VERTICAL TANGENTIAL INVARIANTS ON SOME FOLIATED LAGRANGE SPACES

2013 ◽  
Vol 10 (04) ◽  
pp. 1320002
Author(s):  
CRISTIAN IDA
Keyword(s):  
Tangent Bundle ◽  
Differential Forms ◽  
Characteristic Classes ◽  
Exterior Derivative ◽  
Cohomology Groups ◽  
Foliated Manifold ◽  
Smooth Complex ◽  
De Rham ◽  
Poincaré Lemma ◽  
Natural Decomposition

In this paper we consider a decomposition of tangentially differential forms with respect to the lifted foliation [Formula: see text] to the tangent bundle of a Lagrange space [Formula: see text] endowed with a regular foliation [Formula: see text]. First, starting from a natural decomposition of the tangential exterior derivative along the leaves of [Formula: see text], we define some vertical tangential cohomology groups of the foliated manifold [Formula: see text], we prove a Poincaré lemma for the vertical tangential derivative and we obtain a de Rham theorem for this cohomology. Next, in a classical way, we construct vertical tangential characteristic classes of tangentially smooth complex bundles over the foliated manifold [Formula: see text].

scholarly journals The Double Complex of a Blow-up

2019 ◽  
Author(s):  
Jonas Stelzig
Keyword(s):  
Blow Up ◽  
Differential Forms ◽  
Complex Manifolds ◽  
Double Complex ◽  
Smooth Complex ◽  
Projective Bundles ◽  
De Rham ◽  
Suitable Notion ◽  
Complex Valued ◽  
Compact Complex Manifolds

AbstractWe compute the double complex of smooth complex-valued differential forms on projective bundles over and blow-ups of compact complex manifolds up to a suitable notion of quasi-isomorphism. This simultaneously yields formulas for “all” cohomologies naturally associated with this complex (in particular, de Rham, Dolbeault, Bott–Chern, and Aeppli).

scholarly journals The de Rham theorem for the noncommutative complex of Cenkl and Porter

2002 ◽  
Vol 30 (11) ◽  
pp. 667-696 ◽  
Author(s):  
Luis Fernando Mejias
Keyword(s):  
Finite Type ◽  
Differential Forms ◽  
Natural Isomorphism ◽  
Graded Algebra ◽  
Differential Graded Algebra ◽  
Noncommutative Version ◽  
Simplicial Set ◽  
De Rham ◽  
Poincaré Lemma

We use noncommutative differential forms (which were first introduced by Connes) to construct a noncommutative version of the complex of Cenkl and PorterΩ∗,∗(X)for a simplicial setX. The algebraΩ∗,∗(X)is a differential graded algebra with a filtrationΩ∗,q(X)⊂Ω∗,q+1(X), such thatΩ∗,q(X)is aℚq-module, whereℚ0=ℚ1=ℤandℚq=ℤ[1/2,…,1/q]forq>1. Then we use noncommutative versions of the Poincaré lemma and Stokes' theorem to prove the noncommutative tame de Rham theorem: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, integration of forms induces a natural isomorphism ofℚq-modulesI:Hi(Ω∗,q(X),M)→Hi(X;M)for alli≥0. Next, we introduce a complex of noncommutative tame de Rham currentsΩ∗,∗(X)and we prove the noncommutative tame de Rham theorem for homology: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, there is a natural isomorphism ofℚq-modulesI:Hi(X;M)→Hi(Ω∗,q(X),M)for alli≥0.

scholarly journals The Filtered Poincaré Lemma in Higher Level (With Applications to Algebraic Groups)

2008 ◽  
Vol 191 ◽  
pp. 79-110
Author(s):  
Bernard Le Stum ◽  
Adolfo Quirós
Keyword(s):  
Differential Forms ◽  
Algebraic Groups ◽  
Classical Case ◽  
Group Scheme ◽  
Crystalline Cohomology ◽  
Hodge Filtration ◽  
Rham Complex ◽  
De Rham ◽  
Poincaré Lemma ◽  
Invariant Differential

AbstractWe show that the Poincaré lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincaré lemma for transversal crystals of level m. We interpret the de Rham complex in terms of what we call the Berthelot-Lieberman construction and show how the same construction can be used to study the conormal complex and invariant differential forms of higher level for a group scheme. Bringing together both instances of the construction, we show that crystalline extensions of transversal crystals by algebraic groups can be computed by reduction to the filtered de Rham complexes. Our theory does not ignore torsion and, unlike in the classical case (m = 0), not all invariant forms are closed. Therefore, close invariant differential forms of level m provide new invariants and we exhibit some examples as applications.

