# ctop(CoherentSheaf) -- compute the top Chern class of a coherent sheaf

## Synopsis

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

## Description

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))