Multiplier Ideal Sheaves Research Articles

Nadel-type multiplier ideal sheaves on complex spaces with singularities

Nadel-type multiplier ideal sheaves on complex spaces with singularities

On an injectivity theorem for log-canonical pairs with analytic adjoint ideal sheaves

As an application of the residue functions corresponding to the lc-measures developed by the authors, the proof of the injectivity theorem on compact Kähler manifolds for plt pairs by Matsumura is improved in this article to allow multiplier ideal sheaves of plurisubharmonic functions with neat analytic singularities in the coefficients of the relevant cohomology groups. With the use of a refined version of analytic adjoint ideal sheaves, a plan towards a solution to the generalised version of Fujino’s conjecture (i.e. an injectivity theorem on compact Kähler manifolds for lc pairs with multiplier ideal sheaves) is laid down and, in addition to the result for plt pairs, a proof for lc pairs in dimension 2 2 , which is also an improvement of Matsumura’s result, is given.

Geometric Flow, Multiplier Ideal Sheaves and Optimal Destabilizer for a Fano Manifold

In (Donaldson in J Differ Geom 70(3):453–472, 2005), it was asked whether the lower bound of the Calabi functional is achieved by a sequence of the normalized Donaldson–Futaki invariants. We answer to the question for the Ricci curvature formalism, in place of the scalar curvature. Our principle is that the stability indicator is optimized by the multiplier ideal sheaves of certain weak geodesic ray asymptotic to the geometric flow. We actually obtain the results for the two cases: the inverse Monge–Ampère flow and the Kähler–Ricci flow.

ON A BERNSTEIN–SATO POLYNOMIAL OF A MEROMORPHIC FUNCTION

AbstractWe define Bernstein–Sato polynomials for meromorphic functions and study their basic properties. In particular, we prove a Kashiwara–Malgrange-type theorem on their geometric monodromies, which would also be useful in relation with the monodromy conjecture. A new feature in the meromorphic setting is that we have several b-functions whose roots yield the same set of the eigenvalues of the Milnor monodromies. We also introduce multiplier ideal sheaves for meromorphic functions and show that their jumping numbers are related to our b-functions.

Bogomolov–Sommese type vanishing theorem for holomorphic vector bundles equipped with positive singular Hermitian metrics

In this article, we obtain the Bogomolov–Sommese type vanishing theorem involving multiplier ideal sheaves for big line bundles. We define a dual Nakano semi-positivity of singular Hermitian metrics with L^2-estimates and prove a vanishing theorem which is a generalization of the Bogomolov–Sommese type vanishing theorem to holomorphic vector bundles.

An optimal support function related to the strong openness conjecture

In the present article, we obtain an optimal support function of weighted $L^{2}$ integrations on superlevel sets of psh weights, which implies the strong openness property of multiplier ideal sheaves.

Stability of Multiplier Ideal Sheaves

Stability of Multiplier Ideal Sheaves

Injectivity theorems with multiplier ideal sheaves for higher direct images under Kähler morphisms

The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we obtain a relative vanishing theorem of Kawamata-Viehweg-Nadel type and an extension theorem for holomorphic sections from fibers of morphisms to the ambient space. Our approach is based on transcendental methods and works for Kahler morphisms and singular hermitian metrics with non-algebraic singularities.

On an $$L^2$$ extension theorem from log-canonical centres with log-canonical measures

With a view to prove an Ohsawa-Takegoshi type $L^2$ extension theorem with $L^2$ estimates given with respect to the log-canonical (lc) measures, a sequence of measures each supported on lc centres of specific codimension defined via multiplier ideal sheaves, this article is aiming at providing evidence and possible means to prove the $L^2$ estimates on compact K\"ahler manifolds $X$. A holomorphic family of $L^2$ norms on the ambient space $X$ is introduced which is shown to "deform holomorphically" to an $L^2$ norm with respect to an lc-measure. Moreover, the latter norm is shown to be invariant under a certain normalisation which leads to a "non-universal" $L^2$ estimate on compact $X$. Explicit examples on $\mathbb{P}^3$ with detailed computation are presented to verify the expected $L^2$ estimates for extensions from lc centres of various codimensions and to provide hint for the proof of the estimates in general.

