Canonical Filtration Research Articles

Upper estimates for the expected Betti numbers of random subcomplexes

We observe that every zero-dimensional simplicial cochain defines a canonical filtration of a finite simplicial complex and deduce upper estimates for the expected Betti numbers of codimension one random subcomplexes in its support. Moreover, a monotony theorem improves these estimates given any packing of disjoint simplices.

Moduli of vector bundles on primitive multiple schemes

A primitive multiple scheme is a Cohen–Macaulay scheme Y such that the associated reduced scheme X = Yred is smooth, irreducible, and that Y can be locally embedded in a smooth variety of dimension [Formula: see text]. If n is the multiplicity of Y, there is a canonical filtration [Formula: see text], such that [Formula: see text] is a primitive multiple scheme of multiplicity i. The simplest example is the trivial primitive multiple scheme of multiplicity n associated to a line bundle L on X: it is the nth infinitesimal neighborhood of X, embedded in the line bundle [Formula: see text] by the zero section. The main subject of this paper is the construction and properties of fine moduli spaces of vector bundles on primitive multiple schemes. Suppose that [Formula: see text] is of multiplicity n, and can be extended to [Formula: see text] of multiplicity [Formula: see text], and let [Formula: see text] be a fine moduli space of vector bundles on [Formula: see text]. With suitable hypotheses, we construct a fine moduli space [Formula: see text] for the vector bundles on [Formula: see text] whose restriction to [Formula: see text] belongs to [Formula: see text]. It is an affine bundle over the subvariety [Formula: see text] of bundles that can be extended to [Formula: see text]. In general this affine bundle is not banal. This applies in particular to Picard groups. We give also many new examples of primitive multiple schemes Y such that the dualizing sheaf [Formula: see text] is trivial.

DUALITY FOR GORENSTEIN MULTIPLE STRUCTURES ON SMOOTH ALGEBRAIC VARIETIES

One gives a characterization of the Gorenstein nilpotent scheme structures on a smooth algebraic variety as support, in terms of a duality property of the graded objects associated to two canonical filtrations.

Motive of the SL4-character variety of torus knots

In this paper, we compute the motive of the character variety of representations of the fundamental group of the complement of an arbitrary torus knot into SL4(k), for any algebraically closed field k of zero characteristic. For that purpose, we introduce a stratification of the variety in terms of the type of a canonical filtration attached to any representation. This allows us to reduce the computation of the motive to a combinatorial problem.

Poincaré–Birkhoff–Witt theorem for Poisson enveloping algebra

Let A be a Poisson polynomial algebra and let I be a Poisson ideal of A. It is established that the Poisson algebra satisfies the Poincaré-Birkhoff-Witt theorem. That is, the symmetric algebra of the Kähler differential Ω B of B is isomorphic as a graded B-algebra to the associated graded algebra Be with respect to a canonical filtration of Be , where Be is the Poisson enveloping algebra of B.

On Fredholm boundary conditions on manifolds with corners, I: Global corner cycles obstructions

Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles. Our main theorem is that, for any manifold with corners $X$ of any codimension, there is a natural and explicit morphism $$K_*(\mathcal{K}_b(X)) \stackrel{T}{\longrightarrow} H^{pcn}_*(X,\mathbb{Q})$$ between the $K-$theory group of the algebra $\mathcal{K}_b(X)$ of $b$-compact operators for $X$ and the periodic conormal homology group with rational coeficients, and that $T$ is a rational isomorphism. As shown by the first two authors in a previous paper this computation implies that the rational groups $H^{pcn}_{ev}(X,\mathbb{Q})$ provide an obstruction to the Fredholm perturbation property for compact connected manifold with corners. The difference with respect to the previous article of the first two authors in which they solve this problem for low codimensions is that we overcome in the present article the problem of computing the higher spectral sequence K-theory differentials associated to the canonical filtration by codimension by introducing an explicit topological space whose singular cohomology is canonically isomorphic to the conormal homology and whose K-theory is naturally isomorphic to the $K-$theory groups of the algebra $\mathcal{K}_b(X)$.

Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groups

Il est connu que le groupe d’automorphismes d’une variété de Fano K-polystable est réductif. Codogni et Dervan ont construit une filtration canonique de l’anneau des sections, appelée filtration de Loewy, et ont conjecturé que la filtration déstabilise n’importe quelle variété de Fano avec le groupe d’automorphismes non réductif. Dans cette note, nous fournissons un contre-exemple à leur conjecture.

Primitive multiple schemes

A primitive multiple scheme is a Cohen–Macaulay scheme Y such that the associated reduced scheme \(X=Y_{\mathrm{red}}\) is smooth, irreducible, and Y can be locally embedded in a smooth variety of dimension \(\dim (X)+1\). If n is the multiplicity of Y, there is a canonical filtration \(X=X_1\subset X_2\subset \cdots \subset X_n=Y\), such that \(X_i\) is a primitive multiple scheme of multiplicity i. The simplest example is the trivial primitive multiple scheme of multiplicity n associated to a line bundle L on X: it is the nth infinitesimal neighborhood of X, embedded in the line bundle \(L^*\) by the zero section. Let . The primitive multiple schemes of multiplicity n are obtained by taking an open cover \((U_i)\) of a smooth variety X and by gluing the schemes \(U_i\,{\times }\, \mathbf{Z}_n\) using automorphisms of \(U_{ij}\,{\times }\, \mathbf{Z}_n\) that leave \(U_{ij}\) invariant. This leads to the study of the sheaf of nonabelian groups of automorphisms of \(X\,{\times }\, \mathbf{Z}_n\) that leave the X invariant, and to the study of its first cohomology set. If \(n\geqslant 2\) there is an obstruction to the extension of \(X_n\) to a primitive multiple scheme of multiplicity \(n+1\), which lies in the second cohomology group \(H^2(X,E)\) of a suitable vector bundle E on X. In this paper we study these obstructions and the parametrization of primitive multiple schemes. As an example we show that if \(X={{\mathbb {P}}}_m\) with \(m\geqslant 3\), all the primitive multiple schemes are trivial. If \(X={{\mathbb {P}}}_2\), there are only two nontrivial primitive multiple schemes, of multiplicities 2 and 4, which are not quasi-projective. On the other hand, if X is a projective bundle over a curve, we show that there are infinite sequences \( X=X_1\subset X_2\subset \cdots \subset X_n\subset X_{n+1}\subset \cdots \) of nontrivial primitive multiple schemes.

On the Andreadakis Problem for Subgroups of IA n

Abstract Let $F_n$ be the free group on $n$ generators. Consider the group $IA_n$ of automorphisms of $F_n$ acting trivially on its abelianization. There are two canonical filtrations on $IA_n$: the 1st one is its lower central series $\Gamma _*$; the 2nd one is the Andreadakis filtration $\mathcal A_*$, defined from the action on $F_n$. The Andreadakis problem consists in understanding the difference between these filtrations. Here, we show that they coincide when restricted to the subgroup of triangular automorphisms and to the pure braid group.

On the stable Andreadakis problem

Let Fn be the free group on n generators. Consider the group IAn of automorphisms of Fn acting trivially on its abelianization. There are two canonical filtrations on IAn: the first one is its lower central series Γ⁎; the second one is the Andreadakis filtration A⁎, defined from the action on Fn. In this paper, we establish that the canonical morphism between the associated graded Lie rings L(Γ⁎) and L(A⁎) is stably surjective. We then investigate a p-restricted version of the Andreadakis problem. A calculation of the Lie algebra of the classical congruence group is also included.

Harder–Narasimhan theory for linear codes (with an appendix on Riemann–Roch theory)

In this text we develop some aspects of Harder–Narasimhan theory, slopes, semistability and canonical filtration, in the setting of combinatorial lattices. Of noticeable importance is the Harder–Narasimhan structure associated to a Galois connection between two lattices. It applies, in particular, to matroids.We then specialize this to linear codes. This could be done from at least three different approaches: using the sphere-packing analogy, or the geometric view, or the Galois connection construction just introduced. A remarkable fact is that these all lead to the same notion of semistability and canonical filtration. Relations to previous propositions toward a classification of codes, and to Wei's generalized Hamming weight hierarchy, are also discussed.Last, we study the two important questions of the preservation of semistability (or more generally the behavior of slopes) under duality, and under tensor product. The former essentially follows from Wei's duality theorem for higher weights—and its matroid version—which we revisit in an appendix, developing analogues of the Riemann–Roch, Serre duality, Clifford, and gap and gonality sequence theorems. Likewise the latter is closely related to the bound on higher weights of a tensor product, conjectured by Wei and Yang, and proved by Schaathun in the geometric language, which we reformulate directly in terms of codes. From this material we then derive semistability of tensor product.

