Universally Equidimensional Morphisms and Weakly or Strongly rational Singularities

Abstract

The focus of this article is the study of a certain type of singularities and their transfer properties in a universally equidimensional morphism (i.e. an open morphism with constant pure-dimensional fibers). The singularities of interest are those called weakly rational, characterized by the vanishing of the (m-1)-th direct image of the structural sheaf in a given desingularization of a complex space of dimension $m$. If the singular locus of the considered space has codimension at least two, this condition is equivalent to the equality, in maximal degree, of the Grothendieck dualizing sheaves ω and L (whose sections extend analytically across any resolution of singularities). We show that this type of singularity transfers from the fibers and the base to the total space. Moreover, if the morphism induces local holomorphic traces (meaning it is geometrically flat), there is a transfer from the total space to the base. Finally, under certain conditions ensuring the existence of a simultaneous resolution, there is a natural transfer from the total space to the fibers, provided the base itself has such singularities.