Given a locally-free sheaf $E$ of rank $r$ on a smooth variety such that its Chern class formally factor as chern $E = \prod_{j=1}^r (1 + \alpha_j)$, we define its Todd class to be todd $E := \prod_{j=1}^r \alpha_j / [1- exp(-\alpha_j)]$ written as a polynomial in the elementary symmetric functions chern $(i, E)$ of the $\alpha_j$.
The first few components of the Todd class are easily related to Chern classes.
i1 : X0 = kleinschmidt(4, {1,3,5});
|
i2 : E0 = cotangentSheaf X0
o2 = cokernel {2, 0} | 2x_2x_3 4x_1x_3 5x_0x_3 0 0 2x_1x_2 0 0 0 0 3x_0x_2 0 0 0 0 x_0x_1 0 0 0 0 0 0 0 0 |
{-7, 2} | x_4 0 0 0 x_1 0 0 0 0 3x_0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{-7, 2} | -x_5 0 0 0 0 0 0 x_1 0 0 0 0 3x_0 0 0 0 0 0 0 0 0 0 0 0 |
{-5, 2} | 0 x_4 0 x_2 0 0 0 0 0 0 0 0 0 0 x_0 0 0 0 0 0 0 0 0 0 |
{-5, 2} | 0 -x_5 0 0 0 0 x_2 0 0 0 0 0 0 0 0 0 0 x_0 0 0 0 0 0 0 |
{-4, 2} | 0 0 x_4 0 0 0 0 0 3x_2 0 0 0 0 x_1 0 0 0 0 0 0 0 0 0 0 |
{-4, 2} | 0 0 -x_5 0 0 0 0 0 0 0 0 3x_2 0 0 0 0 x_1 0 0 0 0 0 0 0 |
{-3, 2} | 0 0 0 -2x_3 -x_3 x_4 0 0 0 0 0 0 0 0 0 0 0 0 0 x_0 0 0 0 0 |
{-9, 3} | 0 0 0 -x_5 -x_5 0 -x_4 -x_4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_0 0 |
{-3, 2} | 0 0 0 0 0 -x_5 -2x_3 -x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 x_0 0 0 |
{-2, 2} | 0 0 0 0 0 0 0 0 -5x_3 -2x_3 x_4 0 0 0 0 0 0 0 x_1 0 0 0 0 0 |
{-8, 3} | 0 0 0 0 0 0 0 0 -x_5 -x_5 0 -x_4 -x_4 0 0 0 0 0 0 0 0 0 0 x_1 |
{-2, 2} | 0 0 0 0 0 0 0 0 0 0 -x_5 -5x_3 -2x_3 0 0 0 0 0 0 0 x_1 0 0 0 |
{0, 2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 -5x_3 -4x_3 x_4 0 0 -3x_2 -2x_2 0 0 0 0 |
{-6, 3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_5 -x_5 0 -x_4 -x_4 0 0 0 0 -x_2 -3x_2 |
{0, 2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_5 -5x_3 -4x_3 0 0 -3x_2 -2x_2 0 0 |
{-4, 3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_5 -x_5 -x_4 -x_4 2x_3 5x_3 |
1 2 2 2 1 1 1 1 1 1 1 1 1 1
o2 : coherent sheaf on X0, quotient of OO (-2, 0) ++ OO (7, -2) ++ OO (5, -2) ++ OO (4, -2) ++ OO (3, -2) ++ OO (9, -3) ++ OO (3, -2) ++ OO (2, -2) ++ OO (8, -3) ++ OO (2, -2) ++ OO (0, -2) ++ OO (6, -3) ++ OO (0, -2) ++ OO (4, -3)
X0 X0 X0 X0 X0 X0 X0 X0 X0 X0 X0 X0 X0 X0
|
i3 : A0 = intersectionRing X0; |
i4 : todd E0
13 11 2 145 3 121 2 3
o4 = 1 + (- 2t - --t ) + (--t + ---t t ) + (- t - ---t t ) + t t
3 2 5 6 3 12 3 5 3 12 3 5 3 5
o4 : A0
|
i5 : assert (part (0, todd E0) == 1) |
i6 : assert (part (1, todd E0) === (1/2) * chern (1, E0)) |
i7 : assert (part (2, todd E0) === (1/12)*((chern (1, E0))^2 + chern (2, E0))) |
On a complete smooth normal toric variety, the Todd class of the tangent bundle factors as a product over the irreducible torus-invariant divisors.
i8 : X1 = smoothFanoToricVariety (3, 12); |
i9 : E1 = dual cotangentSheaf X1
o9 = image {0, 0, -2, 0} | 0 0 0 x_2 x_3 |
{0, -2, 0, 0} | 0 -x_6 x_1x_4 0 0 |
{-2, 0, 0, 0} | -x_3x_6 0 -x_0x_3x_5 x_0x_5x_6 0 |
{-2, 0, 0, 0} | x_2x_6 0 x_0x_2x_5 0 x_0x_5x_6 |
{0, 0, 0, -2} | -x_1x_3x_4 -x_0x_3x_5 0 x_0x_1x_4x_5 0 |
{0, 0, 0, -2} | x_1x_2x_4 x_0x_2x_5 0 0 x_0x_1x_4x_5 |
1 1 2 2
o9 : coherent sheaf on X1, subsheaf of OO (0, 0, 2, 0) ++ OO (0, 2, 0, 0) ++ OO (2, 0, 0, 0) ++ OO (0, 0, 0, 2)
X1 X1 X1 X1
|
i10 : A1 = intersectionRing X1; |
i11 : f1 = todd E1
3 1 3 2 2 2 2 3
o11 = 1 + (-t + t + t + -t ) + (-t t - t - -t t - -t ) + t
2 3 4 5 2 6 2 3 4 5 3 5 6 3 6 6
o11 : A1
|
i12 : assert (f1 === product(# rays X1, i -> todd OO (X1_i))) |
Applying todd to a normal toric variety is the same as applying it to the tangent sheaf of the variety.