Expository introduction to multiplier ideals, based on the talk at Algebraic Geometry: Presentations by Young Researchers meeting in Snowbird, July 2004. final version
In this expository introductory text we discuss the multiplier ideals in algebraic geometry. We state Kawamata-Viehweg's and Nadel's vanishing theorems, give a proof (following Ein and Lazarsfeld) of Kollár's bound on the maximal multiplicity of the theta divisor, and explain the asymptotic constructions and the ideas of Siu's proof of the deformation invariance of plurigenera. We also indicate the analytic interpretation of the theory.
In this note we calculate the multiplier ideal associated to an arbitrary monomial ideal in C^n. We discuss applications to the calculation of log canonical thresholds.
This paper provides a formula for the Mather-Jacobian multiplier ideals of torus invariant ideals on (not necessarily normal) toric varieties that generalizes Howald's formula for the multiplier ideal of monomial ideals in a polynomial ring.
We consider the multiplier ideals of the ideal of a reduced union of lines through the origin in C^3. For general arrangements of lines, we calculate the multiplier ideals.
We define a version of multiplier ideals, the Mather multiplier ideals, on a variety with arbitrary singularities, using the Mather discrepancy and the Jacobian ideal. In this context we prove a relative vanishing theorem, thus obtaining restriction theorems and a subadditivity and summation theorems. The Mather multiplier ideals also satisfy a Skoda type result. As an application, we obtain a Briancon-Skoda type formula for the integral closures of ideals on a variety with arbitrary singularities.
We use methods from birational geometry to study the Hodge and weight filtrations on the localization along a hypersurface. We focus on the lowest piece of the Hodge filtration of the submodules arising from the weight filtration. This leads to a sequence of ideal sheaves called weighted multiplier ideals. The last ideal of this sequence is a multiplier ideal (and a Hodge ideal), and we prove that the first is the adjoint ideal. We also study the local and global properties of weighted multiplier ideals and their applications to singularities of hypersurfaces of smooth varieties.
In this paper, we establish several results related to vanishing theorems for Mather-Jacobian multiplier ideals on a Gorenstein projective variety, including an injectivity theorem, a Nadel-type vanishing theorem, a Griffith-type vanishing theorem for vector bundles, and a vanishing theorem for the asymptotic Mather-Jacobian multiplier ideals.
We give a survey on b-function, spectrum, and multiplier ideals together with certain interesting relations among them including the case of arbitrary subvarieties.
We establish a relative Bertini type theorem for multiplier ideal sheaves. Then we prove a relative version of the Kollár--Nadel type vanishing theorem as an application.
We show that the reduction to positive characteristic of the multiplier ideal in the sense of de Fernex and Hacon agrees with the test ideal for infinitely many primes, assuming that the variety is numerically Q-Gorenstein. It follows, in particular, that this reduction property holds in dimension 2 for all normal surfaces.
For a simple complete ideal $\wp$ of a local ring at a closed point on a smooth complex algebraic surface, we introduce an algebraic object, named Poincaré series $P_{\wp}$, that gathers in an unified way the jumping numbers and the dimensions of the vector space quotients given by consecutive multiplier ideals attached to $\wp$. This paper is devoted to prove that $P_{\wp}$ is a rational function giving an explicit expression for it.
In this paper we make a systematic study of the multiplicity of the jumping points associated to the mixed multiplier ideals of a family of ideals in a complex surface with rational singularities. In particular we study the behaviour of the multiplicity by small perturbations of the jumping points. We also introduce a Poincaré series for mixed multiplier ideals and prove its rationality. Finally, we study the set of divisors that contribute to the log-canonical wall.
We use the language of multiplier ideals in order to relate the syzygies of an abelian variety in a suitable embedding with the local positivity of the line bundle inducing that embedding. This extends to higher syzygies a result of Hwang and To on projective normality.
We describe a relation between two invariants that measure the complexity of a hypersurface singularity. One is the Hodge spectrum which is related to the monodromy and the Hodge filtration on the cohomology of the Milnor fiber. The other is the multiplier ideal, having to do with log resolutions.
