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On Poincar\'e lemma or Volterra theorem about differential forms and cohomology groups
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abstract
The Poincar\'{e} lemma (or Volterra theorem) is of utmost importance both in theory and in practice. It tells us every differential form which is closed, is locally exact. In other words, on a contractible manifold all closed forms are exact. The aim of this paper is to present some direct proofs of this lemma and explore some of its numerous consequences. Some connections with Cech-De Rham-Dolbeault cohomologies, $\overline{\partial}$-Poincar\'{e} lemma or Dolbeault-Grothendieck lemma are given.
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The Poincare lemma, antiexact forms, and fermionic quantum harmonic oscillator
The homotopy operator of the Poincaré lemma and the exterior derivative satisfy a fermionic oscillator algebra with eigenvalues ±1, and split on complex manifolds into a pair of operators generating a dual Dolbeault b...
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