Betti numbers
These are the ranks of the cohomologies $\mathrm{H}^i(X,\mathbb{Z})$.
Links to other invariants
In the Hodge diamond the Betti numbers correspond to sums of the rows in the diamond (although by symmetry this is the same as the columns in this special case): we have that $\operatorname{rk}\operatorname{H}^i(X,\mathbb{Z})=\dim_{\mathbb{C}}\operatorname{H}^i(X,\mathbb{C})=\sum_{p+q=i}\dim_{\mathbb{C}}\operatorname{H}^q(X,\Omega_X^p)$.
The alternating sum of the Betti numbers is then the Euler characteristic .
Salamon's identity
The middle Betti number $\mathrm{b}_{2n}$ on a $2n$-dimensional hyperkähler manifold is in an interesting way related to the other Betti numbers:
\[
n\mathrm{b}_{2n}=2\sum_{j=1}^{2n}(-1)^j(3j^2-n)\mathrm{b}_{2n-j}(X)
\]
For instance, in the 4-dimensional case, we have
\[
\begin{aligned}
2\cdot\mathrm{b}_4(\mathrm{K3}^{[2]})&=2\left( 10\cdot\mathrm{b}_2(\mathrm{K3}^{[2]}) + 46\mathrm{b}_0(\mathrm{K3}^{[2]}) \right) \\
&=2\cdot 276 \\
2\cdot\mathrm{b}_4(\mathrm{Kum}^2)&=2\left( -\mathrm{b}_3(\mathrm{Kum}^2) + 10\cdot\mathrm{b}_2(\mathrm{Kum}^2) + 46\mathrm{b}_0(\mathrm{Kum}^2) \right) \\
&=2\cdot 108
\end{aligned}
\]
All Betti numbers
dimension d (2–20) :
Betti number
K3
$\mathrm{b}_{ 0 }(X)$
1
$\mathrm{b}_{ 1 }(X)$
0
$\mathrm{b}_{ 2 }(X)$
22
Betti number
K3[2] -type Kum2 -type
$\mathrm{b}_{ 0 }(X)$
1
1
$\mathrm{b}_{ 1 }(X)$
0
0
$\mathrm{b}_{ 2 }(X)$
23
7
$\mathrm{b}_{ 3 }(X)$
0
8
$\mathrm{b}_{ 4 }(X)$
276
108
Betti number
K3[3] -type Kum3 -type OG6
$\mathrm{b}_{ 0 }(X)$
1
1
1
$\mathrm{b}_{ 1 }(X)$
0
0
0
$\mathrm{b}_{ 2 }(X)$
23
7
8
$\mathrm{b}_{ 3 }(X)$
0
8
0
$\mathrm{b}_{ 4 }(X)$
299
51
199
$\mathrm{b}_{ 5 }(X)$
0
56
0
$\mathrm{b}_{ 6 }(X)$
2554
458
1504
Betti number
K3[4] -type Kum4 -type
$\mathrm{b}_{ 0 }(X)$
1
1
$\mathrm{b}_{ 1 }(X)$
0
0
$\mathrm{b}_{ 2 }(X)$
23
7
$\mathrm{b}_{ 3 }(X)$
0
8
$\mathrm{b}_{ 4 }(X)$
300
36
$\mathrm{b}_{ 5 }(X)$
0
64
$\mathrm{b}_{ 6 }(X)$
2852
168
$\mathrm{b}_{ 7 }(X)$
0
288
$\mathrm{b}_{ 8 }(X)$
19298
1046
Betti number
K3[5] -type Kum5 -type OG10
$\mathrm{b}_{ 0 }(X)$
1
1
1
$\mathrm{b}_{ 1 }(X)$
0
0
0
$\mathrm{b}_{ 2 }(X)$
23
7
24
$\mathrm{b}_{ 3 }(X)$
0
8
0
$\mathrm{b}_{ 4 }(X)$
300
36
300
$\mathrm{b}_{ 5 }(X)$
0
64
0
$\mathrm{b}_{ 6 }(X)$
2875
191
2899
$\mathrm{b}_{ 7 }(X)$
0
344
0
$\mathrm{b}_{ 8 }(X)$
22127
915
22150
$\mathrm{b}_{ 9 }(X)$
0
1312
0
$\mathrm{b}_{ 10 }(X)$
125604
3748
126156
Betti number
K3[6] -type Kum6 -type
$\mathrm{b}_{ 0 }(X)$
1
1
$\mathrm{b}_{ 1 }(X)$
0
0
$\mathrm{b}_{ 2 }(X)$
23
7
$\mathrm{b}_{ 3 }(X)$
0
8
$\mathrm{b}_{ 4 }(X)$
300
36
$\mathrm{b}_{ 5 }(X)$
