762400 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 10 }^{ }$ 14450680 441784 $\mathrm{c}_{ 12 }^{ }$ 1073720 2744
monomial K3[7] -type Kum7 -type $\mathrm{c}_{ 2 }^{ 7 }$ 765374164992 421414305792 $\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 4 }^{ }$ 326732507136 149664301056 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ 2 }$ 139582386432 53149827072 $\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 6 }^{ }$ 79324710912 24230756352 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 3 }$ 59674012416 18874417152 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$ 33935583744 8610545664 $\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ }$ 14528215296 3059945472 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 8 }^{ }$ 12357114624 1914077184 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ 2 }$ 8273055744 1397121024 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$ 5296568832 681332736 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 10 }^{ }$ 1324608768 71909376 $\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$ 1296158976 110853120 $\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 10 }^{ }$ 569044224 25700352 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 12 }^{ }$ 102477312 1198080 $\mathrm{c}_{ 14 }^{ }$ 5930496 7680
monomial K3[8] -type Kum8 -type $\mathrm{c}_{ 2 }^{ 8 }$ 70277256403200 35447947999488 $\mathrm{c}_{ 2 }^{ 6 }\mathrm{c}_{ 4 }^{ }$ 30327407026560 13129602781824 $\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ 2 }$ 13094639681760 4862661530400 $\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 6 }^{ }$ 7517275416000 2332758616128 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 3 }$ 5657019716880 1800797040144 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$ 3249219677760 864167470848 $\mathrm{c}_{ 4 }^{ 4 }$ 2445207931980 666853820172 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ }$ 1405173296520 320117226120 $\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 8 }^{ }$ 1205400258720 215605377504 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 6 }^{ 2 }$ 807925003200 153694101888 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$ 521787430080 79938804096 $\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ 2 }$ 349760996280 56953381608 $\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 8 }^{ }$ 225987046020 29638792620 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 10 }^{ }$ 133823975040 10441752768 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$ 130128762960 14239224576 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 10 }^{ }$ 58033047240 3878495784 $\mathrm{c}_{ 8 }^{ 2 }$ 21049285275 1322820801 $\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 10 }^{ }$ 14525621460 692780364 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 12 }^{ }$ 10767198960 254566800 $\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 12 }^{ }$ 4678568010 94850190 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 14 }^{ }$ 649511820 2685636 $\mathrm{c}_{ 16 }^{ }$ 30178575 9477
monomial K3[9] -type Kum9 -type $\mathrm{c}_{ 2 }^{ 9 }$ 7105044485242880 3297871360000000 $\mathrm{c}_{ 2 }^{ 7 }\mathrm{c}_{ 4 }^{ }$ 3095054052884480 1262135680000000 $\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 4 }^{ 2 }$ 1348811566120960 482990816000000 $\mathrm{c}_{ 2 }^{ 6 }\mathrm{c}_{ 6 }^{ }$ 781347805921280 240910720000000 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ 3 }$ 588050734243840 184814229440000 $\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$ 340787113328640 92197363200000 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 4 }$ 256482451425280 70712975120000 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ }$ 148696308725760 35281909440000 $\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 8 }^{ }$ 128601459097600 25082624000000 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 6 }^{ 2 }$ 86242390425600 17605804800000 $\mathrm{c}_{ 4 }^{ 3 }\mathrm{c}_{ 6 }^{ }$ 64907421320960 13500841600000 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$ 56155350159360 9603236160000 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ 2 }$ 37660572692480 6738177040000 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 8 }^{ }$ 24530800855040 3676588120000 $\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 10 }^{ }$ 14747557928960 1459909120000 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$ 14244457018880 1835380960000 $\mathrm{c}_{ 6 }^{ 3 }$ 9553579524480 1287476640000 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 10 }^{ }$ 6448976952320 559476160000 $\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$ 6227441933120 702799360000 $\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 10 }^{ }$ 2821199089280 214406248000 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 8 }^{ 2 }$ 2360786818560 191623650000 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 10 }^{ }$ 1640647441920 107096280000 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 12 }^{ }$ 1231467509760 46722720000 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 12 }^{ }$ 539392972800 17937420000 $\mathrm{c}_{ 8 }^{ }\mathrm{c}_{ 10 }^{ }$ 273089658720 11208918000 $\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 12 }^{ }$ 137685310240 3443000000 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 14 }^{ }$ 77346804480 774480000 $\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 14 }^{ }$ 33938470560 298344000 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 16 }^{ }$ 3748665600 6090000 $\mathrm{c}_{ 18 }^{ }$ 143184000 18000
