Mircea Mustata, Mihnea Popa
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.
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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.
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 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.
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 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 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.
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…
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…
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 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 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.
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。
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 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.
We bound the generation level of the Hodge filtration on the localization along a hypersurface in terms of its minimal exponent. As a consequence, we obtain a local vanishing theorem for sheaves of forms with log poles. These results are extended to Q-divisors, and are derived from a result of independent interest on the generation level of the Hodge filtration on nearby and vanishing cycles.
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.
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 …
Let $k$ be a perfect field of odd characteristic and $X$ a smooth algebraic variety over $k$ which is $W_2$-liftable. We show that the exponent twisiting of the classical Cartier descent gives an equivalence of categories between the category of nilpotent Higgs sheaves of exponent $\leq p-1$ over $X/k$ and the category of nilpotent flat sheaves of exponent $\leq p-1$ over $X/k$, and it is equivalent up to sign to the inverse Cartier and Cartier transforms for these nilpotent objects constructed in the nonabelian Hodge theory in positive characteristic by Ogus-Vologodsky. In view of the crucial…
We investigate the Lyapunov Exponents of a variation of Hodge structure which has $G_2$ as geometric monodromy group, and discuss formulas for the sum of positive Lyapunov Exponents of variations of Hodge structures of any weight.
We determine the second fundamental form of a variation of Hodge Structure of a smooth projective hypersurface using the classical identification of the Hodge structure and the action of the infinitesimal variation of Hodge structure with pieces of the Jacobian Ideal of the hypersurface. This simple algebraic description also implies that, in this case, the variation of Hodge structure satisfies a new set of second order partial differential equations.Comments are welcome!
Calculating Hodge and weight filtrations on localizations
本文介绍Macaulay2包MixedHodgeModules，用于计算多项式环局部化上的Hodge与权重滤过及相关不变量，并支持计算多种复形。
Limit distribution of Hodge spectral exponents of irreducible plane curve singularities
本文研究不可约平面曲线奇点的Hodge谱指数分布，通过数值不变量给出其与连续分布差值的闭式公式，并刻画极限分布为连续分布的曲线族及占优值区间。
If the monodromy representation of a VHS over a hyperbolic curve stabilizes a rank two subspace, there is a single non-negative Lyapunov exponent associated with it. We derive an explicit formula using only the representation in the case when the monodromy is discrete.
Let $S$ be a smooth algebraic surface in $\mathbb{P}^3(\mathbb{C})$. A curve $C$ in $S$ has a cohomology class $η_C \in H^1 \hspace{-3pt}\left( Ω^1_S \right)$. Define $α(C)$ to be the equivalence class of $η_C$ in the quotient of $H^1 \hspace{-3pt}\left( Ω^1_S \right)$ modulo the subspace generated by the class $η_H$ of a plane section of $S$. In the paper "Reconstructing subvarieties from their periods" the authors Movasati and Sertöz pose several interesting questions about the reconstruction of $C$ from the annihilator $I_{α(C)}$ of $α(C)$ in the polynomial ring $R=H^0_*(\mathcal{O}_{\mathb…
We compute the sum of the positive Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. The computation is based on the analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and hyperbolic Laplacians when the underlying Riemann surface degenerates.
