De Rham Cohomology and the Jordan Curve Theorem

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 $p=2$ .

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Digital Jordan Curves and Surfaces with Respect to a Closure Operator

Fundamenta Informaticae ◽

10.3233/fi-2021-2013 ◽

2021 ◽

Vol 179 (1) ◽

pp. 59-74

Author(s):

Josef Šlapal

Keyword(s):

Direct Product ◽

Jordan Curve ◽

Closure Operator ◽

Jordan Curve Theorem ◽

Closure Operators ◽

Ternary Relation ◽

Digital Space ◽

Jordan Curves ◽

Curves And Surfaces ◽

Digital Plane

In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.

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On the de Rham Cohomology Group of a Compact Kähler Symplectic Manifold

Algebraic Geometry, Sendai, 1985 ◽

10.2969/aspm/01010105 ◽

2018 ◽

Cited By ~ 30

Author(s):

Akira Fujiki

Keyword(s):

Symplectic Manifold ◽

Cohomology Group ◽

De Rham Cohomology ◽

De Rham

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

Compositio Mathematica ◽

10.1112/s0010437x17007679 ◽

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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