- Usage:
`HH^i F``cohomology(i,F)`

- Function: cohomology
- Inputs:
`i`, an integer`F`, a coherent sheaf, on a projective variety`X`

- Optional inputs:
`Degree => ...`(missing documentation),

- Outputs:
- a module, the
`i`-th cohomology group of`F`as a vector space over the coefficient field of`X`

- a module, the

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*x*_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}-5λx_{0}x_{1}x_{2}x_{3}x_{4}=0*h*^{1,1}(V)=1, h^{2,1}(V) = h^{1,2}(V) = 101,

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

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.

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 , x , x , x ]\ | 0 1 2 3 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

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 *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/5 ^{3}*. Then

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 , x , x , x ]), subsheaf of OO 0 1 2 3 4 Proj(QQ[x , x , x , x , x ]) 0 1 2 3 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

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 |

- coherent sheaves
- HH^ZZ SumOfTwists -- coherent sheaf cohomology module
- HH^ZZ SheafOfRings -- cohomology of a sheaf of rings on a projective variety
- hh -- Hodge numbers of a smooth projective variety
- CoherentSheaf -- the class of all coherent sheaves