We present a new approach to the study of multiplier ideals in a local, two-dimensional setting. Our method allows us to deal with ideals, graded systems of ideals and plurisubharmonic functions in a unified way. Among the applications are a formula for the complex integrability exponent of a plurisubharmonic function in terms of Kiselman numbers, and a proof of the openness conjecture by Demailly and Kollar. Our technique also yields new proofs of two recent results: one on the structure of the set of complex singularity exponents for holomorphic functions; the other by Lipman and Watanabe on the realization of ideals as multiplier ideals.
Given a pseudo-effective divisor L we construct the diminished ideal of L, a "continuous" extension of the asymptotic multiplier ideal for big divisors to the pseudo-effective boundary. For most pseudo-effective divisors L the multiplier ideal of the metric of minimal singularities of L is contained in the diminished ideal. We also characterize abundant divisors using the diminished ideal, indicating that in this case the geometric and analytic information should coincide.
In this article, we will characterize the multiplier ideal sheaves with weights of log canonical threshold one by restricting the weights to complex regular surface.
Proofs of two results about a monomial ideal -- describing membership in auxiliary ideals associated to the monomial ideal -- are given which do not invoke resolution of singularities. The AM--GM inequality is used as a substitute for taking a log resolution of the monomial ideal.
We show that the restriction to a smooth transversal section commutes to the computation of multiplier ideals and V-filtrations. As an application we prove the constancy of the spectrum along any stratum of a Whitney regular stratification.
In this paper we establish a Nadel-type vanishing theorem and a Kawamata--Viehweg-type vanishing theorem concerning the asymptotic multiplier ideal sheaf on a compact Kähler manifold $X$. After that, we provide a relative variant.
In this article, we characterize plurisubharmonic functions of Lelong number one at the origin, such that the germ of the associated multiplier ideal sheaf is nontrivial: in arbitrary complex dimension, their singularity must be the sum of a germ of smooth divisor and of a plurisubharmonic function with zero Lelong number. We also present a new proof of the related well known integrability criterion due to Skoda.
Given an effective Q-divisor D on a smooth complex variety, one can associate to D its multiplier ideal sheaf J(D), which measures in a somewhat subtle way the singularities of D. Because of their strong vanishing properties, these ideals have come to play an increasingly important role in higher dimensional geometry. We prove that for two effective Q-divisors D and E, one has the "subadditivity" relation: J(D + E) \subseteq J(D) . J(E) . (We also establish several natural variants, including the analogous statement for the analytic multiplier ideals associated to plurisubharmonic functions.) …
These notes are the write-up of my 2008 PCMI lectures on multiplier ideals. They aim to give an introduction to the algebro-geometric side of the theory, with an emphasis on its global aspects. The focus is on concrete examples and applications. The lectures take into account a number of recent perspectives, including adjoint ideals and the resulting simplifications in Siu's theorem on plurigenera in the general type case. While the notes refer to my book [PAG] and other sources for some technical points, the conscientious reader should arrive at a reasonable grasp of the machinery after worki…
In this expository article we first give an overview on multiplier ideal sheaves and geometric problems in Kählerian and Sasakian geometries. Then we review our recent results on the relationship between the support of the subschemes cut out by multiplier ideal sheaves and the invariant whose non-vanishing obstructs the existence of Kähler-Einstein metrics on Fano manifolds.
In this article, we obtain two sharp equality conditions in the restriction formula on complex singularity exponents: an equality between the codimension of the zero variety of related multiplier ideal sheaves and the relative codimension of the restriction of the variety on the submanifold (in the restriction formula); an equivalence between the transversality (between the variety and the submanifold) and the regularity of the restriction of the variety. As applications, we present sharp equality conditions in the fundamental subadditivity property on complex singularity exponents.
We extend Nadel's results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author. Applying our extension we can determine the multiplier ideal sheaves on toric del Pezzo surfaces which do not admit Kähler-Einstein metrics. We also show that one can define multiplier ideal sheaves for Kähler-Ricci solitons and extend the result of Nadel using the holomorphic invariant defined by Tian and Zhu.
