For a coherent sheaf F on a variety X, the top Chern class is chern (dim X, F).
On projective space, the coefficient of the top Chern class of the tangent bundle is the 1 plus dimension of the variety.
i1 : X0 = toricProjectiveSpace 5; |
i2 : A0 = intersectionRing X0; |
i3 : ctop dual cotangentSheaf X0
5
o3 = 6t
5
o3 : A0
|
i4 : assert all (5, d -> (leadCoefficient ctop dual cotangentSheaf toricProjectiveSpace (d+1)) == d+2) |
i5 : assert all (5, d -> (
F := dual cotangentSheaf toricProjectiveSpace (d+1);
chern (d+1, F) === ctop F
)
)
|
On a complete smooth normal toric variety, the top Chern class is a sum of the classes corresponding to maximal cones in the underlying fan.
i6 : X1 = smoothFanoToricVariety (4, 50); |
i7 : A1 = intersectionRing X1; |
i8 : E1 = cotangentSheaf X1
o8 = cokernel {2, 0, 0, 0} | 0 x_0x_4x_7 x_0x_3x_6x_7 0 0 x_0x_3x_5x_7 0 0 0 0 |
{0, 0, 2, 0} | x_3x_4 0 0 x_0x_2x_3x_7 x_0x_1x_3x_7 0 0 0 0 0 |
{0, 2, 0, 0} | x_5 0 0 0 0 0 x_0x_2x_7 x_0x_1x_7 0 0 |
{0, 2, 0, 0} | -x_6 0 0 0 0 0 0 0 x_0x_2x_7 x_0x_1x_7 |
{0, 0, 0, 2} | 0 x_1 0 0 0 0 x_3x_6 0 x_3x_5 0 |
{0, 0, 0, 2} | 0 -x_2 0 0 0 0 0 x_3x_6 0 x_3x_5 |
{0, 0, 0, 2} | 0 0 x_1 x_5 0 0 -x_4 0 0 0 |
{0, 0, 0, 2} | 0 0 -x_2 0 x_5 0 0 -x_4 0 0 |
{0, 0, 0, 2} | 0 0 0 -x_6 0 x_1 0 0 -x_4 0 |
{0, 0, 0, 2} | 0 0 0 0 -x_6 -x_2 0 0 0 -x_4 |
1 1 2 6
o8 : coherent sheaf on X1, quotient of OO (-2, 0, 0, 0) ++ OO (0, 0, -2, 0) ++ OO (0, -2, 0, 0) ++ OO (0, 0, 0, -2)
X1 X1 X1 X1
|
i9 : f1 = ctop E1
16 4
o9 = --t
3 7
o9 : A1
|
i10 : assert (f1 === sum(max X1, s -> product(s, i -> -A1_i))) |
i11 : assert (f1 === chern (dim X1, E1)) |