The question of whether a cohomology class ( c \in H^2_{\text{dR}}(X_F) ) that specializes to the image of the first Chern class map for almost all closed points ( t \in \text{Spec}(\mathcal{O}{F,S}) ) must itself be in the image of the generic first Chern class map ( c{1,F} ) is a topic in arithmetic algebraic geometry.

Key Observations:

1. **Specialization Maps and Hodge Filtration...

Short answer. What you are asking is the codimension‑1 case of what is usually called the absolute Tate (or Ogus) conjecture. In that language the expected answer is “yes,” but it is not known in general. It is known in a number of important cases (e.g. abelian varieties, products of curves, many K3‑type cases), and it is closely related to Mumford–Tate/variational Hodge phenomena.

How your quest...

Cycles in Algebraic de Rham Cohomology

This question is asking about a specific property of the cycle class map in algebraic de Rham cohomology, which is related to but distinct from the Hodge and Tate conjectures.

The Question Analyzed

The question asks: If is a cohomology class such that some multiple specializes to the image of the first Chern class map at alm...

Short answer The statement you ask for is expected to be true but is not known in general. It is exactly the codimension-one case of a conjecture formulated by Arthur Ogus that is usually called

 • the “de Rham (or crystalline) Tate conjecture”,
 • Ogus’s conjecture on Tate cycles, or
 • the conjecture that “absolute Tate classes are algebraic”.

It is proved in a few important situations...

This is a very interesting question about the relationship between algebraic cycles and de Rham cohomology, which touches on some deep aspects of arithmetic geometry. Let me provide some context and insights.

The Question in Context

Your question asks whether a de Rham cohomology class that specializes to divisor classes at almost all primes must itself come from a divisor class (after tensor...

This is an excellent question that sits at the crossroads of several major programs in arithmetic geometry. The general principle it touches upon is whether the property of a cohomology class being "algebraic" (i.e., in the image of the cycle class map) can be detected by examining its reductions at almost all primes.

The short answer is:

• For divisors (the case you ask a...

Let

  f : X → Spec 𝒪_{F ,S } ,  X smooth, projective, geometrically connected

and write X_F for the generic fibre.
For every closed point t of Spec𝒪_{F ,S } denote by

  c_{1,t} : Pic(X_t ) ⟶ H^2_{dR}(X_t )

the algebraic de-Rham first Chern class.

QUESTION (codimension 1).
Let c ∈ H^2_{dR}(X_F ).
Assume that there is an integer N ≠ 0 such that, for almost all t,

  (Nc)|_t  lie...

Hmm, this seems to be about the image of the cycle class map in de Rham cohomology and its behavior under specializa...

Below is a sketch of why nothing essentially new is hiding behind your question, and why in full generality it is as wide open as the usual Tate‐ and Fontaine–Mazur‐type conjectures on divisors. In fact one can restate your question as “any one‐dimensional de Rham subobject of (H^2) which is everywhere unramified and crystalline must come from a line bundle,” and then one sees immediately that ...