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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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Hodge theory: $\Delta \alpha = 0$ iff $d\alpha = d^* \alpha = 0$ on a noncompact manifold?

Let $M$ be a Riemannian manifold (connected, oriented). One can define the co-differential $d^* : \Omega^k(M, \mathbb{R}) \to \Omega^{k-1}(M, \mathbb{R})$ even if $M$ is not compact (for example use ...
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Real Hodge structures and representations of the Deligne torus

Example 2.31 in these notes of Milne is about real Hodge structures and the fact that they can be seen as representations of the Deligne torus $\mathbb{S}$. The pace is a bit too fast for me. I have ...
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Hodge star operator on differential algebras

Given an arbitrary differential graded algebra $(\Omega,d)$ (over the field $\mathbb{R}$ or $\mathbb{C}$ ), it is posible to define an operator that acts like the Hodge star operator on differential ...
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The cohomological self-dual and anti-self-dual decomposition

The following statement is from Lübke's The Kobayashi-Hitchin Correspondence pp.222: If $a\in A^2(X)$ is harmonic, and $a=a^++a^-$ with $a^{\pm}\in A^2_{\pm}(X)$, then $a^+$ and $a^-$ are also ...
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Induced isomorphism between $H^{1,0}(X)$ and $H^{1,0}(Y)$ implies the induced isomorphism between $H^{0,1}(X)$ and $H^{0,1}(Y)$?

Let $X,Y$ be two compact Kahler manifolds and $f:X\to Y$ is a holomorphic map. If the induced map $f^*:H^1(X)\to H^1(Y)$'s restriction $f^*|_{H^{1,0}}$ an isomorphism from $H^{1,0}(X)\to H^{1,0}(Y)$, ...
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Character group of torus over the real numbers (Theorie de Hodge II)

I am trying to read the article Théorie de Hodge II (which can be found in French here) and in page 24, when Deligne starts discussing Hodge structures, he makes the following claim about the ...
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deformation of Hodge star operator and harmonic forms

Suppose $(M,g)$ is a compact Riemannian manifold, and $*_g$ is the Hodge star operator defined on the de Rham algebra $\Omega^*(M)$ with respect to the metric $g$. Let $\phi:M\to M$ be a ...
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The image of the exterior derivative is closed: Hodge theory

If we have an operator $T$ in a Hilbert space $H$ then one can decmpose $H$ as an orthogonal sum $ker(T^*) \oplus \overline{ran(T)}$. This is purely operator theoretic context. On the other hand there ...
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The homomorphism induced by holomorphic map preserves Hodge decomposition

Let $f:X\to Y$ be a holomorphic map between two compact Kähler manifolds. Then the induced map $f^*:H^1(Y,\mathbb{C})\to H^1(X,\mathbb{C})$ preserves the Hodge decomposition. Is there a reference for ...
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Use of the Rellich Lemma in the proof of the Hodge Theorem

I am reading through the proof of the Hodge theorem that is given in Griffiths' and Harris' Principles of Algebraic Geometry, see pages 84-100. The method of proof is to establish a weak solution of ...
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Hodge-$\star$ operator computation on a smooth two-dimensional manifold

Let $(x,y)$ be the local coordinates on a Riemannian manifold $M$ with $\dim(M) =2$. Let $\star$ denote the Hodge-$\star$ operator, and let $g = g_{ij}$ denote the Riemannian metric on $M$. I am ...
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Weak vs. strong convergence in the proof of the Hodge decomposition theorem in Warner, p.224

I'm reading the proof of the Hodge decomposition theorem in Warner, Foundations of Differentiable Manifolds and Lie Groups. At p.224, it is shown that $$ \lim_{j\to\infty} \langle\beta_j,\psi\rangle = ...
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Reference for Hodge decomposition for flag variety

Any references for proof of the following facts: The cohomology of the (complex) flag variety is always in $(p, p)$-type of Hodge Decomposition. The natural map $G/T → G_\mathbb{C}/B$ is a ...
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Why the $2 \pi i$ in the Tate twist?

I am reading here about Hodge structures and in (1.4) the Tate Hodge-structure $\mathbb{Z}(n)$ and the Tate twist $V(n)$ on a Hodge structure $V$ are defined. I understand that one might want to ...
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Projective algebraic manifold admits a positive line bundle

By the theorem: Let $X$ be a compact Hodge manifold. Then $X$ is a projective algebraic manifold, it follows that any compact complex manifold $X$ is projective algebraic iff it admits a positive line ...
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A symmetry of the Hodge star operator

Theorem 1.3.2(i) on p. 7 of these notes on the Hodge decomposition states $$ * \eta \wedge w = \langle \eta, w \rangle _{\wedge^k V} dV .$$ I don't see how this holds. We defined the Hodge $*$ ...
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Reference request Eigenspace decomposition Hodge Laplacian on forms on manifolds with boundary

