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

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Zampieri G.

Annales Polonici Mathematici

2000

tom 74

nr 1

291-296

Let M be a real-analytic submanifold of $ℂ^n$ whose "microlocal" Levi form has constant rank $s^{+}_{M} + s^{-}_{M}$ in a neighborhood of a prescribed conormal. Then local non-solvability of the tangential ∂̅-system is proved for forms of degrees $s^{-}_{M}$, $s^{+}_{M}$ (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. $s^{-}_{M} + s^{+}_{M} = n - codim M$). 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.

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1 Annales Polonici Mathematici

1 Zampieri G.

1 2000

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