# New tool: twisted Hodge diamonds for complete intersections

Recently Piet Glas, one of my master students, finished his thesis, and part of it was an implementation of the computation of **twisted Hodge numbers**, i.e. the numbers
\begin{equation}
\mathrm{h}_j^{p,q}
=
\dim\mathrm{H}^q(X,\Omega_X^p(j))
\end{equation}
where $X\subseteq\mathbb{P}^n$ is a complete intersection $H_{d_1}\cap\ldots\cap H_{d_c}$, and $\mathcal{O}_X(j)$ is the restriction of $\mathcal{O}_{\mathbb{P}^n}(j)$.

For $j=0$ there is a beautiful formula due to Hirzebruch, which is implemented in Hodge diamond cutter, but for $j\neq 0$ no generating series is known and rather one needs to carefully induct on the Euler sequence and its exterior powers. Luckily, in a little known series of papers by Brückmann ([1], [2], [3]) this analysis is performed, and these are formulae one can implement.

There is extensive documentation in the README and the docstrings, let me just show you an example. After loading, the code `print TwistedHodgeDiamond((5, [3]), 3)`

computes the twisted Hodge diamond of a cubic fourfold where the twist by 3 (which is the *index* of $X$) actually ensures that we are looking at the Hochschild cohomology of $X$:

0 0 0 0 0 0 0 0 0 0 1 20 1 0 0 0 0 0 0 20 0 0 70 0 55

As explained in this blogpost from almost a year ago on polyvector parallelograms I like to visualise this rather as the following parallelogram (via `print PolyvectorParallelogram((5, [3]))`

), giving

1 0 0 0 20 20 0 0 0 70 0 0 1 0 55 0 0 0 0 0 0 0 0 0 0

but maybe you are interested in other twists too. Enjoy!