Hodge star operator

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###Context###
#### Riemannian geometry
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[[!include Riemannian geometry - contents]]
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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# Contents
* table of contents
{: toc}

## Idea

Given a finite dimensional (pseudo)-[[Riemannian manifold]] $(X,g)$, the _Hodge star operator_ "completes" a $k$-[[differential form]] to the [[volume form]] of $(X,g)$.

## Definition

Let $(X,g)$ be an [[orientation|oriented]] $n$-[[dimension]]al [[smooth manifold]] $X$ endowed with a ([[pseudo-Riemannian metric|pseudo]])-[[Riemannian metric]] $g$.  For $0 \leq k \leq n$, write $\Omega^k(X)$ for the [[vector space]] of $k$-[[differential forms|forms]] on $X$.

### Hodge inner product

The metric $g$ naturally induces a nondegenerate symmetric [[bilinear form]]

$$
  (-\mid-)
   \;\colon\;
  \Omega^k(X) \otimes \Omega^k(X)
    \to
  \Omega^0(X)
  \,.
$$

If $X$ is [[compact space|compact]] then the [[integral]] of this against the [[volume form]] $vol_g$ exists. This is the _Hodge inner product_

$$
  \langle - , - \rangle
  \;\colon\;
  \Omega^k(X)\otimes \Omega^k(X)
  \to
  \mathbb{R}
$$

$$
  \langle \alpha, \beta \rangle
  :=
  \int_X (\alpha\mid \beta) vol
 \,.
$$

### Hodge star operator

The **Hodge star operator** is the unique
[[linear function]]

$$
  {\star}\colon \Omega^k (X) \to \Omega^{n-k} (X)
$$

defined by the identity

$$
  \alpha \wedge \star\beta
  =
  (\alpha \mid \beta) vol_g,
  \qquad
  \forall \alpha,\beta \in \bigwedge^k X
  \,,
$$

where $vol_g \in \Omega^n X$ is the [[volume form]] induced by $g$.

Therefore in terms of the Hodge operator the [[Hodge inner product]] reads

$$
  \langle \alpha , \beta\rangle
  =
  \int_X \alpha \wedge \star \beta
  \,.
$$

### Generalizations

The [[Riemannian metric|metric]] $g$ is used in two places in the specification of the Hodge operator: in the inner product on forms and in the volume form.  If $X$ is equipped only with a [[volume form]] (not necessarily coming from a metric), then the Hodge operator still takes $k$-forms to $(n-k)$-[[multivector field|vector fields]].  If the manifold is not oriented, then the metric only gives a [[volume pseudoform]], but the Hodge operator still takes $k$-forms to $(n-k)$-[[pseudoforms]].  Finally, if $X$ is equipped with only a volume pseudoform (which is equivalent to an [[absolutely continuous measure|absolutely continuous]] [[Radon measure]] on $X$), then the Hodge operator takes $k$-forms to $(n-k)$-pseudovector fields.  (Of course, in every case, one might apply the operator to pseudoforms or multivector fields to begin with.)

## Properties

### Component expression
 {#ComponentExpression}

Let $X$ be a ([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]] of [[dimension]] $D$, and locally, on some [[open subset]] $U \subset X$, let

$$
  e^1, \dots, e^D
  \;\in\;
  \Omega^1(U)
$$

be a frame of [[differential 1-forms]] (a [[vielbein]]). For example if $\{x^i\}$ is a [[coordinate chart]] on $U$, then $e^i \coloneqq d x^i$ is such a frame.

With this choice, any [[differential p-form]] $\alpha \in \Omega^p(U)$ has a component expansion

$$
  \alpha
  \;=\;
  \frac{1}{p!}
  \alpha_{i_1 \dots i_p}
  \, e^{i_1} \wedge \cdots \wedge e^{i_p}
$$

for [[smooth function]]-components $\{\alpha_{i_1 \cdots i_p}\}$ (where here and in the following we use the [[Einstein summation convention]]).

In terms of these components, the Hodge dual $\star \alpha$ of $\alpha$ is expressed by the following formula:

\[
  \label{ComponentFormulaForHodgeStar}
  \begin{aligned}
    \star \alpha
    & = \;
    \frac{1}{
      p!
     (D-p)!
    }
    \sqrt{
      \left\vert
       det\big((g_{i j})\big)
      \right\vert
    }
    \,
    \alpha_{
      \color{green} j_1 \dots j_p
    }
    g^{
     {\color{green} j_1 }
     {\color{cyan} i_1 }
    }
    \cdots
    g^{
     {\color{green} j_p }
     {\color{cyan} i_p }
    }
    \epsilon_{
      {\color{cyan} i_1 \dots i_p }
      {\color{orange} i_{p+1} \cdots i_D }
    }
    e^{
      \color{orange} i_{p+1}
    }
     \wedge
     \cdots
     \wedge
    e^{
      \color{orange} i_D
    }
    \\
    & =
    \frac{1}{
      p! (D-p)!
    }
    \sqrt{
      \left\vert
       det\big((g_{i j})\big)
      \right\vert
    }
    \,
    \alpha^{ \color{green} i_1 \dots i_p }
    \epsilon_{
      { \color{green} i_1 \dots i_p }
      { \color{orange} i_{p + 1} \cdots i_D }
     }
     e^{
       \color{orange}
       i_{p + 1}
     }
     \wedge
       \cdots
     \wedge
     e^{
       \color{orange}
       i_{D}
     }
  \end{aligned}
\]

Here

* $p!$, $(D-p)!$ are the [[factorials]] of $p$ and $(D-p)$, respectively,

* $\epsilon_{i_1,\dots,i_n} \in \{+1, ,-1\}$ (the [[Levi-Civita symbol]]) is the [[signature of a permutation|signature]] of the [[permutation]] $(1,2,\dots,D) \mapsto (i_1,i_2,\dots,i_D)$

* $(g_{i j})$ is the [[square matrix]] of components of the [[metric tensor]] in the chosen basis, i.e. such that

  $$
    g \;=\;
    g_{i j} e^i \otimes e^j
  $$

* $det(g)$ is the [[determinant]] of $(g_{i j})$

* $\left\vert g \right\vert$ is the [[absolute value]] of the determinant.

### Basic properties
 {#BasicProperties}

Let $(X,g)$ be a ([[pseudo-Riemannian manifold|pseudo]]-) [[Riemannian manifold]] of [[dimension]] $D$ and let $\omega,\lambda \in \Omega^k(X)$. Then the following holds:

\[
  \label{HodgeSquareOnRiemannian}
  \star(\star\omega)
  \;=\;
  \left\{
  \begin{array}{rc}
    (-1)^{k(D+1)} \omega = (-1)^{k(D-k)} \omega
    & \text{(Riemannian)}
    \\
    -(-1)^{k(D+1)} \omega
      =
    - (-1)^{k(D-k)} \omega
    & \text{(pseudo-Riemannian)}
  \end{array}
  \right.
\]

\[
  \label{HodgeStarPreservesHodgeInnerProduct}
  \langle\star\omega , \star\lambda\rangle
  \;=\;
  \langle \omega | \lambda \rangle
\]

\[
  \label{HodgeStarOfUnityIsVolume}
  \star 1 = dvol
  \,,
\]

where $dvol$ denotes the [[volume form]].

(e.g. [Frankel 1997 (14.9), (14.5)](#Frankel97))

## Examples

### Hodge star operator on a Kähler manifold
 {#OnAKahlerManifold}

On a [[Kähler manifold]] $\Sigma$ of [[dimension]] $dim_{\mathbb{C}}(\Sigma) = n$ the Hodge star operator acts on the [[Dolbeault complex]] as

$$
  \star \;\colon\; \Omega^{p,q}(X) \longrightarrow \Omega^{n-q,n-p}(X)
  \,.
$$

(notice the exchange of the role of $p$ and $q$). See e.g. ([Biquerd-Höring 08, p. 79](#BiquerdHoering08)). See also at _[[Serre duality]]_.

[[!include Hodge star operator on Minkowski spacetime -- section]]

## Related concepts

* [[Laplace-Beltrami operator]]

* [[constitutive equations]], [[pre-metric electromagnetism]]

* [[self-dual higher gauge field]]

## References

Lecture notes:

* _Hodge theory on Riemannian manifolds_ , lecture notes ([pdf](http://math.uh.edu/~minru/Riemann08/hodgetheory.pdf))

Textbook account:

* {#Frankel97} [[Theodore Frankel]], §14.1a in: *[[The Geometry of Physics - An Introduction]]*, Cambridge University Press (1997, 2004, 2012) \[doi:10.1017/CBO9781139061377\]

A unified perspective in terms of [[Berezin integration]]:

* [[Leonardo Castellani]], [[Roberto Catenacci]], [[Pietro Antonio Grassi]], _The Hodge Operator Revisited_ ([arXiv:1511.05105](https://arxiv.org/abs/1511.05105))

Discussion in [[complex geometry]]:

* {#BiquerdHoering08} O. Biquard, A. Höring, _Kähler geometry and Hodge theory_, 2008 ([pdf](http://math.unice.fr/~hoering/hodge/hodge.pdf))

With an eye towards application in [[supergravity]] and [[string theory]]:

* {#BNS04} [[Igor Bandos]], [[Alexei Nurmagambetov]], [[Dmitri Sorokin]], Appendix A of _Various Faces of Type IIA Supergravity_, Nucl. Phys. B676 (2004) 189-228 ([arXiv:hep-th/0307153](https://arxiv.org/abs/hep-th/0307153))

Discussion of the [[Hodge star operator]] on [[supermanifolds]] (in terms of [[picture changing operators]] and integral top-forms for [[integration over supermanifolds]]):

* [[Leonardo Castellani]], [[Roberto Catenacci]], [[Pietro Antonio Grassi]], _Hodge Dualities on Supermanifolds_, Nuclear Physics B Volume 899, October 2015, Pages 570-593 ([arXiv:1507.01421](https://arxiv.org/abs/1507.01421))

[[!redirects Hodge star operators]]

[[!redirects Hodge star-operator]]
[[!redirects Hodge star-operators]]

[[!redirects Hodge-star operator]]
[[!redirects Hodge-star operators]]

[[!redirects Hodge star]]
[[!redirects Hodge stars]]

[[!redirects Hodge inner product]]

[[!redirects Hodge dual]]
[[!redirects Hodge duals]]

[[!redirects Hodge duality]]
[[!redirects Hodge dualities]]

[[!redirects Hodge pairing]]
[[!redirects Hodge pairings]]


Hodge star operator on Minkowski spacetime -- section

### Hodge star operator on Minkowski spacetime
 {#HodgeStarOperatorOnMinkowskiSpacetime}

We spell out component expressions for the [[Hodge star operator]] on $D = d+1$-dimensional [[Minkowski spacetime]].

