scholarly journals The Algebraic de Rham Cohomology of Representation Varieties

2018 ◽  
Vol 70 (3) ◽  
pp. 702-720
Author(s):  
Eugene Z. Xia
Keyword(s):  
De Rham Cohomology ◽  
Natural Family ◽  
De Rham ◽  
The One ◽  
Representation Varieties ◽  
Algebraic De Rham Cohomology ◽  
Natural Families

AbstractThe SL(2, ℂ)-representation varieties of punctured surfaces form natural families parameterized by monodromies at the punctures. In this paper, we compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauß-Manin connection on the natural family of the smooth SL(2, ℂ)-representation varieties of the one-holed torus.

Algebraic de Rham Cohomology

2017 ◽  
pp. 73-96
Author(s):  
Annette Huber ◽  
Stefan Müller-Stach
Keyword(s):  
De Rham Cohomology ◽  
De Rham ◽  
Algebraic De Rham Cohomology

Algebraic De Rham cohomology

1972 ◽  
Vol 7 (2) ◽  
pp. 125-140 ◽  
Author(s):  
Robin Hartshorne
Keyword(s):  
De Rham Cohomology ◽  
De Rham ◽  
Algebraic De Rham Cohomology

Poincar� duality for algebraic de rham cohomology

2004 ◽  
Vol 114 (1) ◽  
pp. 61-116 ◽  
Author(s):  
Luisa Fiorot ◽  
Maurizio Cailotto ◽  
Francesco Baldassarri
Keyword(s):  
De Rham Cohomology ◽  
De Rham ◽  
Algebraic De Rham Cohomology

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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