Hyperkaehler.info

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):