#### Conventions

We use [[Einstein summation convention]] throughout. With this convention, a generic [[differential p-form]] reads

$$
  \alpha
  \;=\;
  \tfrac{1}{p!}
  \alpha_{ \color{green} \mu_1 \cdots \mu_p }
  d x^{ \color{green} \mu_1}
  \wedge
  \cdots
  \wedge
  d x^{\color{green} \mu_p}
  \,.
$$

Here $p! \coloneqq 1 \cdot 2 \cdot 3 \cdots p \,\in \mathbb{N} \subset \mathbb{R}$ denotes the [[factorial]] of $p \in \mathbb{N}$.

We take the [[Minkowski metric]] to be the $D \times D$ [[diagonal matrix]] of the form

$$
  \eta
  \;=\;
  (\eta_{\mu \nu})
  \;=\;
  (\eta^{\mu \nu})
  \;\coloneqq\;
  diag(-1,+1, +1 , \cdots , +1)
  \,.
$$

We normalize the [[Levi-Civita symbol]] as

\[
  \label{LCSymbolDown}
  \epsilon_{0 1 2 \cdots d}
  \;\coloneqq\;
  + 1
\]

which means that

\[
  \label{LCSymbolUp}
  \epsilon^{0 1 2 \cdots d}
  \;=\;
  - 1
  \,.
\]

We normalize the sign of the [[volume form]] as

\[
  \label{VolumeForm}
  \begin{aligned}
  dvol
  & \coloneqq\;
  d x^0 \wedge d x^1 \wedge \cdots \wedge d x^d
  \\
  & =
  \tfrac{1}{D!}
  \epsilon_{
    \color{green}
    \mu_1 \cdots \mu_D
  }
  d x^{\color{green}\mu_1}
   \wedge
   \cdots
   \wedge
  d x^{\color{green}\mu_D}
  \end{aligned}
\]

We write

\[
  \label{GeneralizedKroneckerDelta}
  \delta^{ \mu_1 \cdots \mu_p }_{ \nu_1 \cdots \nu_p }
  \;\coloneqq\;
  \left\{
  \array{
    sgn(\sigma)
     &\vert&
    \underset{ \sigma \in Sym(p) }{\exists}
    \left(
      \underset{1 \leq i \leq p}{\forall}
      \left(
        \nu_{\sigma(i)} = \mu_i
      \right)
    \right)
    \\
    0 &\vert& \text{otherwise}
  }
  \right.
\]

for the generalized [[Kronecker delta]], whose value is the [[signature of a permutation|signature]] of the [[permutation]] that takes the upper indices to the lower indices, if any such exists, and [[zero]] otherwise.

This appears whenever the [[Levi-Civita symbol]] is contracted with itself:

\[
  \label{ContractionofLeviCivitaSymbols}
  \epsilon_{
     { \color{green} \mu_1 \cdots \mu_p }
     {\color{blue} \mu_{p+1} \cdots \mu_{D} }
  }
  \epsilon^{
    { \color{orange} \nu_1 \cdots \nu_p }
    { \color{blue} \mu_{p+1} \cdots \mu_D }
  }
  \;=\;
  { \color{magenta} - }
  (D-p)!
  \;
  \delta_{
    \color{green} \mu_1 \cdots \mu_p
  }^{
    \color{orange} \nu_1 \cdots \nu_p
  }
\]

Notice the minus sign in (eq:ContractionofLeviCivitaSymbols), which comes, via (eq:LCSymbolUp), from the [[Minkowski spacetime|Minkowski]] [[signature of a quadratic form|signature]].

#### Definition

We write $\iota_\mu$ for the operator of contraction of [[differential forms]] with the [[vector field]] $d/d x^\mu$, hence the [[linear operator]] on [[differential forms]] with [[anticommutator]]

$$
  \big\{
    \iota_\mu, d x^\nu \wedge
  \big\}
  \;=\;
  \delta_\mu^\nu
$$

With the [[volume form]] as in (eq:VolumeForm) it follows that
(notice the reversion of the index ordering in the contraction operators $\iota$)

\[
  \label{ContractionIntoVolumeForm}
  \alpha^{
    \color{green}
    \mu_1 \cdots \mu_p
  }
  \iota_{\color{green} \mu_p}
     \cdots
   \iota_{ \color{green} \mu_1}
  dvol
  \;=\;
  \epsilon_{
    { \color{green} \mu_1 \cdots \mu_p }
    { \color{orange} \nu_1 \cdots \nu_{(D-p)}  }
  }
  d x^{\color{orange} \nu_1}
    \wedge
    \cdots
    \wedge
  d x^{\color{orange} \nu_{(D-p)}}
\]

+-- {: .num_defn}
###### Definition

For a [[differential p-form]]

$$
  \alpha
  \;\coloneqq\;
  \tfrac{1}{ \color{green} p! }
  \alpha_{ \color{green} \mu_1 \cdots \mu_p}
  d x^{ \color{green} \mu_1 }
    \wedge \cdots \wedge
  d x^{ \color{green} \mu_p }
$$

its [[Hodge dual]] is:

\[
  \label{HodgeStar}
  \begin{aligned}
  \star
  \alpha
  & \coloneqq
  \tfrac{1}{
    { \color{green} p! }
    { \color{orange} (D-p)! }
  }
  \,
  \alpha^{ \color{green} \mu_1 \cdots \mu_p }
  \iota_{ \color{green} \mu_p }
  \cdots
  \iota_{ \color{green} \mu_1 }
  \,
  dvol
  \\
  & =
  \tfrac{1}{
    { \color{green} p! }
    { \color{orange} (D-p)! }
  }
  \,
  \alpha^{ \color{green} \mu_1 \cdots \mu_p }
  \epsilon_{
     { \color{green} \mu_1 \cdots \mu_p }
     { \color{orange} \mu_{p+1} \cdots \mu_D }
  }
  \,
  d x^{ \color{orange} \mu_{p+1} }
  \wedge
  \cdots
  \wedge
  d x^{ \color{orange} \mu_D }
  \,,
  \end{aligned}
\]

where in the second line we used (eq:ContractionIntoVolumeForm).

=--

#### Properties

+-- {: .num_prop #HodgePairing}
###### Proposition
**(Hodge pairing)**

For a [[differential p-form]]
$
  \alpha
  \;\coloneqq\;
  \tfrac{1}{p!}
  \alpha_{\mu_1 \cdots \mu_p}
  d x^{\mu_1}
    \wedge \cdots \wedge
  d x^{\mu_p}
$
on $D$-dimensional [[Minkowski spacetime]]
its [[wedge product]] with its [[Hodge dual]] (eq:HodgeStar) is

\[
  \label{AlphaWedgeStarAlpha}
  \alpha \wedge \star \alpha
  \;=\;
    \tfrac{ \color{magenta} -1 }{
      { p! }
    }
  \alpha_{ \mu_1 \cdots \mu_p }
  \alpha^{ \mu_1 \cdots \mu_p }
  \,
  dvol
  \,.
\]

=--

+-- {: .proof}
###### Proof

We compute as follows:

$$
  \begin{aligned}
    \alpha \wedge \star \alpha
    & =
    \tfrac{1}{
      { \color{green}  p! }
      { \color{orange} p! }
      { \color{blue} (D-p)! }
    }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p }
    d x^{ \color{green} \mu_1 }
    \wedge \cdots \wedge
    d x^{ \color{green} \mu_p }
    \wedge
    \alpha^{ \color{orange} \nu_1 \cdots \nu_p }
    \iota_{ \color{orange} \nu_p }
    \cdots
    \iota_{ \color{orange} \nu_1 }
    dvol
    \\
    & =
    \tfrac{1}{
      { \color{green}  p! }
      { \color{orange} p! }
      { \color{blue} (D-p)! }
    }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p }
    \alpha^{ \color{orange} \nu_1 \cdots \nu_p }
    \epsilon_{
      { \color{orange} \nu_1 \cdots \nu_p }
      { \color{blue} \nu_{p+1} \cdots \nu_D  }
    }
    d x^{ \color{green} \mu_p }
    \wedge \cdots \wedge
    d x^{ \color{green} \mu_1 }
    \wedge
    d x^{ \color{blue} \nu_{p+1} }
    \wedge
    \cdots
    d x^{ \color{blue} \nu_{D} }
    \\
    & =
    \tfrac{1}{
      { \color{green}  p! }
      { \color{orange} p! }
      { \color{blue} (D-p)! }
    }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p }
    \alpha^{ \color{orange} \nu_1 \cdots \nu_p }
    \epsilon_{
      { \color{orange} \nu_1 \cdots \nu_p }
      { \color{blue} \nu_{p+1} \cdots \nu_D  }
    }
    \epsilon^{
      { \color{green} \mu_p \cdots \mu_1 }
      { \color{blue} \nu_{p+1} \cdots \nu_{D} }
    }
    \,
    dvol
    \\
    & =
    \tfrac{ \color{magenta} -1 }{
      { \color{green}  p! }
      { \color{orange} p! }
    }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p }
    \alpha^{ \color{orange} \nu_1 \cdots \nu_p }
    \delta^{ \color{green} \mu_1 \cdots \mu_p }_{ \color{orange} \nu_1 \cdots \nu_p }
    \,
    dvol
    \\
    & =
    \tfrac{ \color{magenta} -1 }{
      { \color{green}  p! }
    }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p }
    \alpha^{ \color{green} \mu_1 \cdots \mu_p }
    \,
    dvol
  \end{aligned}
$$

Here the sign in the last lines arises from the [[Minkowski spacetime|Minkowski]] [[signature of a quadratic form|signature]] via (eq:ContractionofLeviCivitaSymbols).

