This Python program computes the Hodge numbers of a hypersurface in a projective space.
requires the 'SymPy' module
It can easily be modified to work with complete intersections.
Usage:
$ python3 -i a.py3.py
OR ` $ python2 -i a.py2.py
HS(n,d)
where n is the dimension of V, and d is the degree.
Example:
n = 2, d = 4 (K3 surface)
HS(2,4)
⎡1 0 1⎤
⎢ ⎥
⎢0 20 0⎥
⎢ ⎥
⎣1 0 1⎦
n = 2, d = 5 (quintic surface in P^3)
HS(2,5)
⎡4 0 1⎤
⎢ ⎥
⎢0 45 0⎥
⎢ ⎥
⎣1 0 4⎦
n = 3, d = 5 (3-dimensional Calabi-Yao)
HS(3,5)
⎡1 0 0 1⎤
⎢ ⎥
⎢0 101 1 0⎥
⎢ ⎥
⎢0 1 101 0⎥
⎢ ⎥
⎣1 0 0 1⎦
References:
* Deligne in SGA 7, part 2
Exposé XI.
Cohomologie des intersections complètes
very well written
works in any characteristics
* Hirzebruch's book
"Topological methods in algebraic geometry"
Schwarzenberger's appendix
* https://math.stackexchange.com/questions/606592/hodge-diamond-of-complete-intersections
For the odd dimension of a variety it looks like
| * 1
| * 1
| 1 *
| 1 *
---------
and for the even dimension of the variety it looks like
| * 1
| * 1
| *
| 1 *
| 1 *
---------