Twisted Hodge numbers for complete intersections

The file twisted.py implements the computation of twisted Hodge numbers for complete intersections, due to Brückmann in the following papers

• [MR0399102] Brückmann, Peter: Zur Kohomologie von projektiven Hyperflächen. Beiträge zur Algebra und Geometrie, 2. 4 (1973), 87–101 (1974).

• [MR0417202] Brückmann, Peter: Zur Kohomologie von vollständigen Durchschnitten mit Koeffizienten in der Garbe der Keime der Differentialformen. Math. Nachr. 71 (1976), 203–210.

• [MR0447266] Brückmann, Peter: Zur Kohomologie von vollständigen Durchschnitten mit Koeffizienten in der Garbe der Keime der Differentialformen. II. Math. Nachr. 77 (1977), 307–318.

If you have used this code in any way, please consider citing it as explained on Zenodo. You can choose to cite a specific version, or the library in general.

Getting started

It suffices to put twisted/twisted.py in your directory and load it using load("twisted.py") in Sage to get started.

Alternatively you can install it as follows:

sage --pip install git+https://github.com/pbelmans/twisted-hodge.git

and then you can use

from twisted import *

to use it.

twisted_hodge_number?
TwistedHodgeDiamond?

To get the (untwisted) Hodge diamond of a quartic surface, use

sage: print(TwistedHodgeDiamond((3, 4)))
          1
      0        0
  1       20       1
      0        0
          1

This luckily agrees with the output for

sage: print(TwistedHodgeDiamond((4, [3, 2])))
sage: print(TwistedHodgeDiamond((5, [2, 2, 2])))

If you rather care about twisted Hodge diamonds (otherwise you could also use the Hodge diamond cutter), you can add a twist parameter. For example, to compute the twisted Hodge diamond for projective 3-space, twisted by O(4) (so that we are in fact computing the Hochschild-Kostant-Rosenberg decomposition of Hochschild cohomology) as follows

sage: print(TwistedHodgeDiamond((3, []), 4))
                0
           0         0
      0         0        0
  1        0         0       0
      15        0        0
           45        0
                35

For more information, see the docstrings.

Also check out the documentation for the auxiliary class CompleteIntersection, and if you are into Hochschild cohomology the class PolyvectorParallelogram.

How to cite

If you have used this code in any way, please consider citing it as explained on Zenodo). You can choose to cite a specific version, or always the latest version.

The following BibTeX entry is a good starting point:

@software{twisted-hodge,
  author = {Belmans, Pieter and Glas, Piet},
  title = {Twisted Hodge diamonds of complete intersections},
  url = {https://github.com/pbelmans/twisted-hodge},
  doi = {10.5281/zenodo.7006756},
}

which leads to something like

Pieter Belmans and Piet Glas. Twisted Hodge diamonds of complete intersections. doi:10.5281/zenodo.7006756. url: https://github.com/pbelmans/twisted-hodge.

Instructions to myself

To perform the unit tests:

sage -t twisted/twisted.py

Authors

• Pieter Belmans
• Piet Glas