hh -- Hodge numbers of a smooth projective variety

Usage:

hh^(p,q)(X)
Inputs:
- (p,q), a pair of integers between 0 and $\dim X$
- X, a projective variety,
Outputs:
- an integer, the $(p,q)$-Hodge numbers of $X$

Description

The command computes the Hodge numbers $$ h^{p,q}(X) = \dim H^q(\Omega_X^p) $$ of the smooth projective variety $X$, calculated as HH^q(cotangentSheaf(p, X)).

As an example we compute the Hodge diamond of a smooth K3 surface (Fermat quartic in projective threespace):

i1 : X = Proj QQ[x_0..x_3]/ideal(x_0^4+x_1^4+x_2^4+x_3^4)

o1 = X

o1 : ProjectiveVariety

i2 : assert isSmooth X

i3 : matrix table(toList(0..2), toList(0..2), (p,q) -> hh^(p,q)(X))

o3 = | 1 0  1 |
     | 0 20 0 |
     | 1 0  1 |

              3       3
o3 : Matrix ZZ  <-- ZZ

i4 : euler X

o4 = 24

Ways to use hh:

hh(Sequence,ProjectiveVariety)

For the programmer

The object hh is a scripted functor.

The source of this document is in Varieties/doc-functors.m2:600:0.

hh -- Hodge numbers of a smooth projective variety

Description

See also

Ways to use hh:

For the programmer