All Questions
Tagged with hodge-structure complex-geometry
13 questions
2
votes
1
answer
225
views
Prefactor $2\pi i$ for Tate-Hodge structure
A rather basic question. What was the original reason to consider the underlying $\mathbb{Z}$-module of the - as canonical object regarded - Tate-Hodge structure $\mathbb{Z}(1)$ to be given as $2 \pi ...
- 35.2k
1
vote
0
answers
116
views
Abelian subvarieties corresponding to vector subspaces
Let $S$ be a connected smooth projective surface.
Let $C$ a smooth curve on $S$
In page 9 of the paper "https://arxiv.org/abs/1704.04187v1" a read the following:
Let
\begin{equation*}
r: ...
- 519
3
votes
1
answer
298
views
Automorphism of integral Hodge structures
Let $(V,V^{p,q},Q)$ be a polarized integral Hodge strucutre of weight $n$. I would like to understand the automorphism of this datum better. Specifically, I'm wondering if there are good conditions ...
- 35.2k
1
vote
1
answer
248
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Hodge variation
I am reading Milne's online book of Shimura Varieties https://www.jmilne.org/math/xnotes/svi.pdf, I confused by a Definition of Hodge variation. On page 29, it was said something is called Hodge ...
- 397
5
votes
1
answer
273
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Failure of Borel-Schmid quasi-unipotency theorem in non-algebraic case
Let $\mathcal{X} \to \Delta^{\times}$ be a smooth family of compact Kähler manifolds over a punctured disc.
When this family is algebraic, the celebrated quasi-unipotency theorem (I think, due to ...
- 4,111
3
votes
0
answers
171
views
Duality of Mixed Hodge Structures without compactness
Let $X$ be a smooth separated algebraic variety over $\mathbb{C}$ and $Z \subset X$ a subvariety of codimension $p$. There are no compactness assumptions. I am looking for an isomorphism of mixed ...
- 195
8
votes
1
answer
980
views
Variations of Hodge structures over the line
Let $f\colon X\to \mathbb{A}^1$ be a smooth projective morphism of complex algebraic manifolds, where the target $\mathbb{A}^1$ is the affine line. Are there any restrictions on the Hodge structures ...
- 23k
5
votes
0
answers
185
views
Reference request: If the local system extends, then the variation of Hodge structures extends
I'm looking for a precise reference for the following theorem.
Let $C$ be a smooth curve over $\mathbb{C}$ and let $S$ be a finite set of closed points of $C$. Let $\ V$ be a polarized variation ...
- 51
1
vote
0
answers
440
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Is the category of mixed Hodge modules bi-filtered?
Let $X$ be a smooth complex algebraic variety and let $MHM(X)$ be the category of mixed Hodge modules on $X$, as defined in (Saito, "Mixed Hodge Modules", 1990), (Peters-Steenbrink, "Mixed Hodge ...
- 507
15
votes
1
answer
1k
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Mixed Hodge structure on sheaf cohomology of a variation of Hodge structures
I'm new here. I hope to do it right!
I am interested in studying mixed Hodge structures over complex algebraic surfaces and their generalizations.
Let us take a smooth complex variety $X$ and a ...
- 35.2k
3
votes
1
answer
494
views
Is there a description of variation of (mixed) Hodge Structures in terms of a Deligne operator?
A complex Mixed Hodge Structure is given by a complex vector space $V$ together with a descending filtration $W$ and two ascending filtrations $F,\bar{F}$ that satisfy the condition
\begin{equation}
...
- 35.2k
3
votes
0
answers
141
views
Does the monodromy of such VHS have to be trivial
Consider a variation of polarized Hodge structure on a punctured disk. Suppose that connection preserves Hodge filtration (which is much stronger, than Griffiths transversality). Moreover assume that ...
- 385
5
votes
0
answers
189
views
Real structure in the mixed Hodge structure associated to an isolated singularity
We know that a mixed Hodge structure on a complex vector space $H$ with an integral lattice $H_{\mathbb Z}$ consists of the weight filtration and the Hodge filtration. For an isolated hypersurface ...
- 45.7k