Valuations and log canonical thresholds

Abstract

The goal of this paper is to continue the investigation of valuative quasi-plurisubharmonic functions (qpsh for short) on certain valuation spaces of a regular scheme, in line with the works [4], [5], [6] of Boucksom, Favre, Jonsson, and the works [31], [32] of Jonsson, Mustaţȃ. We divide this paper into two parts. In the first part we mainly discuss those valuations which compute the log canonical thresholds of qpsh functions. We expect them to be useful for the conjecture [[31], Conjecture B] raised by Jonsson and Mustaţȃ. In the second part we define the restriction of a valuative qpsh function to a regular subscheme and prove a number of expected results including the restriction theorem and the inversion of adjunction. We also treat some applications in complex algebraic geometry such as extensions of pluri-canonical forms on a dlt pair under an abundance assumption.

Growth of attraction rates for iterates of a superattracting germ in dimension two

“…The most detailed references for this subject are the monograph [FJ04] and the notes [Jon12]. The former stresses the combinatorial aspects of V, while the latter prefers the geometric approach, a perspective that has been useful in applications, see [FJ11], [Fav10] for dynamical applications, [FJ05b], [BFJ08b] for applications to singularities of plurisubharmonic functions, and [FJ05a], [JM11], [JM12], [Hu12b], [Hu12a] for applications to multiplier ideals in analytic and algebraic geometry.…”