monomial K3[10] -type Kum10 -type $\mathrm{c}_{ 2 }^{ 10 }$ 784015765747670016 336252992654447616 $\mathrm{c}_{ 2 }^{ 8 }\mathrm{c}_{ 4 }^{ }$ 344349868718803968 132107428736160768 $\mathrm{c}_{ 2 }^{ 6 }\mathrm{c}_{ 4 }^{ 2 }$ 151292288348880768 51898082311033728 $\mathrm{c}_{ 2 }^{ 7 }\mathrm{c}_{ 6 }^{ }$ 88352799453985536 26693534659013376 $\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ 3 }$ 66492814703915520 20386379301294336 $\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$ 38843392796682624 10486371945354624 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 4 }$ 29232974793607632 8007472661159664 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ }$ 17082588734970336 4119203015724192 $\mathrm{c}_{ 2 }^{ 6 }\mathrm{c}_{ 8 }^{ }$ 14887462352860800 3051655882366080 $\mathrm{c}_{ 4 }^{ 5 }$ 12856151785953456 3144990890482320 $\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 6 }^{ 2 }$ 9985643035208064 2119158341714304 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 3 }\mathrm{c}_{ 6 }^{ }$ 7515004051819440 1617975749261520 $\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$ 6551210934127872 1199055419079936 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ 2 }$ 4394286954851616 832451953404192 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 8 }^{ }$ 2883767951787984 471105410929296 $\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ 2 }$ 1934365074963120 326987093337168 $\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 10 }^{ }$ 1758703316056704 204371090647680 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$ 1687307749020288 242424490790592 $\mathrm{c}_{ 4 }^{ 3 }\mathrm{c}_{ 8 }^{ }$ 1269802518792480 185086417093248 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ 3 }$ 1131809390142912 168265889899008 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 10 }^{ }$ 774819641550240 80342429404512 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$ 743198906501136 95252580881040 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 10 }^{ }$ 341463574094256 31583103012912 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 8 }^{ 2 }$ 285897881921148 27756335356332 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 10 }^{ }$ 200033938656144 16258455456144 $\mathrm{c}_{ 6 }^{ 2 }\mathrm{c}_{ 8 }^{ }$ 191775038293488 19264369884144 $\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 12 }^{ }$ 152045432439552 8013253087488 $\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ 2 }$ 126041828580756 10909113168228 $\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 10 }^{ }$ 88209449234208 16391906873440 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 12 }^{ }$ 67076166081096 3153305609256 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 8 }^{ }\mathrm{c}_{ 10 }^{ }$ 34013661979068 1864193494284 $\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 12 }^{ }$ 29600340453792 1240853563488 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 12 }^{ }$ 17364913158312 639144656040 $\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 14 }^{ }$ 9924722506512 178626056400 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 14 }^{ }$ 4384872164952 70412082840 $\mathrm{c}_{ 10 }^{ 2 }$ 4065174516348 125480168748 $\mathrm{c}_{ 8 }^{ }\mathrm{c}_{ 12 }^{ }$ 2965017020340 73457352276 $\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 14 }^{ }$ 1138643559096 14310113400 $\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 16 }^{ }$ 501196808844 12116210140 $\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 16 }^{ }$ 221782223484 836469612 $\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 18 }^{ }$ 19976926140 11419980 $\mathrm{c}_{ 20 }^{ }$ 639249300 15972
K3[n ] -type The Chern numbers can be computed using the Bott residue formula, starting from [Theorem 0.1, MR1795551 ]. Kumn The Chern numbers have been computed by Nieper–Wisskirchen in [MR1906063 ]. OG6 The Chern numbers are computed in [Corollary 6.8, MR3798592 ]. OG10 The Chern numbers are computed in [Appendix A, MR4781884 ]. Computations of Chern numbers of K3[n ] - and Kumn IntersectionTheory library written by Jieao Song in Julia.
References MR1795551 Ellingsrud, G., Göttsche, L., & Lehn, M. (2001). On the cobordism class of the Hilbert scheme of a surface. J. Algebraic Geom. , 10 (1), 81–100. MR1795551 MR1906063 Nieper-Wisskirchen, M. A. (2002). On the Chern numbers of generalised Kummer varieties. Math. Res. Lett. , 9 (5–6), 597–606. doi:10.4310/MRL.2002.v9.n5.a3 MR1906063 MR3798592 Mongardi, G., Rapagnetta, A., & Saccà, G. (2018). The Hodge diamond of O’Grady’s six-dimensional example. Compos. Math. , 154 (5), 984–1013. doi:10.1112/S0010437X1700803X MR3798592 MR4781884 Ríos Ortiz, Á. D. (2024). Riemann-Roch polynomials of the known hyperkähler manifolds. Bull. Soc. Math. France , 152 (2), 169–184. MR4781884