I have the following question, which I'm sure must be explored somewhere.

Consider a group of polynomial automorphisms of $\mathbb{A}^2_\mathbb{C}$ preserving a standard hermitian metric. Is there any description of this group?

I know that analogous question for symplectomorphisms has been studied, for example, here the authors prove infinite transitivity of the group of polynomial symplectomorphisms.

Still, I was unable to find anything similar for the group of polynomial isometries.


Yes, this is just $\mathrm{U}(2)\ltimes \mathbf{C}^2$, that is, all such automorphisms are affine.

Indeed, let $f$ belong to your group. After composing by a translation, we can suppose that $f$ fixes zero. The tangent map of $f$ at zero preserves the Hermitian scalar product, and hence, after composing by some element of $\mathrm{U}(2)$, we can suppose that the differential of $f$ at zero is identity. Now we use that for an arbitrary connected Riemannian manifold, if the differential of some isometry at some fixed point is identity, then the isometry is the identity.

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  • 2
    $\begingroup$ (This is not surprising: for an arbitrary connected Riemannian manifold, the isometry group is a [finite-dimensional] Lie group.) $\endgroup$ – YCor Jun 23 at 12:52
  • $\begingroup$ Two good references for finite dimensionality results of symmetry groups of various structures on manifolds: Kobayshi, Transformation Groups, and d'Ambra, Gromov, Lectures on transformation groups: geometry and dynamics. $\endgroup$ – Ben McKay Jun 23 at 19:23

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