Special and general types
The Poincaré Lemma in differential geometry and complex analytic geometry asserts that “every differential form which is closed, , is locally exact, .”
In more detail: if is contractible then for every closed differential form with there exists a differential form such that
Moreover, for a smooth family of closed forms, there is a smooth family of 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
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.
Let be two smooth functions between smooth manifolds and a (smooth) homotopy between them.
Then there is a chain homotopy between the induced morphisms
on the de Rham complexes of and .
In particular, the action on de Rham cohomology of and coincide,
Moreover, an explicit formula for the chain homotopy is given by the “homotopy operator”
Here denotes contraction (see Cartan calculus) with the canonical vector field tangent to , and the integration is that of functions with values in the vector space of differential forms.
where in the integral we used first that the exterior differential commutes with pullback of differential forms, then Cartan's magic formula for the Lie derivative along the cylinder on and finally the Stokes theorem.
The Poincaré lemma proper is the special case of this statement for the case that is a function constant on a point :
If a smooth manifold admits a smooth contraction
then the de Rham cohomology of is concentrated on the ground field in degree 0. Moreover, for any closed form on in positive degree an explicit formula for a form with is given by
In the general situation discussed above we now have in positive degree.
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)