Precise Arrhenius law for p-forms: The Witten Laplacian and   Morse-Barannikov complex
  Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov's presentation of Morse theory. 
Dorian Le Peutrec (LM-Orsay), Francis Nier (IRMAR), Claude Viterbo   (CMLS-EcolePolytechnique)
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