Questions tagged [hodge-theory]

For question about Hodge theory, which is a method for studying the cohomology groups of a smooth manifold using partial differential equations.

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14 views

What is a rotation number in topology in relation to complex numbers? [closed]

I believe this has something to do with algebraic cycles on complex projective planes?
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Does the proof of Hodge Conjecture require the use of Poincare Duals and if so why? [closed]

Wikipedia says that certain de Rham cohomologies should be the sums of the Poincare Duals of their subvarieties. But I haven't seen this mentioned anywhere else on the internet.
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How do I prove that the coordinate independant definitions of the Hodge dual operator are equivalent?

I am trying to show that the different ways of defining the Hodge star operator are equivalent. I started with the following definition from the following lecture notes "An application of ...
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1answer
72 views

The Kähler condition on a Riemann surface

A Hermitian metric $h$ on a complex manifold $X$ is Kähler if the associated $2$-form $\omega=\mathrm{Im} (h)$ is closed. This condition is trivial on compact Riemann surface, implying that every ...
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Are there any useful and comprehensive lectures or videos related to that of the topic of Hodge theory?

I am planning to use Claire Voisin's books for learning Hodge theory. They have a look towards algebraic geometry, which is one of my main areas of study. Notes and papers are ok too.
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Can every cohomology class be represented by an analytic form

Let $M$ be an analytic manifold (you may assume it is equipped with an analytic metric). Must each De Rham cohomology class be representatble by an analytic differential form ? I think Hodge theory ...
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What is the intuitive idea of $rot(A)$ of a matrix $A$?

I am a beginner in this field. Some background knowledge: An $n$ by $n$ antisymmetric matrix $\mathbf{A}$ can be considered as a flow assignment of a fully connected graph with $n$ vertices where the ...
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1answer
128 views

Hodge decomposition seems to say harmonic and exact is zero (and possibly (harmonic and co-exact) and (exact and co-exact))

What I understand. Please correct if wrong: Part of Hodge Decomposition Theorem says that for a compact oriented Riemannian (smooth) $m$−manifold $(M,g)$ (I think M need not be connected, but you may ...
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62 views

Consequence of Hodge decomposition: What is the norm here?

Hodge Decomposition Theorem says smooth $k$-forms on on compact oriented Riemannian (smooth) $m$−manifold $(M,g)$ (I think M need not be connected, but assume connected if need be or you want) ...
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1answer
68 views

Hodge decomposition seems to say co-exact component is zero

Hodge Decomposition Theorem says smooth $k$-forms on on compact oriented Riemannian (smooth) $m$−manifold $(M,g)$ (I think M need not be connected, but assume connected if need be or you want) ...
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74 views

What's the term for elements in (image of codifferential/adjoint exterior differential aka 3rd part of hodge decomposition theorem )?

Part of the Hodge Decomposition Theorem is that smooth $k$-forms on compact oriented Riemannian (smooth) $m-$manifold $(M,g)$ (I think $M$ need not be connected, but assume connected if need be or you ...
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59 views

What is the meaning of $\flat$ and $\sharp$ in this smoothing operator? $M^\sharp(x,\xi)=\sum_k \Psi_0(2^{-k\delta}D)m_k(x)\;\psi_{k+1}(\xi) $

I found this: $$M(x,\xi) = M^\sharp(x,\xi)+M^\flat(x,\xi),$$ in this paper, where $$M^\sharp(x,\xi)=\sum_k \Psi_0(2^{-k\delta}D)m_k(x)\;\psi_{k+1}(\xi) $$ and i'm not sure whether i understand it ...
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How to understand this hermitian vector bundle?

I am reading Huybrechts's complex geometry. In page 166, he gives the definition of hermitian structure. Let $E$ be a complex vector bundle over a real manifold $M$. An hermitian structure $h$ on $E\...
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33 views

How to show that motives classify Weil cohomological theories?

Let $ \mathcal{Sm}_k $ be the category of smooth projective varieties over a field $ k $. Let $ X \in \mathcal{Sm}_k $ of dimension $ n $. A Weil cohomology is a cohomological theory of algebraic ...
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Proposition 4.3 from “Mixed Hodge structures”

In the book "Mixed Hodge structures", Proposition 4.3 reads ($U$ a smooth variety, $X$ a simple normal crossings compactification): In inclusion of complexes $$\Omega_X^\bullet(\log E)\to ...
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Question from $p$-adic HodgeTheory, linear algebra data

I want to understand the definition $2.1.4$ of $\text{Mod}_{\mathscr{O}}^{\varphi}$ category in the pages $4$-$5$ in the $p$-adic Hodge Theory survey paper here, by Brian Conrad. The map $\varphi_{\...
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Why does morphism $P \to X$ imply that intermediate Jacobian $J(X)$ is a direct summand of $J(P)$?

