# Questions tagged [hodge-structure]

The hodge-structure tag has no usage guidance.

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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 ...

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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 \...

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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 $H_\mathbb{C}$ (...

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### 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 ...

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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 ...

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### 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 ...

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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 ...

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### 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 ...

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### 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).
...

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### 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 ...

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### 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 ...

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### 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 ...

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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 ...

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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 $\...

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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 ...

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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 ...

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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^{...

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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 ...

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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 ...

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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 ...

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### 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 ...

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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 ...

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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}
...

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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 ...

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### 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 ...

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### 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 ...

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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?...

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### 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 ...

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### 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 ...

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### 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 ...