The study of relations between, one the one hand, the topology of a smooth manifold as encoded in the cohomology groups, and on the other, the set of solutions to the Laplace operator on differential forms relative to some Riemannian metric on the manifold.

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59 views

Calabi-Yau $3$-fold given as elliptically fibered manifold over $\mathbb{C}P^1 \times \mathbb{C}P^1$

Consider a Calabi-Yau three-fold given as an elliptically fibered manifold over $\mathbb{C}P^1 \times \mathbb{C}P^1$$$y^2 = x^3 + f(z_1, z_2)x + g(z_1, z_2),$$where $z_1$, $z_2$ represent the two $\...
5
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1answer
66 views

Is the contraction of an harmonic form harmonic?

As I'm still a beginner in complex differential geometry, as soon as I tried reading an article I got stuck on what (I think) should be a minor detail. I hope someone can help me a bit. Let $M$ be a ...
2
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1answer
54 views

Hodge theory on semi-Riemannian manifolds [reference request]

I need to learn a bit of Hodge theory on manifolds and I am looking for a reference which covers the case where the metric has arbitrary signature $(p,q)$. Most books I have found seem to focus on the ...
4
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1answer
40 views

Griffiths and Harris $\mu=\mathcal{H}(\mu)+dd^*G(\mu)$

Griffith and Harris state on page $116$ that for a closed form $\mu$ on a Kahler manifold of type $(p,q)$ we have $$\mu=\mathcal{H}(\mu)+dd^*G(\mu)$$ Here $$\mathcal{H}:\Omega^{p,q}(M)\to\mathcal{H}^{...
4
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30 views

Hodge decomposition on Riemann surface

On a compact Riemannian manifold $M$ the Hodge decomposition takes the form $$\Omega^k(M)=d\Omega^{k-1}(M)\oplus\mathcal{H}(M)\oplus d^*\Omega^{k+1}(M)$$ Where $d^*$ is the adjoint of $d$ w.r.t. the ...
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29 views

Inner products on $C_c^\infty$ corresponding to differential forms

Let $M$ be a compact, oriented Riemannian manifold of dimension $n$. We can locally identify smooth $k$-forms by smooth functions $\mathbb{R}^n \to \mathbb{C}^{m}$, where $m = \binom{n}{k}$ (via a ...
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1answer
58 views

Picard rank of K3s — How can it be small?

I must be missing something. How can the Picard rank of a compact Kahler manifold drop below its Hodge number $H^{1,1}(X)$? For simplicity, let $X$ be a K3 surface. After all, we have the standard ...
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45 views

Relationship between invariant and harmonic forms

Let $G$ be a compact real Lie group which is also a complex manifold (note that I do not want $G$ to be a complex Lie group). Let $\Omega^{p , q}(G)$ denote the space of smooth $(p , q)$-forms on $G$, ...
2
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2answers
73 views

Frank Warner's definition of the Hodge star

Frank Warner's book, chapter 2, excercise 13 states the following: If $V$ is an oriented inner product space ($n$ dimensional) there is a linear map $\ast \colon \Lambda (V) \to \Lambda (V)$, ...
3
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26 views

Kodaira decomposition of 1-form on a Real manifold

I recently stumbled across the Hodge decomposition theorem, which states that on any compact orientable manifold, for any form the following holds $$ \omega = \text{d}\alpha + \delta \beta + \gamma $$ ...
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1answer
48 views

Hodge decomposition thm. Why $\Delta(E^p)=d\delta(E^p)\oplus\delta d(E^p)=d(E^{p-1})\oplus\delta(E^{p+1})?$

$\DeclareMathOperator{Img}{Im}$ In Warner's "Foundations of differentiable manifolds and Lie groups" we read that $$E^p\stackrel{(1)}{=}\Delta(E^p)\oplus H^p\stackrel{(2)}{=}d\delta(E^p)\oplus\delta d(...
2
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0answers
92 views

The cohomology of the Dirac operator $d+d^{*}$

Let $(M,g))$ be a Riemannian manifold with the Hodge dual operator $d^{*}$. Is there a name (and some computation in some reference) for the cohomology of the complex of Harmonic forms with ...
5
votes
1answer
67 views

Where is the error in this proof of the Hodge theorem?

Let $(M,g)$ be a closed smooth Riemannian manifold. The following is the decomposition part of the Hodge theorem: Theorem The canonical map $\mathscr{H}^k(M)\to H^k(M)$ from harmonic $k$ ...
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27 views

Constructing the Hodge Laplacian from the Laplace-Beltrami one

I know, in principle, how to construct the Hodge Laplacian with the aid of the Hodge star. Is it possible, though, to construct it only from the Laplace-Beltrami operator? I have already found a way, ...
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15 views

Definition of type of a Hodge structure

In the context of mixed Hodge structures what does one mean by "type $(p,q)$"? Surprisingly I was not able to find a definition. In a particular case, what does it mean "$V$ is the maximal $\mathbb{Q}$...
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43 views

Killing vector field for Witten complex?

