Non-solvability of the tangential ∂̅-system in manifolds with constant Levi rank

Let M be a real-analytic submanifold of ${\displaystyle {ℂ}^n}$ whose "microlocal" Levi form has constant rank ${\displaystyle s^{+}\_\left\{M\right\}+s^{-}\_\left\{M\right\}}$ in a neighborhood of a prescribed conormal. Then local non-solvability of the tangential ∂̅-system is proved for forms of degrees ${\displaystyle s^{-}\_\left\{M\right\}}$, ${\displaystyle s^{+}\_\left\{M\right\}}$ (and 0).  This phenomenon is known in the literature as "absence of the Poincaré Lemma" and was already proved in case the Levi form is non-degenerate (i.e. ${\displaystyle s^{-}\_\left\{M\right\}+s^{+}\_\left\{M\right\}=n-codimM}$). We owe its proof to [2] and [1] in the case of a hypersurface and of a higher-codimensional submanifold respectively. The idea of our proof, which relies on the microlocal theory of sheaves of [3], is new.