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README.md
diamond.sage

README.md

Hodge diamond cutter

A collection of Python classes and functions to deal with Hodge diamonds (and Hochschild homology) of smooth projective varieties, together with many constructions.

Getting started

It suffices to load diamond.sage in Sage to get started. It was written using Sage 8.3, but presumably it works without modification in more recent and not too old versions, as nothing fancy is used.

Examples

  1. As a first example, let us consider the cubic fourfold. There is an intricate connection to K3 surfaces. To see the Hodge diamond of a K3 surface, it suffices to do

    print K3
    

    because K3 surfaces are hardcoded (well, all of them have the same Hodge diamond, which is just that of a quartic surface), which is

              1
          0        0
      1       20       1
          0        0
              1
    

    Now for the cubic fourfold we do

    print complete_intersection(3, 4)
    

    which gives

                      1
                  0        0
              0       1        0
          0       0        0       0
      0       1       21       1       0
          0       0        0       0
              0       1        0
                  0        0
                      1
    

    Removing the primitive part of the cohomology gives you back the Hodge diamond of a K3 surface, and this is the first glimpse at a very interesting story relating the two.

  2. As a second example, let us discuss a conjectural semiorthogonal decomposition for the moduli space of rank 2 bundles with fixed determinant of degree 1 on a curve $C$ of genus $g$. As observed by Kyoung-Seog Lee and myself the Hodge diamond of this variety can be decomposed in terms of symmetric powers of the curve $C$, which can be checked as follows

    for g in range(2, 10):
      assert moduli_vector_bundles(2, 1, g) == sum([symmetric_power(i, g)(i) for i in range(g)]) + sum([symmetric_power(i, g)(3*g - 3 - 2*i) for i in range(g - 1)])
    

Contributing

Please feel free to make suggestions for more examples of Hodge diamonds. Preferably with a link to a closed formula, generating series or method of computation.

Feature requests are also very welcome. And suggestions on improving the documentation are also welcome.