b-divisors, Monge- Ampère measures and tropical geometry

b-divisors encode divisor data across all birational models of a variety and provide a natural framework for studying positivity and intersection theory. Monge–Ampère measures arise from (non-Archimedean) pluripotential theory and capture curvature and volume information associated to metrics on line bundles. Tropical geometry offers a combinatorial language for degenerations via polyhedral complexes and fans.
In this talk, we highlight connections between these notions and apply them to the study of degenerations of natural Hermitian metrics in families of Calabi–Yau and abelian varieties.