# chi(CoherentSheaf) -- compute the Euler characteristic of a coherent sheaf

## Synopsis

• Function: chi
• Usage:
chi F
• Inputs:
• F, , on a complete simplicial normal toric variety
• Outputs:
• , in the coefficient ring of intersection ring of the underlying variety

## Description

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)