=--

+-- {: .num_prop #HodgeStarFollowedByHodgeStar}
###### Proposition
**(double Hodge dual)**

For a [[differential p-form]]
$
  \alpha
  \;=\;
  \tfrac{1}{p!}
  \alpha_{\mu_1 \cdots \mu_p}
  d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p}
$
on $D$-dimensional [[Minkowski spacetime]], its double [[Hodge dual]] (eq:HodgeStar) is

\[
  \label{DoubleHodgeDual}
  \star \star \alpha
  \;=\;
  {\color{magenta} -}
  (-1)^{ p (D - p) }
  \,
  \alpha
  \,.
\]

=--

+-- {: .proof}
###### Proof

We compute as follows:

$$
  \begin{aligned}
    &
    \star \star
    \tfrac{1}{ \color{green} p! }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p}
    d x^{ \color{green} \mu_1 }
      \wedge
      \cdots
      \wedge
    d x^{ \color{green} \mu_p }
    \\
    & =
    \star
    \tfrac{1}{
      { \color{green} p! }
      { \color{orange} (D-p)! }
    }
    \alpha^{ \color{green} \mu_1 \cdots \mu_p}
    \iota_{ \color{green} \mu_p} \cdots \iota_{ \color{green} \mu_1}
    dvol
    \\
    & =
    \star
    \tfrac{1}{
      { \color{green} p! }
      { \color{orange} (D-p)! }
    }
    \alpha^{ \color{green} \mu_1 \cdots \mu_p }
    \epsilon_{
       { \color{green} \mu_1 \cdots \mu_p }
       { \color{orange} \mu_{p+1} \cdots \mu_D }
    }
    d x^{\color{orange} \mu_{p+1}} \wedge \cdots d x^{ \color{orange} \mu_d}
    \\
    & =
    \tfrac{1}{
      { \color{green} p! }
      { \color{orange} (D-p)! }
      { \color{blue} p! }
    }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p }
    \epsilon^{
       { \color{green} \mu_1 \cdots \mu_p }
       { \color{orange} \mu_{p+1} \cdots \mu_D }
    }
    \epsilon_{
      { \color{orange} \mu_{p+1} \cdots \mu_D }
      { \color{blue} \nu_1 \cdots \nu_p  }
    }
    \,
    d x^{ \color{blue} \nu_1}
      \wedge
        \cdots
      \wedge
    d x^{ \color{blue} \nu_D }
    \\
    \\
    & =
    \tfrac{
     (-1)^{ {\color{green} p} { \color{orange}  (D-p) } }
    }{
      { \color{green} p! }
      { \color{orange} (D-p)! }
      { \color{blue} p! }
    }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p }
    \epsilon^{
       { \color{green} \mu_1 \cdots \mu_p }
       { \color{orange} \mu_{p+1} \cdots \mu_D }
    }
    \epsilon_{
      { \color{blue} \nu_1 \cdots \nu_p  }
      { \color{orange} \mu_{p+1} \cdots \mu_D }
    }
    \,
    d x^{ \color{blue} \nu_1}
      \wedge
        \cdots
      \wedge
    d x^{ \color{blue} \nu_D }
    \\
    & =
    {\color{magenta} -}
    \tfrac{
      (-1)^{ {\color{green}p} {\color{orange} (D-p) } }
    }{
      { \color{green} p! }
      { \color{blue} p! }
    }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p }
    \delta^{
       { \color{green} \mu_1 \cdots \mu_p }
    }_{
      { \color{blue} \nu_1 \cdots \nu_p  }
    }
    \,
    d x^{ \color{blue} \nu_1}
      \wedge
        \cdots
      \wedge
    d x^{ \color{blue} \nu_D }
    \\
    & =
    {\color{magenta} -}
    (-1)^{ {\color{green}p} {\color{orange} (D-p) } }
    \,
    \alpha
  \end{aligned}
$$

Here the sign in the last lines arises from the [[Minkowski spacetime|Minkowski]] [[signature of a quadratic form|signature]] via (eq:ContractionofLeviCivitaSymbols).

=--

+-- {: .num_prop #WaveOperatorOnMinkowskiSpacetime}
###### Proposition
**([[Laplace operator]]/[[wave operator]])**

Let
$
  \alpha
  =
  \tfrac{1}{p!}
  \alpha_{\mu_1 \cdots \mu_p}
  d x^{\mu_1}
    \wedge
    \cdots
    \wedge
  d x^{\mu_p}
$
be a [[differential p-form]]
on $D$-dimensional [[Minkowski spacetime]]
such that

$$
  \partial^\nu \alpha_{\nu \mu_1 \cdots \mu_{p-1}}
  \;=\;
  0
$$

(i.e. [[Lorenz gauge]]).

Then the [[Laplace-Beltrami operator]]

$$
  \star d \star d \alpha
  \;=\;
  { \color{magenta} - }
  \partial^\nu \partial_\nu \alpha
$$

is the [[wave operator]] acting on the components of $\alpha$.

=--

+-- {: .proof}
###### Proof

We compute as follows:

$$
  \begin{aligned}
    &
    \star d \star d
    \tfrac{1}{ \color{green} p! }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p }
    d x^{ \color{green} \mu_1 }
      \wedge
      \cdots
      \wedge
    d x^{ \color{green} \mu_p }
    \\
    & =
    \star d \star
    \tfrac{1}{ \color{green} p! }
    \partial_{ \color{magenta} \nu }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p }
    d x^{ \color{magenta} \nu }
    \wedge
    d x^{ \color{green} \mu_1 }
      \wedge
      \cdots
      \wedge
    d x^{ \color{green} \mu_p }
    \\
    & =
    \star d
    \tfrac{1}{
      { \color{green} p! }
      { \color{orange} (D-(p+1))! }
    }
    \partial^{ \color{magenta} \nu }
    \alpha^{ \color{green} \mu_1 \cdots \mu_p }
    \epsilon_{
      { \color{magenta} \nu }
      { \color{green} \mu_1 \cdots \mu_{p} }
      { \color{orange} \mu_{p+2} \cdots \mu_D }
    }
    \,
    d x^{ \color{orange} \mu_{p+2} }
      \wedge
      \cdots
      \wedge
    d x^{ \color{orange} \mu_D }
    \\
    & =
    \star
    \tfrac{1}{
      { \color{green} p! }
      { \color{orange} (D-(p+1))! }
    }
    \partial_{ \color{red} \nu' }
    \partial^{ \color{magenta} \nu }
    \alpha^{ \color{green} \mu_1 \cdots \mu_p }
    \epsilon_{
      { \color{magenta} \nu }
      { \color{green} \mu_1 \cdots \mu_p }
      { \color{orange} \mu_{p+2} \cdots \mu_D }
    }
    \,
    d x^{ \color{red} \nu' }
    \wedge
    d x^{ \color{orange} \mu_{p+2} }
      \wedge
      \cdots
      \wedge
    d x^{ \color{orange} \mu_D }
    \\
    & =
    \tfrac{1}{
      { \color{green} p! }
      { \color{orange} (D-(p+1))! }
      { \color{blue} p! }
    }
    \partial^{ \color{red} \nu' }
    \partial_{ \color{magenta} \nu }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p }
    \epsilon^{
      { \color{magenta} \nu }
      { \color{green} \mu_1 \cdots \mu_p }
      { \color{orange} \mu_{p+2} \cdots \mu_D }
    }
    \epsilon_{
      { \color{red} \nu' }
      { \color{orange} \mu_{p+2} \cdots \mu_D }
      { \color{blue} \kappa_1 \cdots \kappa_p }
    }
    \,
    d x^{\color{blue} \kappa_1} \wedge \cdots d x^{\color{blue}\kappa_p}
    \\
    & =
    { \color{magenta} - }
    \tfrac{1}{
      { \color{green} p! }
      { \color{blue} p! }
    }
    \partial^{ \color{red} \nu' }
    \partial_{ \color{magenta} \nu }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p }
    \delta^{
      { \color{magenta} \nu }
      { \color{green} \mu_1 \cdots \mu_p }
    }
    _{
      { \color{red} \nu' }
      { \color{blue} \kappa_1 \cdots \kappa_p }
    }
    \,
    d x^{\color{blue} \kappa_1} \wedge \cdots d x^{\color{blue}\kappa_p}
    \\
    & =
    { \color{magenta} - }
    \tfrac{1}{
      { \color{green} p! }
      { \color{blue} p! }
    }
    \partial^{ \color{magenta} \nu }
    \partial_{ \color{magenta} \nu }
    \alpha_{ \color{green} \mu_1 \cdots \mu_p }
    \delta^{
      { \color{green} \mu_1 \cdots \mu_p }
    }
    _{
      { \color{blue} \kappa_1 \cdots \kappa_p }
    }
    \,
    d x^{\color{blue} \kappa_1} \wedge \cdots d x^{\color{blue}\kappa_p}
    \\
    & =
    { \color{magenta} - }
    \partial^{ \color{magenta} \nu }
    \partial_{ \color{magenta} \nu }
    \alpha
  \end{aligned}
$$

Here the sign in the last lines arises from the [[Minkowski spacetime|Minkowski]] [[signature of a quadratic form|signature]] via (eq:ContractionofLeviCivitaSymbols).

=--


Hodge structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

### Pure Hodge structure

A _Hodge structure_ (or _pure Hodge structure_, for emphasis) is a (bi-)[[graded object|grading]] structure on [[cohomology groups]] -- called a _Hodge decomposition_ -- of the kind that is exhibited by the [[de Rham cohomology]]/complex-[[ordinary cohomology]] of [[compact topological space|compact]] [[Kähler manifolds]], according to the [[Hodge theorem]]. A Hodge structure is said to be _of weight $d$_ if it behaves like the cohomology of a Kähler manifold of [[dimension]] $d$.

If instead of considering a single [[cohomology group]] one considers the cohomology groups of a parameterized collection of spaces -- hence the cohomology _[[sheaves]]/[[stacks]]_ -- then one speaks of _variation of Hodge structure_ (of a given weight).

By a central theorem of Hodge theory (recalled as theorem \ref{HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure} below) the traditional (and original) [[filtered complex|filtration]] on the complex cohomology of a [[Kähler manifold]] induced by the [[harmonic differential forms]] generalizes to a [[filtered complex|filtration]] of the complex-valued [[ordinary cohomology]] of any [[complex analytic space]] which is simply given by the canonical degree-filtration of the [[holomorphic de Rham complex]].

This means that [[ordinary differential cohomology]] in the guise of [[Deligne cohomology]] is nothing but the [[homotopy pullback]] of a stage of the Hodge filtration along the " [[Chern character]] " map from integral to complex cohomology. (A point of view highlighted for instance in [Peters-Steenbrink 08, section 7.2](#PetersSteenbrink08)). Viewed this way Hodge structures are filtrations of stages of differential form cycle refinements of [[Chern characters]] that appear in the general definition/characterization of [[differential cohomology]], as discussed at  _[[differential cohomology hexagon]]_ starting around the section _[de Rham coefficients](differential+cohomology+diagram#DeRhamCoefficients)_

This modern point of view is also crucial for instance in the characterization of an [[intermediate Jacobian]] (see there) as the subgroup of [[Deligne cohomology]] that is in the [[kernel]] of the map to Hodge-filtering stage of ordinary cohomology. See at _[intermediate Jacobian -- characterization as Hodge-trivial Deligne cohomology](intermediate+Jacobian#CharacterizationAsHodgeTrivialDeligneCohomology)_.