Normalization of Closed Ekedahl—Oort Strata

AbstractWe apply our theory of partial flag spaces developed with W. Goldring to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti–Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl–Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.

On the Hodge–Newton filtration for p-divisible groups of Hodge type

A p-divisible group, or more generally an F-crystal, is said to be Hodge-Newton reducible if its Hodge polygon passes through a break point of its Newton polygon. Katz proved that Hodge-Newton reducible F-crystals admit a canonical filtration called the Hodge-Newton filtration. The notion of Hodge-Newton reducibility plays an important role in the deformation theory of p-divisible groups; the key property is that the Hodge-Newton filtration of a p-divisible group over a field of characteristic p can be uniquely lifted to a filtration of its deformation. We generalize Katz's result to F-crystals that arise from an unramified local Shimura datum of Hodge type. As an application, we give a generalization of Serre-Tate deformation theory for local Shimura data of Hodge type. We also apply our deformation theory to study some congruence relations on Shimura varieties of Hodge type.

Jet Bundles on Projective Space II

In previous papers the structure of the jet bundle as P-module has been studied using different techniques. In this paper we use techniques from algebraic groups, sheaf theory, generliazed Verma modules, canonical filtrations of irreducible SL(V)-modules and annihilator ideals of highest weight vectors to study the canonical filtration Ul (g)Ld of the irreducible SL(V)-module H0 (X, iX(d))* where X = i‡(m, m + n). We study Ul (g)Ld using results from previous papers on the subject and recover a well known classification of the structure of the jet bundle il (i(d)) on projective space i(V*) as P-module. As a consequence we prove formulas on the splitting type of the jet bundle on projective space as abstract locally free sheaf. We also classify the P-module of the first order jet bundle iX1 (iX (d)) for any d ≥ 1. We study the incidence complex for the line bundle i(d) on the projective line and show it is a resolution of the ideal sheaf of I l (i(d)) - the incidence scheme of i(d). The aim of the study is to apply it to the study of syzygies of discriminants of linear systems on projective space and grassmannians.

Erratum - Some Characterizations of Mixed Poisson Processes

Assumption A. Let X := {Xt}t∈R+ be a process with right-continuous paths such that its canonical filtration F := {Ft}t∈R+ is also right-continuous. Sketch of the proof of Lemma 4.1. For the “only if” part, assuming that X is P -conditionally independent, it follows that there exists a PΘ-null set N ∈ B such that, for any θ / ∈ N , {Xt}t∈Q+ is Pθindependent, i.e., the condition (∗) : EPθ [χAf(Xt)] = Pθ(A)EPθ [f(Xt)] is true, for all s, t ∈ Q+ with s < t, all bounded Borel measurable f : R → R and all A ∈ Fs. Using standard arguments, condition (∗) can be proved for any s, t ∈ R+ with s < t. In Lemma 4.6, put F := { Ft}t∈R+ , assume for (i) =⇒ (ii) as well as for Proposition 4.8, thatX is non-decreasing, and insert assertion “In particular, if Θ ∈ L1(P ) then {Nt − tΘ}t∈R+ is a (P, A)-martingale if and only if {Nt − tθ}t∈R+ is a (Pθ, A)-martingale for PΘ-a.a. θ ∈ R”. In Theorem 4.10, the families {Nt − EPθ [Nt]}t∈R+ and {Nt − EP [Nt | Θ]}t∈R+ must be replaced by {Nt − tθ}t∈R+ and {Nt − tΘ}t∈R+ respectively, with the proof remaining unchanged.

