Wolfgang Ebeling, Sabir M. Gusein-Zade
We define a natural filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ of a complex analytic variety $(V,0)$ related with the geometry of arcs on the variety and describe some properties of it.
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In this paper we extend Gazaki's results on the Chow groups of abelian varieties to the higher Chow groups. We introduce a Gazaki type filtration on the higher Chow group of zero-cycles on an abelian variety, whose graded quotients are connected to the Somekawa type $K$-group. Via the étale cycle map, we will compare this filtration with a filtration on the étale cohomology induced by the Hochschild-Serre spectral sequence. As an application over local fields, we obtain an estimate of the kernel of the reciprocity map.
In a previous paper, the authors introduced a filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ $(V,0)$ of a complex analytic variety defined by arcs on the singularity and called the arc filtration. The Poincaré series of this filtration were computed for simple surface singularities in the 3-space. Here they are computed for surface singularities from Arnold's lists including uni- and bimodular ones. The classification of the unimodular singularities by these Poincaré series turns out to be in accordance with their hierarchy defined by E. Brieskorn using the adjacency r…
A Filtration of the Chow Group of Zero-Cycles for a Product of Curves and an Abelian Variety
本文提出了一种针对曲线与阿贝尔簇乘积的零维 Chow 群的滤波方法，并通过 Somekawa K-群描述了其生成元和关系。
In this paper, we give a definition of volume for subsets in the space of arcs of an algebraic variety, and study its properties. Our main result relates the volume of a set of arcs on a Cohen-Macaulay variety to its jet-codimension, a notion which generalizes the codimension of a cylinder in the arc space of a smooth variety.
For a generalized Kummer variety X of dimension 2n, we will construct for each 0 < i < n some co-isotropic subvarieties in X foliated by i-dimensional constant cycle subvarieties. These subvarieties serve to prove that the rational orbit filtration introduced by Voisin on the Chow group of zero-cycles of a generalized Kummer variety coincides with the induced Beauville decomposition from the Chow ring of abelian varieties. As a consequence, the rational orbit filtration is opposite to the conjectural Bloch-Beilinson filtration for generalized Kummer varieties.
The paper surveys several results on the topology of the space of arcs of an algebraic variety and the Nash problem on the arc structure of singularities.
In this paper, we prove the l-independence of monodromy weight filtration for a geometrically smooth variety over an equicharacteristic local field. We also prove the l-independence for the geometric monodromy representation on the associated graded module of weight monodromy filtration.
Earlier, for an action of a finite group $G$ on a germ of an analytic variety, an equivariant $G$-Poincaré series of a multi-index filtration in the ring of germs of functions on the variety was defined as an element of the Grothendieck ring of $G$-sets with an additional structure. We discuss to which extend the $G$-Poincaré series of a filtration defined by a set of curve or divisorial valuations on the ring of germs of analytic functions in two variables determines the (equivariant) topology of the curve or of the set of divisors.
Given an n-dimensional variety Z with rational singularities, we conjecture that for a resolution of singularities whose reduced exceptional divisor E has simple normal crossings, the (n-1)-th higher direct image of the sheaf of differential forms with log poles along E vanishes. We prove this when Z has isolated singularities and when it is a toric variety. We deduce that for a divisor D with isolated rational singularities on a smooth complex n-dimensional variety X, the generation level of Saito's Hodge filtration on the localization of the structure sheaf along D is at most n-3.
We prove that, given a symmetrically distinguished correspondence of a suitable complex abelian variety (which include any abelian variety of dimension atmost 5, powers of complex elliptic curves, etc.) which vanishes as a morphism on a certain quotient of its middle singular cohomology, then it vanishes as a morphism on the deepest part of a particular filtration on the Chow group of 0-cycles of the abelian variety. As a consequence, we prove that given an automorphism of such an abelian variety, which acts as the identity on a certain quotient of its middle singular cohomology, then it acts …
There is a natural descending filtration on the singular cohomology of a complex smooth projective variety called the coniveau filtration. The generalized Hodge conjecture would imply, rather trivially, that the coniveau filtration is compatible with pushforwards, pullbacks and products. The purpose of this paper is to prove this statement unconditionally. This completes the argument in math.AG/0102070.
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…
We describe the standard and Leray filtrations on the cohomology groups with compact supports of a quasi projective variety with coefficients in a constructible complex using flags of hyperplane sections on a partial compactification of a related variety. One of the key ingredients of the proof is the Lefschetz hyperplane theorem for perverse sheaves and, in an appendix, we discuss the base change maps for constructible sheaves on algebraic varieties and their role in a proof, due to Beilinson, of the Lefschetz hyperplane theorem.
