Kähler manifolds with trivial canonical bundle are particularly interesting objects. The Beauville–Bogomolov decomposition tells us in the compact case that (after a finite étale cover) such a variety is isomorphic to the product of
The classification of the third class is the most intriguing: hyperkähler varieties have lots of amazing properties, but there are very few examples. We collect (some) information about the known families. More information on the general concepts is available .
K3 surface
There are two classes of smooth projective surfaces with trivial canonical bundle:
K3 surfaces, with $\mathrm{h}^1(S,\mathcal{O}_S)=0$ and $\pi_1(S)=0$
abelian surfaces, with $\mathrm{h}^1(S,\mathcal{O}_S)=2$ and $\pi_1(S)=\mathbb{Z}^{\oplus4}$
The first class are hyperkähler varieties of dimension 2, and there are 20 moduli (with the algebraic K3 surfaces being given by 19 moduli, in countably many families). Their properties are described below.
They will also be used to construct examples of another infinite family of hyperkählers (Hilbert schemes of points, of K3[n ] -type ) and examples of OG10 -type .
Abelian surfaces, are not hyperkähler themselves, but will be used to construct examples of another infinite family (generalised Kummer varieties, of Kumn ) and examples of OG6 -type .
K3 surface
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
22
$\mathrm{b}_{ 3 }$
0
$\mathrm{b}_{ 4 }$
1
K3[n ] -type
The first examples of this type were constructed by Beauville in 1983 in [§6, MR0730926 ], as Hilbert schemes of $n$ points on K3 surfaces, which explains their name.
Other moduli spaces of sheaves on a K3 surface are, when smooth, also of this type (by Mukai, Göttsche, Huybrechts, O'Grady and finally Yoshioka), and give rise to deformations of Hilbert schemes of $n$ points.
value for n (2–10) :
K3[2] -type
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
23
$\mathrm{b}_{ 3 }$
0
$\mathrm{b}_{ 4 }$
276
$\mathrm{b}_{ 5 }$
0
$\mathrm{b}_{ 6 }$
23
$\mathrm{b}_{ 7 }$
0
$\mathrm{b}_{ 8 }$
1
1
0
0
1
21
1
0
0
0
0
1
21
232
21
1
0
0
0
0
1
21
1
0
0
1
K3[3] -type
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
23
$\mathrm{b}_{ 3 }$
0
$\mathrm{b}_{ 4 }$
299
$\mathrm{b}_{ 5 }$
0
$\mathrm{b}_{ 6 }$
2554
$\mathrm{b}_{ 7 }$
0
$\mathrm{b}_{ 8 }$
299
$\mathrm{b}_{ 9 }$
0
$\mathrm{b}_{ 10 }$
23
$\mathrm{b}_{ 11 }$
0
$\mathrm{b}_{ 12 }$
1
1
0
0
1
21
1
0
0
0
0
1
22
253
22
1
0
0
0
0
0
0
1
21
253
2004
253
21
1
0
0
0
0
0
0
1
22
253
22
1
0
0
0
0
1
21
1
0
0
1
K3[4] -type
monomial
value
$\mathrm{c}_{ 2 }^{ 4 }$
1992240
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }$
813240
$\mathrm{c}_{ 4 }^{ 2 }$
332730
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ }$
182340
$\mathrm{c}_{ 8 }^{ }$
25650
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
23
$\mathrm{b}_{ 3 }$
0
$\mathrm{b}_{ 4 }$
300
$\mathrm{b}_{ 5 }$
0
$\mathrm{b}_{ 6 }$
2852
$\mathrm{b}_{ 7 }$
0
$\mathrm{b}_{ 8 }$
19298
$\mathrm{b}_{ 9 }$
0
$\mathrm{b}_{ 10 }$
2852
$\mathrm{b}_{ 11 }$
0
$\mathrm{b}_{ 12 }$
300
$\mathrm{b}_{ 13 }$
0
$\mathrm{b}_{ 14 }$
23
$\mathrm{b}_{ 15 }$
0
$\mathrm{b}_{ 16 }$
1
1
0
0
1
21
1
0
0
0
0
1
22
254
22
1
0
0
0
0
0
0
1
22
275
2256
275
22
1
0
0
0
0
0
0
0
0
1
21
254
2256
14234
2256
254
21
1
0
0
0
0
0
0
0
0
1
22
275
2256
275
22
1
0
0
0
0
0
0
1
22
254
22
1
0
0
0
0
1
21
1
0
0
1
K3[5] -type
monomial
value
$\mathrm{c}_{ 2 }^{ 5 }$
126867456
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ }$
52697088
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 2 }$
21921408
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 6 }^{ }$
12168576
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$
5075424
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 8 }^{ }$
1774080
$\mathrm{c}_{ 10 }^{ }$
176256
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
23
$\mathrm{b}_{ 3 }$
0
$\mathrm{b}_{ 4 }$
300
$\mathrm{b}_{ 5 }$
0
$\mathrm{b}_{ 6 }$
2875
$\mathrm{b}_{ 7 }$
0
$\mathrm{b}_{ 8 }$
22127
$\mathrm{b}_{ 9 }$
0
$\mathrm{b}_{ 10 }$
125604
$\mathrm{b}_{ 11 }$
0
$\mathrm{b}_{ 12 }$
22127
$\mathrm{b}_{ 13 }$
0
$\mathrm{b}_{ 14 }$
2875
$\mathrm{b}_{ 15 }$
0
$\mathrm{b}_{ 16 }$
300
$\mathrm{b}_{ 17 }$
0
$\mathrm{b}_{ 18 }$
23
$\mathrm{b}_{ 19 }$
0
$\mathrm{b}_{ 20 }$
1
1
0
0
1
21
1
0
0
0
0
1
22
254
22
1
0
0
0
0
0
0
1
22
276
2277
276
22
1
0
0
0
0
0
0
0
0
1
22
276
2530
16469
2530
276
22
1
0
0
0
0
0
0
0
0
0
0
1
21
254
2277
16469
87560
16469
2277
254
21
1
0
0
0
0
0
0
0
0
0
0
1
22
276
2530
16469
2530
276
22
1
0
0
0
0
0
0
0
0
1
22
276
2277
276
22
1
0
0
0
0
0
0
1
22
254
22
1
0
0
0
0
1
21
1
0
0
1
K3[6] -type
monomial
value
$\mathrm{c}_{ 2 }^{ 6 }$
9277276480
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ }$
3910848640
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 2 }$
1650311720
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 6 }^{ }$
927397840
$\mathrm{c}_{ 4 }^{ 3 }$
697106648
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$
392090040
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 8 }^{ }$
139942280
$\mathrm{c}_{ 6 }^{ 2 }$
93495320
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$
59314272
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 10 }^{ }$
14450680
$\mathrm{c}_{ 12 }^{ }$
1073720
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
23
$\mathrm{b}_{ 3 }$
0
$\mathrm{b}_{ 4 }$
300
$\mathrm{b}_{ 5 }$
0
$\mathrm{b}_{ 6 }$
2876
$\mathrm{b}_{ 7 }$
0
$\mathrm{b}_{ 8 }$
22426
$\mathrm{b}_{ 9 }$
0
$\mathrm{b}_{ 10 }$
147431
$\mathrm{b}_{ 11 }$
0
$\mathrm{b}_{ 12 }$
727606
$\mathrm{b}_{ 13 }$
0
$\mathrm{b}_{ 14 }$
147431
$\mathrm{b}_{ 15 }$
0
$\mathrm{b}_{ 16 }$
22426
$\mathrm{b}_{ 17 }$
0
$\mathrm{b}_{ 18 }$
2876
$\mathrm{b}_{ 19 }$
0
$\mathrm{b}_{ 20 }$
300
$\mathrm{b}_{ 21 }$
0
