Zero map between obstruction spaces: subvarieties versus cycles.

In 1970s, for local complete intersections, Spencer Bloch [2] constructed the semi-regularity map $\pi: H^{1}(\mathcal{N}_{Y/X}) \to H^{q+1}(\Omega_{X/k}^{q-1})$. As an analogue, we construct a map $\pi: H^{1}(\mathcal{N}_{Y/X}) \to H^{q+1}(\Omega_{X/\mathbb{Q}}^{q-1})$, without assuming local complete intersections. While semi-regularity asks for injectivity, our map is a zero map. We use this zero map to interpret how to eliminate obstructions of deformation of cycles, an idea by Mark Green and Phillip Griffiths in [7].