HH^i(X,F)
HH^i F
cohomology(i,X,F)
cohomology(i,F)
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.