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。
Hodge properties of Airy moments
本文研究三次指数和对称幂矩的复类似物，证明其 de Rham 上同调具有 Anderson 算术 Hodge 结构，并通过非正则 Hodge 滤过计算 Hodge 数，得出其只能为 0 或 1 的结论。
In this paper, we introduce a coherent subsheaf of Saito's $S$-sheaf, which is a combination of the $S$-sheaf and the multiplier ideal sheaf. We construct its $L^2$-Dolbeault resolution, which generalizes MacPherson's conjecture on the $L^2$ resolution of the Grauert-Riemenschneider sheaf. We also prove various vanishing theorems for the $S$-sheaf (Saito's vanishing theorem, Kawamata-Viehweg vanishing theorem and some new ones like Nadel vanishing theorem) transcendentally. Finally, we discuss some applications of our results on the relative version of Fujita's conjecture (e.g. Kawamata's conj…
We develop a class of uniformizations for certain weight 3 variations of Hodge structure (VHS). The analytic properties of the VHS are used to establish a conjecture of Eskin, Kontsevich, Möller, and Zorich on Lyapunov exponents. Additionally, we prove that the monodromy representations are log-Anosov, a dynamical property that has a number of global consequences for the VHS. We establish a strong Torelli theorem for the VHS and describe appropriate domains of discontinuity. Additionally, we classify the hypergeometric differential equations that satisfy our assumptions. We obtain several mult…
The GIT stability and Hodge structures of hypersurfaces via minimal exponent
本文通过最小指数刻画超曲面的GIT稳定性，证明其与半稳定性的临界条件，并引入m-极限源以推广周期映射行为，从而由奇点类型决定极限混合Hodge结构。
On the invariance of irregular Hodge numbers under crepant birational equivalences
研究朗道 - 金兹堡模型\((U,f)\)中扭曲德拉姆上同调\(\mathrm{H}^k_{\mathrm{dR}}(U,f)\)的非正则霍奇数，证明其在crepant双有理等价下不变，类似巴蒂列夫 - 孔采维奇定理对卡拉比 - 丘流形霍奇数的结论。
We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit integrals over moduli spaces of algebraic curves with additional structures. Moreover, these integrals can be interpreted as correlators in a topological string theory. Also a new analogy arose between ergodic theory and complex algebraic geometry.
We prove a conjecture of Teissier asserting that if $f$ has an isolated singularity at $P$ and $H$ is a smooth hypersurface through $P$, then $\widetildeα_P(f)\geq \widetildeα_P(f\vert_H)+\frac{1}{θ_P(f)+1}$, where $\widetildeα_P(f)$ and $\widetildeα_P(f\vert_H)$ are the minimal exponents at $P$ of $f$ and $f\vert_H$, respectively, and $θ_P(f)$ is an invariant obtained by comparing the integral closures of the powers of the Jacobian ideal of $f$ and of the ideal defining $P$. The proof builds on the approaches of Loeser and Elduque-Mustata. The new ingredients are a result concerning the behav…
The minimal exponent and $k$-rationality for local complete intersections
该研究将光滑复代数簇中超曲面的相关结论推广到局部完全交子簇，给出其具有$k$-有理奇点的极小指数判据，并用相交上同调Hodge模的Hodge滤过刻画该条件，同时推导了局部上同调层Hodge滤过的生成层数及相关上同调群非零的结果。
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…
Multiplier ideals of normal surface singularities
本文研究正规曲面奇点上乘子理想的跳跃数与重数，证明其正参数部分可由解消图和除子组合计算，并借助混合 Hodge 模推广 Hodge 谱对应，对加权齐性与 splice quotient 奇点给出具体恢复公式。
Consider the flat bundle on $\mathrm{CP}^1 - \{0,1,\infty \}$ corresponding to solutions of the hypergeometric differential equation $ \prod_{i=1}^h (\mathrm D - α_i) - z \prod_{j=1}^h (\mathrm D - β_j) = 0$ where $\mathrm D = z \frac {d}{dz}$. For $α_i$ and $β_j$ distinct real numbers, this bundle is known to underlie a complex polarized variation of Hodge structure. Setting the complete hyperbolic metric on $\mathrm{CP}^1 - \{0,1,\infty \}$, we associate $n$ Lyapunov exponents to this bundle. We compute the parabolic degrees of the holomorphic subbundles induced by the variation of Hodge str…
We show that the limit, by rescaling, of the `new supersymmetric index' attached to the Fourier-Laplace transform of a polarized variation of Hodge structure on a punctured affine line is equal to the spectral polynomial attached to the same object. We also extend the definition by Deligne of a Hodge filtration on the de Rham cohomology of a exponentially twisted polarized variation of complex Hodge structure and prove a E_1 degeneration property for it.
We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their L-functions extend meromorphically to the complex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be "classical", we compute their Hodge numbers by means of the irregular Hodge filtration on their realizations as exponential mixed Hodge structures. We show that all Hodge numbers are either zero or one, which implies potential automorphy thanks to recent results of Patrikis and Taylor.
