Hyperkaehler.info

the geography of compact irreducible holomorphic symplectic (or hyperkähler) varieties

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

:

monomial K3
$\mathrm{c}_{ 2 }^{  }$ 24
monomial K3[2]-typeKum2-type
$\mathrm{c}_{ 2 }^{ 2 }$ 828 756
$\mathrm{c}_{ 4 }^{  }$ 324 108
monomial K3[3]-typeKum3-typeOG6
$\mathrm{c}_{ 2 }^{ 3 }$ 36800 30208 30720
$\mathrm{c}_{ 2 }^{  }\mathrm{c}_{ 4 }^{  }$ 14720 6785 7680
$\mathrm{c}_{ 6 }^{  }$ 3200 448 1920
monomial K3[4]-typeKum4-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]-typeKum5-typeOG10
$\mathrm{c}_{ 2 }^{ 5 }$ 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]-typeKum6-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]-typeKum7-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]-typeKum8-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]-typeKum9-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]-typeKum10-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-type
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, 2006.09307].

Computations of Chern numbers of K3[n]- and Kumn-type can be done using the IntersectionTheory library written by Jieao Song in Julia.


References
MR1795551
Ellingsrud, Geir and Göttsche, Lothar and Lehn, Manfred. "On the cobordism class of the Hilbert scheme of a surface." In: J. Algebraic Geom. 10 (2001), pp. 81–100
MR1906063
Nieper-Wisskirchen, Marc A.. "On the Chern numbers of generalised Kummer varieties." In: Math. Res. Lett. 9 (2002), pp. 597–606. doi:10.4310/MRL.2002.v9.n5.a3
MR3798592
Mongardi, Giovanni and Rapagnetta, Antonio and Saccà, Giulia. "The Hodge diamond of O'Grady's six-dimensional example." In: Compos. Math. 154 (2018), pp. 984–1013. doi:10.1112/S0010437X1700803X
2006.09307
Ortiz, Ángel David Ríos. "Riemann-Roch Polynomials of the known hyperkähler manifolds". arXiv:2006.09307