0
64
$\mathrm{b}_{ 6 }(X)$
2876
176
$\mathrm{b}_{ 7 }(X)$
0
352
$\mathrm{b}_{ 8 }(X)$
22426
786
$\mathrm{b}_{ 9 }(X)$
0
1528
$\mathrm{b}_{ 10 }(X)$
147431
2879
$\mathrm{b}_{ 11 }(X)$
0
4496
$\mathrm{b}_{ 12 }(X)$
727606
7870
Betti number
K3[7] -type Kum7 -type
$\mathrm{b}_{ 0 }(X)$
1
1
$\mathrm{b}_{ 1 }(X)$
0
0
$\mathrm{b}_{ 2 }(X)$
23
7
$\mathrm{b}_{ 3 }(X)$
0
8
$\mathrm{b}_{ 4 }(X)$
300
36
$\mathrm{b}_{ 5 }(X)$
0
64
$\mathrm{b}_{ 6 }(X)$
2876
176
$\mathrm{b}_{ 7 }(X)$
0
352
$\mathrm{b}_{ 8 }(X)$
22449
809
$\mathrm{b}_{ 9 }(X)$
0
1584
$\mathrm{b}_{ 10 }(X)$
150283
3327
$\mathrm{b}_{ 11 }(X)$
0
6136
$\mathrm{b}_{ 12 }(X)$
872162
11298
$\mathrm{b}_{ 13 }(X)$
0
16432
$\mathrm{b}_{ 14 }(X)$
3834308
25524
Betti number
K3[8] -type Kum8 -type
$\mathrm{b}_{ 0 }(X)$
1
1
$\mathrm{b}_{ 1 }(X)$
0
0
$\mathrm{b}_{ 2 }(X)$
23
7
$\mathrm{b}_{ 3 }(X)$
0
8
$\mathrm{b}_{ 4 }(X)$
300
36
$\mathrm{b}_{ 5 }(X)$
0
64
$\mathrm{b}_{ 6 }(X)$
2876
176
$\mathrm{b}_{ 7 }(X)$
0
352
$\mathrm{b}_{ 8 }(X)$
22450
794
$\mathrm{b}_{ 9 }(X)$
0
1592
$\mathrm{b}_{ 10 }(X)$
150582
3278
$\mathrm{b}_{ 11 }(X)$
0
6360
$\mathrm{b}_{ 12 }(X)$
894288
12202
$\mathrm{b}_{ 13 }(X)$
0
21704
$\mathrm{b}_{ 14 }(X)$
4684044
36440
$\mathrm{b}_{ 15 }(X)$
0
51640
$\mathrm{b}_{ 16 }(X)$
18669447
67049
Betti number
K3[9] -type Kum9 -type
$\mathrm{b}_{ 0 }(X)$
1
1
$\mathrm{b}_{ 1 }(X)$
0
0
$\mathrm{b}_{ 2 }(X)$
23
7
$\mathrm{b}_{ 3 }(X)$
0
8
$\mathrm{b}_{ 4 }(X)$
300
36
$\mathrm{b}_{ 5 }(X)$
0
64
$\mathrm{b}_{ 6 }(X)$
2876
176
$\mathrm{b}_{ 7 }(X)$
0
352
$\mathrm{b}_{ 8 }(X)$
22450
794
$\mathrm{b}_{ 9 }(X)$
0
1592
$\mathrm{b}_{ 10 }(X)$
150605
3301
$\mathrm{b}_{ 11 }(X)$
0
6416
$\mathrm{b}_{ 12 }(X)$
897141
12571
$\mathrm{b}_{ 13 }(X)$
0
23456
$\mathrm{b}_{ 14 }(X)$
4831451
43043
$\mathrm{b}_{ 15 }(X)$
0
74040
$\mathrm{b}_{ 16 }(X)$
23203208
118672
$\mathrm{b}_{ 17 }(X)$
0
162808
$\mathrm{b}_{ 18 }(X)$
84967890
198270
Betti number
K3[10] -type Kum10 -type
$\mathrm{b}_{ 0 }(X)$
1
1
$\mathrm{b}_{ 1 }(X)$
0
0
$\mathrm{b}_{ 2 }(X)$
23
7
$\mathrm{b}_{ 3 }(X)$
0
8
$\mathrm{b}_{ 4 }(X)$
300
36
$\mathrm{b}_{ 5 }(X)$
0
64
$\mathrm{b}_{ 6 }(X)$
2876
176
$\mathrm{b}_{ 7 }(X)$
0
352
$\mathrm{b}_{ 8 }(X)$
22450
794
$\mathrm{b}_{ 9 }(X)$
0
1592
$\mathrm{b}_{ 10 }(X)$
150606
3286
$\mathrm{b}_{ 11 }(X)$
0
6424
$\mathrm{b}_{ 12 }(X)$
897440
12522
$\mathrm{b}_{ 13 }(X)$
0
23680
$\mathrm{b}_{ 14 }(X)$
4853600
44142
$\mathrm{b}_{ 15 }(X)$
0
79920
$\mathrm{b}_{ 16 }(X)$
24075047
140073
$\mathrm{b}_{ 17 }(X)$
0
232368
$\mathrm{b}_{ 18 }(X)$
107276810
354034
$\mathrm{b}_{ 19 }(X)$
0
471712
$\mathrm{b}_{ 20 }(X)$
364690994
538070