Nero Budur
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.
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On the Hodge and V-filtrations of mixed Hodge modules
本文通过Hodge滤波证明了V滤波的Beilinson型公式，并利用该公式研究了更高乘数理想与Hodge理想的关系，改进了相关结果并回答了若干开放问题。
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.
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.
The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes, generalizing to the quasi-projective case results of Green-Lazarsfeld and Simpson. As an application we show the polynomial periodicity of Hodge numbers of congruence covers in any dimension, generalizing results of E. Hironaka…
We define and study Hodge ideals associated to a coherent ideal sheaf J on a smooth complex variety, via algebraic constructions based on the already existing concept of Hodge ideals associated to Q-divisors. We also define the generic minimal exponent of J, extending the standard invariant for hypersurfaces. We relate it to Hodge ideals, and show that it is a root of the Bernstein-Sato polynomial of J.
Mustaţă a and Popa introduce the notion of Hodge ideals for an effective $\mathbb{Q}$-divisor $D$ and prove a vanishing theorem for Hodge ideals, which generalizes Nadel vanishing for multiplier ideals. However, their proof needs an extra assumption on the existence of $\ell$-roots of the line bundle $\mathscr{O}_X(\ell D)$, which is not necessary for Nadel vanishing. In this paper, we prove that vanishing for Hodge ideals still holds even without this assumption.
This note provides a connection between the Hodge spectrum and the Hodge ideals recently developed by Mustaţǎ--Popa in the case of weighted homogeneous isolated singularities.
Multivariable Alexander invariants of algebraic links calculated in terms of algebro-geometric invariants (polytopes and ideals of quasiadjunction). The relations with log-canonical divisors, the multiplier ideals and a semicontinuity property of polytopes of quasiadjunction is discussed.
We prove results concerning the behavior of Hodge ideals under restriction to hypersurfaces or fibers of morphisms, and addition. The main tool is the description of restriction functors for mixed Hodge modules by means of the $V$-filtration.
We introduce Hodge ideal spectrum for isolated hypersurface singularities to see the difference between the Hodge ideals and the microlocal $V$-filtration modulo the Jacobian ideal. Via the Tjurina subspectrum, we can compare the Hodge ideal spectrum with the Steenbrink spectrum which can be defined by the microlocal $V$-filtration. As a consequence of a formula of Mustata and Popa, these two spectra coincide in the weighted homogeneous case. We prove sufficient conditions for their coincidence and non-coincidence in some non-weighted-homogeneous cases where the defining function is semi-weigh…
We give an explicit formula for the Hodge filtration on the $\mathscr{D}_X$-module $O_X(*Z)f^{1-α}$ associated to the effective $\mathbb{Q}$-divisor $D=α\cdot Z$, where $0<α\le1$ and $Z=(f=0)$ is an irreducible hypersurface defined by $f$, a weighted homogeneous polynomial with an isolated singularity at the origin. In particular this gives a formula for the Hodge ideals of $D$. We deduce a formula for the generating level of the Hodge filtration, as well as further properties of Hodge ideals in this setting. We also extend the main theorem to the case when $f$ is a germ of holomorphic functio…
In this article we construct a Koszul-type resolution of the p-th exterior power of the sheaf of holomorphic differential forms on smooth toric varieties and use this to prove a Nadel-type vanishing theorem for Hodge ideals associated to effective \mathbb{Q}-divisors on smooth projective toric varieties. This extends earlier results of Mustaţă and Popa.
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…
We develop the theory of Hodge ideals for Q-divisors by means of log resolutions, extending our previous work on reduced hypersurfaces. We prove local (non-)triviality criteria and a global vanishing theorem, as well as other analogues of standard results from the theory of multiplier ideals, and we derive a new local vanishing theorem. The connection with the V-filtration is analyzed in a sequel.
Weighted Hodge ideals of reduced divisors
本文基于双有理几何与V-filtration研究超曲面定位上的Hodge与权滤子，定义了包含adjoint ideal和multiplier ideal的weighted Hodge ideals，并分析其性质以应用于光滑簇的超曲面奇点研究。
We explicitly compute the Hodge ideals of Q-divisors in terms of the V-filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties of Hodge ideals in this generality, and relate them to Bernstein-Sato polynomials. As a consequence of our study we establish general properties of the minimal exponent, a refined version of the log canonical threshold, and bound it in terms of discrepancies on log resolutions, addressing a question of Lichtin and Kollár.
Vanishing Theorem for Hodge Ideals on smooth hypersurfaces
本文通过Koszul型消解证明光滑超曲面的弱Bott消失定理，并由此建立有效Cartier除子对应Hodge理想的消失定理，推广了Mustata和Popa的已有工作。
Given a reduced effective divisor D on a smooth variety X, we describe the generating function for the classes of the Hodge ideals of D in the Grothendieck group of coherent sheaves on X in terms of the motivic Chern class of the complement of the support of D. As an application, we compute the generating function for the Hilbert series of Hodge ideals of a hyperplane arrangement in terms of the Poincare polynomial of the arrangement.
