The sheaf $α$ $\bullet$ X

Abstract: We introduce in a reduced complex space, a "new coherent sub-sheaf" of the sheaf $\omega_{X}^{\bullet}$ which has the "universal pull-back property" for any holomorphic map, and which is in general bigger than the usual sheaf of holomorphic differential forms $\Omega_{X}^{\bullet}/torsion$. We show that the meromorphic differential forms which are sections of this sheaf satisfy integral dependence equations over the symmetric algebra of the sheaf $\Omega_{X}^{\bullet}/torsion$. This sheaf $\alpha_{X}^{\bullet}$ is also closely related to the normalized Nash transform. We also show that these $q-$meromorphic differential forms are locally square-integrable on any $q-$dimensional cycle in $X$ and that the corresponding functions obtained by integration on an analytic family of $q-$cycles are locally bounded and locally continuous on the complement of closed analytic subset.