### Mixed Hodge structure

A _mixed Hodge structure_ is a [[filtration]] on [[cohomology groups]] -- called a _[[Hodge filtration]]_ -- such that the [[associated graded object]] has pure Hodge structure of weight $k$ in each degree $k$. The archetypical example exhibiting this is the cohomology of  [[complex varieties]] that have singularities ([Deligne 71](#Deligne71) [Deligne 74](#Deligne74)).

## Definition

Historically, Hodge structures originate in the special structure induced on the [[de Rham cohomology|de Rham]] [[cohomology groups]] of a compact [[Kähler manifold]] by the existence of [[harmonic differential forms]]. Below we first discuss this canonical Hodge structure

* [on the cohomology of a Kähler manifold](#ForAKaehlerManifold)

But it turns out that this Hodge structure only depends on the natural degree-[[filtration]] on the [[holomorphic de Rham complex]] and hence more generally there is canonical Hodge structure

* [on the cohomology of a complex analytic space](#ForAComplexAnalyticSpace).

Abstracting from here one defines Hodges structures

* [generally on abelian groups](#OnAnAbelianGroup).

### On the cohomology of a Kähler manifold
 {#ForAKaehlerManifold}

Let $X$ be a [[compact topological space|compact]] [[Kähler manifold]] and write $H^{p,q}(X)$ for its space of [[harmonic differential forms]], equivalently, via the [[Hodge isomorphism]], its [[Dolbeault cohomology]] in bidegree $(p,q)$.

Notice that by the [[de Rham theorem]] there are canonical maps

$$
  H^{p,q}(X)\to H^{p+q}(X,\mathbb{C})
$$

to [[ordinary cohomology]] of $X$ with complex [[coefficients]].

+-- {: .num_defn #TraditionalHodgeFiltration}
###### Definition

The _Hodge filtration_ on the cohomology of $X$ is the [[filtered complex]] structure given by the [[direct sum]]

$$
  F^p H^k(X, \mathbb{C})
   \coloneqq
  \underset{k-q \geq p}{\oplus}
  H^{k-q,q}(X)
  \,.
$$

=--

+-- {: .num_example}
###### Example

The full Hodge filtration of degree-2 cohomology is

$$
  \begin{aligned}
    F^0 H^2(X,\mathbb{C})
    & =
    H^{0,2}(X) \oplus H^{1,1}(X) \oplus H^{2,0}(X)
    \\
    F^1 H^2(X,\mathbb{C})
    & =
    \;\;\;\;\;\;\;
    H^{1,1}(X)
    \oplus
    H^{2,0}(X)
    \\
    F^2 H^2(X,\mathbb{C})
    & =
    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
    H^{2,0}(X)
 \end{aligned}
$$

=--

+-- {: .num_example}
###### Example

For all $p$ the mid-dimensional Hodge filtration stage in even total degree is

$$
  F^p H^{2p}
  =
  H^{p,p}(X)
  \oplus
  H^{p+1,p-1}(X)
  \oplus
  \cdots
  \oplus
  H^{2p,0}(X)
  \,.
$$

=--

### On the cohomology of a complex analytic space
 {#ForAComplexAnalyticSpace}

+-- {: .num_defn }
###### Definition

For $X$ a [[complex analytic space]], write

$$
  \Omega^\bullet_X
  \coloneqq
  (\mathcal{O}_X \stackrel{\partial}{\longrightarrow} \Omega^1_X \stackrel{\partial}{\longrightarrow} \Omega^2(X) \stackrel{\partial}{\longrightarrow} \cdots)
$$

for its [[holomorphic de Rham complex]].

=--

+-- {: .num_remark }
###### Remark

Notice the [relation to complex cohomology](holomorphic+de+Rham+complex#RelationToComplexCohomology) given by

$$
  H^k(X,\mathbb{C}) \simeq H^k(X,\Omega^\bullet_X)
  \,.
$$

=--

+-- {: .num_remark }
###### Remark

The [[holomorphic de Rham complex]] is naturally [[filtered object|filtered]] by degree with the $p$th filtering stage being

$$
  F^p \Omega^\bullet_X
  \coloneqq
  \big(
    0
     \to
    \cdots
      \to
    0
      \to
    \Omega^p_X
      \stackrel{\partial}{\longrightarrow}
    \Omega^{p+1}
      \stackrel{\partial}{\longrightarrow}
   \cdots
   \big)
  \,.
$$

Notice that here $\Omega^p$ is still regarded as sitting in degree $-p$, one just replaces by 0 the groups of differential forms of degree less than $p$.

=--

+-- {: .num_defn #HodgeFiltrationForComplexSpace}
###### Definition

The _Hodge filtration_ on $H^\bullet(X,\mathbb{C})$ is defined to be the [[filtration]] with $p$th stage the [[image]] of the [[hypercohomology|hyper]]-[[abelian sheaf cohomology]] with coefficients in the $p$th filtering stage of the holomorphic de Rham complex inside that with coefficients the full de Rham complex:

$$
  F^p H^k(X,\mathbb{C})
  \coloneqq
  im
  \left(
    H^k(X, F^p \Omega^\bullet_X)
    \to
    H^k(X, \Omega^\bullet_X)
  \right)
$$

=--

(e.g. [Voisin 02, def. 8.2](#Voisin02), [Peters-Steenbrink 08, def. 2.21](#PetersSteenbrink08)).

+-- {: .num_theorem #HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure}
###### Theorem

When the compact [[complex manifold]] $X$ happens to have the structure of a [[Kähler manifold]] then the [[Frölicher spectral sequence]] degenerates at the $E_1$ page which implies that def. \ref{HodgeFiltrationForComplexSpace} coincides with the traditional definition via [[harmonic differential forms]], def. \ref{TraditionalHodgeFiltration}:

$$
  \underset{k-q \geq p}{\oplus}
  H^{k-q,q}(X)
  \simeq
  im
  \left(
    H^k(X, F^p \Omega^\bullet_X)
    \to
    H^k(X, \Omega^\bullet_X)
  \right)
  \,.
$$

=--

(e.g. [Voisin 02, remark 8.29](#Voisin02); [Voisin 08, 1.1.2](#Voisin08); [Peters-Steenbrink 08, prop 2.22](#PetersSteenbrink08)).

+-- {: .num_remark #FrolicherEquivalenceInComponents}
###### Remark

The equivalence in theorem \ref{HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure} is exhibited by the following morphism.

Write

$$
  tot(\Omega^{\bullet \geq p, \bullet}, \mathbf{d}= \partial + \bar \partial)
$$

for the holomorphically truncated [[de Rham complex]], as indicated, thought of as the [[total complex]] of the [[Dolbeault complex|Dolbeault]] [[double complex]]

$$
  \array{
    \Omega^{p,0} &\stackrel{\bar \partial}{\to}& \Omega^{p-1,1} &\stackrel{\bar \partial}{\to}& \cdots
    \\
    \downarrow^{\mathrlap{\partial }}
    &&
    \downarrow^{\mathrlap{\partial }}
    \\
    \Omega^{p+1,0} &\stackrel{\bar \partial}{\to}& \Omega^{p,1} &\stackrel{\bar \partial}{\to}& \cdots
    \\
    \downarrow^{\mathrlap{\partial }}
    &&
    \downarrow^{\mathrlap{\partial }}
    \\
    \vdots && \vdots
  }
  \,.
$$

Since this is in each row the [[Dolbeault resolution]] of the given sheaf of [[holomorphic differential forms]], this total complex is indeed [[quasi-isomorphism|quasi-isomorphic]] to the (truncated) [[holomorphic de Rham complex]].

The total complex is in degree $-k$ given by
$\underset{k-q \geq p}{\oplus} \Omega^{k-q, q}$
and hence globally defined closed $(k-q \geq p,q)$-forms naturally inject into

$$
  H^k(X, tot(\Omega^{\bullet\geq p, \bullet}, \mathbf{d} = \partial + \bar \partial) )
  \simeq
  H^0(X,tot(\Omega^{\bullet\geq p, \bullet}, \mathbf{d} = \partial + \bar \partial)[-k])
  \,.
$$

Therefore given a representative $\alpha \in \Omega^{p,q}_{cl}(X)$ of $[\alpha] \in H^{p,q}(X)$ it is canonically sent along

$$
  \underset{k-q\geq p}{\oplus} \Omega^{p,q}_{cl}(X)
  \simeq
  \underset{k-q\geq p}{\oplus}
   H^0(X, \Omega^{p,q}_{cl})
  \to
  H^0(X,tot(\Omega^{\bullet\geq p, \bullet}, \mathbf{d} = \partial + \bar \partial)[-k])
  \,.
$$

This map exhibits the equivalence in theorem \ref{HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure} (e.g. [Voisin 08, section 1.1.2](#Voisin08)).

Dually,

$$
  \Omega_{hol}^{\leq k}
  \simeq
  tot(  \Omega^{\bullet \leq k, \bullet})
  \,.
$$

This plays a role in the discussion of [[intermediate Jacobians]],
where for $dim_{\mathbb{C}}(X)= k+1$ we have

$$
  H^{2k+1}(X,\mathbb{R})
  \simeq
  H^{2k+1}(X,\mathbb{C}) / F^{k+1} H^{2k+1}(X,\mathbb{C})
  \simeq
  H^{2k+1}(X, \Omega_{hol}^{\bullet \leq k})
  \,.
$$

Here a real differential $(2k+1)$-form

$$
  \alpha
  =
  \overline{\alpha^{0,2k+1}}
  +
  \overline{\alpha^{1, 2k}}
  +
  \cdots
  +
  \alpha^{1, 2k}
  +
  \alpha^{0,2k+1}
$$

injects via its pieces in

$$
  \Omega^{p \leq k, 2k+1-p}(X)
  \simeq
  H^0(X, \Omega^{p \leq k, 2k+1-p})
  \to
  H^0(X, tot(\Omega^{\bullet \leq k, \bullet})[-k])
  \simeq
  H^k(\Omega_{hol}^{\bullet\leq k})
  \,.
$$

=--

### Generally on an abelian group
 {#OnAnAbelianGroup}

+-- {: .num_defn #HodgeStructureOfWeightk}
###### Definition

For $H_{\mathbb{Z}}$ an [[abelian group]], a _Hodge structure_
of _weight_ $k \in \mathbb{Z}$ on $H_{\mathbb{Z}}$ is a [[direct sum]] decomposition of its [[complexification]]

$$
  H_{\mathbb{C}}\coloneqq H_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{C}
$$

into [[complex vector spaces]] $H^{p,q}$ with $p +q = k$ of the form

$$
  H_{\mathbb{C}} \simeq \underset{p+q = k}{\oplus} H^{p,q}
$$

such that $H^{q,p}$ is the [[complex conjugation|complex conjugate]] of $H^{p,q}$:

$$
  H^{p,q} = \overline{H^{q,p}}
  \,.
$$
This is an equality of the underlying sets of the complex vector spaces.
=--

