Topologies and structures of the Cremona groups

We study the algebraic structure of the $n$-dimensional Cremona group and show that it is not an algebraic group of infinite dimension (ind-group) if $n\ge 2$. We describe the obstruction to this, which is of a topological nature. By contrast, we show the existence of a Euclidean topology on the Cremona group which extends that of its classical subgroups and makes it a topological group.

💡 Research Summary

The paper investigates the algebraic and topological nature of the Cremona group $\operatorname{Cr}_n(k)$, the group of birational self‑maps of the projective space $\mathbb{P}^n$ over a field $k$. After recalling that the one‑dimensional case $\operatorname{Cr}_1(k)$ coincides with $\operatorname{PGL}_2(k)$ and is therefore a finite‑dimensional algebraic group, the authors turn to the higher‑dimensional situation $n\ge2$, where the structure becomes dramatically more intricate.

The first major part of the work attempts to endow $\operatorname{Cr}_n(k)$ with the structure of an ind‑group, i.e. a direct limit of finite‑dimensional algebraic varieties. For each integer $d\ge1$ they consider the space $\operatorname{Rat}_d(\mathbb{P}^n,\mathbb{P}^n)$ of rational maps of degree at most $d$, which is a quasi‑projective variety. The natural inclusion $\operatorname{Rat}d\hookrightarrow\operatorname{Rat}{d+1}$ is examined in detail. The authors prove that this inclusion is not a closed immersion: as the degree grows, new base‑points can appear, and the image fails to be Zariski‑closed. Consequently the direct limit $\varinjlim_d\operatorname{Rat}_d$ does not inherit a well‑defined ind‑variety structure, and $\operatorname{Cr}_n(k)$ cannot be realized as an infinite‑dimensional algebraic group. This failure is identified as a “topological obstruction”: the filtration by degree does not produce a compatible chain of algebraic varieties, and the resulting topology on the limit is too coarse to support an algebraic group law. The obstruction is essentially geometric, reflecting the fact that birational maps of higher degree can contract or blow up subvarieties in ways that cannot be captured by a simple inclusion of parameter spaces.

In the second part the authors abandon the ind‑group approach and construct a genuine topological group structure on $\operatorname{Cr}_n(\mathbb{C})$ by using the Euclidean topology on the space of coefficients of the defining rational functions. Each birational map can be written as $