$\mathrm{b}_{ 22 }$
23
$\mathrm{b}_{ 23 }$
0
$\mathrm{b}_{ 24 }$
1
1
0
0
1
21
1
0
0
0
0
1
22
254
22
1
0
0
0
0
0
0
1
22
276
2278
276
22
1
0
0
0
0
0
0
0
0
1
22
277
2552
16722
2552
277
22
1
0
0
0
0
0
0
0
0
0
0
1
22
276
2552
18977
103775
18977
2552
276
22
1
0
0
0
0
0
0
0
0
0
0
0
0
1
21
254
2278
16722
103775
481504
103775
16722
2278
254
21
1
0
0
0
0
0
0
0
0
0
0
0
0
1
22
276
2552
18977
103775
18977
2552
276
22
1
0
0
0
0
0
0
0
0
0
0
1
22
277
2552
16722
2552
277
22
1
0
0
0
0
0
0
0
0
1
22
276
2278
276
22
1
0
0
0
0
0
0
1
22
254
22
1
0
0
0
0
1
21
1
0
0
1
K3[7] -type
monomial
value
$\mathrm{c}_{ 2 }^{ 7 }$
765374164992
$\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 4 }^{ }$
326732507136
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ 2 }$
139582386432
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 6 }^{ }$
79324710912
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 3 }$
59674012416
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$
33935583744
$\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ }$
14528215296
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 8 }^{ }$
12357114624
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ 2 }$
8273055744
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$
5296568832
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 10 }^{ }$
1324608768
$\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$
1296158976
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 10 }^{ }$
569044224
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 12 }^{ }$
102477312
$\mathrm{c}_{ 14 }^{ }$
5930496
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
23
$\mathrm{b}_{ 3 }$
0
$\mathrm{b}_{ 4 }$
300
$\mathrm{b}_{ 5 }$
0
$\mathrm{b}_{ 6 }$
2876
$\mathrm{b}_{ 7 }$
0
$\mathrm{b}_{ 8 }$
22449
$\mathrm{b}_{ 9 }$
0
$\mathrm{b}_{ 10 }$
150283
$\mathrm{b}_{ 11 }$
0
$\mathrm{b}_{ 12 }$
872162
$\mathrm{b}_{ 13 }$
0
$\mathrm{b}_{ 14 }$
3834308
$\mathrm{b}_{ 15 }$
0
$\mathrm{b}_{ 16 }$
872162
$\mathrm{b}_{ 17 }$
0
$\mathrm{b}_{ 18 }$
150283
$\mathrm{b}_{ 19 }$
0
$\mathrm{b}_{ 20 }$
22449
$\mathrm{b}_{ 21 }$
0
$\mathrm{b}_{ 22 }$
2876
$\mathrm{b}_{ 23 }$
0
$\mathrm{b}_{ 24 }$
300
$\mathrm{b}_{ 25 }$
0
$\mathrm{b}_{ 26 }$
23
$\mathrm{b}_{ 27 }$
0
$\mathrm{b}_{ 28 }$
1
1
0
0
1
21
1
0
0
0
0
1
22
254
22
1
0
0
0
0
0
0
1
22
276
2278
276
22
1
0
0
0
0
0
0
0
0
1
22
277
2553
16743
2553
277
22
1
0
0
0
0
0
0
0
0
0
0
1
22
277
2574
19252
106031
19252
2574
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
1
22
276
2553
19252
122476
583002
122476
19252
2553
276
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
21
254
2278
16743
106031
583002
2417648
583002
106031
16743
2278
254
21
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
276
2553
19252
122476
583002
122476
19252
2553
276
22
1
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2574
19252
106031
19252
2574
277
22
1
0
0
0
0
0
0
0
0
0
0
1
22
277
2553
16743
2553
277
22
1
0
0
0
0
0
0
0
0
1
22
276
2278
276
22
1
0
0
0
0
0
0
1
22
254
22
1
0
0
0
0
1
21
1
0
0
1
K3[8] -type
monomial
value
$\mathrm{c}_{ 2 }^{ 8 }$
70277256403200
$\mathrm{c}_{ 2 }^{ 6 }\mathrm{c}_{ 4 }^{ }$
30327407026560
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ 2 }$
13094639681760
$\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 6 }^{ }$
7517275416000
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 3 }$
5657019716880
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$
3249219677760
$\mathrm{c}_{ 4 }^{ 4 }$
2445207931980
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ }$
1405173296520
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 8 }^{ }$
1205400258720
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 6 }^{ 2 }$
807925003200
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$
521787430080
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ 2 }$
349760996280
$\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 8 }^{ }$
225987046020
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 10 }^{ }$
133823975040
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$
130128762960
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 10 }^{ }$
58033047240
$\mathrm{c}_{ 8 }^{ 2 }$
21049285275
$\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 10 }^{ }$
14525621460
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 12 }^{ }$
10767198960
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 12 }^{ }$
4678568010
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 14 }^{ }$
649511820
$\mathrm{c}_{ 16 }^{ }$
30178575
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
23
$\mathrm{b}_{ 3 }$
0
$\mathrm{b}_{ 4 }$
300
$\mathrm{b}_{ 5 }$
0
$\mathrm{b}_{ 6 }$
2876
$\mathrm{b}_{ 7 }$
0
$\mathrm{b}_{ 8 }$
22450
$\mathrm{b}_{ 9 }$
0
$\mathrm{b}_{ 10 }$
150582
$\mathrm{b}_{ 11 }$
0
$\mathrm{b}_{ 12 }$
894288
$\mathrm{b}_{ 13 }$
0
$\mathrm{b}_{ 14 }$
4684044
$\mathrm{b}_{ 15 }$
0
$\mathrm{b}_{ 16 }$
18669447
$\mathrm{b}_{ 17 }$
0
$\mathrm{b}_{ 18 }$
4684044
$\mathrm{b}_{ 19 }$
0
$\mathrm{b}_{ 20 }$
894288
$\mathrm{b}_{ 21 }$
0
$\mathrm{b}_{ 22 }$
150582
$\mathrm{b}_{ 23 }$
0
$\mathrm{b}_{ 24 }$
22450
$\mathrm{b}_{ 25 }$
0
$\mathrm{b}_{ 26 }$
2876
$\mathrm{b}_{ 27 }$
0
$\mathrm{b}_{ 28 }$
300
$\mathrm{b}_{ 29 }$
0
$\mathrm{b}_{ 30 }$
23
$\mathrm{b}_{ 31 }$
0
$\mathrm{b}_{ 32 }$
1
1
0
0
1
21
1
0
0
0
0
1
22
254
22
1
0
0
0
0
0
0
1
22
276
2278
276
22
1
0
0
0
0
0
0
0
0
1
22
277
2553
16744
2553
277
22
1
0
0
0
0
0
0
0
0
0
0
1
22
277
2575
19274
106284
19274
2575
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2575
19528
125006
599470
125006
19528
2575
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
276
2553
19274
125006
702926
2983928
702926
125006
19274
2553
276
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
