Closed groups of automorphisms of products of hyperbolic Riemann surfaces (Q1711069)

Let $M$ be a complex manifold and let $X$ be an analytic subvariety of $M$. Let $G(X)$ denote the stabiliser of $X$ in $\text{Aut}(M)$. Given any subgroup $G$ of $\text{Aut}(M)$, let $G_e$ denote the connected component of $G$ containing the identity element $e$. If the natural homomorphism from $G_e(X)$ to $\text{Aut}_e(X)$ is injective, then $\text{Aut}(X)$ must contain $G(X)$. \par Given that $G(X)$ is a closed subgroup of $\text{Aut}(M)$, the latter observation suggests a way to obtain information about automorphism groups and stabilisers of analytic subvarieties if one has knowledge of the closed subgroups of $\text{Aut}(M)$. With this programme in mind, the authors provide the complete list of all essentially non-discrete closed subgroups of $\text{Aut}(M)$ when $M$ is the Cartesian product of hyperbolic Riemann surfaces. (A subgroup $G$ of $\text{Aut}(M)$ is said to be essentially non-discrete if every element of $G/G_e$ is of finite order.) \par Thereafter, the authors describe a rather natural condition on the analytic subvarieties of these product manifolds under which they can give applications that follow from knowing the above-mentioned list. These applications rely on the isomorphism between $\text{Aut}_e(R_1)\times\dots\times\text{Aut}_e(R_n)$ and $\text{Aut}_e(R_1\times\dots\times R_n)$ [\textit{K. Peters}, Math. Ann. 208, 343--354 (1974; Zbl 0281.32018)], where $R_1,\dots, R_n$ are compact hyperbolic Riemann surfaces.