Does it exist a computer program which calculates the cohomology of projective algebraic varieties ? For example, smooth surface in $\mathbb{P}^3$? like $\sum_0^3 X_i^3=0$
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1$\begingroup$ If you mean sheaf cohomology, Macauley2 sometimes can do it. $\endgroup$– Francesco PolizziMay 10, 2014 at 7:05
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1$\begingroup$ In that generality, I'm pretty sure the answer is no. If you restrict the field of definition, the sheaf of coefficients, and the dimension of your variety, then something might be possible. Also, it probably helps if the variety is smooth. $\endgroup$– jmcMay 10, 2014 at 8:01
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2$\begingroup$ I think this should be closed as an exact duplicate of mathoverflow.net/questions/81125/… but can't suggest it, as that only had an answer in a comment. $\endgroup$– Allen KnutsonMay 10, 2014 at 14:41
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$\begingroup$ Technically speaking, not an exact duplicate. That question is about affine varieties, and the answer is only about affine opens in $\mathbb{A}^n$. $\endgroup$– David E SpeyerMay 10, 2014 at 17:55
1 Answer
I will work out a couple of examples the Fermat cubic surface using MacAulay2. For instance we may compute the cohomology of the cotangent sheaf and of the tangent sheaf of the cubic surface $Z(x^3+y^3+z^3+w^3)\subset\mathbb{P}^3$ using MacAulay2.
Form this you can figure out how to compute sheaf cohomology for other sheaves using MacAulay2.
Form this example one gets already interesting informations about $X$. For instance $H^{0}(X,T_X) = T_{Id}Aut(X)$, and $h^{0}(X,T_X) = 0$ implies that $Aut(X)$ is finite. Furthermore $H^{1}(X,T_{X})$ parametrizes first order infinitesimal deformations of $X$. We get $h^{1}(X,T_{X}) = 4$. Ideed $X$ is the blow-up of $\mathbb{P}^2$ at six general points. We have $6\cdot 2 = 12$ possible choices for the six points, but we have to subtract the dimension of $Aut(\mathbb{P}^2)$. Finally $12-8 = 4$, as we expected.
Macaulay2, version 1.6
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone
i1 : P3 = QQ[x,y,z,w]
o1 = P3
o1 : PolynomialRing
i2 : I = ideal(x^3+y^3+z^3+w^3)
o2 = ideal(x^3 + y^3 + z^3 + w^3 )
o2 : Ideal of P3
i3 : X = variety(I)
o3 = X
o3 : ProjectiveVariety
i4 : CT = cotangentSheaf(X)
o4 = cokernel {2} | z 0 0 w 0 x2 y2 0 |
{2} | x w 0 0 y2 -z2 0 0 |
{2} | -y 0 w 0 x2 0 z2 0 |
{2} | 0 y x 0 -w2 0 0 z2 |
{2} | 0 -z 0 x 0 -w2 0 y2 |
{2} | 0 0 -z -y 0 0 w2 x2 |
o4 : coherent sheaf on X, quotient of OO^6_X (-2)
i5 : cohomology(0,CT)
o5 = 0
o5 : QQ-module
i6 : cohomology(1,CT)
o6 = QQ^7
o6 : QQ-module, free
i7 : cohomology(2,CT)
o7 = 0
o7 : QQ-module
i8 : T = dual(CT)
o8 = image {-2} | -xz2 yz2 z3+w3 0 yw2 -xw2 |
{-2} | y3+z3 x2y x2z -yw2 0 zw2 |
{-2} | xy2 -y3-w3 -y2z -xw2 zw2 0 |
{-2} | -y2w -x2w 0 z3+w3 x2z y2z |
{-2} | z2w 0 x2w yz2 x2y y3+w3 |
{-2} | 0 z2w -y2w xz2 -y3-z3 xy2 |
o8 : coherent sheaf on X, subsheaf of OO^6_X (2)
i9 : cohomology(0,T)
o9 = 0
o9 : QQ-module
i10 : cohomology(1,T)
o10 = QQ^4
o10 : QQ-module, free
i11 : cohomology(2,T)
o11 = 0
o11 : QQ-module
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$\begingroup$ I really like the usage of macaulay2 in answers on this site. $\endgroup$ Aug 7, 2016 at 23:24