With this the above def. \ref{TraditionalHodgeFiltration} has the following verbatim generalization

+-- {: .num_defn #GeneralHodgeFiltration}
###### Definition

Given a Hodge structure $H_{\mathbb{Z}}, \{H^{p,q}\}$ of weight $k$, def. \ref{HodgeStructureOfWeightk}, then  the associated _Hodge filtration_ on $H_{\mathbb{C}}$ is the [[filtered complex]] structure given by the [[direct sum]]

$$
  F^p H_{\mathbb{C}}
   \coloneqq
  \underset{k-q \geq p}{\oplus}
  H^{k-q,q}
  \,.
$$

=--

## Related concepts

* [[Hodge theory]],

  [[nonabelian Hodge theory]], [[noncommutative Hodge theory]]

* [[Hodge symmetry]]

* [[Hodge cycle]]

* [[Hodge cohomology]]

* [[Hodge-filtered differential cohomology]]

* [[intermediate Jacobian]]

* [[Lefschetz decomposition]]

## References

The basic Hodge filtration:

* {#Voisin08} [[Claire Voisin]], *Hodge theory and the topology of compact Kähler and complex projective manifolds*, Lecture notes (2008) [[pdf](http://www.math.columbia.edu/~thaddeus/seattle/voisin.pdf), [[Voisin-HodgeTheory.pdf:file]]]

* [[Stacks Project]], *The Hodge Filtration* [[tag:0FM7](https://stacks.math.columbia.edu/tag/0FM7)]

Hodge filtration induced by the Hodge-to-de Rham spectral sequence on $H_{dR}(X/S)$ of an $S$-scheme $X$

* The Stacks Project 50.7, tag/[0FM7](https://stacks.math.columbia.edu/tag/0FM7)

Textbook accounts on Hodge structures:

* {#Kulikov98} [[Valentine S. Kulikov]], *Mixed Hodge Structures and Singularities*, Cambridge University Press (1998) [[doi:10.1017/CBO9780511758928](https://doi.org/10.1017/CBO9780511758928)]

* {#Voisin02} [[Claire Voisin]], section 7 of _[[Hodge theory and Complex algebraic geometry]] I,II_,  Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

* {#PetersSteenbrink08} [[Chris Peters]], [[Jozef Steenbrink]], _[[Mixed Hodge Structures]]_, Ergebisse der Mathematik (2008) [[pdf](http://www.arithgeo.ethz.ch/alpbach2012/Peters_Steenbrinck), [doi:10.1007/978-3-540-77017-6](https://doi.org/10.1007/978-3-540-77017-6)]

The notion of mixed Hodge structures was introduced in

* {#Deligne71} [[Pierre Deligne]], _Théorie de Hodge II_, Publ. Math. I.H.E.S **40** (1971) 5-58 [[numdam:PMIHES_1971__40__5_0](http://www.numdam.org/item/PMIHES_1971__40__5_0), [doi:10.1007/BF02684692](https://doi.org/10.1007/BF02684692)]

* {#Deligne74} [[Pierre Deligne]], _Théorie de Hodge III_, Publ. Math., I. H. E. S **44** (1974) 5-77 [[numdam:PMIHES_1974__44__5_0](http://www.numdam.org/item/PMIHES_1974__44__5_0), [doi:10.1007/BF02685881](https://doi.org/10.1007/BF02685881)]

Review:

* [[Alain Connes]], [[Matilde Marcolli]], Section 8.4 of: *[[Noncommutative Geometry, Quantum Fields and Motives]]*

See also

* [[Jozef Steenbrink]], S. Zucker, _Variation of mixed Hodge structure I_, Invent. Math. __80__ (1985), 489-542.

* Wikipedia, _[Hodge structure](http://en.wikipedia.org/wiki/Hodge_structure),

* [[Donu Arapura]], _Mixed Hodge Structures Associated to Geometric Variations_ [[pdf](http://www.math.purdue.edu/~dvb/preprints/tifr.pdf)]

*  [[eom]],

   A.I. Ovseevich _[Hodge structure](http://eom.springer.de/H/h047470.htm)_, _[Period mapping](http://eom.springer.de/p/p072140.htm)_,

   [[Jozef Steenbrink]], _[Variation of Hodge structure](http://eom.springer.de/v/v096170.htm)_

* [[Michael Hopkins]], [[Gereon Quick]], _Hodge filtered complex bordism_ ([arXiv:1212.2173](http://arxiv.org/abs/1212.2173))

For [[orbifold cohomology]]:

* Javier Fernandez, _Hodge structures for orbifold cohomology_, Proc. Amer. Math. Soc. 134 (2006), no. 9, 2511-2520 ([arXiv:math/0311026](https://arxiv.org/abs/math/0311026))

[[!redirects Hodge structures]]

[[!redirects pure Hodge structure]]
[[!redirects pure Hodge structures]]

[[!redirects mixed Hodge structure]]
[[!redirects mixed Hodge structures]]

[[!redirects Hodge filtration]]
[[!redirects Hodge filtrations]]
[[!redirects Hodge decomposition]]
[[!redirects Hodge decompositions]]

[[!redirects Hodge filtering]]
[[!redirects Hodge filterings]]

[[!redirects variation of Hodge structure]]
[[!redirects variation of Hodge structures]]
[[!redirects variations of Hodge structure]]
[[!redirects variations of Hodge structures]]


Hodge symmetry

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Over [[compact topological space|compact]] [[Kähler manifolds]] $X$, _Hodge symmetry_ is the property that the [[Dolbeault cohomology]] groups $H^{p,q}(X)$ are taken into each other under [[complex conjugation]] followed by switching the bidegree:

$$
  H^{p,q}(X) \simeq \overline{H^{q,p}(X)}
  \,.
$$

In particular this means that the [[dimension]] of the cohomology groups in degree $(p,q)$ -- the [[Hodge number]] $h^{p,q}$ -- coincides with that in bidegree $(q,p)$.

By the [[Dolbeault theorem]] this is formulated more generally in terms of [[abelian sheaf cohomology]] as saying that

$$
  dim_{\mathbb{C}} H^q(X,\Omega^p)
  =
  dim_{\mathbb{C}} H^p(X,\Omega^q)
  \,,
$$

where $\Omega^p$ denotes the [[abelian sheaf]] of [[holomorphic p-forms]].

In this form the statement of Hodge symmetry makes sense more generally for [[complex analytic spaces]] and [[schemes]] over the [[complex numbers]] -- but it is no longer generally true in these more general cases

Sufficient conditions for Hodge symmetry to hold more generally are discussed for instance in ([Joshi 14](#Joshi14)).

## References

* {#Joshi14} Kirti Joshi, _Some remarks on Hodge symmetry_ ([arXiv:1402.4176](http://arxiv.org/abs/1402.4176))


Hodge theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[Hodge theorem]] asserts, in particular, that for a [[compact topological space|compact]] [[Kähler manifold]], the canonical $(p,q)$-[[graded object|grading]] of its [[differential forms]] descends to its [[de Rham cohomology]]/[[ordinary cohomology]]. The resulting structure is called a _[[Hodge structure]]_, and is indeed the archetypical example of such.

## Definitions

+-- {: .num_defn}
###### Definition

Let $(X,g)$ be a [[compact topological space|compact]] [[oriented]] [[Riemannian manifold]] of [[dimension]] $n$. Write $\Omega^\bullet(X)$ for the [[de Rham complex]] of smooth [[differential form]]s on $X$ and $\star : \Omega^\bullet(X) \to \Omega^{n-\bullet}(X)$ for the [[Hodge star operator]].

The Hodge [[inner product]]

$$
 \langle -,-\rangle : \Omega^\bullet(X) \otimes \Omega^{\bullet}(X)
  \to
  \mathbb{R}
$$

is given by

$$
  \langle \alpha , \beta\rangle = \int_X \alpha \wedge \star \beta
  \,.
$$

Write $d^*$ for the formal [[adjoint]] of the de Rham differential under this inner product. Then

$$
  \Delta := [d,d^*] := d d^* + d^* d = (d + d^*)^2
$$

is the Hodge [[Laplace operator]] ($d + d^*$ is the corresponding [[Dirac operator]]). A [[differential form]] $\omega$ in the [[kernel]] of $\Delta$

$$
  \Delta \omega = 0
$$

is called a [[harmonic form]] on $(X,g)$.

Write $\mathcal{H}^k(X)$ for the [[abelian group]] of harmonic $k$-forms on $X$.

=--

## For Riemannian manifolds
 {#ResultsOverRiemannianManifold}

+-- {: .num_prop}
###### Observation

Harmonic forms are precisely those in the kernel of $d + d^*$, which are precisely those in the joint kernel of $d$ and $d^*$.

=--

By the fact that the [[bilinear form]] $\langle -,-\rangle$ is non-degenerate.

Therefore we have a canonical map $\mathcal{H}^k(X) \to H_{dR}^k(X)$ of harmonic forms into the [[de Rham cohomology]] of $X$.

+-- {: .num_theorem}
###### Theorem

The canonical map

$$
  \mathcal{H}^k(X) \to H_{dR}^k(X)
$$

is an [[isomorphism]].

=--

This means that every [[de Rham cohomology]] class on $(X,g)$ has precisely one harmonic cocycle reprentative.

But more is true

+-- {: .num_theorem}
###### Theorem

For $(X,g)$ as above, there exists a unique degree-preserving operator (the [[Green operator]] of the [[Laplace operator]] $\Delta$)

$$
  G : \Omega^\bullet(X) \to \Omega^\bullet(X)
$$

such that

* $G$ commutes with $d$ and with $d^*$;

* $G(\mathcal{H}^\bullet(X)) = 0$;

* and

  $$
    Id - \pi_{\mathcal{H}} = [d, d^* G]
    \,,
  $$

  where $\pi_{\mathcal{H}}$ is the orthogonal projection on [[harmonic form]]s and the angular brackets denote the graded commutator $[d, d^* G] = [d,d^*]G = \Delta G$.

=--

See for instance page 6 of ([GreenVoisinMurre](#GreenVoisinMurre)).

## For Kähler manifolds

(...)

