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--- status: draft
--- author(s): Decker, Popescu
--- notes:
document {
Key => cotangentSheaf,
Headline => "cotangent sheaf of a projective variety",
SeeAlso => {tangentSheaf, ProjectiveVariety}
}
document {
Key => {(cotangentSheaf,ProjectiveVariety),[cotangentSheaf,Minimize]},
Usage => "cotangentSheaf X",
Inputs => {
"X",
Minimize => Boolean => {"whether to apply ", TO "minimalPresentation", " to the result before returning it"}
},
Outputs => {CoherentSheaf => "the cotangent sheaf of the projective variety"
},
"This function computes the cotangent sheaf of the projective variety ", TT "X.",
PARA{},
"As an example we verify Gauss-Bonnet's theorem on a plane quartic curve:",
EXAMPLE {
"X=Proj(QQ[x,y,z]/(x^4+y^4+z^4))",
"genus(X)",
"omega = cotangentSheaf(X)",
"degree omega"
},
SeeAlso => {tangentSheaf, ProjectiveVariety}
}
document {
Key => (cotangentSheaf,ZZ,ProjectiveVariety),
Headline => "exterior powers of the cotangent sheaf of a projective variety",
Usage => "cotangentSheaf(p,X)",
Inputs => {"p", "X",
Minimize => Boolean => {"whether to apply ", TO "minimalPresentation", " to the result before returning it"}
},
Outputs => {CoherentSheaf => {"the ", TT "p", "-th exterior power of
the cotangent sheaf"}
},
"This function computes the ", TT "p", "-th exterior power of
the cotangent sheaf of a variety ", TT "X", ".",
PARA{},
"As an example we compute h^11 on a K3 surface (a quartic in projective threespace):",
EXAMPLE {
"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))",
"omega1 = cotangentSheaf(1,K3);",
"HH^1(omega1)"
},
PARA{},
"As a second example we compute Hodge numbers of the Fermat quintic in projective fourspace:",
EXAMPLE {
"FermatQuintic = Proj(QQ[x_0..x_4]/(x_0^5+x_1^5+x_2^5+x_3^5+x_4^5))",
"omega1 = cotangentSheaf(1,FermatQuintic);",
"HH^1(omega1)",
"omega2 = cotangentSheaf(2,FermatQuintic);",
"HH^1(omega2)",
"HH^2(omega1)"
},
SeeAlso => {tangentSheaf}
}
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