Bloch's semi-regularity conjecture for embedded deformations

Let $X$ be a smooth projective variety over a field $k$ of characteristic zero, and let $Y\subset X$ be a closed locally complete intersection subvariety of codimension $p$. Let $N_{Y/X}$ be the normal bundle, and consider the semi-regularity map

$$H^{1}(N_{Y/X})\to H^{p+1}(\Omega_{X/k}^{p-1}).$$

Bloch's semi-regularity conjecture. The semi-regularity map annihilates every obstruction to embedded deformations of $Y$ in $X$. The conjecture concerns the obstruction theory of embedded deformations; the paper gives a different proof of Bloch's theorem establishing this annihilation result.