# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 $$\lim_{j\to\infty} \langle\beta_j,\psi\rangle = ... 0answers 76 views ### Reference for Hodge decomposition for flag variety Any references for proof of the following facts: The cohomology of the (complex) flag variety is always in (p, p)-type of Hodge Decomposition. The natural map G/T → G_\mathbb{C}/B is a ... 0answers 130 views ### Why the 2 \pi i in the Tate twist? I am reading here about Hodge structures and in (1.4) the Tate Hodge-structure \mathbb{Z}(n) and the Tate twist V(n) on a Hodge structure V are defined. I understand that one might want to ... 1answer 60 views ### Projective algebraic manifold admits a positive line bundle By the theorem: Let X be a compact Hodge manifold. Then X is a projective algebraic manifold, it follows that any compact complex manifold X is projective algebraic iff it admits a positive line ... 1answer 128 views ### A symmetry of the Hodge star operator Theorem 1.3.2(i) on p. 7 of these notes on the Hodge decomposition states$$ * \eta \wedge w = \langle \eta, w \rangle _{\wedge^k V} dV . I don't see how this holds. We defined the Hodge $*$ ...
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 \$...