We give a Hodge-theoretic interpretation of the multiplier ideal of an effective divisor on a smooth complex variety. More precisely, we show that the associated graded coherent sheaf with respect to the jumping-number filtration can be recovered from the smallest piece of M. Saito's Hodge filtration of the D-module of vanishing cycles.
We give a proof of the Thom-Sebastiani type theorem for holonomic filtered $D$-modules satisfying certain good conditions (including Hodge modules) by using algebraic partial microlocalization. By a well-known relation between multiplier ideals and $V$-filtrations of Kashiwara and Malgrange, the argument in the proof implies also a Thom-Sebastiani type theorem for multiplier ideals, which cannot be deduced from a already known proof of the Thom-Sebastiani theorem for mixed Hodge modules (since the latter gives only the information of graded pieces of multiplier ideals). We also sketch a more e…
M. Mustata used jet schemes to compute the multiplier ideals of reduced hyperplane arrangements. We give an alternate proof using a log resolution, which is simpler and allows us to consider non-reduced arrangements. By applying the idea of wonderful models introduced by De Concini--Procesi, we also simplify the result.
The main goal of this paper is to construct an algebraic analogue of quasi-plurisubharmonic function (qpsh for short) from complex analysis and geometry. We define a notion of qpsh function on a valuation space associated to a quite general scheme. We then define the multiplier ideals of these functions and prove some basic results about them, such as subadditivity property, the approximation theorem. We also treat some applications in complex algebraic geometry.
We introduce a spectrum for arbitrary varieties. This generalizes the definition by Steenbrink for hypersurfaces. In the isolated complete intersection singularity case, it coincides with the one given by Ebeling and Steenbrink except for the coefficients of integral exponents. We show a relation to the graded pieces of the multiplier ideals by using a relation to the filtration $V$ of Kashiwara and Malgrange. This implies a partial generalization of a theorem of Budur in the hypersurface case. The point is to consider the direct sum of the graded pieces of the multiplier ideals as a module ov…
We generalize to all normal complex algebraic varieties the valuative characterization of multiplier ideals due to Boucksom-Favre-Jonsson in the smooth case. To that end, we extend the log discrepancy function to the space of all real valuations, and prove that it satisfies an adequate properness property, building upon previous work by Jonsson-Mustaţă. We next give an alternative definition of the concept of numerically Cartier divisors previously introduced by the first three authors, and prove that numerically Q-Cartier divisors coincide with Q-Cartier divisors for rational singularities. T…
Let $\mathfrak a \subset \mathscr O_X$ be a coherent ideal sheaf on a normal complex variety $X$, and let $c \ge 0$ be a real number. De Fernex and Hacon associated a multiplier ideal sheaf to the pair $(X, \mathfrak a^c)$ which coincides with the usual notion whenever the canonical divisor $K_X$ is $\mathbb Q$-Cartier. We investigate the properties of the jumping numbers associated to these multiplier ideals. We show that the set of jumping numbers of a pair is unbounded, countable and satisfies a certain periodicity property. We then prove that the jumping numbers form a discrete set of real…
A characterization of nef and good divisors is given: a divisor D on a smooth complex projective variety is nef and good if and only if the asymptotic multiplier ideals of sufficiently high multiples of e(D) D$ are trivial, where e(D) denotes the exponent of the divisor D. Some results of the same kind are proved in the analytic setting.
We present a generalization of the multiplier ideal version of inversion of adjunction, often known as the restriction theorem, to centers of arbitrary codimension. We approach inversion of adjunction from the subadjunction point of view. Let X be a smooth complex projective variety and let Z be an exceptional log-canonical center of an effective Q-divisor D on some dense open subset of X that contains the generic point of Z. Any subvariety of X can be expressed as such a center for some D. We define an adjoint ideal that measures how non-klt (X, D) is outside the generic point of Z. Our main …
In algebraic geometry, Bertini theorems are an extremely important tool. A generalization of the classical theorem to multiplier ideals show that multiplier ideals restrict to a general hyperplane section. In characteristic $p > 0$, the test ideal can be seen to be the characteristic $p > 0$ analog of the multiplier ideal. However, in this paper it is shown that the same type of Bertini type theorem does not hold for test ideals.
