For the regular reader of my blog it's no secret I like numerical invariants of smooth projective varieties. Today I have a new treat for you. At least, if you like Hodge numbers that is.

Varieties with trivial canonical bundle

In the classification of varieties with trivial canonical bundle the Beauville–Bogomolov decomposition theorem transforms such a variety after a finite étale cover into a product of

  • simple abelian varieties
  • (strict) Calabi–Yau varieties of dimension at least 3
  • irreducible holomorphic symplectic (or hyperkähler) varieties

Let's pretend we understand abelian varieties. On the other hand, already a classification of Calabi–Yau 3-folds is completely out of reach, and as far as I know even the finiteness of number of deformation families is not known. So let's ignore those too.

Irreducible holomorphic symplectic varieties

What remains are the irreducible holomorphic symplectic varieties, i.e. varieties which are simply connected and for which there exists a symplectic form spanning $\mathrm{H}^0(X,\Omega_X^2)$. Here the total opposite is the case: although we have examples in every (necessarily) even dimension, the complete list of known irreducible holomorphic symplectic varieties is rather short:

  • Hilbert schemes of $n$ points on K3 surfaces, and things deformation equivalent to it, for all dimensions;
  • generalised Kummer varieties, and things deformation equivalent to it, for all even dimensions;
  • two sporadic examples due to O'Grady: one of dimension 6, another of dimension 10.

As generalised Kummer varieties form the main topic, let's recall their definition. Let $A$ be an abelian surface. Then the $n$th generalised Kummer variety $\mathrm{K}^n(A)$ of $A$ is the fibre at 0 of the Albanese map \begin{equation} \operatorname{Hilb}^nA\to\operatorname{Sym}^nA\overset{\scriptstyle\sum}{\to} A \end{equation}

In some sources this would be the $(n-1)$th generalised Kummer variety. So in my setup, the first generalised Kummer is a point, and the second generalised Kummer variety is the Kummer K3 surface.

Hodge numbers of irreducible holomorphic symplectic varieties

To get a feel for these varieties one can try to compute some of its numerical invariants, and the computation of the second Betti number was what originally distinguished the Hilbert schemes of points on K3 surfaces from the generalised Kummer varieties: the former has $\mathrm{b}_2=23$, whilst the latter has $\mathrm{b}_2=7$ (when $n\geq 3$).

For the Hodge numbers of Hilbert schemes of points on K3 surfaces, there is my blog post on Hodge numbers for $\operatorname{Hilb}^nS$ containing the code to compute it for arbitrary surfaces.

For the sporadic cases the situation is as follows:

For the record, here is the Hodge diamond for exceptional 6-dimensional case.

1
0 0
1 6 1
0 0 0 0
1 12 173 12 1
0 0 0 0 0 0
1 6 173 1144 173 6 1
0 0 0 0 0 0
1 12 173 12 1
0 0 0 0
1 6 1
0 0
1

Hodge numbers of generalised Kummer varieties

We finally get to the main point now. In the paper Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces of Göttsche--Soergel the (signed) Hodge polynomial for $A\times\mathrm{K}^n(A)$ is given by the intimidating-looking formula \begin{equation} \mathrm{h}(A\times\mathrm{K}^n(A),-x,-y) = \sum_{\alpha\in\mathrm{P}(n)} \left( g(\alpha)^4 (xy)^{n-|\alpha|} \prod_{i,\alpha_i\neq 0} \left( \sum_{\beta\in\mathrm{P}(\alpha_i)} \prod_{j} \frac{1}{j^{\beta_j}\beta_j!} \left( (1-x^j)(1-y^j) \right)^{2\beta_j} \right) \right) \end{equation} in corollary 1, where we write the partition $\alpha$ in the exponential notation $n=\alpha_1\cdot 1+\ldots+\alpha_r\cdot r$, and

  • $g(\alpha):=\gcd\{i\mid\alpha_i\neq 0\}$
  • $|\alpha|:=\sum_{i=1}^r\alpha_i$
The (not very pretty) Sage code, which also deals with the signs and gets rid of the extra factor of $A$ is given by

You can also compute the Betti numbers of a Hodge polynomial, and check that they agree with the table in Göttsche's book Hilbert schemes of zero-dimensional subschemes of smooth varieties, on page 50 (with the caveat that the indexing of the generalised Kummers differs by 1).

So the generalised Kummer variety $\mathrm{K}^3(A)$ has Hodge diamond

1
0 0
1 5 1
0 4 4 0
1 5 96 5 1
0 4 4 0
1 5 1
0 0
1

And the generalised Kummer variety $\mathrm{K}^4(A)$ has Hodge diamond

1
0 0
1 5 1
0 4 4 0
1 6 37 6 1
0 4 24 24 4 0
1 5 37 372 37 5 1
0 4 24 24 4 0
1 6 37 6 1
0 4 4 0
1 5 1
0 0
1

Etc.

I think I should collect all my Hodge diamond related code, as I by now have quite a bit of interesting examples, not all of them on my blog, and some helper code for printing and other manipulations. That will happen at some point in the nearish future.