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 Duality and de Rham cohomology for graded D-modules

2018 ◽  
Vol 340 ◽  
pp. 1141-1165 ◽  
Author(s):  
Nicholas Switala ◽  
Wenliang Zhang
Keyword(s):  
De Rham Cohomology ◽  
D Modules ◽  
De Rham

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.

THE DE RHAM COHOMOLOGY OF DIFFERENTIAL SPACES

1989 ◽  
Vol 22 (1) ◽  
pp. 249-272 ◽  
Author(s):  
Wiesław Sasin
Keyword(s):  
De Rham Cohomology ◽  
De Rham ◽  
Differential Spaces

scholarly journals STEENROD OPERATIONS ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC STACKS

2021 ◽  
pp. 1-48
Author(s):  
Federico Scavia
Keyword(s):  
Finite Groups ◽  
Reductive Groups ◽  
Orthogonal Groups ◽  
De Rham Cohomology ◽  
Algebraic Stacks ◽  
Steenrod Operations ◽  
De Rham

Abstract Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .

scholarly journals Cycles in the de Rham cohomology of abelian varieties over number fields

2018 ◽  
Vol 154 (4) ◽  
pp. 850-882
Author(s):  
Yunqing Tang
Keyword(s):  
Endomorphism Ring ◽  
Abelian Varieties ◽  
Number Fields ◽  
De Rham Cohomology ◽  
Tate Conjecture ◽  
De Rham ◽  
Prime Dimension

In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of$\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus’ prediction for some families of abelian varieties. These families include geometrically simple abelian varieties of prime dimension that have non-trivial endomorphism ring. The proof uses a crystalline analogue of Faltings’ isogeny theorem due to Bost and the known cases of the Mumford–Tate conjecture.

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