Geometry of higher rank valuations

Abstract

The aim of this paper is to introduce a certain number of tools and results suitable for the study of valuations of higher rank on function fields of algebraic varieties. This will be based on a study of higher rank quasi-monomial valuations taking values in the lexicographically ordered group R^k. We prove a duality theorem that gives a geometric realization of higher rank quasi-monomial valuations as tangent cones of dual cone complexes. Using this duality, we provide an analytic description of quasi-monomial valuations as multi-directional derivative operators on tropical functions. We consider moreover a refined notion of tropicalization in which we remember the initial terms of power series on each cone of a dual complex, and prove a tropical analogue of the weak approximation theorem in number theory by showing that any compatible collection of initial terms on cones of a dual cone complex is the refined tropicalization of a rational function in the function field of the variety. Endowing the value group R^k with its Euclidean topology, we study then a natural topology on spaces of higher rank valuations that we call the tropical topology. By using the approximation theorem we provide an explicit description of the tropical topology on tangent cones of dual cone complexes. Finally, we show that tangent cones of dual complexes provide a notion of skeleton in higher rank non-archimedean geometry. That is, generalizing the picture in rank one to higher rank, we construct retraction maps to tangent cones of dual cone complexes, and use them to obtain limit formulae in which we reconstruct higher rank non-archimedian spaces with their tropical topology as the projective limit of their higher rank skeleta. We conjecture that these higher rank skeleta provide appropriate bases for the study of variations of Newton-Okounkov bodies.