Let $S$ be a Puiseux series of the germ of an analytically irreducible plane curve $Z$. We provide a new perspective to construct a set of polynomials $F=\{F_1,\ldots, F_{g-1}\}$ associated to $S$, which is a special choice of maximal contact elements constructed in [AMB17] and approximate roots defined in [Dur18], [AM73a], [AM73b]. Using these polynomials as building blocks, we describe a set of generators of multiplier ideals of the form $\mathfrak{I}(αZ)$ with $0<α<1$ a rational number, which recovers the results about irreducible plane curves in [AMB17], [Dur18].
Let X be a smooth variety and J, K two ideal sheaves on X. We prove the following formula relating the multiplier ideals of J, K and J+K: I(X, c(J+K))\subset \sum_{a+b=c} I(X, aJ)\cdot I(X,bK). An analogous formula holds for the asymptotic multiplier ideals of two graded systems of ideals. As a consequence, we show how to approximate at any given point arbitrary multiplier ideals by multiplier ideals of zero dimensional ideals. We apply this to study the invariance of multiplier ideals under different embeddings.
A simple formula computing the multiplier ideal of a monomial ideal on an arbitrary affine toric variety is given. Variants for the multiplier module and test ideals are also treated.
The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic p. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via vanishing theorems. In this note we give several examples to emphasize the different behavior of test ideals and multiplier ideals. Our main result is that every ideal in an F-finite regular local ring can be written as a generalized test ideal. We also prove the semicontinuity of $F$-pure thresholds (though the analogue of the Generic Restriction Theorem for m…
In recent years, multiplier ideals have found many applications in local and global algebraic geometry. Because of their importance, there has been some interest in the question of which ideals on a smooth complex variety can be realized as multiplier ideals. Other than integral closure no local obstructions have been known up to now, and in dimension two it was established by Favre-Jonsson and Lipman-Watanabe that any integrally closed ideal is locally a multiplier ideal. We prove the somewhat unexpected result that multiplier ideals in fact satisfy some rather strong algebraic properties inv…
By using Mather-Jacobian multiplier ideals, we first prove a formula on comparing Grauert-Riemenschneider canonical sheaf with canonical sheaf of a variety over an algebraically closed field of characteristic zero. Then we turn to study Mather-Jacobian multiplier ideals on algebraic curve, in which case the definition of Mather-Jacobian multiplier ideal can be extended to a ground field of any characteristic. We show that Mather-Jacobian multiplier ideal on curves is essentially the same as an integrally closed ideal. Finally by comparing conductor ideal with Mather-Jacobian multiplier ideal, …
For an effective divisor on a smooth algebraic variety or a complex manifold, we show that the associated multiplier ideals coincide essentially with the filtration induced by the filtration V constructed by B. Malgrange and M. Kashiwara. This implies another proof of a theorem of L. Ein, R. Lazarsfeld, K.E. Smith and D. Varolin that any jumping coefficient in the interval (0,1] is a root of the Bernstein-Sato polynomial up to sign. We also give a refinement (using mixed Hodge modules) of the formula for the coefficients of the spectrum for exponents not greater than one or greater than the di…
It is well known that the multiplier ideal $\multr{I}$ of an ideal $I$ determines in a straightforward way the multiplier ideal $\multr{f}$ of a sufficiently general element $f$ of $I$. We give an explicit condition on a polynomial $f \in \CC[x_1,...,x_n]$ which guarantees that it is a sufficiently general element of the most natural associated monomial ideal, the ideal generated by its terms. This allows us to directly calculate the multiplier ideal $\multr{f}$ (for all $r$) of ``most'' polynomials $f$.
We compute the multiplier ideals of hyperplane arrangements via the interpretation of these ideals in terms of spaces of arcs, due to Ein, Lazarsfeld and the author.
We describe a new software package for computing multiplier ideals in certain cases, including monomial ideals, monomial curves, generic determinantal ideals, and hyperplane arrangements. In these cases we take advantage of combinatorial formulas for multiplier ideals given by results of Howald, Thompson, and Johnson. The package uses Normaliz. It is available as a library for Macaulay2.