$p$-adic non-abelian Hodge theory for curves via moduli stacks
本文通过模堆的典范扭转，将曲线上的 $p$-adic 非阿贝尔 Hodge 理论几何化，并恢复 Faltings 对应为同胚，同时将表示堆映射到半稳定 Higgs 丛的开子堆。
Archimedean zeta functions, singularities, and Hodge theory
利用Hodge理论研究全纯函数f的阿基米德zeta函数Z_f的极点与奇点不变量之间的关系，证明Z_f的最大非平凡极点为f的最小指数的相反数，其阶由Bernstein-Sato多项式b_f(s)对应根的重数决定，并给出b_f(s)的根不是Z_f的极点的反例。
Gröbner Bases Native to Term-ordered Commutative Algebras, with Application to the Hodge Algebra of Minors
为深入理解双行列式基，本文为配备单项式概念与项序的交换代数发展了内生的Gröbner基理论与算法，并借此为任意t-子式理想给出了通用Gröbner基的简明证明。
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。
Inspired by Katz-Mazur theorem on crystalline cohomology and by Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon of Lyapunov spectrum lies above (or on) the Harder-Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons and the integral of eigenvalues of the curvature of the Hodge bundle by using Atiyah-Bott, Forni and Möller's works. We obtain several applications to Teichmüller dynamics conditional to the conjecture.
We define the Weierstrass filtration for Teichmuller curves and construct the Harder-Narasimhan filtration of the Hodge bundle of a Teichmuller curve in hyperelliptic loci and low-genus nonvarying strata. As a result we obtain the sum of Lyapunov exponents of Teichmuller curves in these strata.
We give an expression for the Łojasiewicz exponent of a set of ideals which are pieces of a weighted homogeneous filtration. We also study the application of this formula to the computation of the Łojasiewicz exponent of the gradient of a semi-weighted homogeneous function $(\C^n,0)\to (\C,0)$ with an isolated singularity at the origin.
Hodge Theory of Abelian Covers
本文为复代数簇的阿贝尔覆盖的同调有限维商空间建立典范混合霍奇结构，推广亚历山大模理论，并关联Milnor纤维贝蒂数的组合性问题。
We study the Łojasiewicz exponent and the log canonical threshold of ideals of $\mathcal O_n$ when restricted to generic subspaces of $\mathbb C^n$ of different dimensions. We obtain effective formulas of the resulting numbers for ideals with monomial integral closure. An inequality relating these numbers is also proven. We also introduce the notion of bi-Lipschitz equivalence of ideals and we prove the bi-Lipschitz invariance of Łojasiewicz exponents and log canonical thresholds of ideals.
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.
Suppose that R is a two-dimensional normal standard-graded domain over a finite field. We prove that there exists a uniform Frobenius test exponent b for the class of homogeneous ideals in R generated by at most n elements. This means that for every ideal I in this class we have that f^(p^b) belongs to I^([p^b]) if and only if f belongs to the Frobenius closure I^F. This gives in particular a finite test for the Frobenius closure. On the other hand we show that there is no uniform bound for Frobenius test exponent for all homogeneous ideals independent of the number of generators. Under simila…
Filtrations of D-modules along semi-invariant functions
研究连通可约线性代数群G作用下光滑代数簇X上的D-模M的G分解，探讨V-、Hodge和权重滤子。通过半不变函数局部化纯twistor D-模，确定不可约同构分量中元素的权重级别与根函数的乘数有关。
We analyze the sequence $\mathcal L^*_J(I)$ of mixed Łojasiewicz exponents attached to any pair $I,J$ of monomial ideals of finite colength of the ring of analytic function germs $(\mathbb C^n,0)\to \mathbb C$. In particular, we obtain a combinatorial expression for this sequence when $J$ is diagonal. We also show several relations of $\mathcal L^*_J(I)$ with other numerical invariants associated to $I$ and $J$.
The log canonical threshold of products of ideals and mixed Łojasiewicz exponents
本文证明了环 $\mathcal O_n$ 中两理想之积 $IJ$ 的对数典范阈值关于其混合 Łojasiewicz 指数序列的精确下界，并给出了极大理想情形下等式成立的充要条件。
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