By definition, the Euler characteristic of coherent sheaf $F$ on a variety $X$ is $\sum_i (-1)^i$ dim $HH^i (X, F)$. However, this methods uses the Hirzebruch-Riemann-Roch theorem to calculate the Euler characteristic.
For a nef line bundle on a normal toric variety, the Euler characteristic equals the number of lattice points in the corresponding polytope.
i1 : X0 = hirzebruchSurface 2; |
i2 : degrees ring X0
o2 = {{1, 0}, {-2, 1}, {1, 0}, {0, 1}}
o2 : List
|
i3 : chi OO X0_2 o3 = 2 o3 : QQ[] |
i4 : latticePoints X0_2
o4 = | 0 1 |
| 0 0 |
2 2
o4 : Matrix ZZ <--- ZZ
|
i5 : assert all ({{1,0},{0,1},{1,1},{2,1},{1,2}}, p -> (
D := p#0 * X0_2 + p#1 * X0_3;
isNef D and chi OO D == rank source latticePoints D
)
)
|
i6 : chi OO (2 * X0_1) o6 = -3 o6 : QQ[] |
i7 : assert not isNef (2 * X0_1) |