# 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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### Functoriality of crystalline cohomology

Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth projective $k$-variety.
Denote by $(X/W_n(k))_{\rm cris}$ the small crystalline site of $X$.
Let $f : X\to Y$ be a morphism of ...

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### What is Prismatic cohomology?

Prismatic cohomology is a new theory developed by Bhatt and Scholze; see, for instance, these course notes. For the sake of the community, it would be great if the following question is discussed in ...

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### Intermediate Jacobian of abelian varieties

Is the intermediate Jacobian of an abelian variety again an abelian variety?

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### Principal bundle analogue for Hodge bundle

Let $X$ be a connected smooth complex projective variety.
A holomorphic Higgs bundle is a pair $(E, \theta)$ consists of a holomorphic vector bundle $E$ on $X$ together with a Higgs field $\theta \...

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### Does local system cohomology come equiped with a mixed hodge structure?

Let $X$ be a quasi-projective variety over $\mathbb{C}$, and let $\mathcal{L}$ be a rank one $\mathbb{C}$ local system on $X$. Does $H^*(X,\mathcal{L})$ come with some mixed hodge structure in general?...

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### 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 ...

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### Classes of hyperplane sections in cohomology

Let $X$ be a smooth projective variety over the algebraic closure of a finite field with Galois group $G$.
Is it true that the vector space $H^{2k}(X,\mathbf{Q}_{\ell}(k))^G$ has always positive ...

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### What is Kontsevich's Hodge theory of path integrals?

I was reading about the appearance of Calabi-Yau manifolds in Feynman integrals, and I thought to wonder if there is such a thing as "infinite-dimensional Hodge theory". Googling the phrase turned up ...

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### Hodge theory (after Deligne)

In an interview with Deligne on the Simons Foundation website, I heard Robert MacPherson say that at the time Deligne's papers on Hodge theory were being published, the results seemed absolutely ...

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### Two notions of a “nilpotent orbit”

I am wondering about the equivalence of two notions of a "nilpotent orbit".
The first notion, which I am familiar with, is as follows: given a lie group $G$ and a lie algebra $\frak{g}$, the orbit of ...

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### Hodge decomposition and degeneration of the spectral sequence

I am teaching a course on Hodge theory and I realised that I don't understand something basic.
Let first $X$ be a compact Kahler manifold. Let $H^{p,q}(X)=H^q(X,\Omega^p_X)$ where $\Omega^p_X$ is the ...

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### Is there an example to show the Hodge decomposition fails on non-compact case

The theorem of Hodge decomposition is on the compact Kahler manifold, is it generally true for the non-compact kahler manifold or are there examples to show the failure?
Here is my Hodge ...

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### Degeneration of relative Hodge-de Rham spectral sequence

$$\require{AMScd}$$
$$\newcommand{\CC}{\mathbb{C}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\Hdr}{H_{\mathrm{dRh}}}
\newcommand{\tensor}{\otimes}
\newcommand{\Ohol}{\mathcal{O}}$$
Please excuse that ...

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### Generic Mumford Tate group and algebraic points

I will stick with a concrete example for this question, but it should probably be cast in a more general framework.
Let $Sym_g(\mathbf{C})$ be the space of symmetric matrices of order $g$ with ...

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### dual to Hodge theory

Let $(M,g)$ be a closed Riemannian manifold.
In my understanding Hodge theory shows that any de Rham cohomology class can be represented uniquely by a harmonic form. Moreover the harmonic form ...

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### Topological cycles with Lagrangian support

For a compact Kähler manifold of dimension $2n$, is there a classification of the homological $n$-cycles which are supported in a compact Lagrangian submanifold?
The main example for this question ...

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### Does Lefschetz-type theorems imply ampleness?

Let $X$ be a smooth $n$-dimensional complex projective variety and $D \subset X$ a smooth (effective) divisor. Consider the following properties:
$D$ is ample.
(Positivity) For any $k$-dimensional ...

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### Algebraic vs analytic de Rham cohomology

Let $X$ be a smooth projective variety over $\mathbf{C}$, $\Omega^{\bullet}_X$ its algebraic de Rham cohomology.
Let $p : X_{\rm an}\to X_{\rm Zar}$ the obvious morphism of sites.
We have $p^*\Omega^...

