9
votes
Accepted
Variations of Hodge structures over the line
See Theorem 11 on page 191 of these notes. A special case is as follows.
Theorem of the fixed part. Let $S$ be a smooth quasiprojective variety, and $V$ a variation of $\mathbb{Q}$-Hodge structures ...
- 23k
7
votes
Accepted
Two mixed Hodge structures on equivariant cohomology for actions by finite groups
As I said in my comment, the mixed Hodge structures are the same. Here is the outline. From [Hodge III, 6.1.2.1],
$$[X/G]_n = (G^{n+1}\times X)/G$$
One has a descent spectral sequence
$$E_1= H^q([X/...
- 35.2k
7
votes
Accepted
Automorphism of integral Hodge structures
The polarization $Q$ gives rise to a positive definite form $Q'$. Automorphisms preserve $Q'$, so they lie in a compact group. On the other hand, automorphisms preserve the lattice, so they lie in a ...
- 35.2k
6
votes
Accepted
Prefactor $2\pi i$ for Tate-Hodge structure
You are right that it is in some sense a matter of convention, but I claim it's a natural one.
Perhaps the easiest example to explain is $H=H_1(X)$, where $X=\mathbb{C}^*$. By Deligne, this carries a(...
- 35.2k
6
votes
Accepted
$\mathbb{Q}$-Zariski Closure not equal to smallest Q-subgroup
Sure. Take some elements $A_1,\dots, A_n$ that generate a dense subset $\Gamma$ of a non-abelian connected reductive group $G \subseteq GL_n$.
For instance, I can take $$A_1=\begin{pmatrix} 1 &1 \\...
- 148k
6
votes
Super mixed Hodge structures?
I am not sure this is relevant to you, and you may know this already, but in the theory of monodromic ("exponential") mixed Hodge structures, there is a very natural (even) square root of ...
- 3,202
5
votes
Accepted
Abelian varieties corresponding to Hodge substructures
Hodge substructures of $H^1(C,\mathbb Z)$ and Abelian subvarieties of $J(C)$ ar essentially the same thing:
In general, let $V,\omega$ be a free $\mathbb Z$-module of even rank with an integral ...
- 1,908
5
votes
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Failure of Borel-Schmid quasi-unipotency theorem in non-algebraic case
One universal cover of the punctured disk $\Delta^*$ is the upper half plane $\mathcal{H}$, $$\pi:\mathcal{H}\to \Delta^*, \ \ z=\pi(w) = e^{2\pi iw}.$$ There is a natural translation action of the ...
4
votes
How to cook up an Artin motive from a positive-dimensional variety
Tate twists do indeed shift the Hodge filtration. Since the de Rham realization of the Tate motive $\mathbb{Q}(1)$ is concentrated in the degree $-1$ part of the Hodge filtration, we have $F^n M(m)_{...
- 9,061
4
votes
Super mixed Hodge structures?
If you introduce super-structure, it doesn't matter whether $\mathbb{C}(1/2)$ is even or odd. The reason is that your category of motives has a symmetric monoidal automorphism which acts by $-1$ on $\...
- 7,952
4
votes
Abelian varieties corresponding to Hodge substructures
See ncatlab and the reference to the Peters-Steenbrink for more details.
More concretely, consider the complex torus $$J(H):=\dfrac{H_{\mathbb{C}}}{H_{\mathbb{Z}}+F^1}$$
View the torus in the Jacobian ...
- 431
4
votes
Hodge conjecture for generic points
This turned out to be much simpler than what I expected and it just follows from the localization. I will sketch the proof from (S3) to (S2).
We have the following sequence from the localization for ...
- 5,901
3
votes
Accepted
Geometric Interpretation of absolute Hodge cohomology
In fairness, some of the credit for the general construction of absolute cohomology should go to Beilinson (see his Notes on Absolute cohomology).
I think you'll find the homology version, that you ...
- 35.2k
3
votes
Applications of Hodge-Riemann bilinear relations
Since you ask for further applications, does that mean you already know some? The Hodge-Riemann bilinear relations are used in all kinds of ways. Suppose that $M$ is a compact Riemann surface, then ...
- 35.2k
2
votes
Accepted
When does the rational Hodge structure determine the integral Hodge structure?
If the image of the usual period map is an open subset of the period domain, this map is certainly not generically injective: For a typical point in the image, its translate by any element of $GL_n(\...
- 148k
2
votes
Middle cohomology of very general hyperplane sections
I guess that this is not true even for hypersurfaces. Take a smooth quadric in $Y=Q \subset \mathbb{P}^{3} = X$. Then $h^{1,1}(Q) = b_{2}(Q) = 2$, so the second cohomology cannot be generated by $1$ ...
- 6,995
2
votes
Accepted
How can I determine the monodromy of this variation of mixed hodge structures?
This monodromy is trivial because on the set $t\neq 0$ you can make a change of variables $(x,y,z)\to(x,y,z t^{-1})$, and your ideal becomes $x y (x+y+z)$, so it doesn't depend on $t$. So your family ...
- 3,772
2
votes
Accepted
How can I compute the mixed hodge structure for three copies of $\mathbb{P}^1$ intersecting at one point?
Let me expand my previous comment, and place it in a more general context. Suppose that $X$ is an singular connected but possibly reducible projective curve. Let $\tilde X\to X$ be the normalization....
- 35.2k
2
votes
what is the definition of Hodge structure of geometric origin
To answer your last question, I don't think there is a commonly accepted definition. So let me propose a couple right now.
Let $N$ be Nori's category of mixed motives over $k$. There is now (or soon ...
- 35.2k
1
vote
Accepted
Why do we ask Shimura datum to have Hodge weight $(-1,1),(0,0),(1,-1)$?
This is the condition which gives the Shimura variety the (almost) complex structure, which is obviously necessary if you want to view it as a variety over $\mathbb C$. This is explained in Theorem 1....
- 28.2k
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