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Poincare lemma

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Idea

The Poincaré Lemma in differential geometry and complex analytic geometry asserts that “every differential form ω\omega which is closed, d dRω=0d_{dR}\omega = 0, is locally exact, ω| U=d dRκ\omega|_U = d_{dR}\kappa”.

More in detail: if XX is contractible then for every closed differential form ωΩ cl k(X)\omega \in \Omega^k_{cl}(X) with k1k \geq 1 there exists a differential form λΩ k1(X)\lambda \in \Omega^{k-1}(X) such that

ω=d dRλ. \omega = d_{dR} \lambda \,.

Moreover, for ω\omega a smooth smooth family of closed forms, there is a smooth family of λ\lambdas 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

Ω dR \mathbb{R} \to \Omega^\bullet_{dR}
Ω hol \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.) ans see also at GAGA.)

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

Statement

Theorem

Let f 1,f 2:XYf_1, f_2 : X \to Y be two smooth functions between smooth manifold and Ψ:[0,1]×XY\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 *:Ω (Y)Ω (X) f_1^*, f_2^* : \Omega^\bullet(Y) \to \Omega^\bullet(X)

on the de Rham complexes of XX and YY.

In particular, the action on de Rham cohomology of f 1 *f_1^* and f 2 *f_2^* coincide,

H dR (f 1 *)H dR (f 2 *). H_{dR}^\bullet(f_1^*) \simeq H_{dR}^\bullet(f_2^*) \,.

Moreover, an explicit formula for the chain homotopy ψ:f 1f 2\psi : f_1 \Rightarrow f_2 is given by

ψ:ω(x [0,1]ι t(Ψ t *ω)(x))dt. \psi : \omega \mapsto (x \mapsto \int_{[0,1]} \iota_{\partial_t} (\Psi_t^*\omega)(x) ) d t \,.

Here ι t\iota_{\partial t} denotes contraction (see Cartan calculus). with the canonical vector field tangent to [0,1][0,1] and the integration is that of functions with values in the vector space of differential forms.

Proof

We compute

d Yψ(ω)+ψ(d Xω) = [0,1]d Yι tΨ t *(ω)dt+ [0,1]ι tΨ t *(d Xω)dt = [0,1][d Y,ι t]Ψ t *(ω)dt = [0,1] tΨ t *(ω)dt = [0,1] tΨ t *(ω)dt = [0,1]d [0,1]Ψ t *(ω) =Ψ 1 *ωΨ 0 *ω =f 2 *ωf 1 *ω, \begin{aligned} d_{Y} \psi(\omega) + \psi( d_X \omega ) & = \int_{[0,1]} d_Y \iota_{\partial_t} \Psi_t^*(\omega) d t + \int_{[0,1]} \iota_{\partial_t} \Psi_t^*(d_X \omega) d t \\ & = \int_{[0,1]} [d_Y,\iota_{\partial_t}] \Psi_t^* (\omega) d t \\ & = \int_{[0,1]} \mathcal{L}_{t} \Psi_t^* (\omega) d t \\ & = \int_{[0,1]} \partial_t \Psi_t^* (\omega) d t \\ & = \int_{[0,1]} d_{[0,1]} \Psi_t^* (\omega) \\ & = \Psi_1^* \omega - \Psi_0^* \omega \\ & = f_2^* \omega - f_1^* \omega \end{aligned} \,,

where in the integral we used fist that the exterior differential commutes with pullback of differential forms, then Cartan's magic formula [d,ι t]= t[d,\iota_{\partial t}] = \mathcal{L}_t for the Lie derivative along the cylinder on XX and finally the Stokes theorem.

The Poincaré lemma proper is the special case of this statement for the case that f 1=const yf_1 = const_y is a function constant on a point yYy \in Y:

Corollary

If a smooth manifold XX admits a smooth contraction

X (id,0) id X×[0,1] Ψ X (id,1) const x X \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 XX is concentrated on the ground field in degree 0. Moreover, for ω\omega any closed form on XX in positive degree an explicit formula for a form λ\lambda with dλ=ωd \lambda = \omega is given by

λ= [0,1]ι tΨ t *(ω)dt. \lambda = \int_{[0,1]} \iota_{\partial_t}\Psi_t^*(\omega) d t \,.
Proof

In the general situation discussed above we now have f 1 *=0f_1^* = 0 in positive degree.

References

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

Discussion in complex analytic geometry is in

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

following

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

Revised on June 6, 2014 08:37:17 by Urs Schreiber (89.204.139.83)