All Questions
12 questions
8
votes
1
answer
492
views
Motives of complex-analytic spaces
In any setting where we have a notion of space and a notion of cohomology theory, we in principle could ask "what are motives in this setting?". In some settings the question can be interesting (i.e. ...
user138661
2
votes
0
answers
483
views
Absolute Hodge cycles over $\mathbf{Q}$
In the 1986 notes by Milne "Hodge cycles on abelian varieties", Deligne defines the notion of absolute Hodge cycles.
For a smooth projective variety defined over $k\subset\mathbf{C}$ non ...
user120812
9
votes
1
answer
643
views
Torsion in Deligne cohomology
Let $X$ be a smooth projective complex analytic space, $i,p\ge 0$ integers, $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex of $X$, $H^i_{\mathcal{D}}(X,\mathbf{Z}(p))$ its hypercohomology.
What ...
user113393
7
votes
1
answer
474
views
Motivic $\mathbf{Z}(1)$
I know that the Bloch higher Chow complex, $\mathbf{Z}(i)_{\mathcal{M}}$, on a smooth scheme over a field $k$, reads, in degree $1$:
$$\mathbf{Z}(1)_{\mathcal{M}}\simeq\mathbf{G}_m[-1].$$
How to see ...
user113393
16
votes
1
answer
3k
views
Tate twists and cohomology of $\mathbf{P}^1$
I was wondering if anyone could give me some intuition as to why, for a smooth projective variety $X$ over $\mathbf{C}$ of complex dimension $d$, the Tate twist on $H^n(X(\mathbf{C}),\mathbf{Z})$ to ...
user113452
2
votes
1
answer
164
views
Full lattice images and Hodge decomposition
Let $X$ be a smooth and proper complex analytic space, or a Kahler complex manifold. Is the image $\Lambda$ of the finitely generated abelian group $H^{2i}(X,\mathbf{Z})$ into $J := H^{2i}(X,\mathbf{C}...
user97068
8
votes
1
answer
432
views
Finiteness aspects of Deligne cohomology
Say $X$ is a smooth projective variety over $\mathbf{C}$, and $\mathcal{X} = X^{\rm an}$ its $\mathbf{C}$-analytic space.
For what integers $i,d$ is the Deligne cohomology $H^i_{\mathcal{D}}(\mathcal{...
user97068
6
votes
1
answer
1k
views
Intuition for polarized Hodge structures
A Hodge structure can be defined as a real, algebraic representation of the Deligne torus ${Res}^\mathbb{C}_{\mathbb{R}}\mathbb{G}_m$. Coming from Kahler manifolds the intuition for this is clear. The ...
- 7,789
2
votes
0
answers
304
views
Should all cohomology theories have a smooth proper base change
Let $H$ be a cohomology theory with respect to some Grothendieck topology (e.g. Zariski, analytic, etale, fppf, Nisnevich, etc.)
Does H satisfy smooth proper base?
If yes, does this mean that "...
- 119
4
votes
0
answers
306
views
What is the relation between Beilinson's conjectures and Standard conjectures of algebraic cycles?
Do Standard conjectures on the K-theory of varieties over finite field have implications in the motivic cohomology of Z where exist the correct formalism of Beilinson's conjectures?
What is the ...
user55909
10
votes
2
answers
1k
views
What implications would a solution of the *Standard Conjectures* have on the *Hodge Conjecture*?
I'm new to the field, so I just would like to know what implications would have a solution of the Standard Conjectures on the Hodge Conjecture. I read somewhere they are related in some way, but I don'...
- 101
14
votes
1
answer
746
views
Can there exist Chow motives/motivic cohomology for compact Kähler manifolds?
Can there exist a 'reasonable' extension of the (higher) Chow groups of complex smooth projective algebraic varieties to functors on the category of compact Kähler manifolds? Are there any ...
- 16.9k