Zero map between obstruction spaces: subvarieties versus cycles

TL;DR

This paper constructs a new map relating to deformation obstructions of subvarieties, demonstrating it is identically zero, which offers insights into eliminating obstructions in cycle deformation.

Contribution

It introduces a novel map between obstruction spaces that is shown to be zero, extending the semi-regularity map concept without the local complete intersection assumption.

Findings

The map  The constructed map  is identically zero.

The zero map provides a new perspective on how to eliminate obstructions in cycle deformation.

Abstract

For $Y \subset X$ a locally complete intersection of codimension p, Spencer Bloch [2] constructed the semi-regularity map $\pi: H^{1}(\mathcal{N}_{Y/X}) \to H^{p+1}(\Omega_{X/k}^{p-1})$. As an analogue, we construct a map $\tilde{\pi}: H^{1}(\mathcal{N}_{Y/X}) \to H^{p+1}(\Omega_{X/\mathbb{Q}}^{p-1})$, without assuming local complete intersections. While the semi-regularity map $\pi$ is expected to be injective, we show $\tilde{\pi}$ 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].