nLab Poincaré lemma

Idea

The Poincaré Lemma in differential geometry and complex analytic geometry asserts that “every differential form $\omega$ which is closed, $d_{dR}\omega = 0$, is 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

$$\omega = d_{dR} \lambda \,.$$

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 cohomology) vanishes in positive degree.

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

$$\mathbb{R} \to \Omega^\bullet_{dR}$$

$$\mathbb{C} \to \Omega^\bullet_{hol}$$

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 hyper-abelian sheaf cohomology. (The latter statement fails in general in complex algebraic geometry, see (Illusie 12, 1.) and see also at GAGA.) (A variant of such resolutions of constant sheaves for the case over Klein geometries are BGG resolutions.)

The Poincaré lemma is a special case of the more general statement that the pullbacks of differential forms along homotopic smooth function are related by a chain homotopy.

Statement

Theorem

Let $f_1, f_2 : X \to Y$ be two smooth functions 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

$$f_1^*, f_2^* : \Omega^\bullet(Y) \to \Omega^\bullet(X)$$

on the de Rham complexes of $X$ and $Y$.

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

$$H_{dR}^\bullet(f_1^*) \simeq H_{dR}^\bullet(f_2^*) \,.$$

Moreover, an explicit formula for the chain homotopy $\psi : f_1 \Rightarrow f_2$ is given by the “homotopy operator”

$$\psi : \omega \mapsto (x \mapsto \int_{[0,1]} \iota_{\partial_t} (\Psi^*\omega)(x) ) d t \,.$$

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.

Proof

We compute

$$\begin{aligned} d_{Y} \psi(\omega) + \psi( d_X \omega ) & = \int_{[0,1]} d_Y \iota_{\partial_t} \Psi^*(\omega) d t + \int_{[0,1]} \iota_{\partial_t} \Psi^*(d_X \omega) d t \\ & = \int_{[0,1]} (d_Y \iota_{\partial_t} + \iota_{\partial_t}d_Y) \Psi^* (\omega) d t \\ & = \int_{[0,1]} \mathcal{L}_{t} \Psi^* (\omega) d t \\ & = \int_{[0,1]} \partial_t \Psi^* (\omega) d t \\ & = \int_{[0,1]} d_{[0,1]} \Psi^* (\omega) \\ & = \Psi_1^* \omega - \Psi_0^* \omega \\ & = f_2^* \omega - f_1^* \omega \end{aligned} \,,$$

where in the integral we used first that the exterior differential commutes with pullback of differential forms, then Cartan's magic formula $d_Y \iota_{\partial_t} + \iota_{\partial_t}d_Y = \mathcal{L}_t$ for the Lie derivative along the cylinder on $X$ and finally the Stokes theorem.

The Poincaré lemma proper is the special case of this statement for the case that $f_2 = const_y$ is a function constant on a point $y \in Y$:

Corollary

If a smooth manifold $X$ admits a smooth contraction

$$\array{ X \\ \downarrow^{\mathrlap{(id,0)}} & \searrow^{\mathrlap{id}} \\ X \times [0,1] & \stackrel{\Psi}{\to} & X \\ \uparrow^{\mathrlap{(id,1)}} & \nearrow_{\mathrlap{const_x}} \\ X }$$

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

$$\lambda = - \int_{[0,1]} \iota_{\partial_t}\Psi^*(\omega) d t \,.$$

Proof

In the general situation discussed above we now have $f_2^* = 0$ in positive degree.

Related concepts

• Poincaré lemma

  • Kähler potential

• Stokes theorem

• de Rham theorem

• Hodge theorem

• variational sequence

References

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

• Daniel Litt, The Poincaré lemma and de Rham cohomology (pdf)

Discussion in complex analytic geometry is in

• Luc Illusie, Around the Poincaré lemma, after Beilinson, talk notes 2012 (pdf)

following

• Alexander Beilinson, $p$-adic periods and de Rham cohomology, J. of the AMS 25 (2012), 715-738