21
254
2278
16744
106284
599470
2983928
11251487
2983928
599470
106284
16744
2278
254
21
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
276
2553
19274
125006
702926
2983928
702926
125006
19274
2553
276
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2575
19528
125006
599470
125006
19528
2575
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2575
19274
106284
19274
2575
277
22
1
0
0
0
0
0
0
0
0
0
0
1
22
277
2553
16744
2553
277
22
1
0
0
0
0
0
0
0
0
1
22
276
2278
276
22
1
0
0
0
0
0
0
1
22
254
22
1
0
0
0
0
1
21
1
0
0
1
K3[9] -type
monomial
value
$\mathrm{c}_{ 2 }^{ 9 }$
7105044485242880
$\mathrm{c}_{ 2 }^{ 7 }\mathrm{c}_{ 4 }^{ }$
3095054052884480
$\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 4 }^{ 2 }$
1348811566120960
$\mathrm{c}_{ 2 }^{ 6 }\mathrm{c}_{ 6 }^{ }$
781347805921280
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ 3 }$
588050734243840
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$
340787113328640
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 4 }$
256482451425280
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ }$
148696308725760
$\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 8 }^{ }$
128601459097600
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 6 }^{ 2 }$
86242390425600
$\mathrm{c}_{ 4 }^{ 3 }\mathrm{c}_{ 6 }^{ }$
64907421320960
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$
56155350159360
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ 2 }$
37660572692480
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 8 }^{ }$
24530800855040
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 10 }^{ }$
14747557928960
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$
14244457018880
$\mathrm{c}_{ 6 }^{ 3 }$
9553579524480
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 10 }^{ }$
6448976952320
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$
6227441933120
$\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 10 }^{ }$
2821199089280
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 8 }^{ 2 }$
2360786818560
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 10 }^{ }$
1640647441920
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 12 }^{ }$
1231467509760
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 12 }^{ }$
539392972800
$\mathrm{c}_{ 8 }^{ }\mathrm{c}_{ 10 }^{ }$
273089658720
$\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 12 }^{ }$
137685310240
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 14 }^{ }$
77346804480
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 14 }^{ }$
33938470560
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 16 }^{ }$
3748665600
$\mathrm{c}_{ 18 }^{ }$
143184000
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
23
$\mathrm{b}_{ 3 }$
0
$\mathrm{b}_{ 4 }$
300
$\mathrm{b}_{ 5 }$
0
$\mathrm{b}_{ 6 }$
2876
$\mathrm{b}_{ 7 }$
0
$\mathrm{b}_{ 8 }$
22450
$\mathrm{b}_{ 9 }$
0
$\mathrm{b}_{ 10 }$
150605
$\mathrm{b}_{ 11 }$
0
$\mathrm{b}_{ 12 }$
897141
$\mathrm{b}_{ 13 }$
0
$\mathrm{b}_{ 14 }$
4831451
$\mathrm{b}_{ 15 }$
0
$\mathrm{b}_{ 16 }$
23203208
$\mathrm{b}_{ 17 }$
0
$\mathrm{b}_{ 18 }$
84967890
$\mathrm{b}_{ 19 }$
0
$\mathrm{b}_{ 20 }$
23203208
$\mathrm{b}_{ 21 }$
0
$\mathrm{b}_{ 22 }$
4831451
$\mathrm{b}_{ 23 }$
0
$\mathrm{b}_{ 24 }$
897141
$\mathrm{b}_{ 25 }$
0
$\mathrm{b}_{ 26 }$
150605
$\mathrm{b}_{ 27 }$
0
$\mathrm{b}_{ 28 }$
22450
$\mathrm{b}_{ 29 }$
0
$\mathrm{b}_{ 30 }$
2876
$\mathrm{b}_{ 31 }$
0
$\mathrm{b}_{ 32 }$
300
$\mathrm{b}_{ 33 }$
0
$\mathrm{b}_{ 34 }$
23
$\mathrm{b}_{ 35 }$
0
$\mathrm{b}_{ 36 }$
1
1
0
0
1
21
1
0
0
0
0
1
22
254
22
1
0
0
0
0
0
0
1
22
276
2278
276
22
1
0
0
0
0
0
0
0
0
1
22
277
2553
16744
2553
277
22
1
0
0
0
0
0
0
0
0
0
0
1
22
277
2575
19275
106305
19275
2575
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2576
19550
125281
601727
125281
19550
2576
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2575
19550
127558
721902
3087681
721902
127558
19550
2575
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
276
2553
19275
125281
721902
3667602
14129384
3667602
721902
125281
19275
2553
276
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
21
254
2278
16744
106305
601727
3087681
14129384
49079100
14129384
3087681
601727
106305
16744
2278
254
21
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
276
2553
19275
125281
721902
3667602
14129384
3667602
721902
125281
19275
2553
276
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2575
19550
127558
721902
3087681
721902
127558
19550
2575
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2576
19550
125281
601727
125281
19550
2576
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2575
19275
106305
19275
2575
277
22
1
0
0
0
0
0
0
0
0
0
0
1
22
277
2553
16744
2553
277
22
1
0
0
0
0
0
0
0
0
1
22
276
2278
276
22
1
0
0
0
0
0
0
1
22
254
22
1
0
0
0
0
1
21
1
0
0
1
K3[10] -type
monomial
value
$\mathrm{c}_{ 2 }^{ 10 }$
784015765747670016
$\mathrm{c}_{ 2 }^{ 8 }\mathrm{c}_{ 4 }^{ }$
344349868718803968
$\mathrm{c}_{ 2 }^{ 6 }\mathrm{c}_{ 4 }^{ 2 }$
151292288348880768
$\mathrm{c}_{ 2 }^{ 7 }\mathrm{c}_{ 6 }^{ }$
88352799453985536
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ 3 }$
66492814703915520
$\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$
38843392796682624
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 4 }$
29232974793607632
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ }$
17082588734970336
$\mathrm{c}_{ 2 }^{ 6 }\mathrm{c}_{ 8 }^{ }$
14887462352860800
$\mathrm{c}_{ 4 }^{ 5 }$
12856151785953456
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 6 }^{ 2 }$
9985643035208064
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 3 }\mathrm{c}_{ 6 }^{ }$
7515004051819440
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$
