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Authors
Daniel Barlet
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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.
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The graded coherent sheaf $alpha_X^bullet$ constructed in [B.18] for any reduced pure dimensional complex space $X$ is stable by exterior product but not by the de Rham differential. We construct here a new graded coherent sheaf $alpha_X^bullet$ containing $alpha_X^bullet$ and stable both by exterior product and by the de Rham differential. We show that it has again the ``pull-back property for holomorphic maps $f : X to Y$ between irreducible complex spaces such that $f(X)$ is not contained in the singular set of $Y$. Moreover, this graded coherent sheaf $alpha_X^bullet$ comes with a natural coherent exhaustive filtration and this filtration is also compatible with the pull-back by such holomorphic maps. These sheaves define new invariants on singular complex spaces. We show on some simple examples that these invariants are new.
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