On the complex form of the Poincaré lemma

Abstract

dimension n, any differential form can be decomposed into a sum of forms of degree r, 0 < r < n, and the operator d of exterior differentiation (d2 = 0) is an operator of degree + 1. A differential form q satisfying do =0 is called closed, and a form q such that q =dfl is called exact. The Poincare Lemma states that a closed differential form of positive degree is locally exact; that is, if q is defined and satisfies dqb = 0 in an open UCM, and mo is any point of U, then there exists an open neighborhood V of mo, with VC U, and a form 41 defined in V such that ck = d,6 in V. This result is most conveniently proved by choosing a V differentiably contractible to the point mo, for which there can be constructed an operator k of degree -1 satisfying