I'm currently reading the sketch proof of Theorem 1 in Beavuille's notes on the Luroth problem. Let $P$ be the blow-up of $\mathbb{P}^3$ in finitely many points and smooth curves, and let $X \subset \...
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The cycle class map and the Hodge conjecture.

Let $ X $ be a compact complex algebraic manifold of dimension $ n $. For each integer $ p \in \mathbb{N} $, let $ H^{p,p} (X) $ denotes the subspace of $ H^{2p} (X, \mathbb{C} ) $ of type $ (p,p) $. ...
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Why polarization in Torelli theorem?

Are there two non-isomorphic compact Riemann surfaces with isomorphic integral Hodge structure on $H^1(-,\mathbb{Z})$? Recall Torelli theorem for Riemann surface: For two compact Riemann surfaces $X,Y$...
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41 views

Effect of Hodge star operator on electromagnetic field

From looking at similar questions, it seems that I'm far from the only one having problems with the notation of the effects of the Hodge star operator. My question is quite specific, to page 92 of ...
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Hodge Star in Jost's Book

I am currently reading Jost's Book "Riemannian Geometry and Geometric Analysis" (Sixth Edition). In 3.3 on page 103, he writes: From the rules of multilinear algebra, it follows easily that ...
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Where does the coefficient $\frac{1}{2}$ of $(-1)^p\frac{1}{2}\bar{\partial}_E^* = \bar{\partial}^*$ in $A^{0,q}(\Omega_X^p)$ come from?

Consider the holomorphic bundle $E=\Omega_X^p$ over complex manifold $X$. $\bar{\partial}^*$ is the formal adjoint of $\bar{\partial}$ with respect to inner product of differential forms. As in Hodge ...
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Explicit Hodge decomposition on $T^2$

Given a general compact Riemann manifold $(M,g)$, we have the well-known Hodge decomposition $$ \Omega^*(M)\cong d\Omega^*(M) \oplus \delta\Omega^*(M)\oplus \mathcal H_{\Delta}(M) $$ where $\delta$ is ...
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$h^{p,q}$ of a complex torus.

As we know, a compact Kähler surface with trivial canonical bundle is a K3 surface or a torus of dimension 2. I know $h^{0,2}$ of a K3 surface is 1, and I know $h^{0,2}$ of a torus must not be zero (...
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Sum of Dolbeault cohomology and homotopy invariance

If $X$ is a compact Kahler manifold, then the Hodge decomposition is $$H^n(X,\mathbb{C})=\bigoplus_{i+j=n} H^{i,j}(X).$$ Then the left hand side is homotopy invariant and therefore, so is therefore so ...
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Hodge decomposition and homotopy equivalence

For any compact Kahler manifold $X$, we have a decomposition $$H^n(X,\mathbb{C})\cong \bigoplus_{i+j=n}H^{i,j}(X).$$ The cohomology of spaces is a homotopy invariant. Then it seems natural to ask if ...
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Characterization of Simplicial Complex using p-Laplacian/Hodge Laplacian

We know that for a simple graph (i.e. a graph with no self-loops and multiedges), the Graph Laplacian uniquely characterizes it in the sense that if two graphs have the same Graph Laplacian, then the ...
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Local Characterization of Hodge Star Operator on a Vector Bundle

I am reading the proof of Nakano's Identity in Complex Geometry by Huybrechts. This is Lemma 5.2.3, for reference. If $\mathcal{A}_X^{p,q}$ denotes the sheaf of smooth $(p,q)-$forms on a compact ...
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Griffiths Harris Star operator definition.

I was reading harmonic theory from Griffiths Harris Principles of Algebraic geometry. I have difficulty in verifying the definition of the star operator. We have a Hermitian metric on the holomorphic ...
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99 views

Explanation for devissage argument

Let $K$ be a local field of characteristic $0$ with the ring of integers $\mathcal{O}_K$ and uniformizer $\pi$. Let $k$ be the residue field of $K$ with $\text{card}(k)=q$. Let $\mathcal{O}_\mathcal{E}...
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how does a complex structure on a real vector space give rise to a hodge structure

Let $V$ be a finite dimensional vector space over $\mathbb{R}$, with a complex structure, i.e. a $\mathbb{R}$-linear map $J:V \to V$ such that $J^2=-Id$ (which gives $V$ the structure of a $\mathbb{C}$...
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Question Regarding Proof of Hodge Index Theorem

I am reading Voisin's proof of the Hodge Index Theorem on pp. 153-154 of her Hodge Theory and Complex Algebraic Geometry I. The proof is mostly clear except for one technical point. Let $n$ denote an ...
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Hermitian holomorphic Connection and Dolbeault operator

Let $E \to M$ a Hermitian holomorphic vector bundle of rank $m$ over an almost-$\mathbb{C}$ hermitian Kähler manifold $(M,h)$ of dimesnion $n$. By a well known theorem $E \to M$ has a unique ...
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Geometric genus as dimension of a variety

Given a smooth complex algebraic variety $X$ of dimension $n$, Hodge number $h^{1,0}(X)$ (irregularity) is the dimension of Albanese variety $A(X)$ of $X,$ so in particular it is non-zero if and only ...
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Hodge star is conformally invariant on $\Lambda^{n/2}(V)$, for $n$ even

I am studying the Hodge star operator for the first time. I am trying to prove that for $n$ even, then for any $\omega \in \Lambda^{n/2}(V)$ $\star_g \omega= \star_{\tilde{g}} \omega$, where $g$ and $\...
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Inner product of differential forms. Hodge or metric?