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### Bloch Ogus spectral sequence

Let $X$ be a smooth projective variety over $\mathbf{C}$, and $p : X_{\rm an}\to X_{\rm Zar}$ the obvious map of sites.
The Leray spectral sequence
$$H^r(X_{\rm Zar}, R^sp_*\mathbf{C})\Rightarrow H^{...

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### Set theoretic complete intersections in toric varieties

Is it expected that every smooth projective variety over the complex numbers, is a set-theoretic complete intersection into a smooth projective toric variety?
Is there an example of a smooth ...

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### Neron Severi under specialization

Let $X$ be a smooth projective variety over $\mathbf{Q}$, and $\mathcal{X}$ a smooth projective model over $\mathbf{Z}[1/N]$ for $N$ large enough.
Call $\eta$ the generic point $\text{Spec}(\mathbf{Q}...

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### Motives up to homological equivalence

Let $X$ be a smooth projective variety over a field $k$ finitely generated over its prime field, and $M_{hom}(X)$ the category of motives modulo $\ell$-adic homological equivalence.
(1) Is $M_{hom}(...

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### Hard Lefschetz for cycles

Let $X$ be a smooth projective variety over a field $k$. It is known by work of Deligne, that the Lefschetz operator:
$$L^k : H^{2n-2k}(X_{\overline{k}},\mathbf{Q}_{\ell})\to H^{2n+2k}(X_{\overline{k}...

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### Numerical vs Homological equivalence

Does the Hodge Conjecture in codimension $p$ imply that homological and numerical equivalence on codimension $p$ cycles agree?
What is a reference, please?

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### Hodge classes generated in degree $1$

Let $X$ be a smooth projective variety over the complex numbers, and $\text{Hdg}^p(X)_{\mathbf{Q}}$ the abelian group of Hodge classes in $H^p(X,\mathbf{Q}(p))$.
Denote by $\text{Hdg}^*(X)$ the ...

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### Multiplicative structure on Deligne cohomology

Let $X$ be a smooth projective variety over the complex numbers, and $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex on $X$:
$$\mathbf{Z}(p)_{\mathcal{D}} : \ \ \mathbf{Z}(p)\to\mathcal{O}_X\to\...

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### Spectral sequence in Betti cohomology

Let $X$ be a smooth projective algebraic variety over the complex numbers, and let us name
$$f : X_{\rm an}\to X_{\rm Zar}$$
the morphism of sites induced by sending a Zariski open $U\subset X$ to $...

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### Torsion homologically trivial cycles

Is there an example of a smooth projective variety $X$ over the complex numbers, such that
$$\ker(\text{CH}^2(X)\to H^4(X,\mathbf{Z}(2))$$
is not torsion?

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### Filtrations and the Betti cycle map

Let $X$ be a smooth projective complex variety.
Calling $f : X_{\rm an}\to X_{\rm Zar}$ the "change-of-topology" morphism of sites induced by sending a Zariski open $U$ of $X$ to its analytification, ...

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### Integral coniveau spectral sequences in Hodge Theory

Let $X$ be a smooth projective complex analytic space.
We name $\mathcal{H}^*(\mathbf{Z}(n))$ the Zariski sheafification of Betti cohomology with $\mathbf{Z}(n)$ coefficients.
We have a "coniveau" ...

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### A quite puzzling question on Deligne cohomology sheaves and cycle maps

Intro. I would be deeply grateful if someone could please clarify the following to me.
The question. (the main point is (4))
Let $X$ be a smooth projective variety over $\mathbf{C}$, and $\mathbf{Z}(...

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### Cycle maps as edge maps

Given a smooth projective algebraic variety over $\mathcal{C}$, let $X$ be its associated complex analytic space.
The exponential sequence on $X$:
$$0\to\mathbf{Z}(1)\to\mathcal{O}_X\to\mathcal{O}_X^...

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### Reference for flatness in complex-analytic geometry

What is a good reference for flat morphisms of complex-analytic spaces? (The book by Grauert and Remmert doesn't treat them).
Topics I'm interested in: openness of flat maps, descent for coherent ...

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### 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 ...

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### Chern class map and the exponential sequence

Let $X$ be a smooth projective variety over the complex numbers, and
$$c^1_X : \text{NS}(X)\to H^2_{\rm Betti}(X,\mathbf{Z}(1))$$
the first cycle map to Betti cohomology. The cokernel $\text{coker}(...

