HH^i(X,F)
HH^i F
cohomology(i,X,F)
cohomology(i,F)
As examples we compute the Picard numbers, Hodge numbers and dimension of the infinitesimal deformation spaces of various quintic hypersurfaces in projective fourspace (or their CalabiYau small resolutions)
We will make computations for quintics V in the family given by $$x_0^5+x_1^5+x_2^5+x_3^5+x_4^55\lambda x_0x_1x_2x_3x_4=0$$ for various values of $\lambda$. If $\lambda$ is general (that is, $\lambda$ not a 5th root of unity, 0 or $\infty$), then the quintic $V$ is smooth, so is a CalabiYau threefold, and in that case the Hodge numbers are as follows.
$$h^{1,1}(V)=1, h^{2,1}(V) = h^{1,2}(V) = 101,$$
so the Picard group of V has rank 1 (generated by the hyperplane section) and the moduli space of V (which is unobstructed) has dimension 101:





By Hodge duality this is $h^{2,1}$. Directly $h^{2,1}$ could be computed as

The Hodge numbers of a (smooth) projective variety can also be computed directly using the hh command:


Using the Hodge number we compute the topological Euler characteristic of V:

When $\lambda$ is a 5th root of unity the quintic V is singular. It has 125 ordinary double points (nodes), namely the orbit of the point $(1:\lambda:\lambda:\lambda:\lambda)$ under a natural action of $\ZZ/5^3$. Then $V$ has a projective small resolution $W$ which is a CalabiYau threefold (since the action of $\ZZ/5^3$ is transitive on the sets of nodes of $V$, or for instance, just by blowing up one of the $(1,5)$ polarized abelian surfaces $V$ contains). Perhaps the most interesting such 3fold is the one for the value $\lambda=1$, which is defined over $\QQ$ and is modular (see Schoen's work). To compute the Hodge numbers of the small resolution $W$ of $V$ we proceed as follows:




The defect of $W$ (that is, $h^{1,1}(W)1$) can be computed from the cohomology of the ideal sheaf of the singular locus Z of V twisted by 5 (see Werner's thesis):


The number $h^{2,1}(W)$ (the dimension of the moduli space of $W$) can be computed (ClemensGriffiths, Werner) as $\dim H^0({\mathbf I}_Z(5))/JacobianIdeal(V)_5$.


In other words W is rigid. It has the following topological Euler characteristic.
