Hodge and numerical invariants
- Hodge diamond
- Total degree $p+q$ increases downwards. The independent entries are $\mathrm{h}^{1,1}$, $\mathrm{h}^{1,2}$, $\mathrm{h}^{1,3}$, and $\mathrm{h}^{2,2}$.
- Picard rank $\rho$
- For these fourfolds, $\rho=\mathrm{h}^{1,1}$.
- Volume and Chern numbers
- Write $\mathrm{c}_i=\mathrm{c}_i(\mathrm{T}_X)$. The volume is $\operatorname{vol}(X)=\int_X\mathrm{c}_1^4$. The entries $\mathrm{c}_1^4$, $\mathrm{c}_1^2\mathrm{c}_2$, $\mathrm{c}_2^2$, $\mathrm{c}_1\mathrm{c}_3$, and $\mathrm{c}_4$ mean the integrals of those products over $X$. The last equals the topological Euler characteristic.
- Tangent Euler characteristic $\chi(X,\mathrm{T}_X)$
- On these Fano fourfolds, $\chi(X,\mathrm{T}_X)=\mathrm{h}^0(X,\mathrm{T}_X)-\mathrm{h}^1(X,\mathrm{T}_X)$.