I am reading a classical paper by Atiyah Bott "The moment map and equivariant cohomology". In paragraph "Relation with Witten complex" (at the very beginning of this paragraph) they claims that “$W_s$...
2
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0answers
55 views

Hodge theory in general

I know a bit of Hodge theory, and I know that there is an analogue in the symplectic case, where instead of inducing the $\star$-product using the metric we use the symplectic form. Is in true in ...
3
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1answer
269 views

The proof of the abc conjecture

I recently heard that there was a workshop on Inter-universal Teichmuller theory in the Clay institute from 7-11 December 2015.This field of mathematics offers a potential proof of the abc conjecture....
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45 views

paper request about mixed hodge

Does any one have the paper?: the obstruction to splitting a mixed hodge structure over the integers I, Preprint, University of Utah, 1979, by James Carlson. I cannot find it on google.
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0answers
26 views

Is there a description of variation of (mixed) Hodge Structures in terms of a Deligne operator?

A complex Mixed Hodge Structure is given by a complex vector space $V$ together with a descending filtration $W$ and two ascending filtrations $F,\bar{F}$ that satisfy the condition \begin{equation} ...
5
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1answer
92 views

A homological perspective on the Hodge-theorem

Let $M$ be a smooth oriented manifold of dimension $n$. Let $(\mathcal{A} ^*(M),d)$ be the chain complex of differential forms on $M$. Endowing $M$ with an inner product gives us a hodge star operator ...
3
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80 views

Primitive cohomology, example request

$X$ is a compact Kähler manifold or smooth projective variety. is there an example that a primitive class $0\neq [\omega]$ of $H^{p+q}(X, \mathbb{C})$ is wedge product of other two primitive classes: $...
5
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1answer
57 views

Induced bilinear form on exterior powers - Towards a global Hodge Star Operator

In all constructions of the hodge star operator I've seen so far there was a part where an inner product on the exterior power of the tangent space was defined by the ungodly local formula: $$\...
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20 views

Regularity Hodge Laplacian on bounded domains

I need a reference for the $W^{s,p}$ regularity of the Hodge boundary value problem on bounded domains. I need estimates $\lVert \omega \rVert_{W^{s+2,p}} \leq c \lVert f \rVert _{W^{s,p}}$, for $...
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1answer
90 views

About rational Hodge conjecture.

What progress has been made to date on the rational Hodge conjecture ? Can anyone tell us if there is some new books related to Hodge conjecture which explain in detail, the latest development in the ...
2
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2answers
62 views

Repeated application of the gradient on a Riemannian manifold

While reading about Sobolev spaces on manifolds, I encountered the following notation regarding the norm in $H^k$: $\| \nabla ^k f \|_{L^2}$. There are two questions here: 1) What kind of object id $\...
4
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78 views

$h^{0, 1}$ of K3 surface (a priori non-Kahler)

I am trying to understand paper by Siu "Every K3 surface is Kahler". Let $M$ be a K3 surface. Siu wrote $H^1 ( M , \mathscr{O}_M ) =0$ without any references. It is written on fifth page. Maybe I ...
3
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0answers
66 views

Computing the signature of the intersection form on the middle cohomology of compact, symplectic, non-Kaehler manifolds…

For a compact Kaehler manifold, one can compute the signature of the intersection form on the middle-degree cohomology, by taking an alternating sum of the Hodge numbers (this is the Hodge Index ...
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1answer
91 views

Calculating Betti and Hodge number for product of a curve

How can we compute the Betti numbers and Hodge numbers for $S=C\times C/\sigma$? (Where $\sigma$ is swapping the two factors.)
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1answer
63 views

$H^{p,0}$ is isomorphic to the space of holomorphic forms?

In Voisin's "Hodge Theory and Complex Algebraic geometry I" Corollary 7.6: $\textbf{Corollary 7.6}$ For every $p\leq n$, $H^{p,0}(X)$ is isomorphic to the space of holomorphic forms of degree $p$ on $...
3
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67 views

Proofs of Hodge duality: $H^{0,1}(X) = H^{1,0}(X)^*$

I am looking for a proof of this fact, where $H^{1,0} = Ker(d: \mathscr{E}^{1,0} \rightarrow \mathscr{E}^{2})$ and $H^{0,1} = Coker(\overline{\partial}: \mathscr{E} \rightarrow \mathscr{E}^{0,1}$, (...
12
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2answers
247 views

Hodge Theory, intuition?

