This talk gives an overview of the problematic related to projectively induced and balanced Kaehler metrics, and to the existence of a Tian-Yau-Zelditch expansion of the epsilon-function. Differences between compact and noncompact case will be highlighted. The talk is based on joint works with Simone Cristofori (Università di Parma), Andrea Loi and Fabio Zuddas (Università di Cagliari).
