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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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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30 views

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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48 views

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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1answer
40 views

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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43 views

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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31 views

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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142 views

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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1answer
31 views

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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61 views

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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61 views

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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1answer
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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
85 views

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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What does mean by Tate Motive $\mathbb{Z}(1)$ in the p-adic Hodge theory?

What does mean by Tate Motive $\mathbb{Z}(1)$ in the p-adic Hodge theory? From wikipedia(https://en.wikipedia.org/wiki/Cyclotomic_character), I got In terms of motives, the p-adic cyclotomic ...
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29 views

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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69 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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165 views

Albanese map for complex compact manifold

Let $X$ be a compact Kahler manifold. To it, it is possible to associate a complex torus $Alb(X)$ with a map $$alb: X \to Alb(X) .$$ In class, our teacher claimed the image of this map generates the ...
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what is the domain or codomain of $\phi$ in $(\phi, N)$-module?

In the p-adic Hodge-Tate theory, what does the $\phi$-function in the definition of $(\phi, N)$-module mean? I mean what is the definition of $\phi$-function here? I got it that, A $\phi-$ ...
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43 views

Showing that Green operator is linear bounded self-adjoint operator

I tried to solve an exercise : show that Green's operator ${G}$ is linear bounded self-adjoint operator. There is another question about this exercise here. Notation is based on Warner's textbook. ...
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26 views

what does $\mathbb{Z}_p[1]$ or $\mathbb{Z}_p[i]$ represent in p-adic Hodge-Tate theory?

What does it represent by $\mathbb{Z}_p[1]$ in p-adic Hodge-Tate theory? Is it called Tate object or Motives etc something like that? I know it makes some geometric realization of cyclotomic ...
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51 views

Reference Request:Real analyticity of smooth eigenforms of the Laplacian

Let $(M,g)$ be an oriented, real analytic, Riemannian $n$-manifold without boundary. Let $\Delta_k=(d+\delta)^2$ denote the Laplace-Beltrami operator acting on (smooth) $k$-forms. I am looking for a ...
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1answer
49 views

Why $i_{!}\mathbb{C}_{X\backslash \Sigma} \subset \mathcal{O}_{X}(-\Sigma)$?

Let $X$ be a smooth variety and $\Sigma$ be a simple normal crossing divisor. Let $i: X\backslash \Sigma \to X$ be the inclusion. Then it is claimed that $$i_{!}\mathbb{C}_{X\backslash \Sigma} \subset ...
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55 views

Hodge star operator independant from hermitian metric

My problem is on the definition of the Hodge star operator on Riemann surface $S$. In general the definition of an Hodge star operator on a manifold depend of the metric but on Riemann surfaces the ...
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58 views

Splitting of a short exact sequence in cohomology due to torsion

Consider a set $S$ of $n$ torsion points on an elliptic curve $E$, then I find in a paper a claim without any further comment, that the following short exact sequence splits, as a sequence of Mixed ...
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1answer
35 views

Finding the Hodge dual of $dy$

By definition, Hodge dual operator is defined as follows: Let $\sigma = (i_1, i_2, ..., i_n)$ be a permutation of (1, 2, ..., n), then for any $k \in {0,1,...,n}$, the Hodge dual of the corresponding ...
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1answer
164 views

Has anyone got a reference as to why étale Galois representations are de Rham?

I am currently studying $p$-adic Hodge theory and searching for help. If $X$ is a variety over a p-adic field $K$ (should we take it global ? $p$-adic ?), then the étale cohomology $H^i(X_{\bar K}, \...
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27 views

Higgs Bundle with value not in canonical bundle.

I read an old paper by Donagi (Principal bundle over elliptic fibrations), and he mentioned the possibility for Higgs bundle to take value in an elliptic curve or vector bundle rather than the ...
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1answer
25 views

Hard Lefschetz and Hodge Decoposition

Let $M\subset \mathbb{P}^N$ be a compect Kahler manifold of dimension $n$, and let $\omega$ be the associated closed (1,1)-form. $A^{p,q}(M)$ is the set of $C^\infty$ complex differential forms of ...
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1answer
39 views

Hodge star operator of $\frac{\omega^k}{k!}$ is $\frac{\omega^{n-k}}{(n-k)!}$

Let $\omega=\sum_{j=1}^ndx_j\wedge dy_j$ be the standard Kähler form on $\mathbb{C}^n$. I'm trying to prove that $*\frac{\omega^k}{k!}=\frac{\omega^{n-k}}{(n-k)!}$, where $*$ is the Hodge star ...
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2answers
145 views

What is an example of harmonic form on a compact manifold?

I'm doing some self-study on Hodge theory and elliptic operators right now. I'm trying to come up with an example of a harmonic $p$-form $\omega$ on a compact manifold, i.e. a form such that $d\omega =...
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1answer
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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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1answer
196 views

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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1answer
55 views

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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1answer
43 views

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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1answer
125 views

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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1answer
112 views

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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1answer
38 views

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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127 views

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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1answer
193 views

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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27 views

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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130 views

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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1answer
60 views

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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1answer
128 views

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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51 views

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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