For every $n$-dimensional smooth projective variety $X$ over ${\bf C}$, the motive $M(X)$ is expected to admit a Chow-Künneth decomposition $M_0(X)\oplus\cdots \oplus M_{2n}(X)$. Inspired by the slice filtration of $M(X)$ we propose the definitions of $M_2(X)$ and $M_{2n-2}(X)$.
For a smooth projective variety X, let CH(X) be the Chow ring (with rational coefficients) of algebraic cycles modulo rational equivalence. The conjectures of Bloch and Beilinson predict the existence of a functorial ring filtration of CH(X). We want to investigate for which varieties this filtration splits, that is, comes from a graduation on CH(X) -- this occurs for K3 surfaces and, conjecturally, for abelian varieties. We observe that, though the Bloch-Beilinson filtration is only conjectural, the fact that it splits has some simple consequences which can be tested in concrete examples. Nam…
This paper gives a map from the set of families of arcs on a variety to the set of valuations on the rational function field of the variety We characterize a family of arcs which corresponds to a divisorial valuation by this map. We can see that both the Nash map and a certain McKay correspondence are the restrictions of this map. This paper also gives the affirmative answer to the Nash problem for a non-normal variety in a certain category. As a corollary, we get the affirmative answer for a non-normal toric variety.
For the Grothendieck group of a split simple linear algebraic group, the twisted gamma-filtration provides a useful tool for constructing torsion elements in gamma-rings of twisted flag varieties. In this paper, we construct a non-trivial torsion element in the gamma-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline. We use this torsion element to study algebras with orthogonal involutions.
Effective zero-cycles and the Bloch-Beilinson filtration
本文将Voisin关于两点有理等价的猜想推广到高次有效零闭链，在阿贝尔簇与hyper-Kähler流形上给出证据，并对K3曲面上半稳定层模空间给出显式判据，同时提出关于对角线所属对应子代数的第二个猜想。
We prove a duality theorem for the $p$-adic etale motivic cohomology of a variety $U$ which is the complement of a divisor on a smooth projective variety over $\F_p$. This extends the duality theorems of Milne and Jannsen-Saito-Zhao. The duality introduces a filtration on $H^1_{\etl}(U, {\Q}/{\Z})$. We identify this filtration to the classically known Matsuda filtration when the reduced part of the divisor is smooth. We prove a reciprocity theorem for the idele class groups with modulus introduced by Kerz-Zhao and Rulling-Saito. As an application, we derive the failure of Nisnevich descent for…
Let $X$ be a smooth complex cubic fourfold and let $F$ be the variety of lines of $X$. The variety $F$ is known to be a smooth projective hyperkaehler fourfold, which is moreover endowed with a self rational map $ϕ: F -\rightarrow F$ first constructed by C. Voisin. Here we define a filtration of Bloch--Beilinson type on the Chow group of zero-cycles $CH_0(F)$ which canonically splits under the action of $ϕ$, thereby answering in this case a question of A. Beauville. Moreover, we show that this filtration is of motivic origin, in the sense that it arises from a Chow--Kuenneth decomposition of t…
We study the K-stability of a polarised variety with non-reductive automorphism group. We associate a canonical filtration of the co-ordinate ring to each variety of this kind, which destabilises the variety in several examples which we compute. We conjecture this holds in general. This is an algebro-geometric analogue of Matsushima's theorem regarding the existence of constant scalar curvature Kähler metrics. As an application, we give an example of an orbifold del Pezzo surface without a Kähler-Einstein metric.
For a smooth complex projective variety X defined over a number field, we have filtrations on the Chow groups depending of the choice of realizations. If the realization consists of mixed Hodge structure without any additional structure, we can show that the obtained filtration coincides with the filtration of Green and Griffiths, assuming the Hodge conjecture. In the case the realizations contain Hodge structure and etale cohomology, we prove that if the second graded piece of the filtration does not vanish, it contains a nonzero element which is represented by a cycle defined over a field of…
A cohomology class of a smooth complex variety of dimension $n$ has coniveau $\geq c$ if it vanishes in the complement of a closed subvariety of codimension $\geq c$, and has strong coniveau $\geq c$ if it comes by proper pushforward from the cohomology of a smooth variety of dimension $\leq n-c$. We show that these two notions differ in general, both for integral classes on smooth projective varieties and for rational classes on smooth open varieties.