In this article, we will characterize regular points respectively by the local vanishing, positivity of the Ricci curvature and $L^2$-solvability of the $\overline\partial$-equation together with Skoda's theorem for Nadel-Lebesgue multiplier ideal sheaves associated to plurisubharmonic (psh) functions on any (reduced) complex space of pure dimension. As a by-product, we show that any weak Kähler-Einstein metric on \emph{singular} $\mathbb{Q}$-Fano/Calabi-Yau/general type varieties cannot be smooth, and that in general there exists no \emph{singular} normal Kähler complex space such that the Kä…
On certain del Pezzo surfaces with large automorphism groups, it is shown that the solution to the Kähler-Ricci flow with a certain initial value converges in $C^\infty$-norm exponentially fast to a Kähler-Einstein metric. The proof is based on the method of multiplier ideal sheaves.
This article discusses the geometric application of the method of multiplier ideal sheaves. It first briefly describes its application to effective problems in algebraic geometry and then presents and explains its application to the deformational invariance of plurigenera for general compact algebraic manifolds. Finally its application to the conjecture of the finite generation of the canonical ring is explored and the use of complex algebraic geometry in complex Neumann estimates is discussed.
This note is written for the Festschrift in honor of Professor Christer Kiselman. Multiplier ideal sheaves identify the location and the extent of the failure of crucial estimates. In this note we will discuss and explain the historic evolution of the notion of multiplier ideal sheaves, especially the interpretation from the viewpoint of destabilizing subsheaves in the context of terminating or bounding an infinite process. We will also discuss the approach of constructing rational curves in Fano manifolds by using dynamic multiplier ideal sheaves and singularity-magnifying complex Monge-Amper…
In this article, we first establish an $L^2$-type Dolbeault isomorphism for the sheaf of logarithmic differential forms twisted by the multiplier ideal sheaf. By using this isomorphism and $L^2$-estimates equipped with a singular Hermitian metric, we obtain logarithmic vanishing theorems involving multiplier ideal sheaves on compact Kähler manifolds with simple normal crossing divisors.
Let $ϕ$ be a psh function on a bounded pseudoconvex open set $Ω\subset \C^n$, and let ${\cal I}(ϕ)$ be the associated multiplier ideal sheaf. Motivated by resolution of singularities issues, we establish an effective version of the coherence property of ${\cal I}(mϕ)$ as $m\to +\infty$. Namely, we estimate the order of growth in $m$ of the number of generators needed to engender ${\cal I}(mϕ)$ on a fixed compact subset, as well as the growth of the coefficients featuring in the decomposition of local sections as linear combinations over ${\cal O}_Ω$ of finitely many generators. The main idea i…
The goal of this note is to present some recent results of our research concerning multiplier ideal sheaves on complex spaces and singularities of plurisubharmonic functions. We firstly introduce multiplier ideal sheaves on complex spaces (\emph{not} necessarily normal) via Ohsawa's extension measure, as a special case of which, it turns out to be the so-called Mather-Jacobian multiplier ideals in the algebro-geometric setting. As applications, we obtain a reasonable generalization of (algebraic) adjoint ideal sheaves to the analytic setting and establish some extension theorems on Kähler mani…
We extend the study of jumping numbers of multiplier ideals due to Ein-Lazarsfeld-Smith-Varolin from the algebraic case to the case of general plurisubharmonic functions. While many properties from Ein-Lazarsfeld-Smith-Varolin are shown to generalize to the plurisubharmonic case, important properties such as periodicity and discreteness do not hold any more. Previously only two particular examples with a cluster point (i.e. failure of discreteness) of jumping numbers were known, due to Guan-Li and to Ein-Lazarsfeld-Smith-Varolin respectively. We generalize them to all toric plurisubharmonic fu…
We apply Nadel's method of multiplier ideal sheaves to show that every complex del Pezzo surface of degree at most six whose automorphism group acts without fixed points has a Kähler-Einstein metric. In particular, all del Pezzo surfaces of degree $4,5$, or $6$ and certain special del Pezzo surfaces of lower degree are shown to have a Kähler-Einstein metric. This result is not new, but the proofs given in the present paper are less involved than earlier ones by Siu, Tian and Tian-Yau.
