HH^ZZ(ProjectiveVariety,CoherentSheaf) -- cohomology of a coherent sheaf on a projective variety

The command computes the $i$-th cohomology group of F as a vector space over the coefficient field of X. For $i>0$ this is currently done via local duality. More precisely, the code takes advantage of the isomorphism $$H^i (X, \mathcal{F}) = ( H_{\mathfrak{m}}^i ( M) )_0,$$ where the notation $(-)_0$ denotes the degree $0$ part of a module, and $M$ is any module representing $\mathcal F$. Combining this with the local duality isomorphism for the graded polynomial ring $S$ yields $$H_{\mathfrak{m}}^i (M) \cong \operatorname{Ext}^{d-i}_S (M, S(-n-1)),$$ where we are viewing $X$ as being embedded into $\mathbb{P}^n$ via some affine morphism. For $i=0$, things are not as simple since there is not a direct isomorphism between the global sections of a sheaf and the degree 0 part of the $0$th local cohomology. Thus it is instead computed as a limit of Homs.

See cohomologyTable and cohomologyHashTable for alternative sheaf cohomology algorithms via the Bernstein-Gelfand-Gelfand correspondence.

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 Calabi-Yau small resolutions).

We will make computations for quintics V in the family given by $$ x_0^5+x_1^5+x_2^5+x_3^5+x_4^5-5\lambda x_0x_1x_2x_3x_4=0 $$ for various values of $\lambda$. If $\lambda$ is general (that is, $\lambda$ not a 5-th root of unity, 0 or $\infty$), then the quintic V is smooth, so is a Calabi-Yau 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 Calabi-Yau 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 3-fold 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 (Clemens-Griffiths, Werner) as $\dim H^0({\mathbf I}_Z(5))/\mathrm{JacobianIdeal}(V)_5$.

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