6551210934127872
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ 2 }$
4394286954851616
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 8 }^{ }$
2883767951787984
$\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ 2 }$
1934365074963120
$\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 10 }^{ }$
1758703316056704
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$
1687307749020288
$\mathrm{c}_{ 4 }^{ 3 }\mathrm{c}_{ 8 }^{ }$
1269802518792480
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ 3 }$
1131809390142912
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 10 }^{ }$
774819641550240
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$
743198906501136
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 10 }^{ }$
341463574094256
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 8 }^{ 2 }$
285897881921148
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 10 }^{ }$
200033938656144
$\mathrm{c}_{ 6 }^{ 2 }\mathrm{c}_{ 8 }^{ }$
191775038293488
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 12 }^{ }$
152045432439552
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ 2 }$
126041828580756
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 10 }^{ }$
88209449234208
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 12 }^{ }$
67076166081096
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 8 }^{ }\mathrm{c}_{ 10 }^{ }$
34013661979068
$\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 12 }^{ }$
29600340453792
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 12 }^{ }$
17364913158312
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 14 }^{ }$
9924722506512
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 14 }^{ }$
4384872164952
$\mathrm{c}_{ 10 }^{ 2 }$
4065174516348
$\mathrm{c}_{ 8 }^{ }\mathrm{c}_{ 12 }^{ }$
2965017020340
$\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 14 }^{ }$
1138643559096
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 16 }^{ }$
501196808844
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 16 }^{ }$
221782223484
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 18 }^{ }$
19976926140
$\mathrm{c}_{ 20 }^{ }$
639249300
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
23
$\mathrm{b}_{ 3 }$
0
$\mathrm{b}_{ 4 }$
300
$\mathrm{b}_{ 5 }$
0
$\mathrm{b}_{ 6 }$
2876
$\mathrm{b}_{ 7 }$
0
$\mathrm{b}_{ 8 }$
22450
$\mathrm{b}_{ 9 }$
0
$\mathrm{b}_{ 10 }$
150606
$\mathrm{b}_{ 11 }$
0
$\mathrm{b}_{ 12 }$
897440
$\mathrm{b}_{ 13 }$
0
$\mathrm{b}_{ 14 }$
4853600
$\mathrm{b}_{ 15 }$
0
$\mathrm{b}_{ 16 }$
24075047
$\mathrm{b}_{ 17 }$
0
$\mathrm{b}_{ 18 }$
107276810
$\mathrm{b}_{ 19 }$
0
$\mathrm{b}_{ 20 }$
364690994
$\mathrm{b}_{ 21 }$
0
$\mathrm{b}_{ 22 }$
107276810
$\mathrm{b}_{ 23 }$
0
$\mathrm{b}_{ 24 }$
24075047
$\mathrm{b}_{ 25 }$
0
$\mathrm{b}_{ 26 }$
4853600
$\mathrm{b}_{ 27 }$
0
$\mathrm{b}_{ 28 }$
897440
$\mathrm{b}_{ 29 }$
0
$\mathrm{b}_{ 30 }$
150606
$\mathrm{b}_{ 31 }$
0
$\mathrm{b}_{ 32 }$
22450
$\mathrm{b}_{ 33 }$
0
$\mathrm{b}_{ 34 }$
2876
$\mathrm{b}_{ 35 }$
0
$\mathrm{b}_{ 36 }$
300
$\mathrm{b}_{ 37 }$
0
$\mathrm{b}_{ 38 }$
23
$\mathrm{b}_{ 39 }$
0
$\mathrm{b}_{ 40 }$
1
1
0
0
1
21
1
0
0
0
0
1
22
254
22
1
0
0
0
0
0
0
1
22
276
2278
276
22
1
0
0
0
0
0
0
0
0
1
22
277
2553
16744
2553
277
22
1
0
0
0
0
0
0
0
0
0
0
1
22
277
2575
19275
106306
19275
2575
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2576
19551
125303
601980
125303
19551
2576
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2576
19572
127834
724433
3104170
724433
127834
19572
2576
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2575
19551
127834
741153
3790055
14712111
3790055
741153
127834
19551
2575
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
276
2553
19275
125303
724433
3790055
17671980
62609014
17671980
3790055
724433
125303
19275
2553
276
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
21
254
2278
16744
106306
601980
3104170
14712111
62609014
202385236
62609014
14712111
3104170
601980
106306
16744
2278
254
21
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
276
2553
19275
125303
724433
3790055
17671980
62609014
17671980
3790055
724433
125303
19275
2553
276
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2575
19551
127834
741153
3790055
14712111
3790055
741153
127834
19551
2575
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2576
19572
127834
724433
3104170
724433
127834
19572
2576
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2576
19551
125303
601980
125303
19551
2576
277
22
1
0
0
0
0
0
0
0
0
0
0
0
0
1
22
277
2575
19275
106306
19275
2575
277
22
1
0
0
0
0
0
0
0
0
0
0
1
22
277
2553
16744
2553
277
22
1
0
0
0
0
0
0
0
0
1
22
276
2278
276
22
1
0
0
0
0
0
0
1
22
254
22
1
0
0
0
0
1
21
1
0
0
1
References
MR0730926
Beauville, Arnaud. "Varié té s Kä hleriennes dont la première classe de C hern est nulle." In: J. Differential Geom. 18 (1983), pp. 755–782 (1984)
Kumn
The first examples of this type were constructed by Beauville in 1983 in [§7, MR0730926 ], as the fiber over 0 of the summation morphism $A^{[n+1]}\to A$ for an abelian surface $A$. For $n=1$ this becomes the Kummer surface of the abelian surface, which is a K3 surface, for $n\geq 2$ we obtain generalised Kummer varieties
value for n (2–10) :
Kum2 -type
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
7
$\mathrm{b}_{ 3 }$
8
$\mathrm{b}_{ 4 }$
108
$\mathrm{b}_{ 5 }$
8
$\mathrm{b}_{ 6 }$
7
$\mathrm{b}_{ 7 }$
0
$\mathrm{b}_{ 8 }$
1
1
0
0
1
5
1
0
4
4
0
1
5
96
5
1
0
4
4
0
1
5
1
0
0
1
Kum3 -type
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
7
$\mathrm{b}_{ 3 }$
8
$\mathrm{b}_{ 4 }$
51
$\mathrm{b}_{ 5 }$
56
$\mathrm{b}_{ 6 }$
458
$\mathrm{b}_{ 7 }$
56
$\mathrm{b}_{ 8 }$
51
$\mathrm{b}_{ 9 }$