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### Cycles modulo homological equivalence

Let $\text{CH}^p(X)_{\rm hom}$ be the abelian group of codimension $p$ algebraic cycles on a smooth projective variety over a field $k$, modulo homological equivalence. Is $\text{CH}^p(X)_{\rm hom}$ ...

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### Artin $\ell$-adic comparison and Galois action

Let $X_0$ be a smooth projective variety defined over a number field $k$.
Let $\sigma : k\to\mathbf{C}$ be one of the finitely many field embeddings of $k$ into the complex numbers, and call $X := (...

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### Absolute Hodge Cycles over $\mathbf{Q}$

In the 1986 notes by Milne "Hodge cycles on abelian varieties", Deligne defines the notion of absolute Hodge cycles.
For a smooth projective variety defined over $k\subset\mathbf{C}$ non ...

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### Locus of Hodge classes

Let $\pi: X\to S$ be a proper smooth morphism of complex analytic spaces, with connected smooth $X$ and $S$ over $\mathbf{C}$, projective fibers, and $$\mathscr{H}_{X/S}^p := R^p\pi_*\Omega^{\bullet}_{...

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### Absolute Hodge cycles

Let $X$ is a smooth projective variety defined over a finite extension $K/\mathbf{Q}$, $\sigma : K\to\mathbf{C}$ any of the finitely many field embeddings of $K$ into the complex numbers, and call $X^{...

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### How to think about infinite generatedness of motivic cohomology

In this question I previously asked how to think about the motivic complex $\mathbf{Z}(1)_{\mathcal{M}}$, whose Zariski hypercohomology should morally be the "singular cohomology" $H^*((-)\wedge S^{2n}...

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### How to think about $\mathbf{Z}(n)_{\mathcal{M}}$

One definition of motivic cohomology for smooth schemes $X$ over a field, is via Friedlander-Suslin complexes.
A refresher (you may skip to the question at the bottom)
One defines
(1) $z_n(X,d) :=$...

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**1**answer

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### Analytic cycles on complex-analytic spaces

If $X$ is a proper smooth complex analytic space, one can define Chow groups of analytic cycles on $X$ the usual way.
We have a cycle map
$$c^p_X: \text{CH}^p(X) \to \text{H}^{2p}_{D}(X,\mathbf{Z}(...

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### Structure of Deligne cohomology

It is a classical fact that for a smooth and proper complex-analytic space $X$, the Deligne cohomology $H^p_{\mathcal{D}}(X,\mathbf{Z}(q))$, defined as the hypercohomology of the complex
$$\mathbf{Z}(...

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### Hodge cycles defined over algebraic extensions of $\mathbf{Q}$

Is it true that the Hodge conjecture for all smooth projective varieties over the complex numbers, follows from the Hodge conjecture for smooth projective varieties defined over $\overline{\mathbf{Q}}$...

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### Beilinson regulators and Bloch's mythological algebraic intermediate Jacobians

In the paper introducing his motivic cycle complexes, Bloch outlines a project he says he was going to return to in the future:
Towards the end of page 270, he says, given a smooth projective variety ...

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202 views

### About the exponential sequence

For a complex analytic space $X$, we have the exponential sequence
$$0\to\mathbf{Z}(1)_X\to\mathcal{O}_X\to\mathcal{O}_X^{\times}\to 1$$
the last map being the exponential $\text{exp}$.
For $d>0$ ...

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### Which rational cohomology classes on a product of elliptic curves come from subschemes?

Let $X=E_1\times\cdots\times E_n$, where $E_i$ is the elliptic curve $E_i=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\alpha_i)$. In Grothendieck's "The Hodge Conjecture is False for Trivial Reasons," $X$ is ...

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### Hodge theory on cylindrical end

Let $(X,g_0)$ be a compact oriented Riemannian 4-manifold with boundary $S^3$ and $Y=X\cup [0,\infty)\times S^3$ with metric $g$ which is product on the cylindrical part.
Consider the operator $d^++...

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### Is Deligne cohomology the motivic cohomology of analytic spaces?

Let $X$ be a smooth projective complex analytic space.
We can cook up a complex analytic version of Bloch's cycle complex by declaring
$z^n(X^{\rm an}, m)$
is the free abelian group on all ...