scholarly journals On twisted de Rham cohomology

1997 ◽  
Vol 146 ◽  
pp. 55-81 ◽  
Author(s):  
Alan Adolphson ◽  
Steven Sperber
Keyword(s):  
Differential Forms ◽  
Exterior Derivative ◽  
De Rham Cohomology ◽  
De Rham ◽  
Twisted De Rham Cohomology

Abstract.Consider the complex of differential forms on an open affine subvariety U of AN with differential where d is the usual exterior derivative and ø is a fixed 1-form on U. For certain U and ø, we compute the cohomology of this complex.

scholarly journals Effective de Rham cohomology — The general case

2019 ◽  
Vol 21 (05) ◽  
pp. 1850067
Author(s):  
Peter Scheiblechner
Keyword(s):  
Arbitrary Dimension ◽  
Differential Forms ◽  
Affine Variety ◽  
De Rham Cohomology ◽  
Smooth Complex ◽  
Polynomial Coefficients ◽  
Dimension Formula ◽  
De Rham ◽  
Smooth Affine ◽  
Single Exponential

Grothendieck has proved that each class in the de Rham cohomology of a smooth complex affine variety can be represented by a differential form with polynomial coefficients. We prove a single exponential bound on the degrees of these polynomials for varieties of arbitrary dimension. More precisely, we show that the [Formula: see text]th de Rham cohomology of a smooth affine variety of dimension [Formula: see text] and degree [Formula: see text] can be represented by differential forms of degree [Formula: see text]. This result is relevant for the algorithmic computation of the cohomology, but is also motivated by questions in the theory of ordinary differential equations related to the infinitesimal Hilbert 16th problem.

scholarly journals From Sheaf Cohomology to the Algebraic de Rham Theorem

2017 ◽  
Author(s):  
Fouad El Zein ◽  
Loring W. Tu
Keyword(s):  
Projective Variety ◽  
Differential Forms ◽  
Sheaf Cohomology ◽  
Coherent Sheaves ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Singular Cohomology ◽  
De Rham ◽  
Cech Cohomology ◽  
Complex Projective

This chapter proves Grothendieck's algebraic de Rham theorem. It first proves Grothendieck's algebraic de Rham theorem more or less from scratch for a smooth complex projective variety X, namely, that there is an isomorphism H*(Xₐₙ,ℂ) ≃ H*X,Ω‎subscript alg superscript bullet) between the complex singular cohomology of Xan and the hypercohomology of the complex Ω‎subscript alg superscript bullet of sheaves of algebraic differential forms on X. The proof necessitates a discussion of sheaf cohomology, coherent sheaves, and hypercohomology. The chapter then develops more machinery, mainly the Čech cohomology of a sheaf and the Čech cohomology of a complex of sheaves, as tools for computing hypercohomology. The chapter thus proves that the general case of Grothendieck's theorem is equivalent to the affine case.

Vertical Chern Type Classes on Complex Finsler Bundles

2011 ◽  
Vol 57 (2) ◽  
pp. 377-386
Author(s):  
Cristian Ida
Keyword(s):  
Differential Forms ◽  
Linear Connection ◽  
Curvature Form ◽  
Cohomology Groups ◽  
Finsler Manifolds ◽  
Invariance Properties ◽  
Type Classes

Vertical Chern Type Classes on Complex Finsler BundlesIn the present paper, we define vertical Chern type classes on complex Finsler bundles, as an extension of thev-cohomology groups theory on complex Finsler manifolds. These classes are introduced in a classical way by using closed differential forms with respect to the conjugated vertical differential in terms of the vertical curvature form of Chern-Finsler linear connection. Also, some invariance properties of these classes are studied.

scholarly journals The de Rham Cohomology through Hilbert Space Methods

2019 ◽  
pp. 455-465
Author(s):  
Ihsane Malass ◽  
Nikolai Tarkhanov
Keyword(s):  
Hilbert Space ◽  
Elliptic Boundary ◽  
Differential Forms ◽  
Adjoint Operator ◽  
Lagrange Equations ◽  
De Rham Cohomology ◽  
Elliptic Boundary Value ◽  
De Rham ◽  
Canonical Representations ◽  
The Neumann Problem

We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler- Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer

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