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:
i1 : Quintic = Proj(QQ[x_0..x_4]/ideal(x_0^5+x_1^5+x_2^5+x_3^5+x_4^5-101*x_0*x_1*x_2*x_3*x_4)) o1 = Quintic o1 : ProjectiveVariety |
i2 : singularLocus(Quintic) /QQ[x ..x ]\ | 0 4 | o2 = Proj|----------| \ 1 / o2 : ProjectiveVariety |
i3 : omegaQuintic = cotangentSheaf(Quintic); |
i4 : h11 = rank HH^1(omegaQuintic) o4 = 1 |
i5 : h12 = rank HH^2(omegaQuintic) o5 = 101 |
By Hodge duality this is $h^{2,1}$. Directly $h^{2,1}$ could be computed as
i6 : h21 = rank HH^1(cotangentSheaf(2,Quintic)) o6 = 101 |
The Hodge numbers of a (smooth) projective variety can also be computed directly using the hh command:
i7 : hh^(2,1)(Quintic) o7 = 101 |
i8 : hh^(1,1)(Quintic) o8 = 1 |
Using the Hodge number we compute the topological Euler characteristic of V:
i9 : euler(Quintic) o9 = -200 |
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:
i10 : SchoensQuintic = Proj(QQ[x_0..x_4]/ideal(x_0^5+x_1^5+x_2^5+x_3^5+x_4^5-5*x_0*x_1*x_2*x_3*x_4)) o10 = SchoensQuintic o10 : ProjectiveVariety |
i11 : Z = singularLocus(SchoensQuintic) o11 = Z o11 : ProjectiveVariety |
i12 : degree Z o12 = 125 |
i13 : II'Z = sheaf module ideal Z o13 = image | x_3^4-x_0x_1x_2x_4 x_0x_1x_2x_3-x_4^4 x_2^4-x_0x_1x_3x_4 x_1^4-x_0x_2x_3x_4 x_0^4-x_1x_2x_3x_4 x_2^3x_3^3-x_0^2x_1^2x_4^2 x_1^3x_3^3-x_0^2x_2^2x_4^2 x_0^3x_3^3-x_1^2x_2^2x_4^2 x_1^2x_2^2x_3^2-x_0^3x_4^3 x_0^2x_2^2x_3^2-x_1^3x_4^3 x_0^2x_1^2x_3^2-x_2^3x_4^3 x_1^3x_2^3-x_0^2x_3^2x_4^2 x_0^3x_2^3-x_1^2x_3^2x_4^2 x_0^2x_1^2x_2^2-x_3^3x_4^3 x_0^3x_1^3-x_2^2x_3^2x_4^2 | 1 o13 : coherent sheaf on Proj(QQ[x ..x ]), subsheaf of OO 0 4 Proj(QQ[x ..x ]) 0 4 |
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):
i14 : defect = rank HH^1(II'Z(5)) o14 = 24 |
i15 : h11 = defect + 1 o15 = 25 |
The number $h^{2,1}(W)$ (the dimension of the moduli space of W) can be computed (Clemens-Griffiths, Werner) as $dim H^0(I_Z(5))/JacobianIdeal(V)_5$.
i16 : quinticsJac = numgens source basis(5,ideal Z) o16 = 25 |
i17 : h21 = rank HH^0(II'Z(5)) - quinticsJac o17 = 0 |
In other words W is rigid. It has the following topological Euler characteristic.
i18 : chiW = euler(Quintic)+2*degree(Z) o18 = 50 |