Questions tagged [mixed-hodge-structure]
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45 questions
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Are the components of the Leray spectral sequence of pure weight
Let $f:X \to Y$ be a smooth, projective morphism of non-singular $\mathbb{C}$-varieties. We then have the Leray spectral sequence degenerating in $E_2$: $H^k(X,\mathbb{Q}) \cong \oplus_i H^i(Y,R^{k-i}...
- 2,639
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Monodromic Fourier-Sato transform
The question in brief: Is there a simple way to deduce inversion of the Fourier-Sato transform on monodromic mixed Hodge modules from inversion in the non-mixed setting?
In more detail:
I will play ...
- 965
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Notes on absolute Hodge cohomology Lemma 1.7
I have a question on Lemma 1.7 of Beilinson's "Notes on absolute Hodge cohomology" at https://www.ams.org/books/conm/055.1/862628/conm055.1-862628.pdf. The purpose of the lemma is to compute ...
- 221
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Global invariant cycle theorem for singular varieties
Let $f:X\to S$ be a flat proper morphism of integral algebraic varieties over $\mathbb{C}$, with $S$ smooth. Assume that all fibers are geometrically normal. Fix an integer $i\ge0$. Shirinking $S$ to ...
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Poset of equidimensional subspace arrangements
Recall the following inductive definition: a collection $V_i\subset W$ of different linear subspaces is an arrangement of codimension $c$ if the codimension $V_i$ in $W$ is $c$ and for all $i$ the ...
- 1,009
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Admissible variation of mixed Hodge structure and canonical extension
Let $\Delta^*$ be the punctured disc, let $\mathbb{V}$ be a graded-polarisable variation of mixed Hodge structure over $\Delta^*$, and let $\mathcal{V}=\mathcal{O}_{\Delta^*}\otimes\mathbb{V}$ be the ...
- 1,970
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Why are the Hodge filtrations on cohomology canonically bounded?
If $X$ is a complex projective variety of dimension $n$ then the de Rham cohomology $H^{k}(X,\mathbb Q)$ naturally has a mixed Hodge structure with an increasing weight filtration $W_\bullet$ and a ...
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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 ...
- 1,275
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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 ...
- 4,142
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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{...
- 6,063
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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 \...
- 6,063
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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 $\...
- 395
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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 ...
- 16.1k
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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 ...
- 317
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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 ...
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