We have the following theorem of Hodge, as follows: $$\dim \ker \Delta^p = \dim H^p(M) = b_p(M).$$My question is, what is the intuition behind this statement?
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213 views

Hodge laplacian of distance function

Let $p$ be a given point on a Riemaniann manifold $\mathcal{M}$. The distance function to point $p$ is denoted $f_p$ : $$ f_p(q) = \operatorname{dist}(p,q)$$ The exterior derivative is denoted $\...
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0answers
51 views

Exercise about Hodge Star Operator on $\Lambda^{p,q}$

In real case, $$ \ast (e_1\cdots e_k)=e_{k+1}\cdots e_n$$ on $\mathbb{R}^n$, where $e_i$ is $1$-form and $e_1\cdots e_n$ is volume on $\mathbb{R}^n$ We will extend to complex case. Define $$ dz_k:= ...
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2answers
64 views

Hodge self-duality in Minkowski spacetime

I was computing the dual map for $k$-forms in Minkowski spacetime, and I found that any $2$-form is either self-dual or anti-self-dual if and only if it is the null form. Does this result make any ...
3
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2answers
73 views

Automorphisms of manifold vs automorphisms of Hodge structure

Let $X$ be a compact Kahler manifold. Let $H^k = H^k_{prim} (X)$ be primitive cohomology of $X$ considered as an object in category of of polarized Hodge structures. We have a map \begin{equation*} ...
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45 views

Proving that $\phi$ is orthogonal to the harmonic forms given $\int\phi \;d\mathrm{vol}$.

I want to prove that given a connected closed (compact without boundary) oriented Riemmanian manifold $(M,g)$, the condition $$\int_M \varphi \;\mathrm{d}\mathrm{vol}_g=0$$ implies that $\int \varphi ...
1
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1answer
90 views

The adjoint of left exterior multiplication by $\xi$ for hodge star operator

As we know, for $V$ vectoral space and a orientation $\mathcal{O}$ on $V$ and $e_{1},...,e_{n}$, the hodge star operator $\ast:\wedge V^*\rightarrow\wedge V^*$ is defined for $\ast(e_{1}\wedge...\...
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55 views

Cohomology of conic bundle 3-folds

It is known that for a smooth cubic 3fold $X\subset \mathbb{P}^4$ we have $H^3(X,\mathcal{O}_X)$ (or if you prefer $H^{0,3}(X)=0$). Moreover, if I project off a line $l\subset X$ I can resolve the map ...
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82 views

Why are algebraic cycles rational?

Let $X_{/\mathbb{C}}$ be a projective non-singular variety of dimension $n$ and $Z \subset X$ be an irreductible subvariety of dimension $p$. Denote by $\mathrm{H}_{\mathrm{dR}}^i(X,\mathbb{C})$ the ...
0
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1answer
77 views

Hodge star/ Technical question

If we have an equation that looks like $$H=Y$$ and we want to multiply $H$ by either $ReM_{IJ}$ or $ImM_{IJ}$ where $M_{IJ}$ is a complex matrix. But the thing is that $$Y=\star(...)$$ where $\star$ ...
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1answer
17 views

Example of non interchangeability of the order of taking graded moduls with respect to three filtrations

Assume we have an R-Module A, with R a commutative ring and three descending filtrations F,G,H on this. We can take the associated graded module with respect to any of these, say F, by setting $$\...
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1answer
74 views

A simple question about rational Hodge conjecture

Good evening everyone : In the link here I found the following sentence : The Hodge conjecture predicts that the $\mathbb{Q}$ - linear span of the classes of algebraic subvarities in the cohomology ...
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2answers
198 views

Sign problems in complex computations

$\newcommand{\dd}{\mathrm{d}} \newcommand{\eg}{\epsilon} \newcommand{\mg}{\mu} \newcommand{\ng}{\nu} \newcommand{\rg}{\rho} \newcommand{\et}{\wedge} \newcommand{\lbar}{\overline} \newcommand{\ubar}{\...
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1answer
115 views

Why is the $ \mathbb{Z} $ - Hodge conjecture false?

I wish someone well initiated in the area , told me , why the $ \mathbb{Z} $ - Hodge conjecture is wrong, which allowed to change its state by tensoring by $ \mathbb{ Q } $ to become as it is known ...
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35 views

Given a smooth 2-form $\varphi$ on a punctured Riemann surface, is there $\nu$ of type (1,0) such that $\varphi=d\nu$?

Let $X$ be a compact Riemann surface, $p\in X$, and $\varphi$ be a smooth 2-form on a $X-\{p\}$, and hence exact. I'm wondering if it is possible to find a form of type (1,0) whose differential is $\...
12
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1answer
438 views

In what sense is $p$-adic Hodge theory related to ordinary (complex) Hodge theory?

I started reading about $p$-adic Hodge theory in the notes of Brinon and Conrad. I quote (page 7): The goal of p-adic Hodge theory is to identify and study various “good” classes of $p$-adic ...
3
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1answer
553 views

Definition of complex conjugate in complex vector space

I am starting reading about Hodge theory and while reading the definition of abstract Hodge structure a very basic question came to my mind... What is the definition of the conjugate of a subspace of ...
11
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2answers
226 views

General “Hodge theorem”

I know basically zero Hodge theory, so this question might be weird. Let $$A \stackrel{S}{\longrightarrow} B \stackrel{T}{\longrightarrow} C$$ be a sequence of closed, densely defined maps of ...
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vote
1answer
67 views

The Hodge dual and the Moyal product related or just notation?

The Hodge star operator $\star$ is a linear map between $\bigwedge ^pV$ and $\bigwedge ^{n-p}V$ for an inner product space $V$ of dimension $n$. So we can we write; \begin{equation} \lambda\in \...