The sheaf $alpha$ $bullet$ X

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.