On the geometry of spaces of filtrations on local rings

Abstract We study the geometry of spaces of filtrations on a Noetherian local domain. We introduce a metric <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>d</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>L</m:mi> <m:mi>loc</m:mi> <m:mn>1</m:mn> </m:msubsup> </m:math> 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 Demailly–Kollár. 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.