Valuations and asymptotic invariants for sequences of ideals

Abstract

We study asymptotic jumping numbers for graded sequences of ideals, and show that every such invariant is computed by a suitable real valuation of the function field. We conjecture that every valuation that computes an asymptotic jumping number is necessarily quasi-monomial. This conjecture holds in dimension two. In general, we reduce it to the case of affine space and to graded sequences of valuation ideals. Along the way, we study the structure of a suitable valuation space.

Strong openness of multiplier ideal sheaves and optimal L 2 extension

“…Dimension two case of the above theorem was proved by Jonsson and Mustata [28] (I is trivial see [18]). The proof of the general case is based on our solution of Demailly's strong openness conjecture [22] and our solution of the L 2 extension problem with optimal estimates [20,21].…”

“…In [23,25], we prove two conjectures posed by Demailly-Kollár (see [15]) and Jonsson-Mustatȃ (see [28]) respectively by using the following result. have positive lower bounds independent of r ∈ (0, 1).…”

Algebraic bounds on analytic multiplier ideals

“…We may now apply the results of [18] to R-divisors. The following is an immediate consequence of [18,Corollary 6.4].…”

“…We study this behavior in Section 4 using ideas of [17] and the relationship between valuations and asymptotic multiplier ideals described in [18].…”

“…We next give an informal introduction to the log discrepancy function A : Val(X) → R 0 ∪ {∞}, referring to [18] for details. For a valuation v measuring the order of vanishing along a divisor E over X, the log discrepancy is…”