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hh -- Hodge numbers of a smooth projective variety

Synopsis

Usage:

hh^(p,q)(X)
Inputs:
- a pair (p,q) of non negative integers
- X, a projective variety
Outputs:
- an integer

Description

The command computes the Hodge numbers h^{p,q} of the smooth projective variety X. They are calculated as HH^q(cotangentSheaf(p,X))

As an example we compute h^11 of a 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 : hh^(1,1)(X)

o2 = 20

Caveat

There is no check made if the projective variety X is smooth or not.

See also

HH^ZZ SumOfTwists -- coherent sheaf cohomology module
HH^ZZ SheafOfRings -- cohomology of a sheaf of rings on a projective variety
euler(ProjectiveVariety) -- topological Euler characteristic of a (smooth) projective variety

Ways to use hh:

hh(Sequence,ProjectiveVariety)

For the programmer

The object hh is a scripted functor.