Minimal singular metrics of a line bundle admitting no Zariski decomposition

Abstract

We give a concrete expression of a minimal singular metric on a big line bundle on a compact Kähler manifold which is the total space of a toric bundle over a complex torus. In this class of manifolds, Nakayama constructed examples which have line bundles admitting no Zariski decomposition even after modifications. As an application, we discuss the Zariski closedness of non-nef loci.This expression enables us to compute the multiplier ideal sheaf J (h t min ) for each positive number t, whose stalk at x 0 ∈ X is defined bywhere ϕ min is the local weight function of h min around x 0 .Corollary 1.2. J (h t min ) is trivial at any point in X \ P(L 0 ). For a point x 0 ∈ P(L 0 ), the stalk J (h min ) x 0 of the multiplier ideal sheaf is the ideal of O X,x 0 which is generated by the polynomials {x p 1 x q 2 | (p + 1, q + 1) ∈ Int(S t ) ∩ Z 2 }, where we denote by S t the set {(tα, tβ) ∈ R 2 | α, β ≥ 0, a 2 (α + β) 2 ≥ (1 − α) 2 + (1 − β) 2 } ( For the shape of S t in this case, see Figure 1).

On metrics with minimal singularities of line bundles whose stable base loci admit holomorphic tubular neighborhoods

“…Note that the description of the singularity of ϕ min,L as in Theorem 1.3 does not depend on the choice of the coordinates (up to O( 1)). This theorem is a generalization of the main result of [10]. Moreover it is also a generalization of [11] in higher codimensional cases.…”

“…Therefore we can apply Theorem 3.1 to this example. In particular, by using the metrics as in the proof of Theorem 1.4, we can reprove the main result in [10].…”

“…one-codimensional) part of the singularities of a metric with minimal singularities on L. From this point of view, it is important for a study of the Zariski decomposition to investigate the higher-codimensional part of the singularities of metrics with minimal singularities in detail. In [10], the second author explicitly described the metrics with minimal singularities for Nakayama's example we mentioned above.…”

“…In this paper, we investigate the metrics of L with minimal singularities for more general cases than both [10] and [11]. Our main result has the following application: Theorem 1.1.…”

“…where |α| := r λ=1 α λ . In [10], a metric with minimal singularities on Nakayama's example was described explicitly by using L on a neighborhood of Y . It is easily observed that a metric with minimal singularities is locally bounded on the complement X \ Y of Y (see Example 2.4).…”