We study the masses charged by ${\left(d{d}^{c}u\right)}^n$ at isolated singularity points of plurisubharmonic functions u. This is done by means of the local indicators of plurisubharmonic functions introduced in [15]. As a consequence, bounds for the masses are obtained in terms of the directional Lelong numbers of u, and the notion of the Newton number for a holomorphic mapping is extended to arbitrary plurisubharmonic functions. We also describe the local indicator of u as the logarithmic tangent to u.

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. $s_M^{-}+s_M^{+}=n-codimM$). We owe its proof to [2] and [1] in the case of a hypersurface and of a higher-codimensional submanifold respectively....

We prove that for a real analytic generic submanifold $M$ of ${\mathbb{C}}^n$ whose Levi-form has constant rank, the tangential ${\overline{\partial }}_M$-system is non-solvable in degrees equal to the numbers of positive and $M$ negative Levi-eigenvalues. This was already proved in [1] in case the Levi-form is non-degenerate (with $M$ non-necessarily real analytic). We refer to our forthcoming paper [7] for more extensive proofs.