Algebraic bounds on analytic multiplier ideals

Abstract

Given a pseudo-effective divisor L we construct the diminished ideal Jσ(L), a "continuous" extension of the asymptotic multiplier ideal for big divisors to the pseudo-effective boundary. Our main theorem shows that for most pseudo-effective divisors L the multiplier ideal J (hmin) of the metric of minimal singularities on OX (L) is contained in Jσ(L). We also characterize abundant divisors using the diminished ideal, indicating that the geometric and analytic information should coincide.

Effectiveness of Demailly’s strong openness conjecture and related problems

“…In [19,20], the authors solved the strong openness conjecture posed by Demailly in [6] and [7] (see also [9], [11], [3], [29], [24], [22], [4], [25], [44], [30], etc. ):…”

Valuative multiplier ideals

“…Under the above setting, we show that there exists the maximal L-psh function ϕ which can be written explicitly as ϕ(v) = −v( L ), and there exists the maximal pseudo L-psh function φ which can be written explicitly as φ(v) = −σ v ( L ) (see Proposition 6.10 and 6.11). As an immediate corollary we generalize [27,Theorem 6.14] as follows. See [27] for the definition of the perturbed ideal and the diminished ideal.…”

“…As an immediate corollary we generalize [27,Theorem 6.14] as follows. See [27] for the definition of the perturbed ideal and the diminished ideal. Theorem 1.3 (=Theorem 6.16).…”

“…However in general, this correspondence become quite mysterious since many analogue notions cannot be constructed. This has been studied in many relevant references such as [3], [16], [17], [18], [27], [28]. We will discuss the qpsh function associated to a line bundle in detail within this section.…”

“…Given a line bundle L on a smooth projective complex variety, a classical theorem of Kodaira asserts that if L carries on a smooth metric with positive curvature, then L is ample, or equivalently the global sections of a multiple of L give an embedding to a projective space and hence induce such a metric on L. More generally, global sections of a multiple of L induce a semi-positive singular metric. Conversely, given a semi-positive singular metric h, the local weight function ϕ, which is plurisubharmonic (psh for short), should be related to sections of multiples of L, or perhaps of a small perturbation of L. See [27] for more details.…”