Meromorphic limits of automorphisms

Abstract

Let $X$ be a compact complex manifold in the Fujiki class $\mathcal{C}$. We study the compactification of ${\mathrm{Aut}}^0(X)$ given by its closure in Barlet cycle space. The boundary points give rise to non-dominant meromorphic self-maps of $X$. Moreover convergence in cycle space yields convergence of the corresponding meromorphic maps. There are analogous compactifications for reductive subgroups acting trivially on $\mathrm{Alb}X$. If $X$ is Kähler, these compactifications are projective. Finally we give applications to the action of $\mathrm{Aut}(X)$ on the set of probability measures on $X$. In particular we obtain an extension of Furstenberg lemma to manifolds in the class $\mathcal{C}$.