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HH^ZZ CoherentSheaf -- cohomology of a coherent sheaf on a projective variety

Synopsis

Description

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, while for i=0 it is computed as a limmit of Homs. Eventually there will exist an alternative option for computing sheaf cohomology 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

x05+x15+x25+x35+x45-5λx0x1x2x3x4=0

for various values of λ. If λ is general (that is, λ not a 5-th root of unity, 0 or ), then the quintic V is smooth, so is a Calabi-Yau threefold, and in that case the Hodge numbers are as follows.

h1,1(V)=1, h2,1(V) = h1,2(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:

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))
singularLocus(Quintic)
omegaQuintic = cotangentSheaf(Quintic);
h11 = rank HH^1(omegaQuintic)
h12 = rank HH^2(omegaQuintic)

By Hodge duality this is h2,1. Directly h2,1 could be computed as

h21 = rank HH^1(cotangentSheaf(2,Quintic))

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

hh^(2,1)(Quintic)
hh^(1,1)(Quintic)

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

euler(Quintic)

When λ 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:λ:λ:λ:λ) under a natural action of ℤ/53. Then V has a projective small resolution W which is a Calabi-Yau threefold (since the action of ℤ/53 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 λ=1, which is defined over and is modular (see Schoen’s work). To compute the Hodge numbers of the small resolution W of V we proceed as follows:

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))
Z = singularLocus(SchoensQuintic)
degree Z
II'Z = sheaf module ideal Z

The defect of W (that is, h1,1(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):

defect = rank HH^1(II'Z(5))
h11 = defect + 1

The number h2,1(W) (the dimension of the moduli space of W) can be computed (Clemens-Griffiths, Werner) as dim H0(IZ(5))/JacobianIdeal(V)5.

quinticsJac = numgens source basis(5,ideal Z)
h21 = rank HH^0(II'Z(5)) - quinticsJac

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

chiW = euler(Quintic)+2*degree(Z)

See also