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12 votes
1 answer
407 views

Precise formulation of conjectures on orders of vanishing?

Let $X$ be a smooth and proper scheme over $\text{Spec}(\mathbf{Z})$. C. Soulé has conjectures about special values of the completed zeta function of $X$, $\zeta(X,s)$, which were first reformulated ...
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6 votes
0 answers
334 views

Current state of Serre's Motives conjectures in Seattle

It would be worth if we have a current state of the conjectures of Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques. J P Serre. In Motives, Seattle And ...
Jean's user avatar
  • 61
5 votes
1 answer
482 views

Around algebraic equivalence of cycles

Let $k$ be a finitely generated field, $X$ a smooth projective $k$-variety, $\ell$ a prime number, $\ell\in k^{\times}$, $r\ge 0$ an integer. The Tate conjecture asserts surjectivity of the cycle ...
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5 votes
0 answers
397 views

Vector bundles vs algebraic cycles

For integral schemes, the Picard group is isomorphic to the group of Cartier divisors modulo linear equivalence. What is the correct analog of this isomorphism, for higher codimension Chow groups vs ...
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4 votes
0 answers
92 views

Locus of Hodge classes

Let $\pi: X\to S$ be a proper smooth morphism of complex analytic spaces, with connected smooth $X$ and $S$ over $\mathbf{C}$, projective fibers, and $$\mathscr{H}_{X/S}^p := R^p\pi_*\Omega^{\bullet}_{...
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3 votes
0 answers
143 views

A Galois equivariant Weil cohomology theory with coefficients in the rational numbers and a variation of the Tate/Hodge conjecture

A well-known example of Serre shows that there can be no Weil cohomology theory with $\mathbb Q$ coefficients for schemes over $\mathbb F_{p^2}$. However, this example is no obstruction to a Weil ...
Asvin's user avatar
  • 7,746
3 votes
0 answers
81 views

Tate Conjecture birational invariant?

Is the Tate Conjecture stable under birational equivalence? In particular, is the Tate Conjecture for rational varieties known?
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3 votes
0 answers
307 views

Semisimplicity conjecture

In this short note Ben Moonen proves that over fields of characteristic zero that are of finite type over their prime field, the Tate conjecture about surjectivity of cycle maps implies the semi-...
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2 votes
1 answer
172 views

Effective cycles of codimension 1 and field extensions

Let $X$ be a smooth quasi-projective variety over a field $k$, with pure dimension $d$, $K/k$ an arbitrary field extension. For any algebraic cycle $\eta$ of codimension $1$ on $X_K$ ($\eta\in Z^1(...
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2 votes
0 answers
239 views

Group completion of Chow varieties

Let $X$ be a quasi-projective variety over a perfect field $k$. Given a projective embedding $j : X\to \mathbf{P}(\mathscr{E})$, the Chow variety $\text{Chow}_r(X, j)$ is a quasi-projective variety ...
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0 votes
0 answers
88 views

Cycles modulo homological equivalence

Let $\text{CH}^p(X)_{\rm hom}$ be the abelian group of codimension $p$ algebraic cycles on a smooth projective variety over a field $k$, modulo homological equivalence. Is $\text{CH}^p(X)_{\rm hom}$ ...
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