This blogpost is about The Albanese morphism for hyperelliptic varieties. If you just want mathematics and not some of the backstory, you can click the link to immediately get to the preprint.

Background

Back in March 2023, Andreas Demleitner and Pedro Núñez gave seminar talks at the University of Luxembourg, with the talk by Andreas being on hyperelliptic varieties (the talk by Pedro, whilst interesting, has no relevance to this story). These hyperelliptic varieties were introduced by Lange in 2001, as a higher-dimensional version of bielliptic surfaces, one of the four classes of K-trivial surfaces.

They can be defined as the quotient of an abelian variety $A$ by a finite group $G$ acting freely and without translations, and they give interesting higher-dimensional varieties with torsion canonical bundles, for which a classification is somewhat feasible. Whilst we still don't know whether there are finitely many or infinitely many deformation classes of Calabi–Yau 3-folds, we have a good understanding of which groups can occur in dimensions 3 and 4, with many interesting examples known, or possibly to construct, given their explicit description. I'd be interested in knowing a full classification of all deformation families in dimension 3 (and being bold: dimension 4), akin to the Mori–Mukai classification of Fano 3-folds, but that is not the topic of this paper.

Rather, the motivation comes from the fact that the derived categories of bielliptic surfaces are interesting: they are indecomposable by Kawatani–Okawa, making them some kind of atomic objects in the study of derived categories. This brings us to the question that motivated the start of our conversations: is the same true for (higher-dimensional) hyperelliptic varieties? The indecomposability of derived categories of higher-dimensional varieties is a significant open problem, with little known beyond the easy cases where the conclusion comes from having trivial canonical bundle, whereas for hyperelliptic varieties it is torsion and potentially not algebraically trivial.

Overview of the results

Our approach to this problem is, as the title suggests, given by studying the Albanese morphism. From general results in birational geometry, one already knows that $X=A/G$ has an isotrivial Albanese morphism which is étale-locally just the product with a fiber. But the explicit geometry of $X$ in terms of $A$ and $G$ allows us to give an effective description of the Albanese variety and the Albanese fiber.

That this should be possible is not necessarily very surprising, and when the preprint was essentially finished it turned out that a similar but less explicit description was already known to Ueno in the 1970s, although it only appears as an ingredient of a proof.

As it turns out, understanding the Albanese morphism and its role in the geography of hyperelliptic varieties gives rise to some cool phenomena taking place already in low dimensions.

And our explicit approach allows us to often give a positive answer to the question whether the derived category of a hyperelliptic variety is indecomposable! For this we use two methods:

Two highlights are:

  • whenever $G$ is cyclic, $\mathbf{D}^{\mathrm{b}}(X)$ is (stably) indecomposable
  • if $\dim X=3$, then $\mathbf{D}^{\mathrm{b}}(X)$ is indecomposable

We conjecture that it is always indecomposable, and highlight some of the first open cases where none of the existing tools seem to work.

Let me know if you want to chat about this, it has been a fun project, and I'm always looking forward to share the fun!