8
$\mathrm{b}_{ 10 }$
7
$\mathrm{b}_{ 11 }$
0
$\mathrm{b}_{ 12 }$
1
1
0
0
1
5
1
0
4
4
0
1
6
37
6
1
0
4
24
24
4
0
1
5
37
372
37
5
1
0
4
24
24
4
0
1
6
37
6
1
0
4
4
0
1
5
1
0
0
1
Kum4 -type
monomial
value
$\mathrm{c}_{ 2 }^{ 4 }$
1470000
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }$
405000
$\mathrm{c}_{ 4 }^{ 2 }$
111750
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ }$
37500
$\mathrm{c}_{ 8 }^{ }$
750
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
7
$\mathrm{b}_{ 3 }$
8
$\mathrm{b}_{ 4 }$
36
$\mathrm{b}_{ 5 }$
64
$\mathrm{b}_{ 6 }$
168
$\mathrm{b}_{ 7 }$
288
$\mathrm{b}_{ 8 }$
1046
$\mathrm{b}_{ 9 }$
288
$\mathrm{b}_{ 10 }$
168
$\mathrm{b}_{ 11 }$
64
$\mathrm{b}_{ 12 }$
36
$\mathrm{b}_{ 13 }$
8
$\mathrm{b}_{ 14 }$
7
$\mathrm{b}_{ 15 }$
0
$\mathrm{b}_{ 16 }$
1
1
0
0
1
5
1
0
4
4
0
1
6
22
6
1
0
4
28
28
4
0
1
6
33
88
33
6
1
0
4
28
112
112
28
4
0
1
5
22
88
814
88
22
5
1
0
4
28
112
112
28
4
0
1
6
33
88
33
6
1
0
4
28
28
4
0
1
6
22
6
1
0
4
4
0
1
5
1
0
0
1
Kum5 -type
monomial
value
$\mathrm{c}_{ 2 }^{ 5 }$
84478464
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ }$
26220672
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 2 }$
8141472
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 6 }^{ }$
3141504
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$
979776
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 8 }^{ }$
142560
$\mathrm{c}_{ 10 }^{ }$
2592
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
7
$\mathrm{b}_{ 3 }$
8
$\mathrm{b}_{ 4 }$
36
$\mathrm{b}_{ 5 }$
64
$\mathrm{b}_{ 6 }$
191
$\mathrm{b}_{ 7 }$
344
$\mathrm{b}_{ 8 }$
915
$\mathrm{b}_{ 9 }$
1312
$\mathrm{b}_{ 10 }$
3748
$\mathrm{b}_{ 11 }$
1312
$\mathrm{b}_{ 12 }$
915
$\mathrm{b}_{ 13 }$
344
$\mathrm{b}_{ 14 }$
191
$\mathrm{b}_{ 15 }$
64
$\mathrm{b}_{ 16 }$
36
$\mathrm{b}_{ 17 }$
8
$\mathrm{b}_{ 18 }$
7
$\mathrm{b}_{ 19 }$
0
$\mathrm{b}_{ 20 }$
1
1
0
0
1
5
1
0
4
4
0
1
6
22
6
1
0
4
28
28
4
0
1
6
34
109
34
6
1
0
4
32
136
136
32
4
0
1
6
34
170
493
170
34
6
1
0
4
28
136
488
488
136
28
4
0
1
5
22
109
493
2488
493
109
22
5
1
0
4
28
136
488
488
136
28
4
0
1
6
34
170
493
170
34
6
1
0
4
32
136
136
32
4
0
1
6
34
109
34
6
1
0
4
28
28
4
0
1
6
22
6
1
0
4
4
0
1
5
1
0
0
1
Kum6 -type
monomial
value
$\mathrm{c}_{ 2 }^{ 6 }$
5603050432
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ }$
1881462016
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 2 }$
631808744
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 6 }^{ }$
268796752
$\mathrm{c}_{ 4 }^{ 3 }$
212190776
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$
90412056
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 8 }^{ }$
17075912
$\mathrm{c}_{ 6 }^{ 2 }$
12976376
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$
5762400
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 10 }^{ }$
441784
$\mathrm{c}_{ 12 }^{ }$
2744
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
7
$\mathrm{b}_{ 3 }$
8
$\mathrm{b}_{ 4 }$
36
$\mathrm{b}_{ 5 }$
64
$\mathrm{b}_{ 6 }$
176
$\mathrm{b}_{ 7 }$
352
$\mathrm{b}_{ 8 }$
786
$\mathrm{b}_{ 9 }$
1528
$\mathrm{b}_{ 10 }$
2879
$\mathrm{b}_{ 11 }$
4496
$\mathrm{b}_{ 12 }$
7870
$\mathrm{b}_{ 13 }$
4496
$\mathrm{b}_{ 14 }$
2879
$\mathrm{b}_{ 15 }$
1528
$\mathrm{b}_{ 16 }$
786
$\mathrm{b}_{ 17 }$
352
$\mathrm{b}_{ 18 }$
176
$\mathrm{b}_{ 19 }$
64
$\mathrm{b}_{ 20 }$
36
$\mathrm{b}_{ 21 }$
8
$\mathrm{b}_{ 22 }$
7
$\mathrm{b}_{ 23 }$
0
$\mathrm{b}_{ 24 }$
1
1
0
0
1
5
1
0
4
4
0
1
6
22
6
1
0
4
28
28
4
0
1
6
34
94
34
6
1
0
4
32
140
140
32
4
0
1
6
35
166
370
166
35
6
1
0
4
32
168
560
560
168
32
4
0
1
6
34
166
633
1199
633
166
34
6
1
0
4
28
140
560
1516
1516
560
140
28
4
0
1
5
22
94
370
1199
4488
1199
370
94
22
5
1
0
4
28
140
560
1516
1516
560
140
28
4
0
1
6
34
166
633
1199
633
166
34
6
1
0
4
32
168
560
560
168
32
4
0
1
6
35
166
370
166
35
6
1
0
4
32
140
140
32
4
0
1
6
34
94
34
6
1
0
4
28
28
4
0
1
6
22
6
1
0
4
4
0
1
5
1
0
0
1
Kum7 -type
Beauville–Bogomolov–Fujiki form $\mathrm{U}^3\oplus(-16)$
$\int\mathrm{td}_X^{1/2}$
$64/315$
polarisation type of general fiber
$(1,1,1,1,1,1,8),(1,1,1,1,1,2,4)$
$\operatorname{Aut}_0(X)$
$(\mathbb{Z}/8\mathbb{Z})^4\rtimes\mathbb{Z}/2\mathbb{Z}$
monomial
value
$\mathrm{c}_{ 2 }^{ 7 }$
421414305792
$\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 4 }^{ }$
149664301056
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ 2 }$
53149827072
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 6 }^{ }$
24230756352
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 3 }$
18874417152
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$
8610545664
$\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ }$
3059945472
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 8 }^{ }$
1914077184
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ 2 }$
1397121024
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$
681332736
$\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$
110853120
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 10 }^{ }$
71909376
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 10 }^{ }$
25700352
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 12 }^{ }$
1198080
$\mathrm{c}_{ 14 }^{ }$
7680
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
7
$\mathrm{b}_{ 3 }$
8
$\mathrm{b}_{ 4 }$
36
$\mathrm{b}_{ 5 }$
64
$\mathrm{b}_{ 6 }$
176
$\mathrm{b}_{ 7 }$
352
$\mathrm{b}_{ 8 }$
809
$\mathrm{b}_{ 9 }$
1584
$\mathrm{b}_{ 10 }$
3327
$\mathrm{b}_{ 11 }$
6136
$\mathrm{b}_{ 12 }$
11298
$\mathrm{b}_{ 13 }$
16432
$\mathrm{b}_{ 14 }$
25524
$\mathrm{b}_{ 15 }$
16432
$\mathrm{b}_{ 16 }$
11298
$\mathrm{b}_{ 17 }$
6136
$\mathrm{b}_{ 18 }$
3327
$\mathrm{b}_{ 19 }$
1584
$\mathrm{b}_{ 20 }$
809
$\mathrm{b}_{ 21 }$
352
$\mathrm{b}_{ 22 }$
176