The purpose of this note is to give a survey of the algebraic properties of multiplier ideals, and illustrate some of their applications to classical projective geometry.
The aim of this paper is to study mixed multiplier ideals associated to a tuple of ideals in a two-dimensional local ring with a rational singularity. We are interested in the partition of the real positive orthant given by the regions where the mixed multiplier ideals are constant. In particular we reveal which information encoded in a mixed multiplier ideal determines its corresponding jumping wall and we provide an algorithm to compute all the constancy regions, and their corresponding mixed multiplier ideals, in any desired range.
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.
Thompson (2014) exhibits a formula for the multiplier ideal with multiplier lambda of a monomial curve C with ideal I as an intersection of a term coming from the I-adic valuation, the multiplier ideal of the term ideal of I, and terms coming from certain specified auxiliary valuations. This short note shows it suffices to consider only one auxiliary valuation. This improvement is achieved through a more intrinsic approach, reduction to the toric case.
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.
The aim of this paper is to study jumping numbers and multiplier ideals of any ideal in a two-dimensional local ring with a rational singularity. In particular we reveal which information encoded in a multiplier ideal determines the next jumping number. This leads to an algorithm to compute sequentially the jumping numbers and the whole chain of multiplier ideals in any desired range. As a consequence of our method we develop the notion of jumping divisor that allows to describe the jump between two consecutive multiplier ideals. In particular we find a unique minimal jumping divisor that is s…
Demailly, Ein and Lazarsfeld \cite{DEL} proved the subadditivity theorem for multiplier ideals, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals, on non-singular varieties. We prove that, in two-dimensional case, the subadditivity theorem holds on log-terminal singularities. However, in higher dimensional case, we have several counter-examples. We consider the subadditivity theorem for monomial ideals on toric rings, and construct a counter-example on a three-dimensional toric ring.
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.
It was recently established by the first two authors that multiplier ideals on a smooth variety satisfy some special syzygetic properties. The purpose of this note is to show how some of these can be extended to the singular setting.
We prove that certain roots of the Bernstein-Sato polynomial (i.e. b-function) are jumping coefficients up to a sign, showing a partial converse of a theorem of L. Ein, R. Lazarsfeld, K.E. Smith, and D. Varolin. We also prove that certain roots are determined by a filtration on the Milnor cohomology, generalizing a theorem of B. Malgrange in the isolated singularity case. This implies a certain relation with the spectrum which is determined by the Hodge filtration, because the above filtration is related to the pole order filtration. For multiplier ideals we prove an explicit formula in the ca…
As a generalization of formulas given by Demailly-Ein-Lazarsfeld, Eisenstein and Takagi, we prove a subadditivity formula for multiplier ideals associated to log pairs.
We prove a generalization of Demailly-Ein-Lazarsfeld's subadditivity formula and Mustata's summation formula for multiplier ideals to the case of singular varieties, using characteristic $p$ methods. As an application of our formula, we improve Hochster-Huneke's result on the growth of symbolic powers of ideals in singular affine algebras.
We introduce the notion of strong test module and show that a large number of such modules appear in the tight closure theory of complete domains: the test ideal (this has already been known), the parameter test module, and the module of relative test elements. They also appear as certain multiplier ideals, a concept of interest in algebraic geometry.
The goal of this paper is to produce a formula for the multiplier ideals of monomial space curves in the spirit of Howald's formula for the multiplier ideals of monomial ideals. This is achieved by constructing a toric blowup of affine space in such a way that a log resolution of the monomial curve may be constructed from this toric variety in a well controlled manner.
We present an algebro-geometric perspective on some generalizations, due to S. Takagi, of the restriction theorem for multiplier ideals. The first version of the restriction theorem for multiplier ideals was discovered by Esnault and Viehweg. In a series of papers S. Takagi has discovered generalizations of the restriction theorem and some formulas for multiplier ideals that follow from the restriction theorem. He uses the technique of tight closure and reduction to positive characteristic. We are able to provide an algebro-geometric proof of generalizations of his restriction theorem and his …
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.
