Twisted Hodge numbers for complete intersections

The file twisted.sage 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.

Getting started

Make sure that Sage knows about twisted.sage, probably by doing load("twisted.sage"). There is ample documentation in the file, which can be acessed via

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.

Authors

• Pieter Belmans
• Piet Glas