We would like to thank Jean-Benoît Bost, Antoine Chambert-Loir, Antoine Ducros, Christophe Soulé, and Amaury Thuillier for interesting discussions related to the contents of the paper. We are particularly grateful to János Kollár, Osamu Fujino and Mircea Mustaţǎ for their help regarding Appendix B.

Our work was carried out at several institutions including the IHES, the École Polytechnique, and the University of Michigan. We gratefully acknowledge their support. The second author was partially supported by the ANR-grant BERKO. The third author was partially supported by the CNRS and the NSF.