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.
We show that the Hodge ideals in the sense of Mustata and Popa are quite closely related to the induced microlocal V-filtration on the structure sheaf, defined by using the microlocalization of the V-filtration of Kashiwara and Malgrange. More precisely the former coincide, module the ideal of the divisor, with the part of the latter indexed by positive integers, although they are different without modulo the ideal in general. This coincidence implies that the $j$-log-canonicity in their sense is determined by the microlocal log-canonical threshold of the divisor, which coincides with the maxi…
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 determine explicitly the Hodge ideals for the determinant hypersurface as an intersection of symbolic powers of determinantal ideals. We prove our results by studying the Hodge and weight filtrations on the mixed Hodge module O_X(*Z) of regular functions on the space X of n x n matrices, with poles along the divisor Z of singular matrices. The composition factors for the weight filtration on O_X(*Z) are pure Hodge modules with underlying D-modules given by the simple GL-equivariant D-modules on X, where GL is the natural group of symmetries, acting by row and column operations on the matrix…
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.
A formula for the irregularity of abelian coverings of the projective plane is established and some applications are presented.
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.
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.
We use methods from birational geometry to study M. Saito's Hodge filtration on the localization along a hypersurface. This filtration leads to a sequence of ideal sheaves, called Hodge ideals, the first of which is a multiplier ideal. We analyze their local and global properties, and use them for applications related to the singularities and Hodge theory of hypersurfaces and their complements.
On diminished multiplier ideal and the termination of flips
本文发展了奇异簇上的减弱乘子理想理论，并证明了在Cartier指数有界等条件下带缩放的特定翻转的终止。
An algorithm for Hodge ideals
本文提出一种计算与任意有效除子关联的Hodge理想算法，基于Malgrange和Kashiwara的V-滤波算法，并提供了新的计算乘子理想和跳跃数的方法。
We consider the Hodge filtration on the sheaf of meromorphic functions along free divisors for which the logarithmic comparison theorem holds. We describe the Hodge filtration steps as submodules of the order filtration on a cyclic presentation in terms of a special factor of the Bernstein-Sato polynomial of the divisor and we conjecture a bound for the generating level of the Hodge filtration. Finally, we develop an algorithm to compute Hodge ideals of such divisors and we apply it to some examples.
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…
Multiplier ideals of plane curve singularities via Newton polygons
本文提出一种有效方法，利用 Newton 多边形及环面嵌入解消确定光滑曲面曲线奇点的乘子理想与跳跃数，并推广了 Howald 定理。
Higher multiplier ideals
本文研究高乘子理想，通过混合Hodge模和V滤波理论，定义了一类由Hodge级别和有理数参数索引的理想sheaves，探讨其局部和全局性质，并应用于证明关于极化阿贝尔簇上θ除数奇异性的猜想。
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.
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…
Containment problems for fat point schemes via asymptotic multiplier ideals
本研究通过渐近乘子理想方法探究射影空间中点的根基理想的包含问题，提出干净渐近乘子理想概念，推导相关不等式，验证双Hesse构形、线的星构形的干净性并提出线排列奇异点的干净性猜想。
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…
Quasi-ordinary hypersurfaces, multiplier ideals and local tropicalizations
本文利用环面嵌入消解，将拟常序超曲面的乘子理想表示为牛顿多面体序列和广义单项式理想，并证明其局部热带化由嵌入拓扑类型决定的扇支撑。
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.
Multiplier modules, $V$-filtrations and Bernstein-Sato polynomials on singular ambient varieties
本文将乘子理想与$V$-滤子的关系推广至奇异不可约簇，引入乘子模与相交复Hodge模，应用于Skoda定理与Ajit公式证明，并定义簇与理想层对的Bernstein-Sato多项式，建立其根与跳跃数联系。
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].
We show that all integrally closed ideals on log terminal surfaces are multiplier ideals by extending an existing proof for smooth surfaces.
We describe methods for calculation of polytopes of quasiadjunction for plane curve singularities which are invariants giving a Hodge theoretical refinement of the zero sets of multivariable Alexander polynomials. In particular we identify some hyperplanes on which all polynomials in multivariable Bernstein ideal vanish.
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…
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 previous work, we constructed for a smooth complex variety $X$ and for a linear algebraic group $G$ a mixed Hodge structure on the complete local ring $\widehat{\mathcal{O}}_ρ$ to the moduli space of representations of the fundamental group $π_1(X,x)$ into $G$ at a representation $ρ$ underlying a variation of mixed Hodge structure. We now show that the jump ideals $J_k^i \subset \widehat{\mathcal{O}}_ρ$, defining the locus of representations such the the dimension of the cohomology of $X$ in degree $i$ of the associated local system is greater than $k$, are sub-mixed Hodge structures; this …
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, …
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