# Hyperkaehler.info

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

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

• complex tori
• strict Calabi–Yau manifolds
• irreducible holomorphic symplectic (or hyperkähler) manifolds

The classification for the first class is trivial, whilst there are many constructions for the second class, already in dimension 3.

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:

1. K3 surfaces, with $\mathrm{h}^1(S,\mathcal{O}_S)=0$ and $\pi_1(S)=0$
2. 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-type) and examples of OG6-type.

#### K3 surface

complex dimension
2
number of moduli
20
Euler characteristic
$24$
Beauville–Fujiki form
$\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^{\oplus3}$
$\int\mathrm{td}_X^{1/2}$
$1$
polarisation type of general fiber
$(1)$
##### Chern numbers
monomial value
$\mathrm{c}_{ 2 }^{ }$ 24
##### Betti numbers
value
$\mathrm{b}_{ 0 }$ 1
$\mathrm{b}_{ 1 }$ 0
$\mathrm{b}_{ 2 }$ 22
$\mathrm{b}_{ 3 }$ 0
$\mathrm{b}_{ 4 }$ 1
1

0 0

1 20 1

0 0

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.

:

#### K3[2]-type

complex dimension
4
number of moduli
21
Euler characteristic
$324$
Beauville–Fujiki form
$\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^{\oplus3}\oplus(-2)$
$\int\mathrm{td}_X^{1/2}$
$25/32$
polarisation type of general fiber
$(1)$
##### Chern numbers
monomial value
$\mathrm{c}_{ 2 }^{ 2 }$ 828
$\mathrm{c}_{ 4 }^{ }$ 324
##### Betti numbers
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

complex dimension
6
number of moduli
21
Euler characteristic
$3200$
Beauville–Fujiki form
$\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^{\oplus3}\oplus(-4)$
$\int\mathrm{td}_X^{1/2}$
$9/16$
polarisation type of general fiber
$(1)$
##### Chern numbers
monomial value
$\mathrm{c}_{ 2 }^{ 3 }$ 36800
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }$ 14720
$\mathrm{c}_{ 6 }^{ }$ 3200
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
8
number of moduli
21
Euler characteristic
$25650$
Beauville–Fujiki form
$\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^{\oplus3}\oplus(-6)$
$\int\mathrm{td}_X^{1/2}$
$2401/6144$
polarisation type of general fiber
$(1)$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
10
number of moduli
21
Euler characteristic
$176256$
Beauville–Fujiki form
$\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^{\oplus3}\oplus(-8)$
$\int\mathrm{td}_X^{1/2}$
$4/15$
polarisation type of general fiber
$(1)$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
12
number of moduli
21
Euler characteristic
$1073720$
Beauville–Fujiki form
$\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^{\oplus3}\oplus(-10)$
$\int\mathrm{td}_X^{1/2}$
$59049/327680$
polarisation type of general fiber
$(1)$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
14
number of moduli
21
Euler characteristic
$5930496$
Beauville–Fujiki form
$\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^{\oplus3}\oplus(-12)$
$\int\mathrm{td}_X^{1/2}$
$15625/129024$
polarisation type of general fiber
$(1)$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
16
number of moduli
21
Euler characteristic
$30178575$
Beauville–Fujiki form
$\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^{\oplus3}\oplus(-14)$
$\int\mathrm{td}_X^{1/2}$
$214358881/2642411520$
polarisation type of general fiber
$(1)$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
18
number of moduli
21
Euler characteristic
$143184000$
Beauville–Fujiki form
$\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^{\oplus3}\oplus(-16)$
$\int\mathrm{td}_X^{1/2}$
$243/4480$
polarisation type of general fiber
$(1)$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
20
number of moduli
21
Euler characteristic
$639249300$
Beauville–Fujiki form
$\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^{\oplus3}\oplus(-18)$
$\int\mathrm{td}_X^{1/2}$
$137858491849/3805072588800$
polarisation type of general fiber
$(1)$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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. "Variétés hleriennes dont la première classe de Chern est nulle." In: J. Differential Geom. 18 (1983), pp. 755–782 (1984)

### Kumn-type

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

:

#### Kum2-type

complex dimension
4
number of moduli
5
Euler characteristic
$108$
##### Chern numbers
monomial value
$\mathrm{c}_{ 2 }^{ 2 }$ 756
$\mathrm{c}_{ 4 }^{ }$ 108
##### Betti numbers
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

complex dimension
6
number of moduli
5
Euler characteristic
$448$
##### Chern numbers
monomial value
$\mathrm{c}_{ 2 }^{ 3 }$ 30208
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }$ 6785
$\mathrm{c}_{ 6 }^{ }$ 448
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
8
number of moduli
5
Euler characteristic
$750$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
10
number of moduli
5
Euler characteristic
$2592$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
12
number of moduli
5
Euler characteristic
$2744$
Beauville–Fujiki form
$\mathrm{U}^3\oplus(-14)$
$\int\mathrm{td}_X^{1/2}$
$823543/2949120$
polarisation type of general fiber
$(1,1,1,1,1,7)$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
14
number of moduli
5
Euler characteristic
$7680$
Beauville–Fujiki form
$\mathrm{U}^3\oplus(-16)$
$\int\mathrm{td}_X^{1/2}$
$64/315$
polarisation type of general fiber
$(1,1,1,1,1,1,8),(1,1,1,1,1,2,4)$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
16
number of moduli
5
Euler characteristic
$9477$
Beauville–Fujiki form
$\mathrm{U}^3\oplus(-18)$
$\int\mathrm{td}_X^{1/2}$
$43046721/293601280$
polarisation type of general fiber
$(1,1,1,1,1,1,1,9),(1,1,1,1,1,1,3,3)$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
18
number of moduli
5
Euler characteristic
$18000$
Beauville–Fujiki form
$\mathrm{U}^3\oplus(-20)$
$\int\mathrm{td}_X^{1/2}$
$1953125/18579456$
polarisation type of general fiber
$(1,1,1,1,1,1,1,1,10)$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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

complex dimension
20
number of moduli
5
Euler characteristic
$15972$
Beauville–Fujiki form
$\mathrm{U}^3\oplus(-22)$
$\int\mathrm{td}_X^{1/2}$
$285311670611/3805072588800$
polarisation type of general fiber
$(1,1,1,1,1,1,1,1,1,11)$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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. "Variétés hleriennes dont la première classe de Chern 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.

complex dimension
6
number of moduli
6
Euler characteristic
$1920$
Beauville–Fujiki form
$\mathrm{U}^3\oplus(-2)^{\oplus2}$
$\int\mathrm{td}_X^{1/2}$
$2/3$
polarisation type of general fiber
$(1,2,2)$
##### Chern numbers
monomial value
$\mathrm{c}_{ 2 }^{ 3 }$ 30720
$\mathrm{c}_{ 2 }^{ }\mathrm{c}_{ 4 }^{ }$ 7680
$\mathrm{c}_{ 6 }^{ }$ 1920
##### Betti numbers
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
##### Hodge diamond
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.

complex dimension
10
number of moduli
22
Euler characteristic
$176904$
Beauville–Fujiki form
$\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^3\oplus\mathrm{A}_2(-1)$
$\int\mathrm{td}_X^{1/2}$
$4/15$
polarisation type of general fiber
$(1,1,1,1,1)$
##### Chern numbers
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
##### Betti numbers
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
##### Hodge diamond
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