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\begin{document}

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\section*{Poincaré lemma}

[[!redirects Poincare lemma]]

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Statement}{Statement}\dotfill \pageref*{Statement} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \textbf{Poincar\'e{} Lemma} in [[differential geometry]] and [[complex analytic geometry]] asserts that ``every [[differential form]] $\omega$ which is closed, $d_{dR}\omega = 0$, is \emph{locally} exact, $\omega|_U = d_{dR}\kappa$.''

In more detail: if $X$ is contractible then for every closed [[differential form]] $\omega \in \Omega^k_{cl}(X)$ with $k \geq 1$ there exists a differential form $\lambda \in \Omega^{k-1}(X)$ such that

\begin{displaymath}
\omega = d_{dR} \lambda
  \,.
\end{displaymath}
Moreover, for $\omega$ a smooth family of closed forms, there is a smooth family of $\lambda$s satisfying this condition.

This statement has several more abstract incarnations. One is that it says that on a [[Cartesian space]] (or a complex [[polydisc]]) the [[de Rham cohomology]] (the [[holomorphic de Rham complex|holomorphic de Rham cohomology]]) vanishes in positive degree.

Still more abstractly this says that the canonical morphisms of [[sheaves of chain complexes]]

\begin{displaymath}
\mathbb{R} \to \Omega^\bullet_{dR}
\end{displaymath}
\begin{displaymath}
\mathbb{C} \to \Omega^\bullet_{hol}
\end{displaymath}
from the [[locally constant sheaf]] on the [[real numbers]] (the [[complex numbers]]) to the [[de Rham complex]] ([[holomorphic de Rham complex]]) is a [[stalk]]-wise [[quasi-isomorphism]] -- hence an [[equivalence]] in the [[derived category]] and hence induce an equivalence in [[hypercohomology|hyper]]-[[abelian sheaf cohomology]]. (The latter statement \emph{fails} in general in complex [[algebraic geometry]], see (\hyperlink{Illusie12}{Illusie 12, 1.}) and see also at \emph{[[GAGA]]}.) (A variant of such [[resolutions]] of constant sheaves for the case over [[Klein geometries]] are [[BGG resolutions]].)

The Poincar\'e{} lemma is a special case of the more general statement that the pullbacks of differential forms along [[homotopy|homotopic]] [[smooth function]] are related by a [[chain homotopy]].

\hypertarget{Statement}{}\subsection*{{Statement}}\label{Statement}

\begin{theorem}
\label{}\hypertarget{}{}
Let $f_1, f_2 : X \to Y$ be two [[smooth function]]s between [[smooth manifolds]] and $\Psi : [0,1] \times X \to Y$ a (smooth) [[homotopy]] between them.

Then there is a [[chain homotopy]] between the induced morphisms

\begin{displaymath}
f_1^*, f_2^* : \Omega^\bullet(Y) \to \Omega^\bullet(X)
\end{displaymath}
on the [[de Rham complex]]es of $X$ and $Y$.

In particular, the action on [[de Rham cohomology]] of $f_1^*$ and $f_2^*$ coincide,

\begin{displaymath}
H_{dR}^\bullet(f_1^*) \simeq H_{dR}^\bullet(f_2^*)
  \,.
\end{displaymath}
Moreover, an explicit formula for the [[chain homotopy]] $\psi : f_1 \Rightarrow f_2$ is given by the ``[[homotopy operator]]''

\begin{displaymath}
\psi 
   :
  \omega \mapsto (x \mapsto \int_{[0,1]} \iota_{\partial_t} (\Psi^*\omega)(x) ) d t
  \,.
\end{displaymath}
\end{theorem}
Here $\iota_{\partial t}$ denotes contraction (see [[Cartan calculus]]) with the canonical [[vector field]] tangent to $[0,1]$, and the [[integration]] is that of functions with values in the [[vector space]] of differential forms.

\begin{proof}
We compute

\begin{displaymath}
\begin{aligned}
    d_{Y} \psi(\omega)
    + 
    \psi( d_X \omega )
     & =
    \int_{[0,1]} d_Y \iota_{\partial_t} \Psi^*(\omega) d t 
    + 
    \int_{[0,1]} \iota_{\partial_t} \Psi^*(d_X \omega) d t
    \\
    & = \int_{[0,1]} [d_Y,\iota_{\partial_t}] \Psi^* (\omega) d t
    \\
    & = \int_{[0,1]} \mathcal{L}_{t} \Psi^* (\omega) d t
    \\
    & = \int_{[0,1]} \partial_t \Psi^* (\omega) d t
    \\
    & = \int_{[0,1]} d_{[0,1]} \Psi^* (\omega) 
    \\
    & = \Psi_1^* \omega - \Psi_0^* \omega
    \\
    & = f_2^* \omega - f_1^* \omega
  \end{aligned}
  \,,
\end{displaymath}
where in the [[integral]] we used first that the [[exterior differential]] commutes with pullback of [[differential forms]], then [[Cartan's magic formula]] $[d,\iota_{\partial t}] = \mathcal{L}_t$ for the [[Lie derivative]] along the [[cylinder]] on $X$ and finally the [[Stokes theorem]].

\end{proof}
The \textbf{Poincar\'e{} lemma} proper is the special case of this statement for the case that $f_1 = const_y$ is a function constant on a point $y \in Y$:

\begin{prop}
\label{}\hypertarget{}{}
If a [[smooth manifold]] $X$ admits a smooth contraction

\begin{displaymath}
\itexarray{
    X
    \\
    \downarrow^{\mathrlap{(id,0)}} & \searrow^{\mathrlap{id}}
    \\
    X \times [0,1] & \stackrel{\Psi}{\to} & X
    \\
    \uparrow^{\mathrlap{(id,1)}} & \nearrow_{\mathrlap{const_x}}
    \\
    X
  }
\end{displaymath}
then the [[de Rham cohomology]] of $X$ is concentrated on the ground field in degree 0. Moreover, for $\omega$ any closed form on $X$ in positive degree an explicit formula for a form $\lambda$ with $d \lambda = \omega$ is given by

\begin{displaymath}
\lambda = \int_{[0,1]} \iota_{\partial_t}\Psi^*(\omega) d t
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
In the general situation discussed above we now have $f_1^* = 0$ in positive degree.

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item \textbf{Poincar\'e{} lemma}

\begin{itemize}%
\item [[Kähler potential]]

\end{itemize}

\item [[Stokes theorem]]


\item [[de Rham theorem]]


\item [[Hodge theorem]]


\item [[variational sequence]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A nice account collecting all the necessary background (in [[differential geometry]]) is in

\begin{itemize}%
\item Daniel Litt, \emph{The Poincar\'e{} lemma and de Rham cohomology} (\href{http://math.stanford.edu/~dlitt/exposnotes/poincare_lemma.pdf}{pdf})

\end{itemize}
Discussion in [[complex analytic geometry]] is in

\begin{itemize}%
\item [[Luc Illusie]], \emph{Around the Poincar\'e{} lemma, after Beilinson}, talk notes 2012 (\href{http://www.math.u-psud.fr/~illusie/derived-deRham3.pdf}{pdf})

\end{itemize}
following

\begin{itemize}%
\item [[Alexander Beilinson]], \emph{$p$-adic periods and de Rham cohomology}, J. of the AMS 25 (2012), 715-738

\end{itemize}
[[!redirects Poincaré lemma]]



\end{document}