Classifying singularities up to analytic extensions of scalars is smooth

AbstractThe singularity space consists of all germs (X,x), with X a Noetherian scheme and x a point, where we identify two such germs if they become the same after an analytic extension of scalars. This is a complete, separable space for the metric given by the order to which jets (=infinitesimal neighborhoods) agree after base change. In the terminology of descriptive set-theory, the classification of singularities up to analytic extensions of scalars is a smooth problem. Over C, the following two classification problems up to isomorphism are then also smooth: (i) analytic germs; and (ii) polarized schemes

Regulators of canonical extensions are torsion: the smooth divisor case

In this paper, we prove a generalization of Reznikov's theorem which says that the Chern-Simons classes and in particular the Deligne Chern classes (in degrees $>1$) are torsion, of a flat bundle on a smooth complex projective variety. We consider the case of a smooth quasi--projective variety with an irreducible smooth divisor at infinity. We define the Chern-Simons classes of Deligne's canonical extension of a flat vector bundle with unipotent monodromy at infinity, which lift the Deligne Chern classes and prove that these classes are torsion

Perturbations, Deformations, and Variations (and "Near-Misses") in Geometry, Physics, and Number Theory

Mathematic

Torsion Invariants for Families

We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments.Comment: LaTeX, 40 pages; v2: references added, BL = IK announced; v3: reference to DWW = IK adde