## Related concepts

* [[Hodge-Maxwell theorem]]

* [[Hodge theory]], [[Hodge structure]]

* [[nonabelian Hodge theorem]]

* [[Poincaré lemma]]

* [[Stokes theorem]]

* [[de Rham theorem]]

* [[Dolbeault theorem]]

* [[Hodge conjecture]]

## References

The theorem is due to

* [[William Hodge]], (1947)

Textbook accounts include

* {#Voisin02} [[Claire Voisin]], section 5 of _[[Hodge theory and Complex algebraic geometry]] I,II_,  Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

* [[Chris Peters]], [[Jozef Steenbrink]], section 1.1. of _[[Mixed Hodge Structures]]_, Ergebisse der Mathematik (2008) ([pdf](http://www.arithgeo.ethz.ch/alpbach2012/Peters_Steenbrinck))

Lecture notes include

* Xi Yin, _Notes on the Hodge Theorem_ ([pdf](http://www.people.fas.harvard.edu/~xiyin/hodge.pdf))

* [[Jonathan Evans]], _Hodge theorem_ ([pdf](http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture5.pdf))

See also

* Mark Green, [[Claire Voisin]], Jacob Murre, _Algebraic cycles and Hodge theory_ Lecture Notes in Mathematics, 1594 (1993)
{#GreenVoisinMurre}

[[!redirects Hodge's theorem]]
[[!redirects theorem of Hodge]]


Hodge theory

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=--
#### Complex geometry
+--{: .hide}
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Overview

_Hodge theory_ is the study of properties of (complex, rational and integer) [[cohomology]] (and [[motives]]) of [[Kähler manifolds]], induced by a [[Hodge filtration]] -- given in the classical situation by [[harmonic differential forms]] --  and the corresponding [[Hodge theorem]]. Hodge theory also applies in [[combinatorics]], for instance to [[matroids]] ([Huh 22](#Huh22)).

Central aspects of the theory include Hodge decomposition, [[hard Lefschetz theorem]], [[pure Hodge structures]], [[mixed Hodge structures]], [[variations of Hodge structure]], [[Hodge to de Rham spectral sequence]], [[periods]]... This basic setup is however by now vastly generalized.

The nomenclature "nonabelian"/"noncommutative" Hodge theory may be confusing - to clarify:

[[nonabelian Hodge theory|Nonabelian Hodge theory]] refers to the generalization of classical Hodge theory for commutative spaces from abelian coefficients to nonabelian coefficients.

[[Noncommutative Hodge theory]] refers to the generalization of classical Hodge theory from commutative geometry (e.g. algebraic varieties) to [[noncommutative geometry]] (e.g. [[dg-categories]] -- see also the page on [[noncommutative algebraic geometry]]). The coefficients in noncommutative Hodge theory are abelian.

Ultimately, the goal is to develop a Hodge theory for noncommutative spaces with nonabelian coefficients.

## Related entries

* [[p-adic Hodge theory]]

* [[harmonic differential form]], [[Hodge star operator]]

* [[Hodge theorem]]

* [[Hodge structure]]

* [[Hodge conjecture]]

* [[nonabelian Hodge theory]], [[Lefschetz hyperplane theorem]]

* [[motive]]

## References

### Contributors

The main historical contributors to Hodge theory include [[William Hodge]], [[Phillip Griffiths]], [[Alexander Grothendieck]], [[Pierre Deligne]]. More recently a noncommutative generalization and refinement of Hodge theory is emerging in the work of [[Maxim Kontsevich|M. Kontsevich]] and collaborators (D. Kaledin, L. Katzarkov, [[Tony Pantev]]...) with precursors in the works of [[Carlos Simpson]], [[Mikio Sato]], [[Claus Hertling]] and others. Another very interesting and complex picture is developing for a number of years in works of Goncharov.

### General

* Bertin, Demailly, [[Luc Illusie]], [[Chris Peters]], _Introduction to Hodge theory_, AMS (1996)

* [[Claire Voisin]], _Hodge theory and the topology of complex Kähler and complex projective manifolds_ (survey, [pdf](http://www.math.columbia.edu/~thaddeus/seattle/voisin.pdf))

* {#Voisin02} [[Claire Voisin]], _[[Hodge theory and Complex algebraic geometry]] I,II_, Cambridge Stud. in Adv. Math. __76, 77__ (2002/3) [[doi:10.1017/CBO9780511615344](https://doi.org/10.1017/CBO9780511615344)]

* Mark Green, [[Claire Voisin]], Jacob Murre, _Algebraic cycles and Hodge theory_, Lecture Notes in Mathematics, 1594 (1993)
{#GreenVoisinMurre}

* [[Phillip Griffiths]], _Periods of integrals on algebraic manifolds I,II,III_,  Amer. J. Math. __90__, 568–626, 808–865 (1968) ; Publ. Math. de l'IHÉS __38__, 228–296 (1970) [numdam](http://www.numdam.org/item?id=PMIHES_1970__38__125_0)

* P. Griffiths, _Hodge theory - from Abel to Deligne_, talk, Inst. Adv. Studies 2016 [yt](https://www.youtube.com/watch?v=hPyDz5R5YaY);
_Some geometric applications of Hodge theory_, Oct 2020, [yt](https://www.youtube.com/watch?v=MA8FhSY3FlU)

* M. Saito, _Mixed Hodge modules_, Publ. R.I.M.S. Kyoto Univ. __26__  (1990)  pp. 221–333.

* [[L. Katzarkov]], [[Maxim Kontsevich|M. Kontsevich]], [[Tony Pantev]], _Hodge theoretic aspects of mirror symmetry_, [arxiv/0806.0107](http://arxiv.org/abs/0806.0107) Proc. of Symposia in Pure Math. __78__ (2008), "From Hodge theory to integrability  and TQFT: $tt^*$-geometry",  eds. Ron Y. Donagi and Katrin Wendland, 87-174

* [[D. Kaledin]], _Cartier isomorphism and Hodge theory in the non-commutative case_,  Arithmetic geometry,  537--562, Clay Math. Proc. __8__, Amer. Math. Soc. 2009, [arxiv/0708.1574](https://arxiv.org/abs/0708.1574)

* D. Kaledin, _Tokyo lectures "Homological methods in non-commutative geometry"_, [pdf](http://imperium.lenin.ru/~kaledin/tokyo/final.pdf), [TeX](http://imperium.lenin.ru/~kaledin/tokyo/final.tex)

* [[Claus Hertling]], Christian Sevenheck, _Twistor structures, $tt^*$-geometry and singularity theory_, pp. 49--74 in: "From tQFT to tt* and integrability", Proceedings of Symposia in Pure Mathematics __78__ (2007, Augsburg, Germany), AMS 2008; [arxiv:0807.2199](https://arxiv.org/abs/0807.2199)

* [[Claus Hertling]], [[Claude Sabbah]], _Examples of non-commutative Hodge structure_ (v1 title: Fourier-Laplace transform of flat unitary connections and TERP structures), [arxiv/0912.2754](https://arxiv.org/abs/0912.2754)

* A. B. Goncharov, _Hodge correlators_, [arxiv/0803.0297](http://arxiv.org/abs/0803.0297), _Hodge correlators II_, [arxiv/0807.4855](http://arxiv.org/abs/0807.4855)

* [[C. T. C. Wall]], _Periods of integrals and topology of algebraic varieties_, [jstor](http://www.jstor.org/stable/2397500) A course in Hodge theory : with emphasis on multiple integrals

* Hossein Movasati, _A course in Hodge theory: with emphasis on multiple integrals_, Somerville: International Press of Boston 2021.

### Relation to motives

Discussions of Hodge theory in relation to [[motives]] can be found in:

* [[Chris Peters]], _Tata Lectures on Motivic Aspects of Hodge Theory_ ([pdf](http://www-fourier.ujf-grenoble.fr/~peters/Books/motivic.f/Tatalects.pdf))

* [[Chris Peters]], _Lectures on Motivic Aspects of Hodge Theory: The Hodge Characteristic._, Summer School Istanbul 2014, [pdf](http://www-fourier.ujf-grenoble.fr/~peters/Books/motivic.f/MotAspects.pdf)

### Nonabelian Hodge theory

General references on nonabelian Hodge theory include:

* [[Ron Donagi]], [[Tony Pantev]], _Lectures on the geometric Langlands conjecture and non-abelian Hodge theory_, 2009 ([pdf](http://www.icmat.es/seminarios/langlands/school/handouts/pantev.pdf))

* [[Alberto García Raboso]], [[Steven Rayan]], _Introduction to Nonabelian Hodge Theory: flat connections, Higgs bundles, and complex variations of Hodge structure_,  Fields Inst. Monogr. 34 (2015), 131--171 ([arXiv](https://arxiv.org/abs/1406.1693)) ([Springer](http://link.springer.com/chapter/10.1007/978-1-4939-2830-9_5))

### In [[combinatorics]]

* {#Huh22} [[June Huh]], _Combinatorics and Hodge theory_, Proc. Int. Cong. Math. **1** (2022)
 [[doi:10.4171/ICM2022/205](), [pdf](https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/2022/jh.pdf), [[Huh-CombinatoricsAndHodgeTheory.pdf:file]]]


This page collects material related to the textbook

* [[Claire Voisin]] (translated by [[Leila Schneps]]),

  **Hodge theory and Complex algebraic geometry**

  Cambridge Stud. in Adv. Math.

  **76**, **77**, 2002/3

  vol I: [doi:10.1017/CBO9780511615344](https://doi.org/10.1017/CBO9780511615344)

  vol II: [doi:10.1017/CBO9780511615177](https://doi.org/10.1017/CBO9780511615177)

on [[Hodge theory]] and [[GAGA]].

#Contents#
* table of contents
{:toc}

## Volume 1

### I Preliminaries

#### 1 Holomorphic functions of many variables

* [[holomorphic function]]

#### 2. Complex manifold

* [[complex manifold]]

* [[Newlander-Nirenberg theorem]]

* [[Dolbeault complex]]

#### 3. Kähler metric

* [[Kähler manifold]]

#### 4. Sheaves and cohomology

* [[abelian sheaf cohomology]]

### II The Hodge decomposition

#### 5. Harmonic forms and cohomology

* [[harmonic differential form]]

* [[Hodge theorem]]

#### 6. The case of Kähler manifolds

* [[hard Lefschetz theorem]]

* [[Lefschetz decomposition]]

#### 7. Hodge structures and polarization

* [[Hodge structure]]

* [[polarized variety]]

#### 8. Holomorphic de Rham complexes and spectral sequences

* [[hypercohomology]]

* [[holomorphic de Rham complex]]

* [[differential form with logarithmic singularities]]

* [[spectral sequence]]

* [[Frölicher spectral sequence]]

### III Variations of Hodge structures

#### 9. Families and deformations

* [[local system]]

* [[flat vector bundle]]

* [[Gauss-Manin connection]]

* [[twisted de Rham cohomology]]

#### 10. Variations of Hodge structure

* [[variations of Hodge structure]]

### IV. Cycles and cycle classes

#### 11. Hodge classes

#### 12. Deligne-Beilinson cohomology and the Abel-Jacobi map

* [[Abel-Jacobi map]]

* [[Deligne cohomology]]

## Volume 2

(...)

category: reference


Hodge-filtered differential cohomology

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

What is called *Hodge-filtered cohomogy* [[Hopkins & Quick (2014)](#HopkinsQuick14)] is the variant of [[differential generalized cohomology]] obtained by passing from real [[differential geometry]] to [[complex geometry]]:

Where [[differential cohomology]] by default  pairs a given   [[Whitehead-generalized cohomology theory]] $E$ of [[underlying]] [[topological spaces]] with the degree-filtered $E_\bullet \otimes \mathbb{R}$-valued [[de Rham complexes]] of [[differential forms]] on real [[smooth manifolds]], the Hodge-filtered variant pairs instead with the [[Hodge filtration|Hodge filtered]] $E_\bullet \otimes \mathbb{C}$-valued [[Dolbeault complexes]] [[Haus & Quick (2022), p. 3](HausQuick22)], hence equivalently (see [there](Hodge+structure#HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure)) with the degree-filtered [[holomorphic de Rham complexes]] [[Hopkins & Quick (2014), Def. 4.2](#HopkinsQuick14)].