Concavity of Minimal L2 Integrals Related to Multiplier Ideal Sheaves

Concavity of Minimal L2 Integrals Related to Multiplier Ideal Sheaves

Injectivity theorem for pseudo-effective line bundles and its applications

We formulate and establish a generalization of Kollár’s injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár’s torsion-freeness, Kollár’s vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge theoretic but analytic, which enables us to treat singular Hermitian metrics with nonalgebraic singularities. For the proof of the main injectivity theorem, we use L 2 L^{2} -harmonic forms on noncompact Kähler manifolds. For applications, we prove a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems.

Extension of cohomology classes and holomorphic sections defined on subvarieties

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general L 2 L^2 extension theorem obtained by Demailly.

On the hard Lefschetz theorem for pseudoeffective line bundles

In this paper, we obtain a number of results related to the hard Lefschetz theorem for pseudoeffective line bundles, due to Demailly, Peternell and Schneider. Our first result states that the holomorphic sections produced by the theorem are in fact parallel, when the Chern connection associated with the singular metric is computed in the sense of currents, and the corresponding multiplier ideal sheaves are taken into account. Our proof is based on a control of the covariant derivative in the delicate approximation process used in the construction of these sections. Then we show that there is an isomorphism between the space of such parallel sections and the sheaf cohomology group of appropriate degree. As an application, we show that the closedness property of the sections produces a singular holomorphic foliation on the tangent bundle. Finally, we discuss some questions related to the optimality of the multiplier ideal sheaves involved in the generalized hard Lefschetz theorem.

On the Briançon-Skoda theorem for analytic local rings with singularities

In this note, we study validity of the original Briançon-Skoda theorem for analytic local rings. By introducing a new concept of multiplier ideal sheaves associated to plurisubharmonic functions on any (reduced) complex space of pure dimension, we establish the original Briançon-Skoda theorem for normal weakly rational singularities (not essentially of finite type over C and Cohen-Macaulay). Moreover, we also characterize two dimensional rational singularities by the original Briançon-Skoda theorem.

Some recent results on multiplier ideal sheaves

Some recent results on multiplier ideal sheaves

Recent Results in Several Complex Variables and Complex Geometry

We first recall the background and contents of our recent solutions of the optimal $$L^2$$ extension problem and Demailly’s strong openness conjecture on multiplier ideal sheaves and related results, and then present some new related results in several complex variables and complex geometry.

Subadditivity of generalized Kodaira dimensions and extension theorems

In this paper, we introduce the notion of generalized Kodaira dimension with multiplier ideal sheaves, and prove the subadditivity of these generalized Kodaira dimensions for certain Kähler fibrations, which was originally obtained for Kodaira dimensions of algebraic fiber spaces by Kawamata and Viehweg. Our method is analytic and based on some new results in recent years. The crucial step in our proof is to prove an [Formula: see text] extension theorem for twisted pluricanonical sections on compact Kähler manifolds. Moreover, we also discuss the relation between two previous optimal [Formula: see text] extension theorems with singular weights on weakly pseudoconvex Kähler manifolds.

The metric geometry of singularity types

Abstract Let X be a compact Kähler manifold. Given a big cohomology class { θ } {\{\theta\}} , there is a natural equivalence relation on the space of θ-psh functions giving rise to 𝒮 ⁢ ( X , θ ) {\mathcal{S}(X,\theta)} , the space of singularity types of potentials. We introduce a natural pseudo-metric d 𝒮 {d_{\mathcal{S}}} on 𝒮 ⁢ ( X , θ ) {\mathcal{S}(X,\theta)} that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence of positive mass we show that this metric space is complete. As applications, we show that solutions to a family of complex Monge–Ampère equations with varying singularity type converge as governed by the d 𝒮 {d_{\mathcal{S}}} -topology, and we obtain a semicontinuity result for multiplier ideal sheaves associated to singularity types, extending the scope of previous results from the local context.

Roles of Plurisubharmonic Functions

We recall the basics of multiplier ideal sheaves and formulate our recent solution of Demailly’s strong openness conjecture on multiplier ideal sheaves and related results. Then we present some applications in complex geometry, including some new results related to the vanishing and finiteness theorems for analytic cohomology groups with multiplier ideal sheaves in the case of pseudo-effective line bundles over holomorphically convex manifolds, and to the generalized Siu lemma and pseudo-effectiveness of the twisted relative pluricanonical bundles and their direct images.

Decreasing Equisingular Approximations with Analytic Singularities

In this note, for the multiplier ideal sheaves with weights $$\log \sum _{i}|z_{i}|^{a_{i}}$$, we present the sufficient and necessary condition of the existence of decreasing equisingular approximations with analytic singularities.