New tool: twisted Hodge diamonds for Hilbert schemes of points

The functionality outlined below is implemented in Twisted Hodge numbers for Hilbert schemes of points, which can be used in Sage. There is documentation.
If you use it for your research, please cite it using the DOI , as explained in the README.

Last year I wrote about a paper Lie Fu, Andreas Krug, and I wrote, about computing the Hochschild cohomology of Hilbert schemes of points. In this paper we conjectured that the identity \[ \sum_{n\geq 0}\sum_{p=0}^{2n}\sum_{q=0}^{2n}\mathrm{h}^{p,q}(\mathop{\rm Hilb}\nolimits^nS,{L}_n)x^py^qt^n = \prod_{k\ge 1}\prod_{p=0}^2\prod_{q=0}^2\left( 1-(-1)^{p+q}x^{p+k-1}y^{q+k-1}t^k\right)^{-(-1)^{p+q}\mathrm{h}^{p,q}(S,{L}^{\otimes k})}. \] holds. It gives an expression of the twisted Hodge numbers of the line bundle $L_n$ on $\mathop{\rm Hilb}^nS$ induced from a line bundle $L$ on $S$, in terms of the twisted Hodge numbers of the line bundles (plural!) $L^{\otimes k}$ on $S$. The version without $\otimes k$ was a conjecture of Boissière, for which counterexamples had already been found.

I was really happy to see that Lie Fu has found a proof of the conjecture!

It is no secret that I like explicit implementations of results. It's not too hard to implement this formula, except that it is a little cumbersome to deal with all the data required for the formula, because you might need many twisted Hodge diamonds of $S$.

That's why I decided to take my implementation of the (at the time conjectural) formula from a few years back, clean it up, and package it together with various interesting surfaces and their twisted Hodge diamonds. My preferred choice of $L$ is the anticanonical line bundle, because then it computes the Hochschild–Kostant–Rosenberg decomposition of the Hochschild cohomology of Hilbert schemes of points.

Some examples

One of the explicit examples we have computed in our joint paper is $\mathop{\rm Hilb}^2\mathbb{P}^2$. This can now be easily recomputed as

sage: from twisted_hilbert import *
sage: S = CompleteIntersectionSurface([], 3) # anticanonical twist
sage: TwistedHilbertSchemeDiamond(S, 2).as_parallelogram()
  1
  0   8
  0   10   38
  0   0    35   80
  0   0    0    28   55
      0    0    0    0
           0    0    0
                0    0
                     0

Here we use the twisted Hodge diamonds of complete intersection surfaces, as implemented by twisted-hodge-ci, something implemented by Piet Glas and myself a few years back.

Alternative examples are given by bielliptic surfaces, which are discussed at length in our joint paper.

Instructions etc.