On the geometry of spaces of filtrations on local rings

TL;DR

This paper explores the geometric structure of spaces of filtrations on Noetherian local rings, introducing a metric and analyzing properties like geodesics, topology, and lattice structure.

Contribution

It introduces a new metric on saturated filtrations, identifies the space with L^1_loc in the toric case, and studies semi-continuity and lattice properties.

Findings

The metric d_1 makes the space of saturated filtrations a geodesic metric space.

In the toric case, the space corresponds to a subspace of L^1_loc.

The space of saturated filtrations has a natural lattice structure.

Abstract

We study the geometry of spaces of fitrations on a Noetherian local domain. We introduce a metric $d_1$ on the space of saturated filtrations, inspired by the Darvas metric in complex geometry, such that it is a geodesic metric space. In the toric case, using Newton-Okounkov bodies, we identify the space of saturated monomial filtrations with a subspace of $L^1_\mathrm{loc}$. We also consider several other topologies on such spaces and study the semi-continuity of the log canonical threshold function in the spirit of Koll\'ar-Demailly. Moreover, there is a natural lattice structure on the space of saturated filtrations, which is a generalization of the classical result that the ideals of a ring form a lattice.