Semiregularity as a consequence of Goodwillie's theorem

Abstract

We realise Buchweitz and Flenner's semiregularity map (and hence a fortiori Bloch's semiregularity map) for a smooth variety $X$ as the tangent of a generalised Abel--Jacobi map on the derived moduli stack of perfect complexes on $X$. The target of this map is an analogue of Deligne cohomology defined in terms of cyclic homology, and Goodwillie's theorem on nilpotent ideals ensures that it has the desired tangent space (a truncated de Rham complex). Immediate consequences are the semiregularity conjectures: that the semiregularity maps annihilate all obstructions, and that if $X$ is deformed, semiregularity measures the failure of the Chern character to remain a Hodge class. This gives rise to reduced obstruction theories of the type featuring in the study of reduced Gromov--Witten and Pandharipande--Thomas invariants. We also give generalisations allowing $X$ to be singular, and even a derived stack.