On the geometry of the symmetrized bidisc

Abstract: We study the action of the automorphism group of the $2$ complex dimensional manifold symmetrized bidisc $\mathbb G$ on itself. The automorphism group is $3$ real dimensional. It foliates $\mathbb G$ into leaves all of which are $3$ real dimensional hypersurfaces except one, viz., the royal variety. This leads us to investigate Isaev's classification of all Kobayashi-hyperbolic $2$ complex dimensional manifolds for which the group of holomorphic automorphisms has real dimension $3$ studied by Isaev. Indeed, we produce a biholomorphism between the symmetrized bidisc and the domain

\[\{(z_1,z_2)\in \mathbb{C} ^2 : 1+|z_1|^2-|z_2|^2>|1+ z_1 ^2 -z_2 ^2|, Im(z_1 (1+\overline{z_2}))>0\}\]

in Isaev's list. Isaev calls it $\mathcal D_1$. The road to the biholomorphism is paved with various geometric insights about $\mathbb G$.

Several consequences of the biholomorphism follow including two new characterizations of the symmetrized bidisc and several new characterizations of $\mathcal D_1$. Among the results on $\mathcal D_1$, of particular interest is the fact that $\mathcal D_1$ is a ``symmetrization''. When we symmetrize (appropriately defined in the context) either $\Omega_1$ or $\mathcal{D}^{(2)} _1$ (Isaev's notation), we get $\mathcal D_1$. These two domains $\Omega_1$ and $\mathcal{D}^{(2)} _1$ are in Isaev's list and he mentioned that these are biholomorphic to $\mathbb D \times \mathbb D$. We produce explicit biholomorphisms between these domains and $\D \times \D$.

functional analysisK-theory and homologyoperator algebras

Audience: researchers in the topic

Webinars on Operator Theory and Operator Algebras