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In this paper, we prove index theorems for integrable metrized line bundles on projective varieties over complete fields and number fields respectively. As applications, we prove a non-archimedean analogue of the Calabi theorem and a rigidity theorem about the preperiodic points of algebraic dynamical systems.

In this article, we classify all standard invariants that can arise from a composed inclusion of an A 3 with an A 4 subfactor. More precisely, if N P is an A 3 subfactor and P M is an A 4 subfactor, then only four standard invariants can arise from the composed inclusion N M. We answer a question posed by Bisch and Haagerup in 1994. The techniques of this paper also show that there are exactly four standard invariants for the composed inclusion of two A 4 subfactors.