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, 2006.09307 ].
Computations of Chern numbers of K3[n ] - and Kumn IntersectionTheory library written by Jieao Song in Julia.
References
MR1795551
Ellingsrud, Geir and Gö ttsche, Lothar and Lehn, Manfred. "On the cobordism class of the H ilbert scheme of a surface." In: J. Algebraic Geom. 10 (2001), pp. 81–100
MR1906063
Nieper-Wisskirchen, Marc A.. "On the C hern numbers of generalised K ummer 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 H odge diamond of O 'G rady'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