Courbes multiples primitives et déformations de courbes lisses

A primitive multiple curve is a Cohen-Macaulay scheme Y over ℂ such that the reduced scheme C = Y red is a smooth curve, and that Y can be locally embedded in a smooth surface. In general such a curve Y cannot be embedded in a smooth surface. If Y is a primitive multiple curve of multiplicity n, then there is a canonical filtration C = C1 ⊂⋅⋅⋅⊂ Cn = Y such that Ci is a primitive multiple curve of multiplicity i. The ideal sheaf IC of C in Y is a line bundle on Cn-1. Let T be a smooth curve and t0 ∈ T a closed point. Let D→ T be a flat family of projective smooth irreducible curves, and C = Dt0. Then the n-th infinitesimal neighbourhood of C in D is a primitive multiple curve Cn of multiplicity n, embedded in the smooth surface D, and in this case IC is the trivial line bundle on Cn-1. Conversely, we prove that every projective primitive multiple curve Y = Cn such that IC is the trivial line bundle on Cn-1 can be obtained in this way.

An extension of the Clark–Ocone formula under benchmark measure for Lévy processes

The classical Clark–Ocone theorem states that any random variable can be represented as where denotes the conditional expectation, is a Brownian motion with canonical filtration and D denotes the Malliavin derivative in the direction of W. Since many applications in financial mathematics require representation of random variables with respect to risk neutral martingale measure, an equivalent martingale measure version of this theorem was stated by Karatzas and Ocone (Stoch. Stoch. Rep. 34 (1991), 187–220). In this paper, we extend these results to be valid for square integrable pure jump Lévy processes with no drift and for square integrable Itô–Lévy processes using Malliavin calculus and white noise analysis. This extension might be useful for some applications in finance. As an application of our result, we calculate explicitly the closest hedge strategy for the digital option whose pay-off, , is square integrable and the stock price is driven by a Lévy process.

Canonical filtrations and stability of direct images by Frobenius morphisms II

We study the stability of direct images by Frobenius morphisms. We prove that if the cotangent vector bundle of a nonsingular projective surface $X$ is semistable with respect to a numerically positive polarization divisor satisfying certain conditions, then the direct images of the cotangent vector bundle tensored with line bundles on $X$ by Frobenius morphisms are semistable with respect to the polarization. Hence we see that the de Rham complex of $X$ consists of semistable vector bundles if $X$ has the semistable cotangent vector bundle with respect to the polarization with certain mild conditions.

Canonical filtrations and stability of direct images by Frobenius morphisms

We study the stability of direct images by Frobenius morphisms. First, we compute the first Chern classes of direct images of vector bundles by Frobenius morphisms modulo rational equivalence up to torsions. Next, introducing the canonical filtrations, we prove that if $X$ is a nonsingular projective minimal surface of general type with semistable $\Omega_X^1$ with respect to the canonical line bundle $K_X$, then the direct images of line bundles on $X$ by Frobenius morphisms are semistable with respect to $K_X$.

Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space

Let (W, H, μ) be the classical Wiener space. Assume that U = I W + u is an adapted perturbation of identity, i.e., u : W → H is adapted to the canonical filtration of W. We give some sufficient analytic conditions on u which imply the invertibility of the map U. In particular it is shown that if $${u\in {\rm ID}_{p,1}(H)}$$ is adapted and if $${\exp(\frac{1}{2}\|\nabla u\|_2^2-\delta u)\in L^q(\mu)}$$ , where p −1 + q −1 = 1, then I W + u is almost surely invertible. With the help of this result it is shown that if $${\nabla u\in L^\infty(\mu,H\otimes H)}$$ , then the Girsanov exponential of u times the Wiener measure satisfies the logarithmic Sobolev inequality and this implies the invertibility of U = I W + u . As a consequence, if, there exists an integer k ≥ 1 such that $${\|\nabla^k u\|_{H^{\otimes(k+1)}}\in L^\infty(\mu)}$$ , then I W + u is again almost surely invertible under the almost sure continuity hypothesis of $${t\to\nabla^i \dot{u}_t}$$ for i ≤ k − 1.