Questions tagged [hodge-theory]
The study of harmonic differential forms on complex projective varieties, their invariantly defined filtrations, their integrals over topological cycles, especially over subvarieties, the deformations of these integrals and filtrations in families, and a multitude of generalizations.
27 questions from the last 365 days
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The definition of Hodge bundles with metric
A system of Hodge bundles is a direct sum of holomorphic vector bundles $E = \oplus_{p+q=n} E^{p,q}$ with a morphism $\theta : E^{p,q} \rightarrow E^{p-1,q+1} \otimes \Omega_X^1$ such that $\theta^2 = ...
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The relation between Hodge bundles with metric and polarized variation of Hodge structures
Recently I've been reading Simpson's paper "constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, 1988, JAMS". On page 898 he mentioned about ...
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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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Are motives of K3 surfaces of abelian type?
I refer to the article of van Geemen https://arxiv.org/pdf/math/9903146. What van Geemen calls the Kuga-Satake-Hodge conjecture suggests that for a K3 surface $X$ over $\mathbb{C}$, the summand $h^2(X)...
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Tate conjecture for singular varieties in terms of intersection homology
In his book “Mixed motives and algebraic K-theory”, Jannsen generalizes the Tate conjecture to a potentially singular projective variety $X$ over a finitely generated field. The statement is the same ...
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Nonabelian Hodge correspondence for $\mathbb{G}_m$
Please excuse me if this question is too naive. I know very little about the nonabelian Hodge correspondence but I am trying to understand how the correspondence works in the simplest case of the ...
- 3,446
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Exact forms, gauge transformations, and the Hodge decomposition in non-abelian Gauge theory
I am trying to understand how the Hodge decomposition is affected by gauge transformations in non-abelian in gauge theory (eg $\mathrm{SU}(N)$). In particular, I am searching for a way to generalise ...
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Unique continuation of Laplace eigenforms
Let $M$ be a compact Riemannian manifold and $\Delta = d\delta + \delta d$ denote the (positive definite) Hodge Laplacian acting on differential forms. Call a smooth differential form $\omega$ a ...
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Road map and references for combinatorial Hodge theory
I'm a PhD student. I'm familiar with graduate level algebraic geometry and toric varieties.
I wanted to know a road map for getting into combinatorial Hodge theory and other prerequisites that I'll ...
- 839
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Relative Dolbeault cohomology using currents
I need to compute the cohomology groups of some relative holomorphic $i$-forms $H^\bullet(X, \Omega^i_{X/Y})$ for a fibration of complex manifolds $X\to Y$, using a kind of distributional de Rham ...
- 2,044
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Different Hodge numbers arising from different holomorphic structures?
Does anyone have an example or know any references for a complex manifold $M$ with two different holomorphic structures that give rise to different Hodge numbers?
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How does the Torelli theorem behave with respect to cyclic covering?
Let $Y\xrightarrow{2:1}\mathbb{P}^3$ be the double cover, branched over a quartic K3 surface $S$, known as quartic double solid. Assume $S$ is generic, we know that there is a Torelli theorem for $Y$ ...
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Hodge numbers of a complement
Let $Y\subset X$ be an analytic subvariety of codimension $d$ of a smooth compact complex variety $X$. Denote $U = X\setminus Y$. The relative cohomology exact sequence implies that
$$
H^i(X) \to H^i(...
- 2,305
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When does the rational Hodge structure determine the integral Hodge structure?
Take a smooth complex projective variety $X$, consider $H^k(X,\mathbb Z)$, and take the global period domain as described, for example, in Voisin's Hodge theory book, 10.1.3: it's a subset of a flag ...
- 683
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Prefactor $2\pi i$ for Tate-Hodge structure
A rather basic question. What was the original reason to consider the underlying $\mathbb{Z}$-module of the - as canonical object regarded - Tate-Hodge structure $\mathbb{Z}(1)$ to be given as $2 \pi ...
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Understanding the Hodge filtration
Let $X$ be a smooth quasiprojective scheme defined over $\mathbb{C}$, and let $\Omega^{\bullet}_X$ denote its cotangent complex, explicitly, we have:
$\Omega^{\bullet}_X:=\mathcal{O}_X\longrightarrow \...
- 2,907
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Hodge filtration vs Hodge structure on algebraic de Rham cohomology
I have a basic question on the relation between the definitions of the Hodge structure on the algebraic de Rham of a smooth proper scheme defined over a subfield of $\mathbb{C}$ and the Hodge ...
- 2,907
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How does complex conjugation act on the Hodge filtration?
Let $X$ be a $\mathbb{R}$-defined smooth proper scheme, and let $H^i_{\text{dR}}(X)$, denote its algebraic de Rham cohomology. The Hodge filtration gives an $\mathbb{R}$-defined pure Hodge structure ...
- 2,907
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Action of monodromy on the $p$-adic period domain in Lawrence-Venkatesh
In here, I asked various questions related to Lawrence and Venkatesh's work on the Mordell-Weil conjecture, which failed to receive any answers.
This is my attempt to try and focus the question.
In ...
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What is the correct definition of intermediate Jacobian for this singular threefold?
I am considering blow up of $\mathcal{C}\subset(\mathbb{P}^1)^3$, $X=\operatorname{Bl}_{\mathcal{C}}(\mathbb{P}^1)^3$, where $\mathcal{C}$ is a curve given by $$\{s^2u=0\}\subset\mathbb{P}^1_{s:t}\...
- 1,982
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Hodge bundles associated to a family of complex manifolds
I'm reading Voisin's books on Hodge theory. In the first volume she claimed but didn't prove this theorem:
Theorem 10.10 (Voisin) Let $\varphi:\chi\rightarrow B$ be a family of compact complex ...
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Blowup formula for a morphism
Let $f: X\to S$ be a smooth projective morphism between smooth schemes over $\mathbb C$, $i: Z \to X$ a closed subscheme of codimension $c$, also smooth over $S$, and let $g: Y\to S$ be the blowup ...
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What are the Hodge and log Hodge groups of $M_{g,n}$?
I would like to know, ideally with a reference, what the Hodge and log Hodge numbers of the moduli space of stable curves $\bar M_{g, n}$ are. At the very least I'd like to know the genus zero case $g ...
- 1,084
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Original proof of Lefschetz's theorem on $(1,1)$ classes
Is there a "modern" account of Lefschetz proof of his theorem about $(1,1)$ classes for projective surfaces ?
I believe that would be very interesting to understand the original arguments ...
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Using connection form for unknown frame field
I have a way to calculate the connection 1-form $\alpha$ associated to a compact simply connected parallelizable Riemannian surface $(M,g)$ (so, $M$ is topologically a disk) and a special orthonormal ...
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Hodge coniveaux of Calabi-Yau manifolds
Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
- 279
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Is the Leray projection continuous with respect to the Frechet topology of smooth periodic vector fields in $3$ dimensions?
Let $\mathbb{T}^3:=(\mathbb{R}/\mathbb{Z})^3$ be the $3$-torus and $C^\infty(\mathbb{T}^3,\mathbb{R}^3)$ be the Frechet space of smooth periodic vector fields on $\mathbb{T}^3$.
By Helmholtz ...
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