Concretely, for $p \in \mathbb{Z}$, the *Hodge-filtered $E$-cohomology* $E^\bullet_{\mathcal{D}}(p)(\mathcal{X})$ of a [[complex manifold]] $\mathcal{X}$, or more generally of an [[infinity-stack|$\infty$-stack]] over the [[site]] $Mfd_{\mathbb{C}}$ of all [[complex manifolds]] (with [[open covers]]), is the [[cohomology]] in the [[(infinity,1)-category|$\infty$-category]] of [[(infinity,1)-sheaves of spectra|$(\infty,1)$-sheaves of spectra]] $Sp\big(Sh_\infty(Mfd_{\mathbb{C}})\big)$ which is represented by the [[homotopy fiber product]]-spectrum

\[
  \label{DefiningHomotopyPullback}
  E_{\mathcal{D}}(p)
  \;\;
    \coloneqq
  \;\;
  E
  \underset{
    \Omega^{\geq p}(\text{-};\pi_{\bullet}(E)\otimes \mathbb{C})
  }{\times}
    \Omega^{\bullet}(\text{-};\pi_{\bullet}(E)\otimes \mathbb{C})
  \,,
\]
where

* $E$ denotes the [[spectrum]] [[Brown representability theorem|representing]] the given [[Whitehead-generalized cohomology theory]], regarded as a [[locally constant infinity-stack|locally constant]] [[(infinity,1)-sheaf of spectra|$(\infty,1)$-sheaf of spectra]];

* $\Omega^\bullet(\text{-};A_\bullet)$ denotes the [[abelian sheaf]] of [[chain complexes]] given by the [[holomorphic de Rham complex]] with [[coefficients]] in a [[graded abelian group]] $A_\bullet$ and understood as an [[(infinity,1)-sheaf of spectra|$(\infty,1)$-sheaf of]] [[Eilenberg-MacLane spectra]] via the [[stable Dold-Kan correspondence]];

* $\Omega^{\geq p}(A_\bullet)$ denotes the subcomplex of $\geq p$-forms, similarly regarded,

* $E$ is regarded as fibered over $\Omega^\bullet\big(\text{-};A_\bullet\big)$ via the "[[complexification]]" map on $E$ given by [[smash product]] $E \simeq E \wedge \mathbb{S}\longrightarrow E \wedge H \mathbb{C}$ with the [[unit]] of the [[ring spectrum]] $H \mathbb{C}$ (what we may recognize as the [[Chern-Dold character]]) or rather (to be compatible with [[Pierre Deligne]]'s original convention) that map followed by the [[automorphism]] which in degree $2n$ is given by multiplication with $(2 \pi \mathrm{i})^n \,\in\, \mathbb{C}$ .

This is [Hopkins & Quick (2014), Def. 4.2](#HopkinsQuick14), being the direct holomorphic analog of the respective definition of [[differential cohomology]] (cf. the *[[differential cohomology hexagon]]*) in [[Quadratic Functions in Geometry, Topology, and M-Theory|Hopkins & Singer (2005)]] (except for that conventional rescaling by $(2\pi \mathrm{i})^{\bullet/2}$.)

In the special case where $E \,\coloneqq\, H\mathbb{Z}$ is [[integral cohomology|integral]] [[ordinary cohomology]] the above homotopy pullback (eq:DefiningHomotopyPullback) reproduces the [[Deligne complex]] in its original form (see the details spelled out [there](Deligne+cohomology#TheDifferentialHexagonForDeligne); but the key observation may be recognized already in the classical review of [Esnault & Viehweg (1988), Def. 2.6](Deligne+cohomology#EsnaultViehweg88)), whence the subscript "$\mathcal{D}$" in the above definition may be read as being for "generalized Deligne cohomology".

## References

### General

The general concept of Hodge-filtered [[differential cohomology]] and introducing the special case of Hodge-filtered [[complex cobordism cohomology]]:

* {#HopkinsQuick14} [[Michael J. Hopkins]], [[Gereon Quick]], *Hodge filtered complex bordism*, Journal of Topology **8** 1 (2014) 147-183 [[arXiv:1212.2173](https://arxiv.org/abs/1212.2173), [doi:10.1112/jtopol/jtu021](https://doi.org/10.1112/jtopol/jtu021)]

### Hodge-filtered ordinary cohomology

The case of Hodge-filtered [[integral cohomology|integral]]$\;$[[ordinary cohomology]] is [cf. [Haus (2022), §3.2](#Haus22)] the original definition of *[[Deligne cohomology]]*, see [there](Deligne+cohomology#References) for references.

### Hodge-filtered topological K-theory

A Hodge-filtered form of complex [[topological K-theory]] appears (cf. [Quick (2016), p. 2](#Quick16)) in:

* [[Max Karoubi]], *Théorie générale des classes caractéristiques secondaires*, K-Theory **4** 1 (1990) 55-87 [[doi:10.1007/BF00534193](http://dx.doi.org/10.1007/BF00534193), [pdf](https://webusers.imj-prg.fr/~max.karoubi/Publications/43.pdf)]

* [[Max Karoubi]], *Classes Caractéristiques de Fibrés Feuilletés, Holomorphes ou Algébriques*, in: *Proceedings of Conference on Algebraic Geometry and Ring Theory* in honor of Michael Artin, Part II (Antwerp 1992), K-Theory **8** 2 (1994) 153-211 [[doi:10.1007/BF00961455](http://dx.doi.org/10.1007/BF00961455)]

### Hodge-filtered complex cobordism

The case of Hodge filtered [[differential cobordism cohomology theory|differential]] [[MU]]-[[cobordism cohomology theory]]

* [Hopkins & Quick (2014), §5](#HopkinsQuick14)

Introduction and survey:

* [[Gereon Quick]], *Geometric Hodge filtered complex cobordism*, [talk at](Center+for+Quantum+and+Topological+Systems#QuickMar2023) *[[CQTS]]* (March 2023) [video:[YT](https://www.youtube.com/watch?v=pMu0gT5kIBo)]

Refinement of the [[Abel-Jacobi map]] to [[Hodge filtration|Hodge filtered]] [[differential cobordism cohomology theory|differential]] [[MU]]-[[cobordism cohomology theory]]:

* {#Quick16} [[Gereon Quick]], *An Abel-Jacobi invariant for cobordant cycles*, Documenta Mathematica **21** (2016) 1645–1668 [[arXiv:1503.08449](https://arxiv.org/abs/1503.08449)]

A geometric cocycle model by actual [[cobordism]]-classes:

* {#Haus22} [[Knut Bjarte Haus]], *Geometric Hodge filtered complex cobordism*, PhD thesis (2022) [[ntnuopen:3017489](https://ntnuopen.ntnu.no/ntnu-xmlui/handle/11250/3017489)]

* {#HausQuick22} [[Knut Bjarte Haus]], [[Gereon Quick]], *Geometric Hodge filtered complex cobordism* [[arXiv:2210.13259](https://arxiv.org/abs/2210.13259)]

On [[Umkehr maps]] in this context:

* [[Knut Bjarte Haus]], [[Gereon Quick]], *Geometric pushforward in Hodge filtered complex cobordism and secondary invariants* [[arXiv:2303.15899](https://arxiv.org/abs/2303.15899)]

[[!redirects Hodge-filtered differential cohomology theory]]

[[!redirects Hodge-filtered cohomology]]
[[!redirects Hodge-filtered cohomology theory]]


Hodge-Maxwell theorem

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=--
#### Physics
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[Hodge theorem]] in the language of [[electromagnetism]]. Over a [[Riemann surface]] this may be regarded as simple case of the _[[Narasimhan-Seshadri theorem]]_.

## Statement

Let $(X,g)$ be a [[compact topological space|compact]] [[orientation|oriented]] [[Riemannian manifold]] of [[dimension]] $n$. Write $\star$ for the corresponding [[Hodge star]] operator.

Then for every exact [[differential n-form]] $j$ of degree $n-k-1$ there is in each [[de Rham cohomology]] class of degree k a unique representative closed k-form $F$

$$
  \mathbf{d} F = 0
$$

such that

$$
  \mathbf{d}\star F = j
  \,.
$$

Reading this as [[Maxwell's equations]] on $(X,g)$ then $g$ is the [[field (physics)|field]] of [[gravity]], $F$ is the [[Faraday tensor]] measuring the [[field strength]] of the [[electromagnetic field]] and $j$ is the [[electric current]].

## References

The term "Hodge-Maxwell theorem" in the above form appears in

* [[Jonathan Evans]], _[Aspects of Yang-Mills theory](http://www.homepages.ucl.ac.uk/~ucahjde/yangmills.htm), [lecture 4 pdf](http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture4.pdf)


Hodge–de Rham spectral sequence

#Contents#
* table of contents
{:toc}

## Idea

A kind of [[spectral sequence]] for the computation of the algebraic [[de Rham cohomology]] of a [[variety]].

## Related entries

* [[degeneration of Hodge to de Rham spectral sequence]]

* [[Frölicher spectral sequence]]

[[!redirects Hodge to de Rham spectral sequence]]

[[!redirects Hodge-de Rham spectral sequence]]


Hofer's geometry

#Contents#
* table of contents
{:toc}

## Idea

__Hofer's metric__ is a [[Riemannian metric]] on the [[group]] of [[Hamiltonian diffeomorphisms]] of a [[symplectic manifold]]. This group is infinite-dimensional, so that the existence of this metric is nontrivial, due noncompactness.

Hofer's geometry is now a classical subject in [[symplectic geometry]].

