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 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

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).

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].

Variations of Integrals in Diffeology

2013 ◽  
Vol 65 (6) ◽  
pp. 1255-1286
Author(s):  
Patrick Iglesias-Zemmour
Keyword(s):  
Differential Forms ◽  
Moment Map ◽  
De Rham Cohomology ◽  
Homotopy Operator ◽  
Chain Homotopy ◽  
De Rham ◽  
A Chain ◽  
The Moment

AbstractWe establish a formula for the variation of integrals of differential forms on cubic chains in the context of diffeological spaces. Then we establish the diffeological version of Stokes’ theorem, and we apply that to get the diffeological variant of the Cartan–Lie formula. Still in the context of Cartan–De Rham calculus in diffeology, we construct a chain-homotopy operator K, and we apply it here to get the homotopic invariance of De Rham cohomology for diffeological spaces. This is the chain-homotopy operator that is used in symplectic diffeology to construct the moment map.

scholarly journals Tempered D-modules and Borel–Moore homology vanishing

2021 ◽  
Author(s):  
Dario Beraldo
Keyword(s):  
Compact Support ◽  
De Rham Cohomology ◽  
Affine Grassmannian ◽  
D Modules ◽  
De Rham ◽  
Smooth Affine ◽  
Big Cell ◽  
Sigma G ◽  
Concrete Statement

AbstractWe characterize the tempered part of the automorphic Langlands category $$\mathfrak {D}({\text {Bun}}_G)$$ D ( Bun G ) using the geometry of the big cell in the affine Grassmannian. We deduce that, for G non-abelian, tempered D-modules have no de Rham cohomology with compact support. The latter fact boils down to a concrete statement, which we prove using the Ran space and some explicit t-structure estimates: for G non-abelian and $$\Sigma $$ Σ a smooth affine curve, the Borel–Moore homology of the indscheme $${\text {Maps}}(\Sigma ,G)$$ Maps ( Σ , G ) vanishes.

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.

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