Geometry of higher rank valuations and applications

Tropical geometry studies degenerations of algebraic varieties by enriching the theory of semistable models and their dual complexes by polyhedral geometry. This enrichment motivates the development of algebraic geometry for combinatorial and polyhedral spaces.

While the theory has been largely developed over the past two decades and has found diverse applications, the framework has been mostly restricted so far to the case of valuations of rank one. From the geometric point of view, this means considering families of complex manifolds which depend only on one parameter. There are several reasons for a desire to extend the scope of tropical geometry beyond the rank one case. For example, one of the leitmotifs in the development of tropical geometry is to explain large scale limits of complex geometry. From the point of view of moduli spaces and their compactifications, the problem should be understood using higher rank valuation theory since large scale limits can depend on several parameters.

In this talk, we present some interesting features of valuation theory in the higher rank setting. In particular, we introduce an appropriate higher rank notion of dual complexes, and discuss some applications to geometry, in the study of Newton-Okounkov bodies and in asymptotic complex geometry, making a connection to the work of the speaker with Nicolussi.

Based on joint work with Hernan Iriarte.