## References

* [[Helmut Hofer]], _On the topological properties of symplectic maps_, Proceedings of the Royal Society of Edinburgh. Section A: Mathematics, Volume 115 Issue 1-2, 1990 ([doi:10.1017/S0308210500024549](https://doi.org/10.1017/S0308210500024549))

* Lev Buhovsky, Yaron Ostrover, _On the uniqueness of Hofer's geometry_, [arxiv:1008.0661](http://arxiv.org/abs/1008.0661)

* Michael Usher, _Hofer geometry and cotangent fibers_, [arxiv/1301.4484](http://arxiv.org/abs/1301.4484)

[[!redirects Hofer's metric]]
[[!redirects Hofer metric]]
[[!redirects Hofer geometry]]


Hoi-Kwong Lo

* [research group](https://www.comm.utoronto.ca/~hklo/)

* [GoogleScholar page](https://scholar.google.ca/citations?user=sJkQETYAAAAJ&hl=en)

## Selected writings

On [[vortex]]$\;$[[anyons]]:

* [[Hoi-Kwong Lo]], [[John Preskill]], *Non-Abelian vortices and non-Abelian statistics*, Phys. Rev. D **48** (1993) 4821 $[$[doi:10.1103/PhysRevD.48.4821](https://doi.org/10.1103/PhysRevD.48.4821)$]$

category: people


HOL

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#### Constructivism, Realizability, Computability
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_HOL_ (short for [[higher-order logic]]) is a kind of [[simply typed lambda calculus]]. There are various [[proof assistant|proof assistants]] that implement this language. The most known one is _[[Isabelle]]_, that has HOL or Isabelle/HOL as its main application.

Other proof assistants of this kind are HOL4, [[HOL Light]], HOL Zero, and ProofPower.

## Related entries

[[!include proof assistants and formalization projects -- list]]

## References

* T. Nipkow, L. Paulson, M. Wenzel. _Isabelle/HOL — A Proof Assistant for Higher-Order
Logic_, Springer (2002)

* Wikipedia, _[HOL (proof assistant)](http://en.wikipedia.org/wiki/HOL_%28proof_assistant%29)_

* Florian Rabe and Mihnea Iancu, _A Formalized Set-Theoretical Semantics of
Isabelle/HOL_ ([pdf](https://kwarc.info/people/frabe/Research/RI_isabelle_10.pdf))

*  M. J. C. Gordon and T. F. Melham (editors), _Introduction to HOL: A
Theorem Proving Environment for Higher Order Logic_, 1993

[[!redirects HOL light]]


HOL Light

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=--
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## Idea

_HOL Light_ is a [[proof assistant]] which is a variant of the _[[HOL]]_ system.

## Applications

The [[Kepler conjecture]] was formalized and proven using _HOL Light_ (in the _[[Flyspeck project]]_).

## References

* [[John Harrison]], _[The HOL Light theorem prover](http://www.cl.cam.ac.uk/~jrh13/hol-light/index.html)_

[[!redirects HOL light]]


hole argument

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=--
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=--

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{:toc}

## Idea

When the [[theory]] of [[gravity]] in the form of _[[general relativity]]_ was developed at the beginning of the 20th century, the abstract notion of a [[smooth manifold]] independently of its [[coordinate charts]] was still unfamiliar. Theories like that of [[electromagnetism]] were traditionally all formulated with respect to what today would be called a choice of [[coordinate patch]]. The development of [[general relativity]] by [[Albert Einstein]] went through a phase in which what is now called the principle of _[[general covariance]]_ was only gradually understood.

The "hole argument" or "hole [[paradox]]", which was put forward in ([Einstein-Grossmann](#EinsteinGrossmann)), is the observation that given a [[pseudo-Riemannian metric|pseudo]]-[[Riemannian metric]] $g$ on some [[spacetime]] $X$ -- hence a field configuration of [[gravity]], and given a point $x : * \to X$ and a [[diffeomorphism]] $\phi : X \stackrel{\simeq}{\to} X$, in general the value of $g$ around $x$ is entirely unrelated to the value of the [[pullback of a cotensor|pullback]] metric $\phi^* g$ at $x$.

This was regarded as a source of puzzlement for a while -- a "[[paradox]]" -- because if one thinks that the point $x$ in $X$ is absolutely fixed (the locus of an "observer" in spacetime) and that two metrics related by a [[diffeomorphism]] as above are equivalent field configurations, it would seem that the field of gravity "at $x$" is undetermined and undeterminable. For a while this reasoning was regarded as an obstacle to the idea of [[general covariance]] and indeed in ([Einstein-Grossmann](#EinsteinGrossmann)) the notion of general covariance was still rejected in view of this.

But of course the value of the metric $g$ at $x$ _does_ coincide with that of $\phi^* g$... at the transformed point $\phi(x)$. An attempt at more words on this is below in [Discussion](#Discussion).

## The "hole argument"
 {#TheArgument}

The original argument was formulated more in detail along the following lines.

In [[general relativity]], every chart of the [[atlas]] of the given spacetime represents the viewpoint of a physical observer.

Let $(X,\mu)$ be a [[spacetime]]. Let $\phi: V \subset X \to \mathbb{R}^4$ be a chart of $X$ such that there is an open, simply connected subset $U \subset \V$ with vanishing [[stress-energy tensor]] $T$ in $U$.

Assume that there are two points $a, b \in U$ such that the curvature vanishes in a neighborhood of $a$ and does not vanish near $b$. This means in terms of the [[differential geometry]] describing the situation that the [[pseudo-Riemannian metric]] $\mu$ on $X$ is [[Ricci-flat metric|Ricci flat]] when restricted to $U$ and flat when restricted to a neighborhood of $a$.

Given these assumptions, there is a [[diffeomorphism]] $\psi: X \stackrel{\simeq}{\to} X$ (an [[isomorphism]] in [[Diff]]) reducing to the identity outside of $U$, with $\psi(a) = b$. Let $\mu' = \psi^*(\mu)$ be the pullback of $\mu$ along $\psi$. Since the field equations of [[general relativity]] are covariant (respect [[isomorphism]]s in the category of [[pseudo-Riemannian manifold]]s), both $\mu$ and $\mu'$ are solutions to the field equations. So one observer will say that there is no gravitation at the spacetime point $a$, while another will say there is.

## Discussion
 {#Discussion}

As is often the case when confused about automorphisms, it clarifies the issue to talk about more general *isomorphisms* first.  Thus, consider two different manifolds $X$ and $Y$ and a diffeomorphism $\psi:Y\cong X$, with metric $\mu$ on $X$ and pulled back metric $\mu' = \psi^*(\mu)$ on $Y$.  Then $(X,\mu)$ and $(Y,\mu')$ represent the same physical reality, with $\psi$ as a witness of this fact.  In particular, if $\psi(a)=b$, then the gravitational field at $a$ in $Y$ will coincide with the gravitational field at $b$ in $X$, as it should be since both represent the same point of spacetime.

Now suppose we have another diffeomorphism $\phi:Y\cong X$.  Then in general $\mu'$ will be different from $\phi^*(\mu)$, and therefore there is no reason that the gravitational field at $a$ in $Y$ will coincide with the gravitational field at $\phi(a)$ in $X$.  This should still be obvious.  But the "hole paradox" now arises when we take $X$ and $Y$ to be the same and $\phi$ to be the identity map.

In other words, if we have used some isomorphism $\psi$ to witness that two manifolds represent the same reality, then we can't "switch halfway through" our argument and start identifying them along some different isomorphism $\phi$.  This is true even if the two manifolds are the same and $\phi$ is the identity.

Notice that this argument has really nothing specifically to do with physics or general relativity. The analogue of what is said here about pseudo-Riemannian manifolds could be said about, say, [[symplectic manifold]]s or whatever.

So from the [[nPOV]] there is no mystery here, but the above argument originally troubled Einstein, because at his time it was felt that it violates the demand that the statement "at a certain region in time and space, there is (or isn't) gravitation" should have an objective, observer independent meaning, whether or not there is matter present that "feels" the influence of gravitation. This assumption is based on the Newtonian notions of the absolute, objective existence of space and time. For Newton's physics, space and time exist independently of any observers and of any objects that are present in time and space. If one adds a structure that models gravitation to space and time in this sense, the statement "at a certain region in time and space, there is (or isn't) gravitation" is independent of observers and of the presence of further content (or structures) like matter.

The conclusion of Einstein and therefore of [[general relativity]] was however that the statement "at a certain region in time and space that contains no matter, there is (or isn't) gravitation" is _not_ independent of the observer. This means that the _physical_ notions of space and time do not have the same objective meaning as in Newton's physics.

On the other hand, the conclusion that one draws from the [[nPOV]] is a simpler and much more general one: when identifing isomorphic objects in a category, we have to remember which isomorphism we used. For a formulation of the situation in terms of [[homotopy type theory]] see the relevant section in [general covariance](general+covariance#InHomotopyTypeTheory).

## Related concepts

* [[general covariance]]

## References

* [[Albert Einstein]], M. Grossmann, _Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation_ Zeitschrift für Math. Phys. 62, 225–259 (1914)
 {#EinsteinGrossmann}

For a treatment in terms of [[HoTT]], see

* [[John Dougherty]], _The Hole Argument, take n_, Foundations of Physics (2019), ([doi](https://doi.org/10.1007/s10701-019-00291-x)).

* Wikipedia, _[hole argument] (http://en.wikipedia.org/wiki/Hole_argument)_

[[!redirects hole paradox]]


Holger Bech Nielsen

* [webpage](https://en.wikipedia.org/wiki/Holger_Bech_Nielsen)

## Selected writings

Introducing [[Nielsen-Olesen vortex strings]] inspired from [[Abrikosov vortices]] in [[type II superconductors]]:

* {#NielsenOlesen73} [[Holger Bech Nielsen]], [[Poul Olesen]], _Vortex-line models for dual strings_, Nuclear Physics B Volume 61, 24 September 1973, Pages 45-61 (doi:10.1016/0550-3213(73)90350-7)

On the [[Cheshire cat principle]] of the [[quark bag model]]:

* {#NNZ85} S. Nadkarni, [[Holger Bech Nielsen]], [[Ismail Zahed]], _Bosonization Relations As Bag Boundary Conditions_, Nucl. Phys. B 253 (1985) 308 (doi:10.1016/0550-3213(85)90533-4)

and derivation from the [[AdS/QCD correspondence]]:

* [[Holger Bech Nielsen]], [[Ismail Zahed]], _The Cheshire Cat Principle from Holography_ ([doi:10.1142/9789814280709_0017](https://doi.org/10.1142/9789814280709_0017), [arXiv:0904.0429](https://arxiv.org/abs/0904.0429)), Chapter 17 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

category: people