New singularity invariants : the sheaf ${\beta }_X^{\bullet }$

Abstract

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.