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Macaulay2Doc :: cotangentSheaf(ZZ,ProjectiveVariety)

cotangentSheaf(ZZ,ProjectiveVariety) -- exterior powers of the cotangent sheaf of a projective variety

Synopsis

Description

This function computes the p-th exterior power of the cotangent sheaf of a variety X.

As an example we compute h^11 on a K3 surface (a quartic in projective threespace):

i1 : K3 = Proj(QQ[x_0..x_3]/(x_0^4+x_1^4+x_2^4+x_3^4-11*x_0*x_1*x_2*x_3))

o1 = K3

o1 : ProjectiveVariety
i2 : omega1 = cotangentSheaf(1,K3);
i3 : HH^1(omega1)

       20
o3 = QQ

o3 : QQ-module, free

As a second example we compute Hodge numbers of the Fermat quintic in projective fourspace:

i4 : FermatQuintic = Proj(QQ[x_0..x_4]/(x_0^5+x_1^5+x_2^5+x_3^5+x_4^5))

o4 = FermatQuintic

o4 : ProjectiveVariety
i5 : omega1 = cotangentSheaf(1,FermatQuintic);
i6 : HH^1(omega1)

       1
o6 = QQ

o6 : QQ-module, free
i7 : omega2 = cotangentSheaf(2,FermatQuintic);
i8 : HH^1(omega2)

       101
o8 = QQ

o8 : QQ-module, free
i9 : HH^2(omega1)

       101
o9 = QQ

o9 : QQ-module, free

See also