Zero map between obstruction spaces: subvarieties versus cycles

Abstract

For $Y\subset X$ a locally complete intersection of codimension p, Spencer Bloch [2] constructed the semi-regularity map $\ddot{I}€:H^1({\mathcal{N}}_{Y/X})\to H^{p+1}(\hat{I}{\text{c◯}}_{X/k}^{p-1})$. As an analogue, we construct a map $\tilde{\ddot{I}}€:H^1({\mathcal{N}}_{Y/X})\to H^{p+1}(\hat{I}{\text{c◯}}_{X/\mathbb{Q}}^{p-1})$, without assuming local complete intersections. While the semi-regularity map $\ddot{I}€$ is expected to be injective, we show $\tilde{\ddot{I}}€$ is a zero map. We use this zero map to interpret how to eliminate obstructions to deforming cycles, an idea by Mark Green and Phillip Griffiths in [9].

Minor change, polish language