I know that on a connected, compact, oriented Riemannian manifold without boundary the Hodge Laplacian $\Delta_k=(d+\delta)^2$ (acting on $k$-forms) admits an orthogonal eigenspace decomomposition of $...
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Laplacian on 1-forms on $(\mathbb R^m,\delta)$

I'm trying to show that if $\omega$ is a 1-form on $(\mathbb R^m,\delta)$, the action of the Laplacian is given by $$\Delta\omega=-\sum_{\mu=1}^m\frac{\partial^2\omega_\nu}{\partial x^\mu\partial x^\...
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Hodge duals of $k$-forms, in a non $g$-orthonormal basis.

The Hodge star $*$ can be defined in various ways. One of them is by its action on an arbitrary wedge product of basis elements $\{e^i\}$ of $V^*$. This definition is the following: $$*(e^{i_1}\wedge.....
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Are Hodge numbers topological invariants for manifolds that admit a Kähler structure?

I know that all fibers in a analytic fibration (proper, holomorpic) are homeomorphic, and if the fibers are Kählerian manifolds, then they have equal Hodge numbers. Could it happen however that a ...
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Wrong sign in the Hodge Laplacian of a product of functions

If $f,g : \mathbb R \to \mathbb R$ are smooth, then $(fg)^{\prime \prime} = f^{\prime \prime} g + 2 f^\prime g^\prime + f g^{\prime \prime}$ and, if we define a scalar product on $1$-forms by $\langle ...
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Do Maxwell's equations (generalized) apply to _every_ $k$-form on a pseudo-Riemannian manifold?

Given a pseudo-Riemannian $n$-manifold and a $k$-form $F$ on the manifold, I will call its exterior derivative $J=dF$ the source of $F$ and the differential $K=dG$ the dual source of $F$, where $G=​{\...
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Deligne's theorem on the Leray spectral sequence and weights

Motivation : If $f : X \to Y$ is a smooth projective map between algebraic varieties, then there is a theorem by Deligne which says that the Leray spectral sequence degenerates at $E_2$. The proof I ...
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Analogies between Hodge conjecture and Tate conjecture

I hear sometimes that there is many analogies between the Hodge conjecture and the Tate conjecture. If we take a look at the statements of this two conjectures, we have the followings : The Tate ...
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Star operator, exterior product

Let $\ast$ be the star operator $$\ast:\Lambda^p(V)\to \Lambda^{d-p}(V)$$ so that we have $$\ast(e_{i_1}\wedge...\wedge e_{i_p})=e_{j_1}...\wedge e_{j_{d-p}}$$ where $$e_{i_1},...,e_{i_p},e_{j_1},...,...
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Systematic application of algebraic topology to energy minimization problems?

I have come across two different occurrences of energy minimization problems which find an interpretation using notions from algebraic topology, and I was wondering whether analogous situations have ...
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Path to learning Hodge Theory via Voisin's text.

I'm currently in the process of learning a bit of Homological Algebra and Smooth Manifold theory via Tu's book. Being curious I searched around and ran into Voisin's Book on "Hodge Theory and Complex ...
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Sign of codifferential

I've read at least two conventions for the sign of the codifferential. The first of them \begin{align} \delta &= (-1)^{k}\star^{-1}\operatorname{d}\star\\ &= (-1)^{kn+n+1}\operatorname{sgn}(g)\...
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Local invariant cycles with integer coefficients

Let $X$ be a smooth complex alegebraic variety and $f: X \to \mathbb{C}$ a proper morphism that is smooth away from $0 \in \mathbb{C}$. Let $0 \neq b \in \mathbb{C}$. The pullback map $H^n(X, \...
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Lefschetz (1,1) theorem for quasi-projective varieties

I was reading about Hodge conjecture on Wikipedia but it started with the assumption that $X$ is smooth projective. If $X$ is a smooth quasi-projective variety, then corresponding to smooth sub-...
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Question in proof of Hodge decomposition theorem

I am reading this set of notes on Hodge theory: https://math.unice.fr/~hoering/hodge/hodge.pdf In particular the proof of 4.2.6 where the author proves the Hodge decomposition for a compact ...
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Every $L^2$ function is the divergence of a $L^2$ vector field

As in the title, I am struggling with the following statement: For any $f\in L^2(\mathbb R^n)$ there exists a vector field $F\in L^2(\mathbb R^n, \mathbb R^n) $ such that $f=div F$. Apparently ...
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Commutation of the covariant Hodge Laplacian with the covariant derivative