In the present article, we establish an equality condition in the restriction formula on jumping numbers by giving a sharp lower bound of the dimension of the support of a related coherent sheaf. As applications, we obtain equality conditions in the restriction formula on complex singularity exponents by giving the dimension, the regularity and the transversality of the support, and we also obtain some sharp equality conditions in the fundamental subadditivity property on complex singularity exponents. We also obtain two sharp relations on jumping numbers.
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 K"ahler morphisms and singular hermitian metrics with non-algebraic…
The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the harmonic forms with respect to a family of regularized metrics, and give a method to obtain L2-estimates of solutions of the dbar-equation by using the de Rham-Weil isomorphism between the dbar-cohomology and the check{C}ech cohomology.
In this note, we reveal that our solution of Demailly's strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Kollár and Jonsson-Mustată implies the truth of twisted versions of the strong openness conjecture; our optimal $L^{2}$ extension implies Berndtsson's positivity of vector bundles associated to holomorphic fibrations over a unit disc.
We study the ideals of the closure of the polynomial multipliers on the Drury-Arveson space. Structural results are obtained by investigating the relation between an ideal and its weak-$*$ closure, much in the spirit of the corresponding classical facts for the disc algebra. Zero sets for multipliers are also considered and are deeply intertwined with the structure of ideals. Our approach is primarily based on duality arguments.
The goal of this note is to survey some recent results on the Grauert-Riemenschneider multiplier ideal sheaves on any (reduced) complex space of pure dimension. In particular, we obtain the Briançon-Skoda number for any Noetherian ring of weakly holomorphic functions with weakly rational singularities (\emph{not} essentially of finite type over $\mathbb{C}$ and Cohen-Macaulay local rings), which will partially answer a question of Huneke.
We study in this paper some local invariants attached via multiplier ideals to an effective divisor or ideal sheaf on a smooth complex variety. First considered (at least implicitly) by Libgober and by Loeser and Vaquie, these jumping coefficients consist of an increasing sequence of rational numbers beginning with the log canonical threshold of the divisor or ideal in question. They encode interesting geometric and algebraic information, and we show that they arise naturally in several different contexts. Given a polynomial f having only isolated singularities, results of Varchenko, Loeser an…
We show that a Skoda complex with a general plurisubharmonic weight function is exact if its 'degree' is sufficiently large. This answers a question of Lazarsfeld and implies that not every integrally closed ideal is equal to a multiplier ideal even if we allow general plurisubharmonic weights for the multiplier ideal, extending the result of Lazarsfeld and Lee \cite{LL}.
The purpose of this survey is to present analytic versions of the injectivity theorem and their applications. The proof of our injectivity theorems is based on a combination of the L^2-method for the dbar-equation and the theory of harmonic integrals. As applications, we obtain Nadel type vanishing theorems and extension theorems for pluri-canonical sections of log pairs. Moreover, we give some results on semi-ampleness related to the abundance conjecture in birational geometry (the minimal model program).
S. K. Donaldson asked whether the lower bound of the Calabi functional is achieved by a sequence the normalized Donaldson-Futaki invariants. We answer the question for the Ricci curvature formalism, in place of the scalar curvature. The 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 prove it in the two case: the inverse Monge-Ampere flow and the Kahler-Ricci flow.
The purpose of this note is threefold. (i) To explain the effective Kohn algorithm for multipliers in the complex Neumann problem and its difference with the full-real-radical Kohn algorithm, especially in the context of an example of Catlin-D'Angelo concerning the ineffectivness of the latter. (ii) To extend the techniques of multiplier ideal sheaves for the complex Neumann problem to general systems of partial differential equations. (iii) To present a new procedure of generation of multipliers in the complex Neumann problem as a special case of the multiplier ideal sheaves techniques for ge…
A unitarily invariant, complete Nevanlinna--Pick kernel $K$ on the unit ball determines a class of operators on Hilbert space called $K$-contractions. We study those $K$-contractions that are constrained, in the sense that they are annihilated by an ideal of multipliers. Our overarching goal is to identify various joint spectra of these constrained $K$-contractions through the vanishing locus of their annihilators. Our methods are based around a careful analysis of a subset of the ball associated to the annihilator, which we call its support. For the functional models, we show how this support…
