Topologies and structures of the Cremona groups. (English) Zbl 1298.14020

The aim of the present paper is discuss the existence of topologies with special geometric properties on the Cremona group of two or more variables. Recall that for any base field \(\mathbf{k}\), the Cremona group \(\mathrm{Cr}_n(\mathbf{k})\) is the group of field automorphisms of \(\mathbf{k}(x_1, \ldots, x_n)\), or more geometrically the group of birational transformations of the projective space \(\mathbb{P}^n(\mathbf{k})\).
First the authors show that it is impossible to provide the group \(\mathrm{Cr}_n(\mathbf{k})\) with the structure of an ind-algebraic group or even an ind-stack, such that it becomes with this structure the universal object for families of birational transformations.
Next, for any locally compact field, the authors provide a natural topology called the Euclidean topology on \(\mathrm{Cr}_n(\mathbf{k})\), which is compatible this the group structure and with the usual topology of closed algebraic subgroups (of finite dimension), such as \(\mathrm{PGL}(n+1, \mathbf{k})\). This topology is not locally compact.


14E07 Birational automorphisms, Cremona group and generalizations
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