Abstract:
The subject of the talk is resolution of singularities of differential forms on an algebraic or analytic variety. We address the problem of finding a resolution of singularities $\sigma: X \to X_0$ of a singular algebraic or analytic variety $X_0$ such that the pulled back cotangent sheaf of $X_0$ (i.e., the pull-back of the differential forms defined in $X_0$) is given, locally in $X$, by monomial differential forms (with respect to a suitable coordinate system). This problem is related with monomialization of maps, the $L^2$ cohomology of singular varieties and reduction of singularities of vector-fields. In a work in collaboration with Bierstone, Grandjean and Milman, we give a positive answer to the problem when $\mbox{dim } X_0 \leq 3$.