Questions tagged [mixed-hodge-structure]

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Confusion about notations in limit mixed Hodge structure

I am reading the paper Monodromy at infinity and Fourier transform by Claude Sabbah and got some confusions about notations. (note first that I am not specialized in mixed Hodge theory but, I am ...
Alexey Do's user avatar
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Example motivating mixed Hodge structures

The suggested intuition behind mixed Hodge structures - developed in particular to generalize Hodge decomposition of cohomology groups from complex smooth complete varieties to more general algebraic ...
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Hodge conjecture for generic points

I was reading the following paper: "Beilinson’s Hodge Conjecture For Smooth Varieties". They study the cycle class map $cl_{m,r}: H^{2r-m}_{\mathcal{M}}(U, \mathbb{Q}(r))\rightarrow \text{...
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Beilinson-Hodge conjecture and generation of cohomology ring by $H^1$

Beilinson's version of Hodge conjecture has the following form. For any quasi-projective smooth complex variety $X$ the following map is surjective: $$H^i_{\mathcal{M}}(X, \mathbb{Q}(j))\rightarrow \...
user127776's user avatar
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3 votes
1 answer
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What does does the monodromy weight filtration represent?

I'm trying to understand variations of Hodge structure. I understand that this is a very broad field, and that many of the concepts have been extended to algebraic geometry over fields other than $\...
Quaere Verum's user avatar
4 votes
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325 views

Voevodsky's motives and Deligne's systems of realizations

$\newcommand{\gm}{\mathrm{gm}}$Let $\mathbf{DM}_{\gm}(\mathbb{Q},\mathbb{Z})$ be Voevodsky's category of geometric motives over $\mathbb{Q}$ with coefficients in $\mathbb{Z}$ (e.g. as on p.124 of ...
David Corwin's user avatar
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2 votes
1 answer
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Representation-induced relations in the Grothendieck of varieties

Let $X$ be a variety over $k = \mathbb{C}$ with an action of a finite group $G$. According to this paper (Section 4), the induced action of $G$ on the cohomology of $X$ respects the mixed Hodge ...
jessetvogel's user avatar
4 votes
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Signed number of pieces in a decomposition in the Grothendieck ring of varieties

Let $X/k$ be a (geometrically integral and connected) variety over $k$ either a field of characteristic $0$ or a finite field. Let $[X] = \sum_{i\in I}[Y_i] - \sum_{j\in J}[Z_j]$ be a decomposition ...
Asvin's user avatar
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4 votes
1 answer
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What is motivic sheaf intuitively?

I am not very familiar with motif theory, but I do know a little about Hodge theory. I view (mixed) motif theory as an enhancement of (mixed) Hodge structures. Q1. Is (mixed) motivic sheaf theory an ...
D.Namrebod's user avatar
2 votes
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115 views

Nilpotent orbits and mixed Hodge structures

Let $(H_\mathbb{Z}, \{h^{p,q}\}_{p+q=n}, \phi)$ be the datum to define weight $n$ polarized Hodge structures on $H_\mathbb{Z}$, $h^{p,q}$ is the Hodge numbers, $\phi$ is a polarization. Let $D$ be the ...
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Super mixed Hodge structures?

It's common in subjects that have some version of the "yoga of weights" that you have a functor called "Tate twist" and that the most natural version of it seems like it should be ...
Ben Webster's user avatar
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Motives based on Hodge cycles vs algebraic cycles

I am not a specialist of motives. I am afraid my questions are rather naive. We have the category of (pure) motives based on Hodge cycles by Deligne. In his articles with Milne, morphisms between ...
Takahiro Matsuda's user avatar
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Hodge's conjecture as a quasi-isomorphism between two complexes of sheaves

A version of Hodge's conjecture due to Beilinson, expects that the Betti cycles class map $H_{\mathcal{M}}^i(X,\mathbb{Q}(j))\rightarrow hom_{MHS}(\mathbb{Q}(0),H^{i}(X,\mathbb{Q}(j) ))$ is surjective ...
user127776's user avatar
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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 $X$?
rumpi123's user avatar
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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 ...
Tommaso Scognamiglio's user avatar
4 votes
1 answer
394 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_{\...
Tommaso Scognamiglio's user avatar
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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 ...
Georgy's user avatar
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3 votes
1 answer
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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 ...
user45397's user avatar
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2 answers
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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, ...
Tommaso Scognamiglio's user avatar
9 votes
2 answers
337 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 ...
César Iglesias's user avatar
3 votes
1 answer
342 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 ...
asv's user avatar
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19 votes
0 answers
1k 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-...
Daniel Litt's user avatar
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5 votes
1 answer
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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?
J.D.Chern's user avatar
3 votes
1 answer
158 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 $$ p\colon \pi_1(\mathcal{M}_{0,n})\longrightarrow H_1(\mathcal{M}_{0,n},\mathbb{Z}/2\mathbb{Z}) $$ ...
Daniil Rudenko's user avatar
3 votes
0 answers
150 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, ...
Jesse Wolfson's user avatar
1 vote
0 answers
154 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 ...
Stabilo's user avatar
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21 votes
1 answer
1k 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
663 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 ...
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40 votes
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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 ...
14 votes
0 answers
842 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, ...
m_for_motive's user avatar
3 votes
0 answers
120 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 \...
Chen's user avatar
  • 1,573
1 vote
2 answers
317 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 ...
stjc's user avatar
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3 votes
0 answers
162 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 ...
BnPrs's user avatar
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13 votes
0 answers
259 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 ...
David E Speyer's user avatar
2 votes
1 answer
266 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 ...
user avatar
4 votes
0 answers
197 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 ...
54321user's user avatar
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3 votes
1 answer
167 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 ...
54321user's user avatar
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2 votes
1 answer
176 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 ...
54321user's user avatar
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