Let $(M, g)$ be a Riemannian manifold, $E$ a Hermitian vector bundle and $A$ a unitary connection over $E$ (i.e. the covariant derivative $d_A$ respects the inner product). The action of $d_A$ is ...
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Let $M$ is compact Riemann surface, if $\omega$ is a 2-form and $\int_{M} \omega =0$ then there exists a smooth function $f$ such that $\omega=d*df$

I want to show that: $(*)$If $\omega \in \Omega^{2}(M)$, which $M$ is compact Riemann surface and $\Omega^{2}(M)$ means 2-form, and $\int_{M} \omega =0$, then there exists a smooth function $f$(i.e....
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Regarding Hodge's theorem

We have $*$ the Hodge operator, and $d $ the exterior derivative. We define $\delta=\pm *d*$ and $\triangle=d\delta+\delta d $. Warner (pp. 223) says that we have $$ \triangle (E^p (M))=d\delta (E^p (...
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Why $H_{dR}^1(M) \simeq \mathbb R^n$ when $H_1(M,\mathbb Z)$ has $n$ generators?

Let $M$ be compact, connected and orientable manifold without boundary. Let $\sigma_1,\ldots, \sigma_n$ be generators of $H_1(M,\mathbb Z)$. Why is the map $I:H_{dR}^1(M) \to \mathbb R^n$ given by $$I(...
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How are rational algebraic Hodge classes of type $ (p,p) $ defined?

Definition ( Hodge classes ) : 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) $. The group of rational $ (p,p) $- ...
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Lefschetz operator identity, local proof

Let $M$ denote a Kähler manifold with Kähler form $\omega$. Let $L$ denote the Lefschetz operator acting on on $A^k(M,\mathbb{C}) \to A^{k+2}(M,\mathbb{C})$ such that $L(\eta)=\omega \wedge \eta$. ...
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Adjoint of coboundary map

I am confused with the adjoint of the coboundary map as described in this article (https://magnus.ece.gatech.edu/Papers/MuhammadEgerstedtMTNS06.pdf, page 4). The adjoint of the coboundary map is ...
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Hodge Theory and the genus of Riemann surface

I just begin to learn about Hodge Theory. The following statement is heard from somewhere, which I know it is true. But I don't understand the exact detail of how and why. Let $M$ be a closed ...
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Hodge duality in relating wedge and cross products?

I've been reading some quantum mechanics papers which involve Clifford Algebra. I am investigating it for an undergrad project but none of my professors seem to know anything about Clifford Algebras. ...
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Computing the second Chern character on Kähler manifold

Let $(X, \omega)$ be a compact Kähler manifold and $E$ a vector bundle on $X$ with hermitian metric $h$. Let also $F_h$ be the curvature of Chern connection on $(E, h)$. It is a $(1,1)$-form with ...
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Hodge star of a space acting on the volume of element of a sub-space

Good evening everyone. This is not an exercise. I am Ph.D student in Physics and this is brief chat I had with a postdoc. Suppose that we have a product space-time defined by $AdS_5 \times S^5$, ...
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How does the hodge codifferential operator act over a wedge product?

We have that the exterior derivative acts over a wedge product in the following manner. Let $\alpha,\beta$ be $p,q$ forms respectively. Then we have that \begin{equation} d(\alpha\wedge \beta) = (d\...
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Computation of exterior derivative of an $(n-1)$ form

Suppose $(M,g)$ is a Lorentzian manifold of dimension $n$. Let $V$ be a one-form on $M$ and define the $(n-1)$ form $\omega = \ast V$ where $\ast$ is the Hodge dual. In a chart $(U,x)$, if we have $$...
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Reference to the Hodge conjecture

My knowledge on Hodge conjecture is very low. I want to start to study related topics to the Hodge conjecture . Is it possible to understand this conjecture by differential geometry and without ...
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Is there a relation between the cohomology ring of a blowup with the base scheme and blowup locus?

Let $$ Y \subset X $$ be a codimension 2 or greater smooth subvariety of a smooth projective variety $X$. Is there a relation between the cohomology of the blowup $Bl_Y(X)$ and the cohomology rings of ...
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Check of Hodge dual identity in a paper

I'm reading a paper (https://arxiv.org/pdf/gr-qc/0012037.pdf) and I have a question about equation H3: $$\star(e_a \rfloor \psi) = (-1)^{p-1}\vartheta_a\wedge\star \psi$$ How can it be possible for ...
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Hodge Theory for Smooth Projective Varieties

Given a smooth complex projective variety $X$, I know that by the "Hodge Theory" we can compute the dimensions of the cohomology groups of the structural sheaf $\mathcal{O}_X$ by: $$H^p(X,\mathcal{O}...
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Some questions about the hodge star operator.

I am a beginner of Hodge Theory. When I read the notes, I find the following which makes me feel uncomfortable: Below is an example: First, from my understanding, $V$ is a vector space and $V^{*}$ ...