$\mathrm{b}_{ 23 }$
64
$\mathrm{b}_{ 24 }$
36
$\mathrm{b}_{ 25 }$
8
$\mathrm{b}_{ 26 }$
7
$\mathrm{b}_{ 27 }$
0
$\mathrm{b}_{ 28 }$
1
1
0
0
1
5
1
0
4
4
0
1
6
22
6
1
0
4
28
28
4
0
1
6
34
94
34
6
1
0
4
32
140
140
32
4
0
1
6
35
167
391
167
35
6
1
0
4
32
172
584
584
172
32
4
0
1
6
35
178
722
1443
722
178
35
6
1
0
4
32
172
736
2124
2124
736
172
32
4
0
1
6
34
167
722
2424
4590
2424
722
167
34
6
1
0
4
28
140
584
2124
5336
5336
2124
584
140
28
4
0
1
5
22
94
391
1443
4590
12432
4590
1443
391
94
22
5
1
0
4
28
140
584
2124
5336
5336
2124
584
140
28
4
0
1
6
34
167
722
2424
4590
2424
722
167
34
6
1
0
4
32
172
736
2124
2124
736
172
32
4
0
1
6
35
178
722
1443
722
178
35
6
1
0
4
32
172
584
584
172
32
4
0
1
6
35
167
391
167
35
6
1
0
4
32
140
140
32
4
0
1
6
34
94
34
6
1
0
4
28
28
4
0
1
6
22
6
1
0
4
4
0
1
5
1
0
0
1
Kum8 -type
Beauville–Bogomolov–Fujiki form $\mathrm{U}^3\oplus(-18)$
$\int\mathrm{td}_X^{1/2}$
$43046721/293601280$
polarisation type of general fiber
$(1,1,1,1,1,1,1,9),(1,1,1,1,1,1,3,3)$
$\operatorname{Aut}_0(X)$
$(\mathbb{Z}/9\mathbb{Z})^4\rtimes\mathbb{Z}/2\mathbb{Z}$
monomial
value
$\mathrm{c}_{ 2 }^{ 8 }$
35447947999488
$\mathrm{c}_{ 2 }^{ 6 }\mathrm{c}_{ 4 }^{ }$
13129602781824
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ 2 }$
4862661530400
$\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 6 }^{ }$
2332758616128
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 3 }$
1800797040144
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$
864167470848
$\mathrm{c}_{ 4 }^{ 4 }$
666853820172
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ }$
320117226120
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 8 }^{ }$
215605377504
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 6 }^{ 2 }$
153694101888
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$
79938804096
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ 2 }$
56953381608
$\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 8 }^{ }$
29638792620
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$
14239224576
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 10 }^{ }$
10441752768
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 10 }^{ }$
3878495784
$\mathrm{c}_{ 8 }^{ 2 }$
1322820801
$\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 10 }^{ }$
692780364
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 12 }^{ }$
254566800
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 12 }^{ }$
94850190
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 14 }^{ }$
2685636
$\mathrm{c}_{ 16 }^{ }$
9477
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
7
$\mathrm{b}_{ 3 }$
8
$\mathrm{b}_{ 4 }$
36
$\mathrm{b}_{ 5 }$
64
$\mathrm{b}_{ 6 }$
176
$\mathrm{b}_{ 7 }$
352
$\mathrm{b}_{ 8 }$
794
$\mathrm{b}_{ 9 }$
1592
$\mathrm{b}_{ 10 }$
3278
$\mathrm{b}_{ 11 }$
6360
$\mathrm{b}_{ 12 }$
12202
$\mathrm{b}_{ 13 }$
21704
$\mathrm{b}_{ 14 }$
36440
$\mathrm{b}_{ 15 }$
51640
$\mathrm{b}_{ 16 }$
67049
$\mathrm{b}_{ 17 }$
51640
$\mathrm{b}_{ 18 }$
36440
$\mathrm{b}_{ 19 }$
21704
$\mathrm{b}_{ 20 }$
12202
$\mathrm{b}_{ 21 }$
6360
$\mathrm{b}_{ 22 }$
3278
$\mathrm{b}_{ 23 }$
1592
$\mathrm{b}_{ 24 }$
794
$\mathrm{b}_{ 25 }$
352
$\mathrm{b}_{ 26 }$
176
$\mathrm{b}_{ 27 }$
64
$\mathrm{b}_{ 28 }$
36
$\mathrm{b}_{ 29 }$
8
$\mathrm{b}_{ 30 }$
7
$\mathrm{b}_{ 31 }$
0
$\mathrm{b}_{ 32 }$
1
1
0
0
1
5
1
0
4
4
0
1
6
22
6
1
0
4
28
28
4
0
1
6
34
94
34
6
1
0
4
32
140
140
32
4
0
1
6
35
167
376
167
35
6
1
0
4
32
172
588
588
172
32
4
0
1
6
35
179
718
1400
718
179
35
6
1
0
4
32
176
768
2200
2200
768
176
32
4
0
1
6
35
179
787
2696
4794
2696
787
179
35
6
1
0
4
32
172
768
2844
7032
7032
2844
768
172
32
4
0
1
6
34
167
718
2696
8026
13144
8026
2696
718
167
34
6
1
0
4
28
140
588
2200
7032
15828
15828
7032
2200
588
140
28
4
0
1
5
22
94
376
1400
4794
13144
27377
13144
4794
1400
376
94
22
5
1
0
4
28
140
588
2200
7032
15828
15828
7032
2200
588
140
28
4
0
1
6
34
167
718
2696
8026
13144
8026
2696
718
167
34
6
1
0
4
32
172
768
2844
7032
7032
2844
768
172
32
4
0
1
6
35
179
787
2696
4794
2696
787
179
35
6
1
0
4
32
176
768
2200
2200
768
176
32
4
0
1
6
35
179
718
1400
718
179
35
6
1
0
4
32
172
588
588
172
32
4
0
1
6
35
167
376
167
35
6
1
0
4
32
140
140
32
4
0
1
6
34
94
34
6
1
0
4
28
28
4
0
1
6
22
6
1
0
4
4
0
1
5
1
0
0
1
Kum9 -type
monomial
value
$\mathrm{c}_{ 2 }^{ 9 }$
3297871360000000
$\mathrm{c}_{ 2 }^{ 7 }\mathrm{c}_{ 4 }^{ }$
1262135680000000
$\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 4 }^{ 2 }$
482990816000000
$\mathrm{c}_{ 2 }^{ 6 }\mathrm{c}_{ 6 }^{ }$
240910720000000
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ 3 }$
184814229440000
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$
92197363200000
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 4 }$
70712975120000
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ }$
35281909440000
$\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 8 }^{ }$
25082624000000
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 6 }^{ 2 }$
17605804800000
$\mathrm{c}_{ 4 }^{ 3 }\mathrm{c}_{ 6 }^{ }$
13500841600000
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$
9603236160000
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ 2 }$
6738177040000
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 8 }^{ }$
3676588120000
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$
1835380960000
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 10 }^{ }$
1459909120000
$\mathrm{c}_{ 6 }^{ 3 }$
1287476640000
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$
702799360000
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 10 }^{ }$
559476160000
$\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 10 }^{ }$
214406248000
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 8 }^{ 2 }$
