Questions tagged [hodge-structure]
The hodge-structure tag has no usage guidance.
5
votes
1answer
105 views
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 ...
3
votes
0answers
128 views
Principal bundle analogue for Hodge bundle
Let $X$ be a connected smooth complex projective variety.
A holomorphic Higgs bundle is a pair $(E, \theta)$ consists of a holomorphic vector bundle $E$ on $X$ together with a Higgs field $\theta \...
0
votes
0answers
193 views
Group Action from $\mathbb{C}^*$ and Hodge decomposition
In Claire Voisin's book (Hodge Theory and complex algebraic geometry), at the first exercise of chapter 6, the author claims that if we have a continuous action from $\mathbb{C}^*$ on $H_\mathbb{C}$ (...
2
votes
0answers
104 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 ...
7
votes
1answer
306 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 ...
2
votes
0answers
91 views
For a mixed hodge structure, what is the exact condition on the graded pieces?
A mixed ($\mathbb{Q}$)-hodge structure is defined to be a vector space $V/\mathbb{Q}$ with an increasing "weight" filtration of $\mathbb{Q}$- vector spaces $0\subset W_0\subset \dots$ and a ...
3
votes
1answer
122 views
How can I determine the monodromy of this variation of mixed hodge structures?
Consider the variation of mixed hodge structures which generates at the origin:
$$
f:X = \text{Proj}\left( \frac{\mathbb{C}[t][x,y,z]}{(xy(x + y + tz))} \right) \to \mathbb{A}^1_t
$$
How can I compute ...
2
votes
1answer
121 views
How can I compute the mixed hodge structure for three copies of $\mathbb{P}^1$ intersecting at one point?
I know there is a spectral sequence for a variety with normal crossing singularities $X$ which gives a tool for making the computation of the mixed hodge structure computable. How can I compute the ...
5
votes
0answers
149 views
Mixed Hodge modules of product spaces
Let $X$ be an algebraic varietiy (as good as you want, say affine and smooth) and let us denote by $MHM(X)$ the category of mixed Hodge modules as descrived by Saito (see for example this or this).
...
5
votes
0answers
114 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 ...
7
votes
1answer
275 views
Two mixed Hodge structures on equivariant cohomology for actions by finite groups
The answer to the following question might be obvious but I haven’t found a full proof yet (neither by myself nor in the literature). So my apologies if it is trivial.
Let $X$ be a (for simplicity ...
4
votes
0answers
354 views
Why is the Hodge conjecture equivalent to the assertion that $ \mathcal{R}_{ \mathrm{Hodge} } $ is fully faithfull?
On pages 17 and 18 of the following document: https://www.math.tifr.res.in/~sujatha/ihes.pdf, we find the following paragraph:
Let $ \mathbb{Q} \mathrm{HS}$ be the category of pure Hodge structures ...
6
votes
0answers
199 views
Cohomology theories from Saito's mixed Hodge complexes
The definition of mixed Hodge complexes by Saito is a very interesting one, since it's more a cohomology theoretic than geometric generalization of Hodge structures. Since Saito's motivation for mixed ...
3
votes
1answer
255 views
what is the definition of Hodge structure of geometric origin
Let $H$ be a mixed Hodge structure or, more generally, a mixed Hodge structure over a subfield $k$ of $\mathbb{C}$, by which I mean a $k$-vector space with two filtrations (Hodge and weight), a $\...
8
votes
0answers
181 views
Mumford-Tate group of the Fermat curve
Let $C$ be the Fermat curve of degree $d$, defined by the equation $x^d+y^d=z^d$ in $\mathbb{P}^2$. The first cohomology group $H^1(C, \mathbb{Q})$ carries a pure Hodge structure, so it has an ...
4
votes
1answer
324 views
The compatibility of the Gysin sequence with mixed Hodge structures
Let $X$ be a compact complex $n$-manifold and $D$ be a smooth comdimension $1$ submanifold. Also let $U:= X\setminus D$ and $j$ be the inclusion of $U$ in $X$.
Then it is well known that the ...
3
votes
0answers
404 views
Reference for the Hodge polynomial or the Hodge Characteristic
What is the first work that studies, refers to, or mentions the Hodge characteristic?
The Hodge polynomial is the unique ring homomorphism
$$
P_{hdg}:K_0(\mathbf{Var}/\mathbb{C)}\to \mathbb{Z}[u,v,u^{...
3
votes
1answer
314 views
Monodromy theorem of degeneration of smooth projective varieties to non-reduced central fiber
Given a family $\pi: \mathcal{X}\rightarrow\Delta$ smooth away from $0\in\Delta$, Where $\mathcal{X}$ is a smooth complex manifold, $\Delta$ is a small disk, the general fiber of $\pi$ is smooth ...
5
votes
0answers
111 views
Nonabelian Hodge structure for noncompact curves and Hodge structure on the fundamental group
Nonabelian Hodge theory, introduced by C. Simpson and others, may be interpreted as a description of the (real) Hodge structure on the fundamental group (say, of a compact curve) in terms of some ...
1
vote
0answers
336 views
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 ...
18
votes
0answers
429 views
Does the “holomorphic spheres-to-continuous spheres” forgetful function respect the mixed Hodge structures on homotopy groups?
For each smooth, projective, complex variety $X$ that is simply connected, John Morgan constructed a natural mixed Hodge structure on the homotopy group $\pi_k(X,x)\otimes \mathbb{Q}$. This was ...
15
votes
1answer
679 views
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 ...
3
votes
1answer
364 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}
...
10
votes
0answers
468 views
Intrinsic definition of the weight filtration
Let $X$ be a smooth quasiprojective complex variety. Then Deligne (Theorie de Hodge II) defined a weight filtration on the Betti cohomology of $X$. The general philosophy is quite simple: express the ...
2
votes
0answers
111 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 ...
3
votes
2answers
184 views
Variation of Hodge structures associated to a hermitian symmetric domain
Let $D$ be an irreducible hermitian symmetric domain. Then there exists a variation of Hodge structures $(h_s)_{s\in D}$ on a vector space $V$ satisfying specific conditions which depend on $D$ such ...
6
votes
1answer
352 views
Mixed Hodge structure and cup product
I'm looking for a reference for the answer to the following questions.
Let $X$ be an algebraic variety over C. When is the cup product a morphism of Mixed Hodge structures? Does $X$ have to be smooth?...
1
vote
0answers
122 views
Hodge structures generated by cohomology groups of varities with dimension less than $n$
Let $X$ be a smooth projective variety over $\mathbb{C}$ with dimension $n$. Is it true that for every $i<n$, the Hodge structure on $\mathrm{H}^i(X,\mathbb{Q})$ is generated by Hodge structures of ...
7
votes
1answer
610 views
periods of Mixed Hodge Structures
Two Questions:
First. As I know the notion of periods comes when one has two vector spaces over a subfield $k$ of $\mathbb{C}$ (usually given by two cohomology theories) and an isomorphism between ...
5
votes
0answers
161 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 ...
