# Questions tagged [mixed-hodge-structure]

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25
questions

**5**

votes

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218 views

### Existence of an affine variety with homotopy type of suspension of another affine variety

Let $X$ be an affine variety. My question is does there exist another affine variety with the homotopy type of the suspension of $X$?

**5**

votes

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209 views

### Character stack and character variety

Let $\Sigma$ be a Riemann surface of genus $g$. We can consider two different type of objects associated to it parametrising representations of its fundamental groups. On one side we have the ...

**3**

votes

**1**answer

204 views

### Comparison of weight filtration on cohomology of complex manifold

Let $X$ be a smooth scheme of finite type over $\mathbb{Z}$ (or let's say a finitely generated $\mathbb{Z}$ algebra). To each prime $p \in \mathbb{Z}$ we can consider the $\mathbb{F}_p$ variety $$X_{\...

**2**

votes

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135 views

### Is the dimension of the pieces of a mixed Hodge structure constant under smooth deformations?

In the case of a family of compact complex manifolds we have the following:
Theorem. Let $f:X→B$ be a family of complex manifolds and assume that $X_0$ is Kähler for some $0\in B$. Then for $b$ in a ...

**3**

votes

**1**answer

218 views

### Cycle class map for singular varieties

I am reading the cycle class map for singular projective varieties as mentioned by Laterveer in this article (see Definition $1$). The article does not define the map but refers to an article of ...

**2**

votes

**2**answers

728 views

### Mixed Hodge structure cohomology of fibration

Let $X$ be a smooth complex algebraic variety. From Deligne's work, we know that the have a Mixed Hodge structure over its (rational) compactly supported cohomology $H^{*}_c(X,\mathbb{Q})$. With this, ...

**9**

votes

**2**answers

261 views

### Simplicial spaces internally to simplicial sets

I am a master’s student with interest in topos theory and its applications (motivated by Ingo Blechschmidt’s thesis, as seems to be usual).
After finding out about some of the uses of simplicial ...

**3**

votes

**1**answer

168 views

### Is the abelian category of pure Hodge modules semi-simple?

I am a beginner in the subject, and at the moment I am trying to understand basic properties of the main objects of the M. Saito's theory of the mixed Hodge modules in general.The question in the ...

**19**

votes

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928 views

### Mumford-Tate conjecture for mixed Tate motives

Let $X$ be a (not necessarily smooth or proper) variety over a number field $k$. Suppose we are given
A subquotient $V_{dR}$ of the algebraic de Rham cohomology $H_{dR}^i(X)$ (defined in the non-...

**5**

votes

**1**answer

258 views

### mixed Hodge structure of general linear group

Is there any literature about mixed Hodge structure on $H^*(GL(n,\mathbb{C}))$ ? I think it is a vary basic problem in mathematics, but there are no literature about it?

**3**

votes

**1**answer

107 views

### Cohomology of the moduli space of rational curves with $n$ marked points with spin structure

Consider $\mathcal{M}_{0,n}$- the moduli space of rational curves with $n$ marked points. A map
$$
p\colon \pi_1(\mathcal{M}_{0,n})\longrightarrow H_1(\mathcal{M}_{0,n},\mathbb{Z}/2\mathbb{Z})
$$
...

**3**

votes

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117 views

### Mixed Hodge structures on (infinite) covers of complex varieties?

Let $X$ be a complex variety, and let $\tilde{X}\to X$ be a covering map. Does the singular cohomology $H_\ast(\tilde{X};\mathbb{Z})$ carry a natural mixed Hodge structure?
If the cover is finite, ...

**1**

vote

**0**answers

146 views

### Mixed Hodge structures over $F\otimes \mathbb{R}$

Let $F$ be a number field. Nekovàř, on page 18 of Values of L-functions and p-adic cohomology, is referring to the category of mixed Hodge structures over $F\otimes_{\mathbb{Q}} \mathbb{R}$. Can ...

**21**

votes

**1**answer

952 views

### When simple cohomological computations predict ingenious algebro-geometric constructions?

Classical algebraic geometry is full of ingenious constructions and miraculous coincidences: 27 lines on a cubic surface are related to Weyl lattice of type $E_6,$ lines on an intersection of four-...

**3**

votes

**0**answers

446 views

### Deligne's Mixed Hodge Theory

Deligne constructs Mixed Hodge Structures (MHS) on the cohomology, $H^{*}(X)$, of an algebraic variety $X$, in his papers Hodge II and Hodge III. Really the question below is rather vague, and is ...

**35**

votes

**0**answers

1k views

### Are we better in computing integrals than mathematicians of 19th century?

When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of ...

**12**

votes

**0**answers

760 views

### On mixed $p$-adic Hodge theory

Does mixed $p$-adic Hodge theory exist? Can we extend the scope of comparison theorems using simplicial resolutions a la Deligne? Do we get 3 opposite filtrations as in classical mixed Hodge theory, ...

**3**

votes

**0**answers

112 views

### Degeneration of cycle class map

Let $f:\mathcal{X} \to \Delta$ be a flat family of projective varieties, smooth over the punctured disc $\Delta^*$ and the central fiber is a simple normal crossings divisor. Let $\mathcal{Z} \subset \...

**1**

vote

**2**answers

259 views

### Differential construction of mixed Hodge structure on smooth open varieties

Let $\bar{X}$ be a complete smooth variety over $\mathbb{C}$ and $D$ be a simple normal crossing divisor. Denote $X:=\bar{X}\backslash D$. Then it is known that $H^\ast(X,\mathbb{C})$ admits a ...

**3**

votes

**0**answers

132 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 ...

**13**

votes

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249 views

### Alexander modules and weight filtrations

$\def\Ext{\mathrm{Ext}}$I have the following situation: I have a complex algebraic variety $X$. I want to compute the weight filtration on various abelian covers $Y \to X$. As a concrete example, take ...

**2**

votes

**1**answer

194 views

### Limiting Hodge structure

Let $\mathcal{X}\rightarrow B$ be a family of projective varities ($B$ is DVR say) whose generic fibre is smooth, but the closed fibre is divisor with normal crossing singulairty. Is there some ...

**4**

votes

**0**answers

170 views

### How can I describe the monodromy of this variation of singular curves?

Consider the family of singular hyperelliptic curves
$$
y^2 - x(x-1)^2(x-2)(x-3)(x-4)(x-t)
$$
over $\mathbb{A}^1_t$. Over a generic point the fiber is a genus three curve where one of the genera comes ...

**3**

votes

**1**answer

146 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

**1**answer

159 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 ...