191623650000
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 10 }^{ }$
107096280000
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 12 }^{ }$
46722720000
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 12 }^{ }$
17937420000
$\mathrm{c}_{ 8 }^{ }\mathrm{c}_{ 10 }^{ }$
11208918000
$\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 12 }^{ }$
3443000000
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 14 }^{ }$
774480000
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 14 }^{ }$
298344000
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 16 }^{ }$
6090000
$\mathrm{c}_{ 18 }^{ }$
18000
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
7
$\mathrm{b}_{ 3 }$
8
$\mathrm{b}_{ 4 }$
36
$\mathrm{b}_{ 5 }$
64
$\mathrm{b}_{ 6 }$
176
$\mathrm{b}_{ 7 }$
352
$\mathrm{b}_{ 8 }$
794
$\mathrm{b}_{ 9 }$
1592
$\mathrm{b}_{ 10 }$
3301
$\mathrm{b}_{ 11 }$
6416
$\mathrm{b}_{ 12 }$
12571
$\mathrm{b}_{ 13 }$
23456
$\mathrm{b}_{ 14 }$
43043
$\mathrm{b}_{ 15 }$
74040
$\mathrm{b}_{ 16 }$
118672
$\mathrm{b}_{ 17 }$
162808
$\mathrm{b}_{ 18 }$
198270
$\mathrm{b}_{ 19 }$
162808
$\mathrm{b}_{ 20 }$
118672
$\mathrm{b}_{ 21 }$
74040
$\mathrm{b}_{ 22 }$
43043
$\mathrm{b}_{ 23 }$
23456
$\mathrm{b}_{ 24 }$
12571
$\mathrm{b}_{ 25 }$
6416
$\mathrm{b}_{ 26 }$
3301
$\mathrm{b}_{ 27 }$
1592
$\mathrm{b}_{ 28 }$
794
$\mathrm{b}_{ 29 }$
352
$\mathrm{b}_{ 30 }$
176
$\mathrm{b}_{ 31 }$
64
$\mathrm{b}_{ 32 }$
36
$\mathrm{b}_{ 33 }$
8
$\mathrm{b}_{ 34 }$
7
$\mathrm{b}_{ 35 }$
0
$\mathrm{b}_{ 36 }$
1
1
0
0
1
5
1
0
4
4
0
1
6
22
6
1
0
4
28
28
4
0
1
6
34
94
34
6
1
0
4
32
140
140
32
4
0
1
6
35
167
376
167
35
6
1
0
4
32
172
588
588
172
32
4
0
1
6
35
179
719
1421
719
179
35
6
1
0
4
32
176
772
2224
2224
772
176
32
4
0
1
6
35
180
799
2785
4959
2785
799
180
35
6
1
0
4
32
176
800
3028
7688
7688
3028
800
176
32
4
0
1
6
35
179
799
3126
9537
15677
9537
3126
799
179
35
6
1
0
4
32
172
772
3028
10096
22916
22916
10096
3028
772
172
32
4
0
1
6
34
167
719
2785
9537
25842
40490
25842
9537
2785
719
167
34
6
1
0
4
28
140
588
2224
7688
22916
47816
47816
22916
7688
2224
588
140
28
4
0
1
5
22
94
376
1421
4959
15677
40490
72180
40490
15677
4959
1421
376
94
22
5
1
0
4
28
140
588
2224
7688
22916
47816
47816
22916
7688
2224
588
140
28
4
0
1
6
34
167
719
2785
9537
25842
40490
25842
9537
2785
719
167
34
6
1
0
4
32
172
772
3028
10096
22916
22916
10096
3028
772
172
32
4
0
1
6
35
179
799
3126
9537
15677
9537
3126
799
179
35
6
1
0
4
32
176
800
3028
7688
7688
3028
800
176
32
4
0
1
6
35
180
799
2785
4959
2785
799
180
35
6
1
0
4
32
176
772
2224
2224
772
176
32
4
0
1
6
35
179
719
1421
719
179
35
6
1
0
4
32
172
588
588
172
32
4
0
1
6
35
167
376
167
35
6
1
0
4
32
140
140
32
4
0
1
6
34
94
34
6
1
0
4
28
28
4
0
1
6
22
6
1
0
4
4
0
1
5
1
0
0
1
Kum10 -type
monomial
value
$\mathrm{c}_{ 2 }^{ 10 }$
336252992654447616
$\mathrm{c}_{ 2 }^{ 8 }\mathrm{c}_{ 4 }^{ }$
132107428736160768
$\mathrm{c}_{ 2 }^{ 6 }\mathrm{c}_{ 4 }^{ 2 }$
51898082311033728
$\mathrm{c}_{ 2 }^{ 7 }\mathrm{c}_{ 6 }^{ }$
26693534659013376
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ 3 }$
20386379301294336
$\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$
10486371945354624
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 4 }$
8007472661159664
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ }$
4119203015724192
$\mathrm{c}_{ 4 }^{ 5 }$
3144990890482320
$\mathrm{c}_{ 2 }^{ 6 }\mathrm{c}_{ 8 }^{ }$
3051655882366080
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 6 }^{ 2 }$
2119158341714304
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 3 }\mathrm{c}_{ 6 }^{ }$
1617975749261520
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ }$
1199055419079936
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ 2 }$
832451953404192
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 8 }^{ }$
471105410929296
$\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 6 }^{ 2 }$
326987093337168
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$
242424490790592
$\mathrm{c}_{ 2 }^{ 5 }\mathrm{c}_{ 10 }^{ }$
204371090647680
$\mathrm{c}_{ 4 }^{ 3 }\mathrm{c}_{ 8 }^{ }$
185086417093248
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ 3 }$
168265889899008
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 8 }^{ }$
95252580881040
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 10 }^{ }$
80342429404512
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 10 }^{ }$
31583103012912
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 8 }^{ 2 }$
27756335356332
$\mathrm{c}_{ 6 }^{ 2 }\mathrm{c}_{ 8 }^{ }$
19264369884144
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 10 }^{ }$
16391906873440
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 10 }^{ }$
16258455456144
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 8 }^{ 2 }$
10909113168228
$\mathrm{c}_{ 2 }^{ 4 }\mathrm{c}_{ 12 }^{ }$
8013253087488
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 12 }^{ }$
3153305609256
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 8 }^{ }\mathrm{c}_{ 10 }^{ }$
1864193494284
$\mathrm{c}_{ 4 }^{ 2 }\mathrm{c}_{ 12 }^{ }$
1240853563488
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 12 }^{ }$
639144656040
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 14 }^{ }$
178626056400
$\mathrm{c}_{ 10 }^{ 2 }$
125480168748
$\mathrm{c}_{ 8 }^{ }\mathrm{c}_{ 12 }^{ }$
73457352276
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 14 }^{ }$
70412082840
$\mathrm{c}_{ 6 }^{ }\mathrm{c}_{ 14 }^{ }$
14310113400
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 16 }^{ }$
12116210140
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 16 }^{ }$
836469612
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 18 }^{ }$
11419980