In a Riemannian manifold, its inner product: $\left( \cdot,\cdot \right)_g : T_pM \times T_pM \to K$ Can be defined as by pairing $\alpha^i$ with $\beta^j$ dual by means of the metric tensor: $\...
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What is the right definition of the n dimensional Hodge dual and cross product?

Could you help me with those geometric algebra confusions: Which one is the right definiton of pseudovector ? $$(n-1)vector\\ \text{or}\\ *((n-1)vector)$$ I found one can express the hodge dual this ...
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A high road to the Kähler identities?

Let $(X, \omega)$ be a compact Kähler manifold. The Kähler identities express the commutator relations between the operators $$\partial, \ \ \overline{\partial}, \ \ L,$$ and their adjoints. To be ...
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Can you help me to get the previous lectures of this same lecture series?

I got a lecture from the link-https://www.math.uni-bonn.de/people/ja/thecurve/the_curve_lecture_15_1_2020.pdf. Unfortunately, there are previous lectures of this series which I could not find. Can ...
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A bound for a differential operator in Sobolev norms

Let $s$ be an integer and $L$ a periodic linear partial differential operator $L=\{L_{ij}\}$ of order $l$ on $\mathcal{P}$, the space of $2\pi$-periodic functions $R^n\longrightarrow C^m$. The sobolev ...
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$\mathbb{Z}_2-$grading by Hodge star operator.

Consider the algebra of exterior forms $\Lambda T^*M$ on an even dimensional $n-$manifold $M$. We can form an operator $\sigma=\bigoplus_{k=1}^ni^{k(n-k)}*_k:\Lambda T^*M\rightarrow\Lambda T^*M$ (...
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Definition of Complex Integral in Warner's Foundations of Differentiable Manifolds and Lie Groups

In chapter 6 of Warner's Foundations of Differentiable Manifolds and Lie Groups, in the section subtitled Some Calculus, the author introduces the complex vector space $\mathcal{P}$ of smooth ...
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What is a Punctured disk?

What is a punctured disk? I am reading Voisin Book (Hodge Theory and Complex Algebraic Geometry) and I found this term and I would like to know what is it. Thank you!!
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definition of of Breuil-Kisin modules

I want to closely understand the definition of category of Breuil-Kisin modules and its category . I am a new guest in the topic. So need your help
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Fiber of direct image of relative de Rham complex in non-proper case

Let $f: X \rightarrow B$ be a smooth family of complex varieties ($X$ and $B$ are also smooth). If $f$ is proper, then the direct image of the relative de Rham complex, $Rf_*\Omega_{X/B}^{\bullet}$,...
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Cohomology computation

Let $\Sigma_2$ be the second Hirzebruch surface and $f : \Sigma_2 \to X$ that contracts the exceptional section $E$. Since $E^2=-2$, $X$ has a quadratic singularity at $p := f(E)$. If $U'$ is a ...
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1answer
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Does the Lie derivative of a harmonic form with respect to a killing vector field vanish

On the wikipedia page for Killing vector fields under 'properties' it is stated that if $X$ is a Killing vector field and $\omega$ is a harmonic differential form then $\mathcal{L}_{X}\omega=0$. https:...
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Does the symbol $(B \otimes_{\mathbb{Q}_p} V)^{G_K}$ mean the set of elements of $(B \otimes_{\mathbb{Q}_p} V)$ fixed by the Galois action of $G_K$?

My question is rather symbolic meaning. In p-adic Hodge theorem (https://en.wikipedia.org/wiki/P-adic_Hodge_theory), it has been defined as follows: $$ D_B(V)=(B \otimes_{\mathbb{Q}_p} V)^{G_K}, \ \...
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Line bundles on projective space and disk

I'm having a difficult time solving some exercice. I should prove the following : Show that any holomorphic line bundle on a disc $\Delta\subset\mathbb{C}$ is trivial. Deduce that any holomorphic ...
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242 views

What is the definition of $\text{weight or Hodge-Tate weight}$ in the p-adic Hodge theory?

My question is about p-adic Hodge-Tate theory and p-adic Galois representation. What is the definition of $\text{weight or Hodge-Tate weight}$ in the above theory? For example, I have the following ...

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