We construct a Harder-Narasimhan filtration for rank $2$ tensors, where there does not exist any such notion a priori, as coming from a GIT notion of maximal unstability. The filtration associated to the 1-parameter subgroup of Kempf giving the maximal way to destabilize, in the GIT sense, a point in the parameter space of the construction of the moduli space of rank $2$ tensors over a smooth projective complex variety, does not depend on certain integer used in the construction of the moduli space, for large values of the integer. Hence, this filtration is unique and we define the Harder-Nara…
We express the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs.
We study the first step of the weight filtration on the cohomology of a proper complex algebraic variety, which we call the combinatorial part. We obtain a natural upper bound on its size, which gives rather strong information about the topology of rational singularities.
We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the rationality of the Poincare series associated to p-adic points on a p-adic variety. The main tools which are used are semi-algebraic geometry in spaces of power series and motivic integration (a notion introduced by M. Kontsevich). In particular we develop the theory of motivic integration for semi-algebraic sets of formal arcs on singular algebraic varieties, we prove a change of variable formula …
We define a filtration on the Chow groups of a smooth projective variety X over a field k by using the cycle map into continuous l-adic etale cohomology. The main theorem says that if k is a function field in one variable over a finite field, this filtration for zero cycles is of length at most one modulo the kernel of the cycle map.
For a regular function f on a smooth complex quasi-projective variety, J.-D. Yu introduced in arXiv:1203.2338 a filtration (the irregular Hodge filtration) on the de Rham complex with twisted differential d+df, extending a definition of Deligne in the case of curves. In this article, we show the degeneration at E_1 of the spectral sequence attached to the irregular Hodge filtration, by using the method of arXiv:0804.4328. We also make explicit the relation with a complex introduced by M. Kontsevich and give details on its proof of the corresponding E_1 degeneration, by reduction to characteris…
We give a dimension bound on the irreducible components of the characteristic variety of a system of linear partial differential equations defined from a suitable filtration of the Weyl algebra $A_{n}(k)$. This generalizes an important consequence of the fact that a characteristic variety defined from the order filtration is involutive. More explicitly, we consider a filtration of $A_{n}(k)$ induced by any vector $(u,v) \in {\mathbb Z}^{n}\times {\mathbb Z}^{n}$ such that the associated graded algebra is the commutative polynomial ring in $2n$ indeterminates. The order filtration is the specia…
Given an integral p-adic variety, we observe that if the integral Hodge--de Rham spectral sequence behaves nicely, then the special fiber knows the Hodge numbers of the generic fiber. Applying recent advancements of integral p-adic Hodge theory, we show that such a nice behavior is guaranteed if the p-adic variety can be lifted to an analogue of second Witt vectors and satisfies some bound on dimension and ramification index. This is a (ramified) mixed characteristic analogue of results due to Deligne--Illusie and Fontaine--Messing. Lastly, we discuss an example illustrating the necessity of t…
Let X be a complex algebraic variety, and L(X) be the scheme of formal arcs in X. Let f be an arc whose image is not contained in the singularities of X. We show that the formal neighborhood of f in L(X) admits a decomposition into a product of an infinite-dimensional smooth piece, and a piece isomorphic to the formal neighborhood of a closed point of a scheme of finite type.