$\mathrm{c}_{ 20 }^{ }$
15972
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
7
$\mathrm{b}_{ 3 }$
8
$\mathrm{b}_{ 4 }$
36
$\mathrm{b}_{ 5 }$
64
$\mathrm{b}_{ 6 }$
176
$\mathrm{b}_{ 7 }$
352
$\mathrm{b}_{ 8 }$
794
$\mathrm{b}_{ 9 }$
1592
$\mathrm{b}_{ 10 }$
3286
$\mathrm{b}_{ 11 }$
6424
$\mathrm{b}_{ 12 }$
12522
$\mathrm{b}_{ 13 }$
23680
$\mathrm{b}_{ 14 }$
44142
$\mathrm{b}_{ 15 }$
79920
$\mathrm{b}_{ 16 }$
140073
$\mathrm{b}_{ 17 }$
232368
$\mathrm{b}_{ 18 }$
354034
$\mathrm{b}_{ 19 }$
471712
$\mathrm{b}_{ 20 }$
538070
$\mathrm{b}_{ 21 }$
471712
$\mathrm{b}_{ 22 }$
354034
$\mathrm{b}_{ 23 }$
232368
$\mathrm{b}_{ 24 }$
140073
$\mathrm{b}_{ 25 }$
79920
$\mathrm{b}_{ 26 }$
44142
$\mathrm{b}_{ 27 }$
23680
$\mathrm{b}_{ 28 }$
12522
$\mathrm{b}_{ 29 }$
6424
$\mathrm{b}_{ 30 }$
3286
$\mathrm{b}_{ 31 }$
1592
$\mathrm{b}_{ 32 }$
794
$\mathrm{b}_{ 33 }$
352
$\mathrm{b}_{ 34 }$
176
$\mathrm{b}_{ 35 }$
64
$\mathrm{b}_{ 36 }$
36
$\mathrm{b}_{ 37 }$
8
$\mathrm{b}_{ 38 }$
7
$\mathrm{b}_{ 39 }$
0
$\mathrm{b}_{ 40 }$
1
1
0
0
1
5
1
0
4
4
0
1
6
22
6
1
0
4
28
28
4
0
1
6
34
94
34
6
1
0
4
32
140
140
32
4
0
1
6
35
167
376
167
35
6
1
0
4
32
172
588
588
172
32
4
0
1
6
35
179
719
1406
719
179
35
6
1
0
4
32
176
772
2228
2228
772
176
32
4
0
1
6
35
180
800
2781
4916
2781
800
180
35
6
1
0
4
32
176
804
3060
7764
7764
3060
804
176
32
4
0
1
6
35
180
811
3191
9816
16062
9816
3191
811
180
35
6
1
0
4
32
176
804
3220
10892
24832
24832
10892
3220
804
176
32
4
0
1
6
35
179
800
3191
11223
30888
47427
30888
11223
3191
800
179
35
6
1
0
4
32
172
772
3060
10892
32944
68308
68308
32944
10892
3060
772
172
32
4
0
1
6
34
167
719
2781
9816
30888
76701
111808
76701
30888
9816
2781
719
167
34
6
1
0
4
28
140
588
2228
7764
24832
68308
131964
131964
68308
24832
7764
2228
588
140
28
4
0
1
5
22
94
376
1406
4916
16062
47427
111808
173836
111808
47427
16062
4916
1406
376
94
22
5
1
0
4
28
140
588
2228
7764
24832
68308
131964
131964
68308
24832
7764
2228
588
140
28
4
0
1
6
34
167
719
2781
9816
30888
76701
111808
76701
30888
9816
2781
719
167
34
6
1
0
4
32
172
772
3060
10892
32944
68308
68308
32944
10892
3060
772
172
32
4
0
1
6
35
179
800
3191
11223
30888
47427
30888
11223
3191
800
179
35
6
1
0
4
32
176
804
3220
10892
24832
24832
10892
3220
804
176
32
4
0
1
6
35
180
811
3191
9816
16062
9816
3191
811
180
35
6
1
0
4
32
176
804
3060
7764
7764
3060
804
176
32
4
0
1
6
35
180
800
2781
4916
2781
800
180
35
6
1
0
4
32
176
772
2228
2228
772
176
32
4
0
1
6
35
179
719
1406
719
179
35
6
1
0
4
32
172
588
588
172
32
4
0
1
6
35
167
376
167
35
6
1
0
4
32
140
140
32
4
0
1
6
34
94
34
6
1
0
4
28
28
4
0
1
6
22
6
1
0
4
4
0
1
5
1
0
0
1
References
MR0730926
Beauville, Arnaud. "Varié té s Kä hleriennes dont la première classe de C hern est nulle." In: J. Differential Geom. 18 (1983), pp. 755–782 (1984)
OG6 -type
The first example of this type was constructed by O'Grady in 2003 in [MR1966024 ]. It takes the Jacobian $J$ of a genus 2 curve as a principally polarised abelian variety, and considers a symplectic desingularisation of the moduli space of semistable sheaves with Mukai vector $(2,0,2)$ on the Jacobian (which is singular because the Mukai vector is divisible). There exists a locally trivial fibration over $J\times\widehat{J}$, and the fiber over $(0,0)$ is a new deformation type of hyperkähler varieties of dimension 6.
O'Grady's 6-dimensional sporadic type
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
8
$\mathrm{b}_{ 3 }$
0
$\mathrm{b}_{ 4 }$
199
$\mathrm{b}_{ 5 }$
0
$\mathrm{b}_{ 6 }$
1504
$\mathrm{b}_{ 7 }$
0
$\mathrm{b}_{ 8 }$
199
$\mathrm{b}_{ 9 }$
0
$\mathrm{b}_{ 10 }$
8
$\mathrm{b}_{ 11 }$
0
$\mathrm{b}_{ 12 }$
1
1
0
0
1
6
1
0
0
0
0
1
12
173
12
1
0
0
0
0
0
0
1
6
173
1144
173
6
1
0
0
0
0
0
0
1
12
173
12
1
0
0
0
0
1
6
1
0
0
1
References
MR1966024
O'Grady, Kieran G.. "A new six-dimensional irreducible symplectic variety." In: J. Algebraic Geom. 12 (2003), pp. 435–505. doi:10.1090/S1056-3911-03-00323-0
OG10 -type
The first example of this type was constructed by O'Grady in 1999 in [MR1703077 ]. It takes a K3 surface and some polarisation on it, and considers a symplectic desingularisation of the moduli space of semistable sheaves with Mukai vector $(2,0,4)$ (which is singular because the Mukai vector is divisible). This is a new deformation type of hyperkähler varieties of dimension 10.
O'Grady's 10-dimensional sporadic type
monomial
value
$\mathrm{c}_{ 2 }^{ 5 }$
127370880
$\mathrm{c}_{ 2 }^{ 3 }\mathrm{c}_{ 4 }^{ }$
53071200
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ 2 }$
22113000
$\mathrm{c}_{ 2 }^{ 2 }\mathrm{c}_{ 6 }^{ }$
12383280
$\mathrm{c}_{ 4 }^{ }\mathrm{c}_{ 6 }^{ }$
5159700
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 8 }^{ }$
1791720
$\mathrm{c}_{ 10 }^{ }$
176904
value
$\mathrm{b}_{ 0 }$
1
$\mathrm{b}_{ 1 }$
0
$\mathrm{b}_{ 2 }$
24
$\mathrm{b}_{ 3 }$
0
$\mathrm{b}_{ 4 }$
300
$\mathrm{b}_{ 5 }$
0
$\mathrm{b}_{ 6 }$
2899
$\mathrm{b}_{ 7 }$
0
$\mathrm{b}_{ 8 }$
22150
$\mathrm{b}_{ 9 }$
0
$\mathrm{b}_{ 10 }$
126156
$\mathrm{b}_{ 11 }$
0
$\mathrm{b}_{ 12 }$
22150
$\mathrm{b}_{ 13 }$
0
$\mathrm{b}_{ 14 }$
2899
$\mathrm{b}_{ 15 }$
0
$\mathrm{b}_{ 16 }$
300
$\mathrm{b}_{ 17 }$
0
$\mathrm{b}_{ 18 }$
24
$\mathrm{b}_{ 19 }$
0
$\mathrm{b}_{ 20 }$
1
1
0
0
1
22
1
0
0
0
0
1
22
254
22
1
0
0
0
0
0
0
1
23
276
2299
276
23
1
0
0
0
0
0
0
0
0
1
22
276
2531
16490
2531
276
22
1
0
0
0
0
0
0
0
0
0
0
1
22
254
2299
16490
88024
16490
2299
254
22
1
0
0
0
0
0
0
0
0
0
0
1
22
276
2531
16490
2531
276
22
1
0
0
0
0
0
0
0
0
1
23
276
2299
276
23
1
0
0
0
0
0
0
1
22
254
22
1
0
0
0
0
1
22
1
0
0
1
References
MR1703077
O'Grady, Kieran G.. "Desingularized moduli spaces of sheaves on a $K3$ ." In: J. Reine Angew. Math. 512 (1999), pp. 49–117. doi:10.1515/crll.1999.056