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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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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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1answer
110 views

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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1answer
517 views

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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2answers
1k views

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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0answers
103 views

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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1answer
608 views

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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2answers
277 views

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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0answers
195 views

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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2answers
241 views

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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0answers
180 views

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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0answers
81 views

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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0answers
248 views

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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0answers
213 views

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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0answers
277 views

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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0answers
108 views

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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0answers
160 views

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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208 views

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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137 views

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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1answer
170 views

Lefschetz standard conjecture under specialization/generization

Let $S$ be a smooth connected noetherian scheme (not necessarily over a field) with residue fields that are all of finite type over their prime field. Let $f: \mathcal{X}\to S$ be a smooth projective ...
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0answers
23 views

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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0answers
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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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93 views

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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1answer
258 views

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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0answers
106 views

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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0answers
97 views

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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1answer
383 views

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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108 views

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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1answer
295 views

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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0answers
96 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 ...
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235 views

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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146 views

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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160 views

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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1answer
282 views

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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1answer
779 views

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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1answer
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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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1answer
213 views

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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143 views

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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126 views

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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323 views

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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0answers
197 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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1answer
379 views

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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2answers
881 views

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 ...
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3answers
921 views

Motivic vs Deligne cohomology

Where can I find the construction of the cycle class map from motivic cohomology to Deligne cohomology of smooth projective varieties over the complex numbers? It should be a construction by Bloch ...
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0answers
294 views

Homotopical enhancements of cycle class maps

Fix a smooth projective variety $X$ over the complex numbers. We write $H^n(X,\mathbf{Z}(d)) = \text{CH}^d(X, 2d-n)$ for Bloch's higher Chow groups. Notation For a field $k$, recall $\Delta^n_{k} :=...
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1answer
181 views

Mumford-Tate groups of abelian surfaces

For elliptic curves, one may easily compute Mumford-Tate groups; there are just two cases: 1) $E$ has no complex multiplication, and the Mumford-Tate group of $E$ is $GL_2$ 2) $E$ has complex ...
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0answers
144 views

Explicit algebraic cycles

Fix a smooth sextic curve curve $C = \{f_6(x,y,z) = 0\}$ in $\mathbb{P}^2$, and consider the double cover $X_{f_6}$ defined by $z^2 = f_6$ in the appropriate weighted projective space. This is known ...