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.
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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 ...
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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 ...
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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 ...
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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(...
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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 ...
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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 \...
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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 ...
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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 ...
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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}\...
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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 ...
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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 ...
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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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On Simpson's motivicity conjecture
Simpson's motivicity conjecture says that for any rigid, flat irreducible connection $(V,\nabla)$ on a smooth complex variety $M$, there exists a proper smooth morphism $f:X \to M$ s.t. $(V,\nabla)$ ...
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Torelli theorem for veronese double cone(reference needed)
Let $Y$ be a smooth Veronese double cone, which is a smooth del Pezzo threefold of degree one, which can be regarded as a weighted hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$. I was wondering ...
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Simpson correspondence for perverse sheaves
Let $X$ be a projective complex manifold. Then Simpson's correspondence from nonabelian Hodge theory shows that the category of semisimple local systems on $X$ is equivalent to the category of ...
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Fourier transform and Hodge-$*$ operator
Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says
$$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$
...
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Hodge conjecture as the equality of arithmetic and algebraic weights of motivic L-functions
Recently I became aware of the following statement given on page 13 of this paper. First, let us recall the following definitions:
Definition 4.1. Suppose $L(s)$ is an analytic $L$-function with ...
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Hodge symmetry without $\mathbb{C}$ [duplicate]
If $k$ is a field of characteristic zero and $X$ is a smooth irreducible projective variety over $k$, then $X$ satisfy Hodge symmetry, meaning that
$$\dim H^p(X, \Omega_{X/k}^q) = \dim H^q(X, \Omega_{...
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Intermediate Jacobian for small resolution of a singular Fano threefold?
I am mainly interested in the nodal Gushel-Mukai threefold. Let $X$ be a Gushel-Mukai threefold with one node, then by page 21 of the paper https://arxiv.org/pdf/1004.4724.pdf there is a short exact ...
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Is $H^*($vanishing cycles$)$ computed by the twisted de Rham complex?
In notes by Sabbah (Theorem 3), it is stated that the cohomology
$$\text{H}^*(X,\varphi_f)$$
of the vanishing cycle sheaf of a function $f:X\to \mathbf{A}^1$ for certain $X$ is expected to be the same ...
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Eigenforms of the Laplacian on Lie groups
I am a bit rusty in my differential geometry and I would like to confirm that my reasoning below holds, and I have some related questions (and all references to related concepts are of interest to me)....
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Cohomology of singular curves
Suppose $X$ is a singular quasi-projective curve over the complex numbers, and $X'$ is a good nonsingular compactification of a resolution of singularities $Y\to X$. Let $D$ be the complement of $Y$ ...
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Geometric Interpretation of absolute Hodge cohomology
$\quad$Let $\mathcal{Sch}/\mathbb C$ denote the category of schemes over $\mathbb C$. For an arbitrary $X\in\mathcal{Ob}(\mathcal{Sch}/\mathbb C)$, Deligne in his Article defined a polarizable Hodge ...
user508433
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Does Poincaré duality preserve algebraic cycles?
Let $X$ be a smooth, projective (complex) variety of dimension $n$ and $Z \subset X$ be a subvariety of codimension $k$ (if necessary assume $Z$ is non-singular). We know the cohomology class $[Z]$ of ...
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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 ...
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Clausen's modified Hodge Conjecture
In a recent talk at the University of Geneve, Dustin Clausen presented a "modified Hodge Conjecture". I found the abstract intriguing but couldn't find videos or notes available online.
If I'...
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Triangulated structure on complexes of mixed Hodge structures
I'm trying to read parts of the Peters Tata Lectures on "Motivic Aspects of Mixed Hodge structures"
One aspect I don't really understand is the construction of the ''mixed cone'' for ...
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Formula in non-Abelian Hodge theory - Hodge-Riemann bilinear relations
I am currently reading about the non-Abelian Hodge correspondence. Let $(X,\omega)$ be a compact Kahler manifold. Given a Higgs bundle $(E, D_0)$ on $X$, we want to construct the corresponding flat ...
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Given a smooth hyperplane section Y of a variety X there exists a Lefschetz pencil of hyperplane sections of X containing Y
Let $X$ be a variety contained in $\mathbb{P}^N$ and let $Y$ be a smooth hyperplane section of $X$. I have read in page 54 of Voisin's book "Hodge theory and complex algebraic geometry II" ...
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Hodge-Helmholtz decomposition for 1-form of strategic game
This question is motivated by the attempt of decomposing (in a direct sum sense) a strategic game into components in the spirit of Hodge decomposition.
Preamble
Combinatorial setting
Candogan et al. (...
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Balanced manifolds and the $dd^c$-lemma
Let $X$ be a compact complex manifold. A Hermitian metric $\omega$ is balanced if $d\omega^{n-1}=0$, where $n=\dim_{\mathbf{C}} X$. By a theorem of Alessandrini-Basanelli, this class of Hermitian ...
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Hodge decomposition on non-compact manifolds
Let $(\mathcal{M},g)$ be a compact Riemannian manifold without boundary. Then we have the well-known Hodge decomposition
$$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\...
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Cohomology classes fixed by algebraic automorphism subgroups
Let $X$ be a smooth projective complex variety and $t\in H^{2p}(X,\mathbf{Q})(p)$ a rational cohomology class.
Assume that there exist
$$t_1,\ldots,t_N\in H^{2*}(X,\mathbf{Q})(*)$$
algebraic classes (...
user480741
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Period calculation for elliptic curve
In the paper "Hodge cycles on Abelian Varieties" (Proposition 1.5), Deligne proves the following theorem:
Let $X$ be a smooth projective variety over $\overline{\mathbf{Q}}$ of dimension $n$...
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Hodge decomposition for non-elliptic complexes
It is a well-known result that there is a bijective correspondence between harmonic sections and cohomology classes of an elliptic complex in a Riemannian/Hermitian manifold. Now consider Riemannian/...
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