# Meromorphic limits of automorphisms

## TL;DR

This paper investigates the compactification of automorphism groups of compact complex manifolds in the Fujiki class, revealing how limits relate to meromorphic maps and applying these results to dynamics on probability measures.

## Contribution

It introduces a new compactification of automorphism groups via cycle space, connecting limits to meromorphic maps and extending dynamical results to Fujiki class manifolds.

## Key findings

- Boundary points correspond to non-dominant meromorphic self-maps.
- Cycle space convergence implies meromorphic map convergence.
- Extension of Furstenberg lemma to Fujiki class manifolds.

## Abstract

Let $X$ be a compact complex manifold in the Fujiki class $\mathscr{C}$. We study the compactification of $\operatorname{Aut}^0(X)$ 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 $\operatorname{Alb} X$. If $X$ is K\"ahler, these compactifications are projective. Finally we give applications to the action of $\operatorname{Aut}(X)$ on the set of probability measures on $X$. In particular we obtain an extension of Furstenberg lemma to manifolds in the class $\mathscr{C}$.