Does it exist a computer program which calculates the cohomology of projective algebraic varieties ? For example, smooth surface in $\mathbb{P}^3$? like $\sum_0^3 X_i^3=0$

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 blowup 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 $128 = 4$, as we expected.