Suppose that $X$ is a smooth projective variety and that $C$ is a general member of a family of free rational curves on $X$. We prove several statements showing that the Harder-Narasimhan filtration of $T_{X}|_{C}$ is approximately the same as the restriction of the Harder-Narasimhan filtration of $T_{X}$ with respect to the class of $C$. When $X$ is a Fano variety, we prove that the set of all restricted tangent bundles for general free rational curves is controlled by a finite set of data. We then apply our work to analyze Peyre's "freeness" formulation of Manin's Conjecture in the setting o…
Let $(X, ξ)$ be a polarized affine variety, i.e. an affine variety $X$ with a (possibly irrational) Reeb vector field $ξ$. We define the volume of a filtration of the coordinate ring of $X$ in terms of the asymptotics of the average of jumping numbers. When the filtration is finitely generated, it induces a Fubini-Study function $φ$ on the Berkovich analytification of $X$. In this case, we define the Monge-Ampère energy for $φ$ using the theory of forms and currents on Berkovich spaces developed by Chambert-Loir and Ducros, and show that it agrees with the volume of the filtration. In the spec…
We give a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. The interest of this proof is that it provides simple effective bounds (depending on the degree of the canonical line bundle) for the degree of the isotrivial finite cove…
Let X be a smooth variety over an algebraically closed field of characteristic p > 0, Z a smooth divisor, and j : U = X\Z --> X the natural inclusion. An axiomatizing of the properties of a V -filtration on a unit F-crystal is proposed and is proven to determine a unique filtration. It is shown that if M is a tame unit F-crystal on U then such a V -filtration along Z exists on j_*M. The degree zero component of the associated graded module is proven to be the (unipotent) nearby cycles functor of Grothendieck and Deligne under the Emerton-Kisin Riemann-Hilbert correspondence. A few applications…
Let ${\cal L}$ be a local system on the complement $X^{\star}$ of a normal crossing divisor (NCD) $ Y$ in a smooth analytic variety $X$ and let $ j: X^{\star} = X - Y \to X $ denotes the open embedding. The purpose of this paper is to describe a weight filtration $W$ on the direct image ${\bf j}_{\star}{\cal L} $ and in case a morphism $f: X \to D$ to a complex disc is given with $Y = f^{-1}(0)$, the weight filtration on the complex of nearby cocycles $Ψ_f ({\cal L})$ on $Y$. A comparison theorem shows that the filtration coincides with the weight defined by the logarithm of the monodromy and …
We study projective completions of affine algebraic varieties induced by filtrations on their coordinate rings. In particular, we study the effect of the 'multiplicative' property of filtrations on the corresponding completions and introduce a class of projective completions (of arbitrary affine varieties) which generalizes the construction of toric varieties from convex rational polytopes. As an application we recover (and generalize to varieties over algebraically closed fields of arbitrary characteristics) a 'finiteness' property of divisorial valuations over complex affine varieties proved…
Let $A$ be an abelian variety over a discretely valued field. Edixhoven has defined a filtration on the special fiber of the Néron model of $A$ that measures the behaviour of the Néron model under tame base change. We interpret the jumps in this filtration in terms of lattices of logarithmic differential forms in the case where $A$ is the Jacobian of a curve $C$, and we give a compact explicit formula for the jumps in terms of the combinatorial reduction data of $C$.
In this paper, we consider $\mathbb{Z}^{r}-$graded modules on the $\mathrm{Cl}(X)$ $-$graded Cox ring $\mathbb{C}[x_{1},\dotsc,x_{r}]$ of a smooth complete toric variety $X$. Using the theory of Klyachko filtrations in the reflexive case, we construct a collection of lattice polytopes codifying the multigraded Hilbert function of the module. We apply this approach to reflexive $\mathbb{Z}^{s+r+2}$-graded modules over non-standard bigraded polynomial rings $\mathbb{C}[x_{0},\dotsc,x_{s},y_{0},\dotsc,y_{r}]$. In this case, we give sharp bounds for the multigraded regularity index of their multig…
Families of m-jet spaces and arc spaces. Let V be an algebraic variety defined over an algebraically closed field of characteristic zero. The m-jet spaces and the arc space provide the information on the geometry of the variety V, therefore it is natural to ask the following questions: When will a deformation of V induce a deformation of the m-jet spaces and the arc space? If one considers a deformation of V which admits a flat simultaneous resolution, how will the image of the Nash application vary? In section 3 some partial answers to these questions can be found. In section 4 two families o…
Filtrations of the Chow group of zero-cycles on abelian varieties and behavior under isogeny
本文研究阿贝尔簇零维Chow群上的Bloch-Beilinson型滤过在同源下的行为，证明乘以n映射在商群上诱导乘以n^r，并在椭圆曲线乘积情形与已知滤过一致，同时为Raskind-Spiess和Colliot-Thélène猜想提供新证据。
Our main goal is to show that the Gelfand--Tsetlin toric degeneration of the type A flag variety can be obtained within a degenerate representation-theoretic framework similar to the theory of PBW degenerations. In fact, we provide such frameworks for all Gröbner degenerations intermediate between the flag variety and the GT toric variety. These degenerations are shown to induce filtrations on the irreducible representations and the associated graded spaces are acted upon by a certain associative algebra. To achieve our goal, we construct embeddings of our Gröbner degenerations into the projec…
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 describe mirror symmetry for weak toric Fano manifolds as an equivalence of D-modules equipped with certain filtrations. We discuss in particular the logarithmic degeneration behavior at the large radius limit point, and express the mirror correspondence as an isomorphism of Frobenius manifolds with logarithmic poles. The main tool is an identification of the Gauss-Manin system of the mirror Landau-Ginzburg model with a hypergeometric D-module, and a detailed study of a natural filtration defined on this differential system. We obtain a solution of the Birkhoff problem for lattices defined …
Let ${\cal L}$ be a variation of Hodge structures on the complement $X^{*}$ of a normal crossing divisor (NCD) $ Y$ in a smooth analytic variety $X$ and let $ j: X^{*} = X - Y \to X $ denotes the open embedding. The purpose of this paper is to describe the weight filtration $W$ on a combinatorial logarithmic complex computing the (higher) direct image ${\bf j}_{*}{\cal L} $, underlying a mixed Hodge complex when $X$ is proper, proving in this way the results in the note [14] generalizing the constant coefficients case. When a morphism $f: X \to D$ to a complex disc is given with $Y = f^{-1}(0)…
To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincaré series, the generalized Poincaré series, and the generalized semigroup Poincaré series. The Hilbert function and the generalized Poincaré series are equivalent in the sense that each of them determines the other one. We show that for a filtration on the ring of germs of holomorphic functions in two variables defined by a collection of plane valuations both of them are equivalent to the generalized semigroup Poincaré s…
We provide an explicit presentation of the equivariant cohomology ring of the compactified Jacobian $J_{q/p}$ of the rational curve $C_{q/p}$ with planar equation $x^{q}=y^{p}$ for $(p,q)=1$. We also prove analogous results for the closely related affine Springer fiber $Sp_{q/p}$ in the affine flag variety of $SL_{p}$. We show that the perverse filtration on the cohomology of $J_{q/p}$ is multiplicative, and the associated graded ring under the perverse filtration is a degeneration of the ring of functions on a moduli space of maps $\mathbb{P}^{1}\to C_{q/p}$. We also propose several conjectur…
Let $ f_i: X \rightarrow {\bf C}$, for $i$ integer between $ 1$ and $ p $, be analytic functions defined on a complex analytic variety $X$. Let us consider $ {\cal D}_X $ the ring of linear differential operators and $ {\cal D}_X [s_1, \ldots, s_p] = {\bf C}_X [s_1, \ldots, s_p] \otimes_ {\bf C} {\cal D}_X$. Let $ m $ be a section of a holonomic $ {\cal D}_X $-Module. We denote $ {\cal B}(m, x_0, f_1, \ldots, f_p) $ the ideal of $ { \bf C} [s_1, \ldots, s_p] $ constituted by the polynomials $ b $ satisfying in the neighborhood of $ x_0 \in X$ : $$ B (s_1, \ldots, s_p) m f_1^{ s_1} \ldots f_p ^…
Filtrations and asymptotic geometry of non-Archimedean norms on section rings
本文研究了非阿基米德范数在截面环上的滤子作用及其渐近几何特性，推广了之前的成果并提供了新的测度收敛结果。
For an affine toric variety X we compute the Poincare series of the multi-index filtration defined by a finite number of monomial divisorial valuations on the ring O_{X,0}. We give an alternative description of the Poincare series as an integral with respect to the Euler characteristic over the projectivization of the space of germs O_{X,0}. In particular we study divisorial valuations on the ring O_{C^d,0} that arise by considering toric constellations. We give an explicit formula for the Poincare series and a nice geometric description. This generalizes an expression of the Poincare series f…
Given a finitely-generated group $π$ and a linear algebraic group $G$, the representation variety Hom$(π,G)$ has a natural filtration by the characteristic varieties associated to a rational representation of $G$. Its algebraic counterpart, the space of $\mathfrak{g}$-valued flat connections on a commutative, differential graded algebra $(A,d)$ admits a filtration by the resonance varieties associated to a representation of $\mathfrak{g}$. We establish here a number of results concerning the structure and qualitative properties of these embedded resonance varieties, with particular attention t…
Let $n$ be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $GL(n,{\mathbb{C}})/B$ such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $GL(n-1,{\mathbb{C}})/B$, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations…
Filtrations on D-modules and multiplicities of roots of Bernstein-Sato polynomials
本文关联Bernstein-Sato多项式根的重数与奇点不变量，引入模b-函数并给出其算法，解决Gelfand问题的渐近解，还将结果应用于Torelli问题并给出肯定答案。
We construct a Z_3-slice to the natural action of PGL(V) on the variety of smooth arcs in the projective space P(V).
We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power series of bounded complexity lying on an algebraic variety defined over the field of power series.
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